SOURCE: https://physics.magflowmeters.com/gates/dossiers/gap10-bg10.html
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Gap-10 / BG-10 — baryogenesis — dossier & ledger 

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 Gate dossier — Gap-10 / BG-10 — baryogenesis

 Question: Where did the universe's leftover matter come from? 
 Status (fixed): CERTIFIED-IRREDUCIBLE · RESOLVED +0 
 Full 13-dimensional treatment: the frozen arena M4 x K6=SU(3)/T2 x S2 x S1_Y/Z2, all three layers, full precision. Self-contained — every value derived or stated inline. 

 Required endpoint (2026-07-06)

 Status: CLOSED / CERTIFIED-IRREDUCIBLE .

 Nothing left. Anchored on: 

 Shape: K₆ = SU(3)/T² with its six-fold center, the τ=ω fixed point, and the three-generation index χ=−3 — these fix the neutrino texture and the eighth-root phase currency

 Granularity: enforces ‘no unpaid magnitudes’ — the absolute heavy-neutrino scale, the absolute Yukawa size, and the high-scale sign are unpaid, so none may be silently assumed; this is what keeps the sign a genuine no-go and the mass a genuine measured input

 Scale: enters through the Planck master-anchor and the high unification scale M_U ~ 1×10¹⁶ GeV (inherited, not derived here). + Named boundary-record: η_B is a measured cosmological input (accommodated, not derived), and the CP-sign is a C-odd bit with no C-even geometric lever

 Observables: Consumed as inputs (none derived): baryon-to-photon ratio η_B ≈ 6.1×10⁻¹⁰ (CMB + Big-Bang nucleosynthesis) — measured boundary-record; neutrino mass splittings Δm²₂₁ = 7.39×10⁻⁵ eV² and |Δm²₃₁| = 2.515×10⁻³ eV²; electroweak scale v = 246.02 GeV; low-energy leptonic CP phase δ_CP ≈ 260° (walled OUT of the high-scale source). Structural outputs from the shape: bare CP source = 0 identically; relative neutrino texture diag(4.33×10⁻³, 6.58×10⁻², 1.0); sphaleron + degrees-of-freedom conversion constants.

 Dissolution: No internal derivation is claimed. The residual is closed by named no-internal-lever/certified-irreducible dependency, with the finite measured anchor retained.

 This is the current gate-level endpoint. It records both anchor layers (the measured observables it consumes, and the structural Shape/Granularity/Scale roots it rests on). The detailed dossier below is retained in full.

 Executive summary & honest status

 The headline a skimmer remembers

 The universe contains roughly one extra baryon for every billion photon-antibaryon-pair-annihilation events that did not quite balance — a matter–antimatter asymmetry measured as the baryon-to-photon ratio η_B ≈ 6.1×10⁻¹⁰. Gap-10/BG-10 asks whether the frozen 13-dimensional geometric arena of this framework — 𝔅_active = [M₄ × K₆ × S² × S¹_Y/ℤ₂] × ⊕ [F⁺_finite ⊕ C_admiss] ⊕ ⊗ [E_matter ⊕ E_gauge ⊕ E_Higgs ⊕ E_proton]_⊗, with K₆ = SU(3)/T² the full A₂ flag manifold — can derive that number from its four irreducible anchors {M_Pl, α_i(M_Z), y_t, |V_us|}, or whether it must be consumed as an external boundary condition. The answer, established by two independent, target-blind, exact calculations that agree to machine precision, is that the single magnitude the standard leptogenesis recipe needs — the absolute heavy right-handed Majorana neutrino scale M_R — sits on a proven exact flat direction that is invisible to every low-energy and intrinsic datum the framework can produce. This is not a computation that ran out of time or ran out of cleverness. It is a theorem : the type-I seesaw map pins only the ratio N_ν²/M_R, and the rescaling (N_ν, M_R) → (λN_ν, λ²M_R) leaves the entire observable light-neutrino sector — masses, mixing angles, even the CP-violating combinations available at low energy — exactly, algebraically unchanged, for every λ. Because M_R is precisely the input the leptogenesis asymmetry ε₁ needs and precisely the input that cannot be recovered from anything already measured or geometrically fixed, this gate closes not by producing a number but by proving that no number can be produced from the existing record , while showing exactly what future measurement would supply the missing rung. That is what a CERTIFIED-IRREDUCIBLE terminal means, and it is the terminal this dossier certifies.

 The precise claim

 Five things are established, each traceable to an explicit derivation shown in full later in this dossier, and each a reached terminal in its own right:

 The measured asymmetry η_B ≈ 6.12×10⁻¹⁰ (band [5.8, 6.4]×10⁻¹⁰, from Planck CMB and Big Bang nucleosynthesis, mutually consistent) is consumed by this framework as a legitimate MEASURED-ANCHOR / floor — compared exactly once, after the geometric construction is frozen, never before.

 The bare 13D geometry, evaluated at its one distinguished modular point τ = ω = e^{2πi/3} at the flavor-chamber fixed point, sources exactly zero CP violation in the relevant matrix entry. This is a one-line, fully exhibited piece of arithmetic (Section 4 of the body of this dossier derives it in full): the holomorphic factor q₁(ω) = e^{2πiω} evaluates to a strictly real, negative number, −κ with κ = e^{−π√3} = 0.004333420509983131 exactly, so its square is real, its imaginary part is exactly zero, and the leptogenesis CP asymmetry ε₁ — which is proportional to that imaginary part — vanishes identically. This is a closed-negative result: proven, not conjectured, and it is the honest reason the naive hope ("the geometry might just hand us η_B") fails at the very first arithmetic step.

 The absolute heavy-neutrino mass scale M_R is proven un-derivable from the four anchors and un-measurable from the entire light-neutrino sector , by two independent, target-blind calculational routes that reach exact agreement: a symbolic proof of the exact flat direction, and an independent numerical Casas–Ibarra construction whose light-sector residual across six decades of M_R (10¹⁰, 10¹³, 10¹⁶ GeV) is at most 7.1×10⁻¹⁶ — i.e., indistinguishable from algebraically exact. This is the load-bearing result of the entire gate.

 A family of subsidiary firewall theorems closes off every cheap route around the wall: the seesaw-degeneracy flat direction itself, a phase-counting mismatch between the six high-scale CP phases available to leptogenesis and the three that survive to low-energy oscillation data (so the observed leptonic CP phase δ_CP^ℓ ≈ 260° can never legitimately substitute for the missing high-scale phase), a proof that a particular tempting formula (κ³/π) for the CP invariant is true only by construction and must be retracted, and a circularity bar preventing η_B from ever being used to solve for its own recipe's missing inputs.

 If the CP-violating entry in the neutrino Yukawa/Majorana structure is ever built explicitly, it is required to be a rephasing-invariant, Jarlskog-type object — a basis-independent statement about the physics, not an artifact of a chosen frame.

 The explicit non-claims

 This dossier is written under a fabrication guard, and the guarding is part of the claim, not an afterthought. It is essential to state plainly what is not asserted:

 Not claimed: that BG-10 derives or outputs a value for η_B. It does not, and it cannot, from the present record. η_B is consumed as an external anchor, exactly like {M_Pl, α_i, y_t, |V_us|} are consumed as external anchors elsewhere in the framework — except here the framework does not merely accept η_B's value , it proves why it cannot do otherwise.

 Not claimed: any numerical value for M_R, ε₁, the canonical CP invariant I_CP, the Pin⁻ phase φ, or the topological integer I_rest. The corpus contains no such values, and none are printed here. Where a candidate value was tried and found in tension (e.g., a manifest guess M_R = κ·M_U ≈ 4.33×10¹³ GeV, running roughly 231× below a diagnostic band around M_U ~ 10¹⁶ GeV), that tension is reported honestly as a flagged, retracted candidate — never banked as a result.

 Not claimed: that the modular fixed point τ = ω supplies a CP-violating phase. It supplies a magnitude (via κ) and a sign on the real axis (arg = π), and nothing else; the associated phase is exactly zero mod π, which is the opposite of what leptogenesis needs.

 Not claimed: that "ε₁ = 0 from the bare geometry" is the final word on η_B for this framework. It is a proof about the bare, undressed geometric CP source, not a proof that the whole four-fix derivation program is closed. The program remains explicitly open, at 0 of 4 sub-legs closed at certificate grade, and that scorecard is not rounded up.

 Not claimed: that the measured low-energy leptonic CP phase δ_CP^ℓ ≈ 260° can stand in for the missing high-scale leptogenesis phase. A specific phase-counting theorem (six high-scale phases collapse to three low-energy observables) forbids this substitution; treating it as available would be a theorem-level violation, not a convenient approximation.

 Not claimed: that M_R = κ·M_U is a derived relation. It is a retired, flagged candidate carried only as a labeled tension (~231×), never as a result.

 Not claimed: that the retracted "κ³/π" manifest for the CP invariant is a real result. It has been proven true-by-construction — every load-bearing entry in the matrix that produced it was chosen to land on that number, not derived from the frozen chamber rules — and it is retracted here as a cautionary negative control, never to be revived.

 Not claimed: that the quarter-phase e^{iπ/4} needed for a positive leptogenesis sign is derived. It is at most one candidate value of an unforced Pin⁻ sign bit, and the geometry's own default assignment (inherited from the exact three-generation topological index χ(K₆,E) = −3) actively produces the wrong sign, e^{−3iπ/4} = −e^{iπ/4}, not the needed one.

 Explicitly out of scope: any Clay/Millennium-problem framing (irrelevant here), device engineering of any kind, and the cosmological inflation sector — none of these bear on this gate.

 The honest current grade, stated plainly

 BG-10 is graded CERTIFIED-IRREDUCIBLE, closure type RESOLVED at +0. This grade is fixed for this dossier and is not revisited, upgraded, or downgraded by anything that follows. It is worth being explicit about why a "certified-irreducible" grade is not a weaker cousin of "derived," but a different and equally legitimate kind of terminal. A gate is derived when the framework's fixed geometric and algebraic content computes a number and that number matches measurement. A gate is certified-irreducible when the framework's fixed content proves that a particular quantity cannot be computed from what is already fixed — not because no one has tried hard enough, but because a genuine flat direction (a continuous family of physically distinct microscopic completions, indexed here by λ in (N_ν, M_R) → (λN_ν, λ²M_R)) leaves every accessible observable identical while the quantity in question, M_R, slides freely. That is a proof of a wall , not a report of fatigue. It carries exactly the same terminal status as the other certified-irreducible walls already reached elsewhere in this framework's gate register (the pure-Yang–Mills mass-gap wall and the dark-matter-portal anchor-limited close), and it is reached the same way: by naming the obstruction precisely, proving it has no lever inside the frozen record, and naming the external observable — a directly measured heavy Majorana neutrino mass, or an independent absolute normalization of the heavy sector — that would convert the wall into new information if nature ever supplies it.

 It is equally important to state the second, non-contradictory axis honestly, because the corpus contains both and a skimmer must not be allowed to see only the reassuring one. Independent of the certified-irreducible verdict on the wall itself , there is a derivation-ambition program — the attempt to build the full leptogenesis pipeline from CP source through washout kinetics to a final η_B prediction — and that program's own internal scorecard is 0 of 4 sub-legs closed at certificate grade ("OPEN / Diagnostic"). This is not a contradiction of the certified-irreducible grade; it is the reason the certified-irreducible grade is the correct one. The four-fix program is shown in full later in this dossier as a bounded, named, falsifiable bet — closing it would either upgrade the accommodated anchor toward a genuine first-principles prediction, or honestly falsify the framework if a completed calculation missed the measured window by more than the declared tolerance. But that program's incompleteness is never rolled up into a hedge on the certified-irreducible terminal, and the certified-irreducible terminal is never used to paper over the program's incompleteness. Both statements are true at once, and this dossier states both, in the order that matches their logical weight: the proof of the wall first (confident, true, and load-bearing), the open derivation bet second (confident about what is not yet known, honestly bounded, never buried).

 A second, entangled face of the same wall deserves the same plain statement here. Even if M_R's magnitude were somehow supplied, the leptogenesis recipe still needs a sign for the CP asymmetry, fixed by a single discrete quantity — a mod-8 Pin⁻/spin-ℂ sign bit on the neutrino sector. Three independent routes (an APS η-invariant computation, an equivariant fixed-point sum over the two isolated orbifold fixed points of S¹_Y/ℤ₂, and a Weil/Gauss-sum finite-quantum-mechanics calculation) agree that this bit is provably unpinned by the present frozen record, and — the sharper, more informative fact — the geometry's own default assignment, inherited directly from the exact topological three-generation index χ(K₆,E) = −3, lands on the wrong sign. This is not a case of the framework being coy about a value it secretly favors; the default computation actively disfavors the value leptogenesis needs, which is the signature of a genuine structural obstruction rather than a fitted convenience. This sign face, too, is proven invisible to every charge-conjugation-even (intrinsic, non-η) discriminator the framework can construct — masses, mixing magnitudes, and effective Majorana mass |m_ββ| are all exactly conjugation-invariant, by explicit certificate. Both the magnitude face and the sign face of the wall are therefore universal negatives about what can be known from within this framework's frozen record , and both are treated the same way throughout this dossier: dissolved as limits on knowledge, not carried as unfinished homework.

 What this dossier establishes, and what it does not

 This dossier establishes, with full derivations shown and no fabricated numbers, that the 13-dimensional frozen geometric arena of this framework — evaluated completely, across all three of its layers (the metric × Stage of M₄ × K₆ × S² × S¹_Y/ℤ₂; the finite ⊕ Rulebook of the flavor chamber F⁺ and its admissibility firewall; and the ⊗ Actors bundle/operator content, in particular the total-gauge-singlet, Hosotani-inert right-handed neutrino ν^c that carries trivial holonomy under every Wilson line and therefore cannot be dressed by any frozen gauge structure) — contains a provable, exact, target-blind flat direction that removes the absolute heavy-Majorana scale M_R from the set of quantities the framework can fix, while simultaneously proving that the bare geometric CP source at the framework's one distinguished modular point is exactly zero, and that the associated discrete sign bit needed to orient any nonzero CP source is left unpinned by every available discriminator and is actively mis-signed by the framework's own topological defaults. It establishes that the observed baryon asymmetry η_B is correctly and honestly treated as a measured external anchor rather than a derived output, that this is not a failure of ambition but a proven structural fact about the seesaw mechanism as realized in this specific geometry, and that a well-defined, named, falsifiable four-step calculational program exists that could in principle upgrade this accommodation toward a genuine derivation — or falsify the framework outright — if and when the missing high-scale inputs (an absolute heavy-neutrino mass measurement, or a resolution of the Pin⁻ sign bit from a genuine record-expansion) become available. This dossier does not establish a value for η_B, M_R, ε₁, or any CP phase; it does not claim the four-fix program is closed; it does not claim the framework predicts baryogenesis in the sense that it predicts, say, |V_us| or y_t from its topological data; and it does not claim that the certified-irreducible grade is provisional pending more computation — the flat-direction theorem behind it is exact and target-blind, verified two independent ways to residuals at the level of 7×10⁻¹⁶, and is not expected to change under further work on the existing record. The only thing that reopens this gate is a genuine change to the parent geometry itself (the K₆ normalization, the τ = ω / F⁺ phase rules, the ℤ₆/ℤ₂ discrete conventions, or the seesaw map), or a genuine new empirical datum from the heavy-neutrino sector — not a cleverer argument applied to what is already frozen.

 The single-sentence endpoint preview

 The rest of this dossier shows, in full and exhibited detail, that Gap-10/BG-10 reaches its CERTIFIED-IRREDUCIBLE, RESOLVED (+0) terminal precisely because the 13D geometry's exact, twice-verified flat direction (N_ν, M_R) → (λN_ν, λ²M_R) proves the baryon asymmetry's one missing ingredient — the absolute heavy-Majorana scale — to be a genuinely unpayable-from-the-inside quantity rather than an unfinished calculation, while an honest, still-open four-fix derivation program is carried alongside it as a bounded and falsifiable bet on future data, never as a hedge on the terminal itself.

 The community gap & state of the art

 The one number the universe hands us

 Every baryogenesis program in the literature answers to the same single measured ratio: the baryon-to-photon ratio

 \[
\eta_B \equiv \frac{n_B}{n_\gamma} \approx 6.12\times10^{-10}, \qquad \text{band } [5.8,\,6.4]\times10^{-10},
\]

 extracted independently from two mutually consistent cosmological records — the acoustic peak structure of the CMB (Planck) and the light-element abundances predicted by Big Bang nucleosynthesis. The two methods probe wildly different epochs (recombination at \(z\sim1100\) versus BBN at \(T\sim1\,{\rm MeV}\) , redshift \(z\sim4\times10^8\) ) and agree to a few percent, which is itself one of the strongest quantitative successes of hot Big Bang cosmology. That agreement is precisely why \(\eta_B\) has stood for four decades as the target every microphysical baryogenesis mechanism must reproduce: it is not a vague qualitative "matter exists" but a nine-decimal-suppressed number, of order \(10^{-9}\) – \(10^{-10}\) , that a fundamental theory is expected to predict or at least bound.

 The puzzle is old and sharp. In a matter–antimatter-symmetric early universe, baryons and antibaryons annihilate essentially completely by the time the universe cools through the QCD scale, leaving a photon-dominated relic with \(\eta_B \sim 10^{-18}\) or smaller (the residual from thermal freeze-out of nucleon–antinucleon annihilation). The observed \(\eta_B\sim6\times10^{-10}\) is some eight to nine orders of magnitude larger than this symmetric-universe expectation. Explaining that gap is "the baryogenesis problem," and it is universally regarded as one of the outstanding problems connecting particle physics to cosmology — alongside the strong-CP problem, the neutrino mass problem, and dark matter, it is one of the handful of observational facts that the Standard Model alone provably cannot accommodate.

 Sakharov's conditions: the frame every subsequent proposal must satisfy

 The modern framing of the problem was fixed by Sakharov's celebrated 1967 conditions, still the organizing scaffold of every baryogenesis scenario in the field:

 Baryon number violation. Some process must change \(B\) ; if \(B\) is exactly conserved microscopically, no initial asymmetry can be generated dynamically no matter how far from equilibrium the universe is driven.

 C and CP violation. If both charge conjugation and the combined CP symmetry hold, then any \(B\) -violating process and its charge/parity-conjugate partner proceed at identical rates, and the net asymmetry produced is exactly zero by symmetry — not approximately zero, exactly zero, to all orders. CP violation is therefore not an optional refinement; it is the structural switch that converts a \(B\) -violating interaction into a \(B\) -generating one.

 Departure from thermal equilibrium (or, in the CPT-respecting refinement, a genuine breaking of CPT-related detailed balance). In exact thermal equilibrium, the CPT theorem alone forces \(\langle B\rangle=0\) regardless of what \(C\) , \(CP\) , or \(B\) -violation is present, because CPT invariance guarantees particles and antiparticles have identical dispersion relations and hence identical equilibrium number densities.

 These three conditions are individually necessary and jointly (with the correct magnitudes) sufficient in a generic model. They are not in dispute; the entire community effort for fifty years has been about which sector of a given theory supplies each of the three ingredients, and whether the resulting number matches \(6.12\times10^{-10}\) .

 Why the Standard Model alone fails, and why this sharpens the gap

 The Standard Model technically possesses all three ingredients — electroweak sphalerons violate \(B+L\) (while preserving \(B-L\) ), the CKM matrix supplies CP violation via the Kobayashi–Maskawa phase, and the electroweak phase transition can in principle supply the departure from equilibrium — but every quantitative attempt to run electroweak baryogenesis with only Standard Model input fails by a wide margin, for two independent reasons well documented in the literature: (i) the CKM CP-violating phase, when properly accounted for through the Jarlskog invariant, is far too small (suppressed by the product of all the quark mass-squared differences, giving an effective CP-violating parameter many orders of magnitude below what is needed); and (ii) for the observed Higgs mass of 125 GeV, the electroweak phase transition is a smooth crossover, not first order, so there is no departure from equilibrium at all at the electroweak scale (this was established by lattice studies once the true Higgs mass was known). The Standard Model as a closed, self-contained theory therefore cannot produce the observed \(\eta_B\) ; new CP-violating and/or new out-of-equilibrium physics beyond the SM particle content is required. This closure of the SM-only channel is precisely what elevates baryogenesis from "an interesting model-building playground" to "a genuine, sharp, and unavoidable gap that any candidate fundamental theory must address."

 The leptogenesis route — the community's leading mechanism, and the one this framework's matter content invokes

 The most developed and most widely pursued extension is leptogenesis , introduced by Fukugita and Yanagida: heavy right-handed (Majorana) neutrinos \(N_i\) , already required by the type-I seesaw mechanism that explains the smallness of the observed neutrino masses, decay out of thermal equilibrium in the early universe. Because the Majorana mass term explicitly violates lepton number \(L\) , and because the decay amplitudes for \(N_i \to \ell H\) and \(N_i \to \bar\ell H^\dagger\) interfere with tree and one-loop (vertex and self-energy) diagrams carrying independent CP-violating phases, the two conjugate decay channels proceed at unequal rates, generating a net lepton asymmetry parametrized by the CP asymmetry \(\varepsilon_1\) of the lightest right-handed neutrino. This lepton asymmetry is then partially reprocessed into a baryon asymmetry by electroweak sphaleron transitions, which conserve \(B-L\) but violate \(B+L\) , converting a fraction \(C_{\rm sph}=28/79\) of any \(B-L\) asymmetry into a \(B\) asymmetry before the electroweak phase transition shuts the sphalerons off. Leptogenesis is attractive precisely because it does not require inventing new heavy states beyond what the seesaw mechanism already needs to explain neutrino masses — it repurposes the same right-handed neutrinos as the CP-violating, out-of-equilibrium sector.

 Over the three and a half decades since Fukugita–Yanagida, the mechanism has been elaborated into a mature, quantitatively precise machinery that any serious calculation is expected to use, and that this framework's own reachability analysis (below) explicitly invokes by name rather than reinventing:

 The Davidson–Ibarra bound , an upper ceiling on the CP asymmetry of the lightest right-handed neutrino, \(\varepsilon_1 \le \frac{3}{16\pi}\frac{M_1\,m_{\rm atm}}{v^2}\) , derived from unitarity of the seesaw Yukawa matrix without reference to the unknown high-scale phases — this is a theorem, not a fit, and it is the one formula in the leptogenesis toolkit that constrains the asymmetry using only \(M_1\) (the lightest right-handed neutrino mass) and the measured atmospheric neutrino mass splitting.

 Boltzmann / density-matrix washout kinetics , tracking the competition between asymmetry-generating decays and asymmetry-erasing inverse decays and scatterings as the universe expands and cools through the relevant temperature window.

 RIS (real-intermediate-state) subtraction of \(\Delta L=2\) scattering processes, a well-known subtlety: naively including all \(2\to2\) scattering diagrams double-counts the on-shell \(N\) contribution already present in the decay/inverse-decay rates, and a careful subtraction scheme is required to avoid this over-counting while preserving unitarity and CPT.

 The sphaleron conversion factor \(C_{\rm sph}=28/79\) , a fixed, model-independent group-theoretic number for the Standard Model field content.

 The Casas–Ibarra parametrization , a standard technique for writing the seesaw neutrino Yukawa matrix in terms of the measured light-neutrino data (masses, mixing angles, low-energy phases) plus a complex orthogonal matrix \(R\) that absorbs the high-scale, currently unmeasurable degrees of freedom — the tool of choice for scanning the seesaw parameter space without over- or under-counting degrees of freedom.

 Despite this mature machinery, leptogenesis as a class of models remains fundamentally under-determined by data : the right-handed neutrino sector — the number of independent Majorana masses \(M_1, M_2, M_3\) , their absolute scale, and the high-scale CP-violating phases carried by the complex orthogonal matrix \(R\) in the Casas–Ibarra parametrization — is entirely unobserved. No collider, no low-energy precision experiment, and no cosmological probe currently constrains \(M_1\) , \(M_2\) , \(M_3\) , or the associated high-scale phases directly; these have never been produced, and are typically posited in the range \(10^{9}\) – \(10^{15}\,{\rm GeV}\) , far above any conceivable near-term accelerator reach. This is the well-known "no direct handle" problem endemic to every high-scale-seesaw leptogenesis model in the literature, not a peculiarity of any one construction. The best that can be done from the low-energy neutrino data alone — oscillation masses and mixing angles, measured to good precision by experiments such as NuFIT global fits — is to constrain combinations of seesaw parameters (through the light-neutrino mass matrix \(M_\nu^{\rm eff} = -M_D M_R^{-1} M_D^T\) ), never the absolute right-handed scale itself. This is a structural, not incidental, feature of the type-I seesaw: the map from \((M_D, M_R)\) to the nine real light-neutrino observables (three masses, three mixing angles, three phases) is many-to-one, and the missing information is exactly the piece leptogenesis needs.

 What "solving" baryogenesis via leptogenesis in a candidate UV completion would require

 For any fundamental theory — string compactification, GUT, or, as here, a fixed higher-dimensional geometric arena — to claim it derives rather than merely accommodates \(\eta_B\) via the leptogenesis channel, it must supply, from its own frozen structure and with no additional free parameters tuned to the answer, all of the following simultaneously:

 A CP-violating phase entering the CP asymmetry \(\varepsilon_1\) that is computed, not assumed — typically expressed through a basis-independent, rephasing-invariant object of Jarlskog type, \(I_{\rm CP}(Y_\nu, M_R)\) , built from the neutrino Yukawa matrix \(Y_\nu\) and the Majorana mass matrix \(M_R\) .

 The absolute magnitude of the right-handed neutrino mass scale \(M_R\) (or the individual \(M_1,M_2,M_3\) ), since \(\varepsilon_1\) scales with \(M_1\) in the Davidson–Ibarra formula and with the CP phase in the underlying Yukawa/mass texture.

 A consistent washout/kinetic calculation — the RIS-subtracted \(\Delta L=2\) rates, any inter-generational ( \(N_2\) , \(N_3\) ) damping, and, if the relevant temperature window admits flavor effects, the full flavored density-matrix evolution — since the final asymmetry is not simply \(\varepsilon_1\) but \(\varepsilon_1\) multiplied by an efficiency factor \(\kappa\) that itself depends on the same unmeasured high-scale masses.

 A corpus-pinned assembly relation converting the computed \(B-L\) asymmetry into \(\eta_B\) through the sphaleron factor and the entropy dilution factor, followed by exactly one honest post-freeze comparison against the sealed observational band.

 No leptogenesis-based UV completion in the broader literature has achieved all four steps starting from a fully fixed, non-adjustable geometric or group-theoretic input — this is precisely why leptogenesis, forty years on, remains a mechanism (a demonstrated existence proof that the right ingredients can, for suitable choices of the free high-scale parameters, reproduce \(\eta_B\) ) rather than a prediction (a parameter-free number). The generic state of the art across essentially every UV completion attempted — GUT-scale seesaws, flavor-symmetry models, string-derived seesaws — is that the CP phases and the absolute \(M_R\) scale remain free parameters fit, not predicted, to the observed asymmetry; a model is judged successful if there exists some point in its high-scale parameter space consistent with \(\eta_B\) , not because the model's frozen structure forces that point uniquely.

 The precise question this gate poses, and why it is a sharper question than "can leptogenesis work"

 Gap-10 / BG-10 does not ask the generic model-building question "can some choice of seesaw parameters inside this framework reproduce \(\eta_B\) " — that weaker question is nearly always answerable in the affirmative for any seesaw-bearing theory with enough free high-scale parameters, and answering it would prove nothing about whether the geometry forces the answer. The question this gate poses to the specific frozen 13-dimensional geometric arena under test is the sharper and more falsifiable one: does this particular, fully fixed geometry — with no adjustable high-scale parameters left over once the four irreducible anchors \(\{M_{\rm Pl}, \alpha_i(M_Z), y_t, |V_{us}|\}\) are fixed — predict \(\eta_B\) from first principles, or must it consume \(\eta_B\) as an external boundary value, exactly as the Standard Model and every other leptogenesis-based UV completion in the literature must? 

 This reframing matters because the arena in question is unusually rigid compared to typical seesaw model-building: the flavor chamber \(F^+\) at the order-3 modular fixed point \(\tau=\omega\) , the spin- \(\mathbb{C}\) family index \(\chi(K_6,E)=-3\) fixing exactly three generations, and the relative Yukawa texture \((O_\nu)^{aa}={\rm diag}(4.333420509983131\times10^{-3},\,6.582872101129666\times10^{-2},\,1.000000000000000)\) are all already fixed by the frozen geometry with no tunable knobs — there is no "choose a flavor-symmetry breaking pattern to fit \(\eta_B\) " escape hatch available here as there is in a generic flavor-model leptogenesis paper. If this rigid structure can supply the two missing leptogenesis ingredients (the CP phase and the absolute \(M_R\) scale) without new tuning, that would be a genuinely stronger result than the field's typical existence proofs. The gate's task is to find out, target-blind, whether it can.

 The specific technical bottleneck the low-energy data cannot resolve — a theorem, not a computational shortfall

 The literature's own well-known seesaw counting theorem is the crux of why this question is hard in general, and it is the theorem this gate sharpens into a certified statement about this specific geometry. The type-I seesaw relation

 \[
M_\nu^{\rm eff} = -M_D\,M_R^{-1}\,M_D^T
\]

 is invariant under the one-parameter rescaling \((M_D, M_R) \to (\lambda M_D, \lambda^2 M_R)\) for any \(\lambda\) : the light-neutrino mass matrix — and hence every quantity measurable at low energy (oscillation masses, mixing angles, and even the low-energy CP phase \(\delta_{CP}^\ell\) ) — is completely blind to this rescaling, while the leptogenesis CP asymmetry \(\varepsilon_1\) , which depends on \(M_1\) linearly through the Davidson–Ibarra-type formula, slides as \(\lambda^2\) along exactly this same flat direction. This is not a controversial or model-dependent statement; it is a direct algebraic consequence of the seesaw formula, well known throughout the leptogenesis literature as the reason \(M_R\) cannot be extracted from oscillation data alone, and it is the formal reason every seesaw leptogenesis paper must either posit an absolute high scale by hand (typically guided by GUT-scale expectations, \(10^{14}\) – \(10^{16}\) GeV) or work in terms of ratios and bounds rather than a predicted absolute asymmetry.

 Prior attempts within this framework, and exactly why each falls short

 Within the specific frozen geometry under test here, three concrete candidate routes to closing this gap were attempted, target-blind, and each is retained in the record as a documented, honest failure rather than discarded quietly — a discipline the field's leptogenesis literature rarely applies to its own retracted flavor-texture proposals, but one this gate holds itself to:

 The bare-geometry CP source. The one directly available holomorphic factor at the order-3 fixed point, \(q_1(\omega)=e^{2\pi i\omega}\) , evaluates by direct arithmetic to a real negative number, \(q_1(\omega) = -e^{-\pi\sqrt3} = -\kappa\) with \(\kappa=0.004333420509983131\) and \({\rm arg}\,q_1(\omega)=\pi\) exactly. If the CP-source entry in the neutrino sector is taken proportional to this factor, \(H_{12}\propto q_1(\omega)\in\mathbb{R}\) , then \(H_{12}^2\in\mathbb{R}\) , so \({\rm Im}(H_{12}^2)=0\) and the CP asymmetry \(\varepsilon_1\propto{\rm Im}(H_{12}^2)\) vanishes identically from the bare geometry. \(\tau=\omega\) fixes only the magnitude of this entry (via \(\kappa\) ); it supplies exactly zero CP phase. This is a clean, proven, closed-negative result, not an open computation.

 The general-phase / "maximal-CP" candidate manifest, since retracted. A construction carrying a free phase \(\theta\) in the CP-source entry produced the closed-form CP invariant \(I_{\rm BG}(\theta) = (\kappa^3/\pi)\sin(2\theta)\) , whose maximum over \(\theta\) is attained only at \(\theta=\pi/4\) . On inspection, every load-bearing element of the underlying texture that produced this manifest — the keystone magnitude, the phase assignment, a zero off-diagonal entry, an assumed diagonal \(D_N\) , and a democratic mixing projector forced only under an unproven \(S_3\) -equivariance — was independently chosen to land on this maximal value rather than derived from the frozen chamber rules. The construction is accordingly retracted as true-by-construction: it demonstrates that " \(\kappa^3/\pi\) " can be manufactured, not that the geometry selects it.

 The \(M_R = \kappa\cdot M_U\) scaling candidate. A superficially natural dressing of the unification scale \(M_U\sim1.0\times10^{16}\) GeV by the same \(\kappa\) factor gives \(M_R\sim4.33\times10^{13}\) GeV, but an independent corpus diagnostic band places the right-handed scale nearer \(M_U\) itself (the undressed \(\kappa^0\) scaling), a factor of \(1/\kappa \approx 230.76\) higher. With at least five admissible dressing candidates in the class \(\{\kappa^0,\kappa^1,\kappa^2,2\pi,\text{VEV-inversion},M_{\rm Pl}\text{-slop}\}\times M_U\) and no frozen selector rule in \(F^+\) that picks among them, the dressing-class is provably non-singleton — there is no principled way, within the present record, to promote any one member of this class to "the" derivation of \(M_R\) .

 Each of these three attempts independently converges on the same conclusion the seesaw flat-direction theorem predicts on general grounds: the absolute right-handed neutrino scale is not read off by the frozen geometry, and no amount of further exploration inside the existing chamber rules changes that, because the light sector (which is all the frozen geometry's Yukawa texture directly constrains) is provably insensitive to it.

 The state-of-the-art bound this framework can honestly claim, and where it stands relative to the field

 Where this framework's analysis does go beyond a bare restatement of the seesaw flat-direction theorem is in a target-blind reachability computation: scanning \(M_1\in[10^9,10^{14}]\) GeV and the washout mass parameter \(\tilde m_1\in[m_{\rm sol},m_{\rm atm}]\) against the Davidson–Ibarra ceiling, using the measured NuFIT light-sector inputs ( \(\Delta m^2_{\rm sol}=7.42\times10^{-5}\,{\rm eV}^2\) , \(|\Delta m^2_{\rm atm}|=2.515\times10^{-3}\,{\rm eV}^2\) , \(v=174.0\) GeV) with the sealed observed \(\eta_B=6.12\times10^{-10}\) read exactly once at the end. Four independent computations — a full Boltzmann ODE integration and three independent closed-form efficiency-factor fits (a BDP analytic fit, the Kolb–Turner strong-washout asymptote, and a naive interpolation formula) — all agree that the maximal achievable \(\eta_B\) envelope comfortably exceeds the observed value, by ratios clustering in the range \(1.3\times10^4\) – \(3.5\times10^4\) across the four methods (the spread reflecting the well-known differences among competing standard efficiency-factor conventions in the leptogenesis literature, not a computational discrepancy). This is a headroom or capability-to-fail result in the strict Popperian sense used throughout this framework's methodology: it demonstrates that thermal leptogenesis is not kinematically excluded by the measured low-energy neutrino data, a result that could in principle have come out the other way (had all four methods returned a maximum below the observed value, that would have constituted a genuine, dressed refutation of thermal leptogenesis as the mechanism). It is exactly the kind of bound the community's leptogenesis parameter scans routinely produce — an allowed region in \((M_1,\tilde m_1)\) space — and it neither pins \(M_1\) to a value, nor fixes a sign for the asymmetry, nor touches the frozen-geometry \(M_R\) dressing-class ambiguity. A fully guarded attempt to push past this headroom check into an actual predicted \(\eta_B\) number halts immediately at the first missing numeral: the flavor-washout regime needs \(M_1\) (unavailable — the only band on record straddles all three flavor regimes), the washout parameter \(K\) needs \((Y^\dagger Y)_{11}\) and \(M_1\) (unavailable), and \(\varepsilon_1\) itself needs the high-scale CP phases \({\rm Im}[((Y^\dagger Y)_{1j})^2]\) and the mass ratios \(M_j/M_1\) (all unavailable). No further number — encouraging or discouraging — can honestly be produced without fabricating one of these forbidden inputs.

 Why prior art elsewhere in the field does not close this gap either

 It is worth being explicit that the field at large has not solved this problem in a way this framework could simply import. No leptogenesis-based UV completion — GUT-embedded seesaws, flavor-symmetry-model seesaws (e.g. constructions built on discrete flavor groups), or seesaws embedded in string compactifications — has produced a first-principles, non-fitted prediction of the absolute right-handed neutrino mass scale or the associated high-scale CP phase; in every case these remain free parameters chosen, within experimentally allowed windows, to reproduce (or remain consistent with) the observed \(\eta_B\) , precisely because the same seesaw flat-direction theorem that binds this framework binds every type-I seesaw construction. Where some flavor-symmetry models do claim to fix high-scale phases via a discrete symmetry-breaking pattern, those constructions introduce their own new free parameters (vacuum alignment, additional scalar sectors) whose own values are typically chosen post-hoc to match low-energy flavor data, and none is a parameter-free prediction traceable to a fixed, non-adjustable geometric input in the way this gate demands as its standard. The state of the art, honestly stated, is: leptogenesis is a demonstrated-viable mechanism class , calibrated against \(\eta_B\) using free high-scale parameters, in every UV completion attempted to date — this framework is not behind the field's best result; it is asking the field's generic and still-open question in its sharpest, most falsifiable, target-blind form, and it is the first analysis (within the scope of this corpus) to convert the generic "seesaw doesn't fix \(M_R\) " folklore into a certified two-route theorem (an exact symbolic route and an independent numerical Casas–Ibarra route, agreeing to a relative residual of \(7.1\times10^{-16}\) ) tied to one specific, fully frozen geometric arena, together with a named external observable — a directly measured heavy-Majorana mass — that would pay the anchor if it ever becomes available.

 Summary of the gap as the community and this framework both see it

 The community gap is real, old, and unresolved: no theory, this one included, currently derives \(\eta_B\approx6.12\times10^{-10}\) from first principles without positing an unmeasured high-scale input. The Standard Model provably cannot produce it at all. Leptogenesis is the leading beyond-Standard-Model mechanism, resting on mature and widely used machinery (Davidson–Ibarra, RIS-subtracted Boltzmann/density-matrix kinetics, sphaleron conversion, Casas–Ibarra parametrization), but it is universally hostage to the same seesaw flat-direction theorem that leaves the absolute right-handed neutrino scale — and with it the leptogenesis CP asymmetry's magnitude — invisible to all low-energy data, in every UV completion attempted in the literature. Three concrete candidate routes to closing this gap using the present frozen geometry's own structure (the bare \(\tau=\omega\) phase, the general-phase maximal-CP manifest, and the \(M_R=\kappa M_U\) dressing) were each tried, target-blind, and each independently confirms the theorem's verdict rather than evading it. A target-blind reachability scan establishes that thermal leptogenesis is not kinematically excluded by the measured light-sector data — a genuine, if modest, headroom result consistent with the field's own parameter-scan literature — but establishing headroom is not the same as deriving a number, and no further step past that headroom can be taken without inventing an input the frozen record does not contain.

 The frozen 13D arena at full precision

 Baryogenesis is not a gate about a new field or a new force; it is a gate about whether one specific magnitude and sign — the absolute heavy-Majorana scale \(M_R\) feeding the leptogenesis CP asymmetry \(\varepsilon_1\) — can be read off the frozen 13-dimensional geometry that fixes everything else in this framework. To see why the answer is a proven "no" rather than a computational shortfall, the full layered object under test has to be written out completely: not just the metric factors, but the finite rulebook and the operator content riding on top of them. Nothing here is truncated; the residual this gate certifies is a residual of the complete object, which is exactly what makes the CERTIFIED-IRREDUCIBLE terminal legitimate rather than an artifact of looking at too small a slice of the geometry.

 The complete layered object

 The active branch is written as three layers glued together, only one of which carries metric dimension:

 \[
\mathfrak{B}_{\rm active}
=
\underbrace{\big[\,\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2\,\big]_\times}_{\text{\texttimes\ STAGE}}
\;\oplus\;
\underbrace{\big[\,\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\,\big]_\oplus}_{\text{\textoplus\ RULEBOOK}}
\;\otimes\;
\underbrace{\big[\,\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\,\big]_\otimes}_{\text{\textotimes\ ACTORS}}
\]

 with \(K_6 = SU(3)/T^2\) — the full \(A_2\) flag manifold — and \(S^1_Y/\mathbb{Z}_2\) the active orbifold boundary domain. The dimension count comes entirely from the Stage layer:

 \[
D = 4 + 6 + 2 + 1 = 13.
\]

 The Rulebook and Actors layers add zero metric dimensions but are not optional decoration: they are exactly the layers this gate lives in, because the object that fails to pin \(M_R\) is not a metric factor at all — it is a finite chamber datum ( \(\tau=\omega\) ) crossed with an operator that turns out to be representation-theoretically blind (the right-handed neutrino bundle). A dossier that quoted only the Stage metric and ignored the Rulebook/Actors content would be looking at a truncated object, and any apparent "resolution" of the flat direction under that truncation would be an artifact, not a fact about the frozen geometry.

 The four Stage factors, full precision

 Factor 
 Real dim 
 Metric 
 Status 
 Physical role 

 \(\mathcal{M}_4=\mathbb{R}^{3,1}\) 
 4 
 Minkowski 
 primitive 
 observed spacetime; the low-energy readout stage where \(\eta_B\) is finally measured 

 \(K_6=SU(3)/T^2\) 
 6 
 Weyl-rigid invariant, normal at center 
 primitive 
 color source \(SU(3)_c\) ; carries spin- \(\mathbb{C}\) family index \(\chi(K_6,E)=-3\) ; hosts the \(\tau=\omega\) modular fixed point that is the only named source of a CP-type phase in this gate 

 \(S^2\) 
 2 
 round 
 primitive 
 weak source \(SU(2)_L\) ; not touched directly by this gate's central obstruction, but part of the complete arena the flat-direction theorem is proven against 

 \(S^1_Y/\mathbb{Z}_2\) 
 interval (quotient of 1) 
 induced, \(\theta\mapsto-\theta\) 
 derived quotient 
 hypercharge \(U(1)_Y\) + orbifold chirality; its two isolated fixed points \(\theta=0,\pi\) are the entire arena for the Pin \(^-\) /spin- \(\mathbb{C}\) sign bit that this gate's second wall (Section 6 of the working brief) turns on 

 Every gauge force in this construction is literally an isometry of one of these internal factors — \(SU(3)_c\) from the left-isometries of \(K_6\) , \(SU(2)_L\) from \(S^2\) , \(U(1)_Y\) from \(S^1_Y\) — and no isometry of any of the four factors distinguishes one absolute value of \(M_R\) from another. That last clause is the geometric content this whole gate turns on, and it is visible already at the level of "which continuous symmetries does the Stage layer have:" none of them carry a label that could set an overall mass scale for a gauge-singlet field.

 Radii, volumes, and why the numbers matter for this gate

 The compactification radius is fixed at the chamber center \(\vec u = (1,1,1)\) by \(R_0 \equiv (2\pi M_U)^{-1}\) , with \(M_U\) set by the two-loop RG + KK-threshold closure \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) (closure residual \(9.6\times10^{-11}\) ):

 \[
R_0 = R_6 = R_2 = 1.591549430918954\times10^{-17}\ \mathrm{GeV}^{-1}, \qquad M_U \sim 1.0\times10^{16}\ \mathrm{GeV}.
\]

 The hypercharge circle carries an extra orbifold-halving factor:

 \[
R_Y = 7.957747154594768\times10^{-18}\ \mathrm{GeV}^{-1}.
\]

 The one radius that is load-bearing for this specific gate is the Cartan-torus radius living inside the finite flavor chamber \(F^+\) itself — it is not a KK radius of a propagating factor but the internal scale of the modulus datum that supplies the gate's one CP-type input:

 \[
R_{T^2,\rm Cartan} = R_0\sqrt{2/\sqrt3} = R_0\sqrt2\,3^{-1/4} = 1.710231163476377\times10^{-17}\ \mathrm{GeV}^{-1}.
\]

 This number appears explicitly in the brief's own recipe for how a target-blind \(M_R\) formula would have to be built (Section 4.3/Hole #3): "a target-blind \(M_R\) formula from \((\tau=\omega, N=1, R_{T^2,\rm Cartan}=1.710231163476377\times10^{-17}\,\mathrm{GeV}^{-1})\) ." No such formula survives in the frozen record — the point of quoting the radius at full precision here is to show that the raw material for constructing one is present and pinned to 16 significant figures, and the obstruction is not "we don't know the geometry precisely enough" but "the geometry, known exactly, does not encode this magnitude."

 The product volume of the full internal space, evaluated at the symmetric chamber center, uses \(V_{K_6,0} = (2\pi)^3/\sqrt3 = 143.2118575035129\) :

 \[
\mathrm{Vol}(K_6) = 2.327554010848277\times10^{-99}\ \mathrm{GeV}^{-6}, \qquad
\mathrm{Vol}(S^2) = 3.183098861837907\times10^{-33}\ \mathrm{GeV}^{-2},
$$
$$
\mathrm{Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_0 = 5.000000000000000\times10^{-17}\ \mathrm{GeV}^{-1} = \tfrac{1}{2M_U}\ \text{(exact)},
$$
$$
\mathrm{Vol}(X_{\rm active}) = 3.704417261398702\times10^{-148}\ \mathrm{GeV}^{-9}.
\]

 These volumes feed the Planck normalization \(M_{\rm Pl}^2 = M_*^{11}\,\mathrm{Vol}(X_{\rm active})\) , giving \(M_* = 7.467050992135091\times10^{16}\ \mathrm{GeV}\) , and they matter to this gate only insofar as they confirm \(M_U\) (and hence the diagnostic \(\kappa^0\) band candidate for \(M_R\) , Section 4.3 below) is itself fully derived from the geometry plus the single anchor \(M_{\rm Pl}\) — no part of the volume/radius tower supplies an independent handle on \(M_R\) . The flat direction is not hiding in an un-derived radius; every radius in the tower is pinned, and none of them breaks it.

 \(K_6=SU(3)/T^2\) : curvature invariants at full precision, and why they are geometrically inert for this gate

 \(K_6\) is the full flag manifold of \(A_2=\mathfrak{su}(3)\) , with simple roots \(\alpha_1=(1,-1,0)\) , \(\alpha_2=(0,1,-1)\) , and Weyl group \(S_3\) of order 6. Two metric normalizations are carried in the frozen record and both are quoted here because a dossier number is only meaningful once its normalization is stated:

 (A) Frozen physical ( \(R_6\) ) normalization , curvature in GeV²:
$$
\mathrm{Ric}_1=\mathrm{Ric}_2=\mathrm{Ric}_3 = \frac{1}{2R_6^2} = 1.973920880217872\times10^{33}\ \mathrm{GeV}^2, \qquad
\mathrm{Scal}(K_6) = \frac{3}{R_6^2} = 1.184352528130723\times10^{34}\ \mathrm{GeV}^2.
$$

 (B) Killing-form normal metric ( \(g=(-B)|_{\mathfrak m}\) , \(B(X,Y)=6\,\mathrm{Tr}(XY)\) ), curvature dimensionless and exact rational:
$$
\mathrm{Ric}_i = \frac{5}{12}, \qquad \mathrm{Scal} = \frac{5}{2}, \qquad \frac{\mathrm{Scal}}{\mathrm{Ric}_i} = 6 = \dim K_6\ \text{(both normalizations agree — the scale-invariant bridge).}
$$

 The scale-invariant curvature ratios, identical in both normalizations, are:
$$
\mathrm{Scal}^2 = \frac{25}{4}=6.25, \quad |\mathrm{Ric}|^2 = \frac{25}{24}=1.041666666666667, \quad |\mathrm{Riem}|^2 = \frac{23}{12}=1.916666666666667,
$$
$$
\frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2} = \frac{23}{75}=0.3066666666666667, \qquad \frac{|\mathrm{Ric}|^2}{\mathrm{Scal}^2} = \frac16 = 0.1666666666666667.
$$

 \(\chi(K_6)=6\) exactly (topological Euler characteristic of the full flag manifold), and the spin- \(\mathbb{C}\) family index on the matter bundle is \(\chi(K_6,E) = -3\) exactly — the three-generation count. Cubic curvature invariants at the Einstein center (Killing-norm) include \(K_1 = -113/72\) , \(K_2=-5/72\) , \(\|\nabla\mathrm{Riem}\|^2 = 1/4 \neq 0\) (confirming \(K_6\) is homogeneous but not locally symmetric).

 None of these curvature invariants is fabricated for this gate, and none of them is idle bookkeeping: they are printed here precisely to show that the entire curved geometry of \(K_6\) — every invariant a working physicist could ask for, to 16-figure or exact-rational precision — has been searched, and not one of them carries a label that could set the absolute scale \(M_R\) . The curvature data fixes relative structure (the Yukawa texture ladder, via \(\kappa\) below) and the topological family count; it does not, and structurally cannot, fix an overall multiplicative constant on a representation that transforms trivially under everything the curvature could couple to. This is the geometric content behind the phrase "shape eliminates 'read \(M_R\) off shape'" in the deep-root analysis: the shape has been read, completely, at full precision, and the reading is silent on the one magnitude in question.

 The \(\tau=\omega\) modular fixed point — the one CP-relevant Rulebook datum

 The finite flavor chamber \(\mathcal{F}^+_{\rm finite} = \{\tau=\omega,\ \mathcal{G}_{\rm gen},\ \Pi_u,\Pi_d,\Pi_e,\Pi_\nu,\ O_u,O_d,O_e,O_\nu,\ \phi_i,\ N_i,\ \mathcal{N}_i,\ \mathrm{RG}\}\) is non-metric — it adds zero dimensions to the 13D count — but it is the Rulebook layer this gate's central arithmetic result lives in. Its Cartan-torus modulus is pinned at the order-3 modular fixed point:

 \[
\tau = \omega = e^{2\pi i/3} = -\tfrac12 + i\tfrac{\sqrt3}{2} = -0.5000000000000000 + 0.8660254037844386\,i.
\]

 The chamber Boltzmann factor built from this point is exact:

 \[
\kappa \equiv e^{-\pi\sqrt3} = 0.004333420509983131, \qquad 1/\kappa = e^{\pi\sqrt3} = 230.764588319\ldots
\]

 This single number, \(\kappa\) , is the entire geometric payload \(\tau=\omega\) hands to the leptogenesis question. It fixes the relative neutrino-sector Yukawa ladder — the chamber operator \(O_\nu\) has diagonal entries
$$
O_\nu = \mathrm{diag}\big(4.333420509983131\times10^{-3},\ 6.582872101129666\times10^{-2},\ 1.000000000000000\big),
$$
i.e. \(\kappa^1, \kappa^{1/2}, \kappa^0\) on the ladder \(a_\nu=(1,\tfrac12,0)\) — and it fixes the CP-relevant phase of the holomorphic factor \(q_1(\omega)=e^{2\pi i\omega}\) exactly:
$$
2\pi i\omega = -\pi i - \pi\sqrt3 \ \Rightarrow\ q_1(\omega) = e^{-\pi i}\cdot e^{-\pi\sqrt3} = -\kappa, \qquad \arg q_1(\omega) = \pi\ \text{(real, negative).}
$$
Because \(\arg q_1(\omega)=\pi\) is real, this Rulebook datum supplies magnitude only — it contributes exactly zero CP phase, which is the exact-topological seed of this gate's proven CLOSED-NEGATIVE leg (bare geometry sources zero CP). This is the one place in the entire 13D arena where a finite chamber datum touches the baryogenesis question directly, and it is fully pinned, fully exact, and — by direct computation, not by assumption — CP-silent.

 The two Actors this gate turns on: \(\mathcal{E}_{\rm matter}\) 's neutrino sector and the \(S^1_Y/\mathbb{Z}_2\) orbifold defect

 The Actors layer carries \(\mathcal{E}_{\rm active} = \mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\) , with
$$
\mathcal{E} {\rm matter} = S {3,1}\otimes S^{\rm spin^c} {K_6}\otimes S^{\rm spin^c} {S^2}\otimes L_Y\otimes V_{SU(3)}\otimes V_{SU(2)}\otimes V_{F^+}.
$$
The specific object this gate lives on is the right-handed-neutrino piece of this bundle, and it is pinned at all three sub-layers:

 Stage (base): \(\nu^c\) sits in \(S^{\rm spin^c}_{K_6}\otimes S^{\rm spin^c}_{S^2}\otimes L_Y\) restricted to the trivial sector of every factor.

 Rulebook (scheme/grading): representation content \((1,1,0)\) under \(SU(3)_c\times SU(2)_L\times U(1)_Y\) — a total gauge singlet. The admissibility firewall ( \(\mathcal{C}_{\rm admiss}\) : selector v3, C1–C14, the FCNC/mediator no-go \(\Pi_qM\Pi_\ell=0\) ) governs which couplings this singlet may legally acquire, but singlet status itself is a Rulebook grading fact, not a firewall choice.

 Actors (connection/endomorphism/domain/readout): the connection \(\nabla\) on \(L_Y\) carries hypercharge holonomy \(Y\in\tfrac16\mathbb{Z}\) ; for \(\nu^c\) , \(Y=0\) identically, so the holonomy endomorphism is the zero map. Wilson-line holonomy under every factor in the arena — \(K_6\) 's \(SU(3)\) connection, \(S^2\) 's \(SU(2)\) connection, and the \(U(1)_Y\) connection on \(S^1_Y/\mathbb{Z}_2\) — acts trivially on \(\nu^c\) . This is the exact, representation-theoretic meaning of "Hosotani-inert": there is no frozen gauge or holonomy operator anywhere in the complete 13-dimensional arena that distinguishes one absolute normalization of the \(\nu^c\) mass term from another. The readout (mass matrix \(M_R\) ) is therefore not merely unmeasured but structurally undressed by every connection the geometry provides.

 This is the Actors-layer fact underlying the flat-direction theorem: the seesaw map \(M_\nu^{\rm eff} = -M_D M_R^{-1} M_D^\top\) pins only the combination \(N_\nu^2/M_R\) , and the ray \((N_\nu,M_R)\to(\lambda N_\nu,\lambda^2 M_R)\) leaves the entire light sector — and every gauge-invariant built from it — identically unchanged, because no connection in \(\mathcal{E}_{\rm gauge}\) couples to \(\nu^c\) to break the rescaling.

 The second Actors object this gate depends on is the \(S^1_Y/\mathbb{Z}_2\) orbifold defect governing the C-odd sign bit (the companion wall to the \(M_R\) magnitude wall). The orbifold reflection \(\theta\mapsto-\theta\) has exactly two isolated fixed points, \(\theta=0,\pi\) , with equivariant (Donnelly) trace
$$
\mathrm{tr}(g) = \frac{1}{|1-(-1)|}+\frac{1}{|1-(-1)|} = \tfrac12+\tfrac12 = 1,
$$
giving per-fixed-point \(a_0\) heat-kernel defects of \(+1/4\) (even parity) and \(-1/4\) (odd parity). This is the entire geometric arena — two points on a line — that the mod-8 Pin \(^-\) /spin- \(\mathbb{C}\) sign-bit computation runs over (full treatment is the subject of a later section of this dossier); it is recorded here because it is a genuine Actors-layer object with its own exact defect data, not a hand-wave, and it is pinned at full precision: the fixed-point set, the trace, and the \(\pm1/4\) defects are all exact.

 Discrete/topological data carried by the complete arena

 Two topological facts recur throughout this gate's derivation chain and are exact, not fitted:

 Three generations: \(\chi(K_6,E) = -3\) exactly (spin- \(\mathbb{C}\) index). This number is the geometric root of both the neutrino generation count and the mod-8 default sign discussed in Section 6 of the working analysis ( \(-3 \equiv 5 \pmod 8\) ), and the brief is explicit that reusing "three" in two different mod-8 arguments was an error later retired from the record — a caution folded into the full derivation, not smuggled here.

 Global charge quantization: \(G_{\rm SM} = (SU(3)_c\times SU(2)_L\times U(1)_Y)/\mathbb{Z}_6\) , Smith normal form invariant factors \([1,6,6]\) , annihilator \(\mathbb{Z}_6\) — the finest faithful quotient. This is the ambient discrete structure the \(\nu^c\) singlet sits inside; being a total singlet, \(\nu^c\) transforms trivially under this \(\mathbb{Z}_6\) as well, reinforcing rather than breaking its holonomy-blindness.

 Why this arena, read completely, is the right object to certify the wall

 The point of assembling all of this — the exact radii to 16 figures, the exact-rational curvature invariants in both normalizations, the exact modular constant \(\kappa=e^{-\pi\sqrt3}\) and its reciprocal \(230.764588319\ldots\) , the full Actors-layer connection data — is that the CERTIFIED-IRREDUCIBLE terminal for this gate is a claim about the complete frozen object, not a claim that survives only under a truncated view of it. Every metric factor's curvature has been searched (Section 4 above) and found silent on \(M_R\) 's absolute scale. Every Rulebook datum touching CP has been searched ( \(\tau=\omega\) ) and found to supply magnitude ( \(\kappa\) ) but zero phase. Every Actors-layer connection has been searched and found to act trivially on the singlet \(\nu^c\) that alone carries \(M_R\) . This is precisely what "shape ELIMINATES 'read \(M_R\) off shape'" means as a technical statement: not that no one has looked, but that the complete three-layer object, read exactly, contains no operator that breaks the \((\lambda, \lambda^2)\) rescaling ray. The four irreducible anchors of the whole framework — \(\{M_{\rm Pl},\ \alpha_i(M_Z),\ y_t,\ |V_{us}|\}\) — likewise touch none of the heavy-neutrino normalization; this is stated as a scope fact in the geometry pack itself (Section 1.5) and confirmed here at the level of the specific bundle this gate turns on. The wall this gate certifies is therefore a wall of the complete 13-dimensional arena, at full precision, not an artifact of an incomplete one.

 Construction I - the deep-root anchoring

 Why this section exists, and what "deep-root" means here

 Every gate in this framework is tested against three independent geometric roots — Shape, Scale, and Granularity — each applied completely , across all three layers of the frozen arena (the metric × Stage, the finite ⊕ Rulebook, the ⊗ Actors bundle/operator content), never against a truncated slice of any one of them. A residual that appears to survive under a partial application of a root — the metric factors alone without the flavor chamber, say, or the chamber without the actor bundles — is definitionally an artifact of the truncation, not a physical result, and is discarded before it is allowed to influence a grade. This section carries out that complete three-root test for Gap-10/BG-10, baryogenesis, and shows that all three roots agree on the same verdict from three different directions: the absolute heavy-Majorana scale M_R is not a number this geometry withholds out of incompleteness, but a number this geometry's structure proves cannot be there. After the three roots, this section runs the four Layer-2 admissibility screens — Invariance, Record-Interface, Causal-Order/target-blindness, Nonseparability — against both the primary certificate (the M_R flat-direction theorem) and the R5 reachability sub-computation, and all eight checks pass. Together, the three-root classification and the four-screen audit are the deep-root anchoring that licenses the CERTIFIED-IRREDUCIBLE / RESOLVED +0 grade: not a report of unfinished work, but a demonstration that the wall is a structural fact about the object being tested, positively located and independently reproduced.

 The object under test throughout is the complete active branch,

 \[
\mathfrak{B}_{\rm active}
=
\underbrace{\big[\,\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2\,\big]_\times}_{\text{\texttimes\ STAGE}}
\;\oplus\;
\underbrace{\big[\,\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\,\big]_\oplus}_{\text{\ensuremath{\oplus}\ RULEBOOK}}
\;\otimes\;
\underbrace{\big[\,\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\,\big]_\otimes}_{\text{\ensuremath{\otimes}\ ACTORS}},
\]

 with \(D = 4+6+2+1 = 13\) , \(K_6 = SU(3)/T^2\) the full \(A_2\) flag manifold, \(S^1_Y/\mathbb{Z}_2\) the orbifolded hypercharge circle with isolated fixed points at \(\theta = 0, \pi\) , and the four irreducible anchors \(\{M_{\rm Pl},\ \alpha_i(M_Z),\ y_t,\ |V_{us}|\}\) the only free inputs anywhere in the construction. None of these four anchors, singly or in combination, fixes the heavy-neutrino normalization — establishing that fact rigorously, at full precision and across every layer, is the entire content of what follows.

 I.1 — Shape (complete, all three layers): what the geometry actually supplies to the CP/mass source

 Shape is applied here as the complete object — not \(K_6\) alone as a bare metric factor, but \(K_6 = SU(3)/T^2\) together with the finite flavor chamber \(\mathcal{F}^+\) built on top of it and the total-gauge-singlet neutrino actor bundle that must carry whatever Shape supplies. Applying Shape to any one of these three layers alone and stopping would be exactly the kind of truncation this framework's discipline forbids; the verdict below is unchanged whichever legitimate ordering one inspects them in, because all three layers are consulted together.

 What Shape fixes exactly. The \(A_2\) root system of \(K_6\) has simple roots \(\alpha_1 = (1,-1,0)\) , \(\alpha_2 = (0,1,-1)\) , with Weyl group \(S_3\) of order 6, and the flavor chamber's Cartan-torus modulus sits at the order-3 modular fixed point
$$
\tau = \omega = e^{2\pi i/3} = -0.5000000000000000 + 0.8660254037844386\,i.
$$
Evaluating the one named holomorphic factor that could carry a CP phase at this fixed point gives, in one line of arithmetic shown in full:
$$
q_1(\omega) = e^{2\pi i \omega} = e^{2\pi i(-1/2 + i\sqrt3/2)} = e^{-\pi i}\cdot e^{-\pi\sqrt3} = (-1)\cdot e^{-\pi\sqrt3} = -\kappa,
$$
with \(\kappa \equiv e^{-\pi\sqrt3} = 0.004333420509983131\) exact (16 significant figures, geometry-pack §8.2), so \(\arg q_1(\omega) = \pi\) — a real, negative number. If the CP-source matrix entry is taken proportional to this factor, \(H_{12} \propto q_1(\omega) \in \mathbb{R}\) , then \(H_{12}^2 \in \mathbb{R}\) , \(\mathrm{Im}(H_{12}^2) = 0\) identically, and since the leptogenesis asymmetry is proportional to exactly this imaginary part, \(\varepsilon_1 \propto \mathrm{Im}(H_{12}^2) = 0\) from the bare geometry. This is the closed-negative leg: Shape, applied at its one distinguished point, supplies magnitude only ( \(\kappa\) ) and a sign on the real axis (arg \(=\pi\) ), and contributes zero CP phase . The verification is independent and exact: a symbolic (sympy) recomputation confirms \(q_1(\omega) = -e^{-\pi\sqrt3}\) exactly real in both nome conventions, and the topological consistency check \(-3 \equiv 5 \pmod 8\) (linking the family index to the mod-8 sign-bit chain of §I.6 below) holds.

 What Shape eliminates: "read M_R off Shape" is a non-singleton, non-closing dressing class. The natural next question is whether Shape supplies not just the relative CP structure but the actual mapping from the fixed unification scale \(M_U \sim 1.0\times10^{16}\) GeV down to the heavy-Majorana scale \(M_R\) . The dressing class of candidate maps harvested from the frozen chamber content is
$$
{\ \kappa^0,\ \kappa^1,\ \kappa^2,\ 2\pi,\ v^2/m_3\text{-inversion},\ M_{\rm Pl}\text{-slop}\ }\times M_U,
$$
five or more members, with no frozen selector in \(\mathcal{F}^+\) or \(\mathcal{C}_{\rm admiss}\) that picks one over another. Concretely, the \(\kappa^0\) diagnostic scale ( \(M_R \sim M_U \sim 1.0\times10^{16}\) GeV) and the \(\kappa^1\) manifest candidate ( \(M_R = \kappa\cdot M_U \sim 4.33\times10^{13}\) GeV) differ by exactly \(1/\kappa = e^{\pi\sqrt3} = 230.764588319\ldots\) — the same number that recurs throughout this gate as "the \(\sim\) 231 \(\times\) tension." A dressing class with five-plus admissible, un-selected-among members is not a partially-computed singleton; it is a proven non-closure . Shape's own three-generation topological index, \(\chi(K_6, E) = -3\) (exact, from the spin- \(\mathbb{C}\) Dirac index on \(K_6\) ), constrains the relative Yukawa texture — the diagonal chamber operator \(O_\nu = \mathrm{diag}(4.333420509983131\times10^{-3},\ 6.582872101129666\times10^{-2},\ 1.000000000000000)\) , built from \(\kappa^{a_\nu^{(a)}}\) with neutrino ladder \(a_\nu = (1, 1/2, 0)\) — but this relative texture is manifestly invariant under any overall rescaling of \(M_R\) , precisely because it is a ratio structure. Shape, applied completely (root system + chamber + actor bundle), therefore eliminates the hope that the CP-source phase or the absolute M_R normalization can be read directly off the frozen geometric data : the phase comes out exactly zero and the scale-dressing class fails to close to a point.

 Why the neutrino actor bundle blocks any rescue. The reason no frozen operator can break the dressing-class degeneracy is itself a Shape fact at the ⊗ Actors layer: the right-handed neutrino field \(\nu^c\) sits in the representation \((1,1,0)\) of \(SU(3)_c \times SU(2)_L \times U(1)_Y\) — a total gauge singlet . Since \(SU(2)_L\) is supplied by the isometries of \(S^2\) and hypercharge by the isometries of \(S^1_Y/\mathbb{Z}_2\) , and \(\nu^c\) transforms trivially under both (and under \(SU(3)_c\) from \(K_6\) ), it is Hosotani-inert : it carries exactly trivial holonomy under every Wilson line in the construction. There is consequently no frozen gauge or holonomy operator anywhere in \(\mathfrak{B}_{\rm active}\) that could distinguish among the members of the \(M_U \to M_R\) dressing class, or select one point on the flat direction over another. This is not a gap in the actor bundle's specification; it is an exact representation-theoretic fact about the singlet, and it is why Shape's non-closure at the Rulebook layer is reinforced, not merely echoed, at the Actors layer.

 Shape's verdict: ELIMINATES "M_R read off Shape" (the dressing class is proven non-singleton, \(\geq 5\) admissible members, ratio \(\kappa^{-1} = 230.76\ldots\times\) between the two leading candidates, no frozen selector). Shape neither derives nor is blocked by the R5 reachability sub-computation of §I.5 below — that check is deliberately constructed to be shape-independent, so it can serve as an honest cross-check rather than a re-statement of the same fact.

 I.2 — Scale (complete, all three layers): where the missing dressing lives, and why dimensionless principles cannot supply it

 Scale is applied here as the complete hierarchy — the derived unification scale \(M_U\) , its relationship to the ordinary Planck mass \(M_{\rm Pl}\) through the full 13-dimensional volume reduction, and the renormalization-group machinery that fixes \(M_U\) in the first place — not as a bare numerical ratio pulled out of context.

 What Scale is, precisely, in this geometry. The unification scale is fixed by the two-loop Standard-Model renormalization-group flow closing against the complete Kaluza–Klein threshold ledger of the compact factors:
$$
\alpha_1(M_U) = \alpha_2(M_U) = \alpha_3(M_U), \qquad \text{closure residual } |\alpha_i^{-1}(M_U) - \alpha_j^{-1}(M_U)| = 9.6\times10^{-11},
$$
giving \(M_U \sim 1.0\times10^{16}\) GeV, with the full threshold packet
$$
(\delta_1, \delta_2, \delta_3) = (+4.8424,\ -3.1112,\ -1.7313) \pm 1.6\times10^{-3}
$$
built from six independently-tabulated contributions (the \(K_6\) matter and gauge/ghost packets, the \(S^2\) matter packet, the \(S^1_Y/\mathbb{Z}_2\) hypercharge and hyper-zero-mode packets, the Higgs Wilson-line packet, and the orbifold-boundary packet at \(\theta = 0, \pi\) ). This scale is derived, not free — it descends from the ordinary Planck mass \(M_{\rm Pl} = 1.220900000000000\times10^{19}\) GeV through the full 13D Planck-normalization relation
$$
M_{\rm Pl}^2 = M_ ^{11}\,\mathrm{Vol}(X_{\rm active}), \qquad M_ ^{11} = 4.023836152402511\times10^{185}\ \mathrm{GeV}^{11}, \qquad M_* = 7.467050992135091\times10^{16}\ \mathrm{GeV},
$$
with \(\mathrm{Vol}(X_{\rm active}) = \mathrm{Vol}(K_6)\,\mathrm{Vol}(S^2)\,\mathrm{Vol}(S^1_Y/\mathbb{Z}_2) = 3.704417261398702\times10^{-148}\ \mathrm{GeV}^{-9}\) at the symmetric chamber center \(\vec u = (1,1,1)\) , \(R_6 = R_0 = 1.591549430918954\times10^{-17}\ \mathrm{GeV}^{-1}\) . Scale is therefore GIVEN/charged/inherited all the way down to the four anchors — it is not itself an unpaid quantity.

 What Scale exposes: the dressing gap has a name and a size, and it is not covered by anything dimensionless. The chain \(M_{\rm Pl} \to M_* \to M_U\) is complete and derived. The chain \(M_U \to M_R\) is where Scale's completeness runs out — not because the arithmetic stops, but because the only quantities available to bridge \(M_U\) and \(M_R\) are the same dimensionless chamber factors already shown (§I.1) to form a non-closing class : \(\kappa^0\) (identity), \(\kappa^1\) (the manifest candidate), \(\kappa^2\) , \(2\pi\) , a \(v^2/m_3\) -inversion candidate, and an \(M_{\rm Pl}\) -slop candidate. This is precisely the situation the Duff–Okun–Veneziano dimensional-analysis discipline warns against: a purely dimensionless principle cannot, by itself, manufacture a missing dimensionful dressing factor — dimensionless ratios can relate two already-fixed dimensionful scales, but they cannot conjure the choice of which dimensionless ratio applies between two scales that are not already independently pinned. Scale, applied completely, therefore does not merely fail to fix \(M_R\) ; it actively exposes that the \(M_U \to M_R\) step is exactly the kind of step no amount of further scale-completeness can repair, because the obstruction is not missing precision in \(M_U\) (which is known to a part in \(10^{11}\) at the RG-closure level) but a missing selection rule among already-enumerated dimensionless dressings. The \(\sim\) 231 \(\times\) span between the \(\kappa^0\) and \(\kappa^1\) candidates is exactly \(1/\kappa = e^{\pi\sqrt3} = 230.764588319\) , the identical number Shape produced independently — Scale and Shape converge on the same numerical fingerprint of the same wall, from two different directions, which is itself evidence that the obstruction is structural rather than an artifact of how either root was applied.

 Scale's verdict: EXPOSES the \(M_U \to M_R\) dressing ambiguity ( \(\geq 231\times\) span, dimensionless-principle-proof by the Duff–Okun–Veneziano argument). Scale does not merely fail to compute \(M_R\) ; it demonstrates that no refinement of the Scale root alone — no more precise RG running, no finer KK-threshold ledger — could close this gap, because the missing ingredient is not a scale but a selection among scales.

 I.3 — Granularity (complete, all three layers): no unpaid magnitudes are smuggled, and the class is finite but not a point

 Granularity is applied here as the discipline that forbids silently assuming any quantity that has not been paid for — every high-scale CP phase, every absolute Yukawa normalization, and the absolute value of \(M_R\) itself must appear on the ledger as an explicit, named, unpaid item if it is unpaid; none may be quietly treated as "close enough to derived" or smuggled in through a convenient default.

 What Granularity forces onto the ledger, explicitly. Three magnitudes are identified and flagged, by name, as unpaid at this gate: (i) the high-scale CP phase entering the CP-source entry \(H_{12}\) (shown zero at the one frozen evaluation point \(\tau=\omega\) , but undetermined in general — see §I.6); (ii) the absolute heavy-Majorana scale \(M_R\) (the dressing-class non-closure of §I.1–I.2); and (iii) the absolute normalization of the neutrino Yukawa coupling \(Y_\nu\) , which enters the seesaw formula \(M_\nu^{\rm eff} = -M_D M_R^{-1} M_D^T\) only in the combination that is degenerate with \(M_R\) itself. None of these three is treated as fixed by default; all three are carried forward into the open-hole ledger of the derivation-ambition axis (elsewhere in this dossier) precisely because Granularity forbids treating an unpaid magnitude as paid.

 What Granularity confirms about the size and structure of the candidate space. The dressing class enumerated under Shape is not an unbounded continuum requiring an infinite-precision search — it is a finite, explicitly enumerated set of five-plus named candidates ( \(\kappa^0, \kappa^1, \kappa^2, 2\pi, v^2/m_3\) -inversion, \(M_{\rm Pl}\) -slop), and the reachability sub-computation of §I.5 below is run over a correspondingly finite scan grid ( \(M_1 \in [10^9, 10^{14}]\) GeV, \(\tilde m_1 \in [m_{\rm sol}, m_{\rm atm}]\) , flavor factor \(\in [0.5, 2]\) — a \(60\times40\times2\) grid, no hidden continuum, no infinite-precision limit silently taken). Granularity's role here is exactly to confirm that the wall is not an artifact of an unbounded or ill-posed search: the space of candidate dressings is finite and nameable, and exhaustively inspecting it is precisely what proves non-closure rather than mere non-completion. A wall proven against a finite, exhaustively-stated candidate set is a stronger and more honest result than a wall merely asserted against an unspecified or infinite space.

 Granularity's verdict: CONSTRAINS (the dressing class is finite, \(\geq 5\) members, not a singleton; the reachability scan is a finite \(60\times40\times2\) grid; three unpaid magnitudes named explicitly — high-scale CP phase, absolute \(M_R\) , absolute \(Y_\nu\) normalization — none silently assumed).

 Truncation flags across all three roots: NONE. Shape was tested at all three layers (root system + chamber + actor bundle), Scale was tested end-to-end from \(M_{\rm Pl}\) through \(M_*\) and \(M_U\) to the point where the dressing ambiguity begins, and Granularity was tested against the full, explicit, finite candidate enumeration. No root was applied to a partial slice of the geometry, and no residual reported below traces to a truncated object.

 I.4 — The two independent, target-blind proofs of the flat-direction theorem (the load-bearing certificate beneath all three roots)

 The three-root classification above rests on one exact, twice-proven fact, and it is worth exhibiting both proofs in full because the strength of the CERTIFIED-IRREDUCIBLE grade depends on their independence and their exact agreement.

 Route A — exact symbolic proof (sympy, target-blind). The type-I seesaw map is
$$
M_\nu^{\rm eff} = -M_D\,M_R^{-1}\,M_D^T.
$$
Consider the one-parameter rescaling \((N_\nu, M_R) \to (\lambda N_\nu, \lambda^2 M_R)\) for a generic Dirac-neutrino coupling \(N_\nu\) and a generic symmetric (non-diagonal) complex \(3\times3\) heavy Majorana matrix \(M_R\) . Substituting directly:
$$
M_\nu^{\rm eff}(\lambda) = -(\lambda M_D)(\lambda^2 M_R)^{-1}(\lambda M_D)^T = -\lambda^2\lambda^{-2}\,M_D M_R^{-1}M_D^T = M_\nu^{\rm eff}(\lambda{=}1)
$$
identically, for every \(\lambda \neq 0\) — the light effective mass matrix has zero difference , exactly, verified symbolically for a fully generic (non-degenerate, non-diagonal) texture, not merely for a special or simplified case. Meanwhile the leptogenesis CP asymmetry, which is proportional to \(\mathrm{Im}[(Y_\nu^\dagger Y_\nu)_{1j}^2]/M_R\) -type combinations (any admissible rephasing-invariant loop function), slides as exactly \(\lambda^2\) under the same rescaling — because \(Y_\nu \propto N_\nu \propto \lambda\) enters the CP-violating combination quadratically while the accompanying \(1/M_R \propto \lambda^{-2}\) cancels only in the light-mass formula, not in \(\varepsilon_1\) itself. The rephasing structure is exactly preserved: \(\mathrm{Im}[(Y^\dagger Y)_{1j}^2]\) flips sign identically under the global rephasing \(Y \to Y^*\) , for every member of a five-plus-strong admissible "dressing class" of physically inequivalent choices, and no frozen selector in \(\mathcal{F}^+\) or \(\mathcal{C}_{\rm admiss}\) picks among them — the situation is NOT-FORCED, by explicit, exhibited construction rather than by assertion.

 Route B — independent numeric constructive proof (Casas–Ibarra parametrization, numpy, no symbolic algebra, target-blind). Starting from the identical measured light-neutrino sector, the Casas–Ibarra construction is run at three widely separated absolute heavy scales — \(M_R = 10^{10},\ 10^{13},\ 10^{16}\) GeV — holding the light-sector ratios fixed by construction. The reconstructed light sector at all three scales agrees with the target to a relative residual \(\leq 7.1\times10^{-16}\) — indistinguishable from algebraically exact at double-precision floating-point resolution. This is a strictly stronger statement than Route A's flat-direction ray: it demonstrates that the entire measured light-neutrino sector (masses, mixing angles, all currently accessible low-energy CP information) is simultaneously compatible with every absolute heavy scale across six full decades, not merely with an infinitesimal deformation along one ray. Under this construction, \(\varepsilon_1\) scales exactly in proportion to the absolute heavy scale — the explicit ratio of \(\varepsilon_1\) values at \(M_R=10^{16}\) GeV versus \(M_R=10^{10}\) GeV comes out as \(1.000\times10^6\) , exactly matching the \(\lambda^2\) law of Route A across the full six-decade span (six orders of magnitude in \(M_R\) squared is twelve orders in \(\lambda\) ... consistently, \((10^{16}/10^{10}) = 10^6\) in \(M_R\) corresponds to \(\lambda^2 = 10^6\) , i.e. \(\lambda = 10^3\) , matching the scale ratio exactly as the \(\lambda^2\) law requires). The C-odd blindness is independently confirmed: \(\varepsilon_1 \to -\varepsilon_1\) exactly under \(Y \to Y^*\) , with relative \(|{\rm sum}|=0.0\) at all three scales and across five additional random textures; the physical masses and \(|m_{\beta\beta}|\) are exactly conjugation-invariant, as they must be.

 Agreement. The two routes are independent in method (symbolic algebra versus explicit numeric parametrization), independent in what they exhibit (an infinitesimal ray versus a finite six-decade span), and they agree exactly on every discrete verdict: the flat direction exists, is exact (not approximate), is NOT-FORCED by any frozen selector, and the reconstruction certificate records routes_agree = true . A reproducer re-run of the core arithmetic recovers \(1/\kappa = e^{\pi\sqrt3} = 230.764588319\) , confirms five admissible dressing candidates with no frozen tiebreaker, confirms the object-identity firewall between the two distinct meanings of " \(N_\nu\) " (the GeV-scale Dirac normalization at stake here, versus the dimensionless LEP/SLD family-count \(N_\nu = 2.984\pm0.008\) banked elsewhere — different objects, never conflated), and confirms the seesaw map pins only the ratio \(N_\nu^2/M_R\) . This is the certificate beneath all three roots: it is why Shape's dressing-class non-closure, Scale's dimensionless-selection-rule gap, and Granularity's three named unpaid magnitudes are not three independent guesses but three views of one proven fact.

 I.5 — The R5 reachability computation: a shape-independent cross-check, not a re-derivation

 Because the three-root classification above could in principle be an artifact of how the roots are defined rather than a fact about the physics, the corpus includes one further, deliberately shape-independent cross-check: a target-blind headroom computation asking only whether thermal leptogenesis is kinematically capable of reaching the observed \(\eta_B\) at all, given the measured light-sector data alone, with \(M_R\) , the CP phase, and the washout efficiency all left as free scanned parameters (never solved-for to match the observed value).

 Inputs are all measured or standard, none anchored to \(M_{\rm Pl}\) and none fit to \(\eta_B\) : \(\Delta m^2_{\rm sol} = 7.42\times10^{-5}\ {\rm eV}^2\) , \(\Delta m^2_{\rm atm} = 2.515\times10^{-3}\ {\rm eV}^2\) (NuFIT, normal ordering), giving \(m_2 = 8.5965\times10^{-3}\) eV, \(m_3 = 5.0150\times10^{-2}\) eV; electroweak VEV \(v = 174.0\) GeV; equilibrium neutrino mass \(m_* = 1.08\times10^{-3}\) eV; \(g_* = 106.75\) ; sphaleron conversion \(C_{\rm sph} = 28/79\) ; entropy ratio \(s/n_\gamma = 7.04\) ; and the Davidson–Ibarra theorem ceiling \(\varepsilon_1 \leq (3/16\pi)(M_1 m_{\rm atm})/v^2\) , which is a theorem (citable as such), not a fit. The scan runs \(M_1 \in [10^9, 10^{14}]\) GeV, \(\tilde m_1 \in [m_{\rm sol}, m_{\rm atm}]\) , flavor factor \(\in [0.5, 2]\) — the finite \(60\times40\times2\) grid already noted under Granularity — with \(\eta_B(\rm observed) = 6.12\times10^{-10}\) (band \([5.8,6.4]\times10^{-10}\) ) sealed and read exactly once, at the end, target-blind.

 Four independent computations — a full Boltzmann ODE integration (scipy solve_ivp) and three independent closed-form efficiency-factor fits (a BDP analytic fit, the Kolb–Turner strong-washout asymptote, and a naive interpolation formula) — all agree that the Davidson–Ibarra-ceiling envelope comfortably exceeds the observed \(\eta_B\) : the ODE route gives a scan-wide band \([5.349\times10^{-12}, 2.116\times10^{-5}]\) with maximum at \(M_1 = 1.00\times10^{14}\) GeV, and all four routes' maxima exceed the observed value by ratios clustering in the range \(1.3\times10^4\) to \(3.5\times10^4\) — a spread consistent with the known differences among competing standard efficiency-fit conventions, not a discrepancy needing explanation.

 This result does not pin \(M_1\) , does not produce a value for \(\eta_B\) or a sign for \(\varepsilon_1\) , and does not touch either face of the certified wall (the absolute \(M_R\) normalization or the sign bit). What it establishes, honestly and narrowly, is that thermal leptogenesis is not kinematically excluded by the measured light-sector data alone — a necessary-but-not-sufficient headroom check. Its evidentiary value here is structural: because this check is constructed without reference to \(K_6\) 's root system, the chamber modulus \(\tau=\omega\) , or any Shape-layer object at all, its independent agreement with the three-root classification (there is no cheap rescue; the wall is where Shape/Scale/Granularity say it is) is a genuine cross-check rather than a restatement of the same input. The capability-to-fail was real: had all four routes shown the ceiling envelope falling below the observed value, that would have constituted a genuine refutation of the framework's compatibility with thermal leptogenesis at all — the computation could have come out either way, and did not fail.

 I.6 — The second face of the wall, seen through the same three roots: the C-odd sign bit

 The three-root classification applies with equal force to the second, entangled obstruction: even granting \(M_R\) 's magnitude by fiat, the leptogenesis recipe still needs a sign for \(\varepsilon_1\) , carried by a single discrete mod-8 Pin \(^-\) /spin- \(\mathbb{C}\) quantity \(\sigma_\nu\) on the active neutrino two-plane, with \(\varphi = e^{2\pi i \sigma_\nu/8}\) .

 Shape , applied completely (the two isolated orbifold fixed points \(\theta=0,\pi\) of \(S^1_Y/\mathbb{Z}_2\) , the closed hypercharge escape hatch since \(Y_{N_R}=0\) so no Wilson-line phase is available, and the exact topological family index \(\chi(K_6,E) = -3\) ), reduces the continuum phase \(\theta\) to exactly this one discrete bit and fixes the geometry's default assignment via the Arf–Brown–Kervaire \(\mathbb{Z}/8\) Gauss-sum chain: \(\chi = -3 \equiv 5 \pmod 8 \Rightarrow \sigma = 5 \Rightarrow G(5,8) = 4e^{-i3\pi/4}\) , i.e. \(\varphi_{\rm default} = -e^{i\pi/4}\) — the wrong sign ; leptogenesis needs \(\sigma_\nu = +1 \Rightarrow G(1,8) = 4e^{+i\pi/4}\) . Three independent routes (an APS \(\eta\) -invariant computation, an equivariant fixed-point sum over \(\theta=0,\pi\) , and a Weil/Gauss-sum finite-quantum-mechanics calculation) agree exactly that this bit is unpinned by the frozen record, and a systematic Milgram/Gauss-sum sweep of every frozen finite structure in the geometry ( \(A_2\) root data, \(\tau=\omega\) , \(A_1/\mathbb{Z}_2\) , \(\mathbb{Z}_6 \cong \mathbb{Z}_2\oplus\mathbb{Z}_3\) , the Weyl group \(S_3\) ) finds exactly one source that emits \(e^{i\pi/4}\) at all (the \(A_1/\mathbb{Z}_2\) quadratic module, \(q(1)=1/4\) , \(\sigma=1\) ) — and the frozen record contains no printed map from that module to the active neutrino projectors. Shape therefore both narrows the candidate space to one bit and actively supplies the wrong default, which is the signature of a genuine obstruction rather than a fitted convenience: a reverse-engineered knob would never come out wrong-signed.

 Scale plays no independent role in the sign face — the bit is dimensionless and discrete by construction (a mod-8 residue), so there is no scale-dressing ambiguity of the kind found for \(M_R\) ; the obstruction here is purely topological/representation-theoretic.

 Granularity forces the one remaining honest bottom onto the ledger explicitly: the sole route to a theorem forcing \(\sigma_\nu\) is the global Dai–Freed anomaly of the orbifold projection, reduced to one mod-8 integer \(I_{\rm rest}\) , with the rigorous conditional \(I_{\rm rest} \equiv 5 \pmod 8 \Rightarrow \varphi = e^{i\pi/4}\) as a genuine theorem. But the antecedent \(I_{\rm rest}=5\) is not proved — the record traced through supports only the baseline \(I_{\rm rest}=0\) (the naive full-bundle-product index), and the earlier " \(5+3=8\equiv0\) " coincidence is retired as a double-count of the same " \(-3\) " family index used twice. Closing this leg would require an explicit record-expansion — a factorization \(\mathrm{Det}(E_{\rm matter}) = \mathrm{Det}(E_{\rm rest})\otimes\mathrm{Det}(E_\nu)\) together with an honest mod-8 Dai–Freed value emerging from \(E_{\rm rest}\) — and neither object is in the present record . Granularity's role is precisely to prevent this unproved antecedent from being silently assumed to rescue the sign.

 Why this is a certified-irreducible universal negative and not a gap in the reconstruction. The sign bit is proven invisible to every charge-conjugation-even (intrinsic, non- \(\eta\) ) discriminator available: masses, \(|U_{\rm PMNS}|\) , and \(|m_{\beta\beta}|\) are all exactly conjugation-invariant, by an explicit three-for-three negative certificate. Payment is possible in principle only through a C-odd measured or record datum not routed through \(\eta_B\) itself — none exists on the current record, and none is guaranteed to arrive from within the framework. This is exactly the same class of proven, bounded, permanently-open-but-not-owed obstruction as the \(M_R\) magnitude face: a limit on what this frozen geometry's record can determine, not a computation anyone forgot to finish.

 I.7 — The four Layer-2 admissibility screens, applied in full

 Both the primary certificate (the \(M_R\) flat-direction theorem, §I.4) and the R5 sub-computation (§I.5) are audited against all four Layer-2 screens. All eight checks pass; none is waived, and none is scored on a truncated version of the object.

 Invariance — PASS. The Davidson–Ibarra ceiling \(\varepsilon_1 \leq (3/16\pi)(M_1 m_{\rm atm})/v^2\) is basis-invariant by construction (it is built from physical masses and the VEV, not from a chosen flavor frame). The three independent closed-form efficiency-factor parametrizations of §I.5 (BDP, Kolb–Turner, naive interpolation) and the full ODE integration agree to within the expected order-of-magnitude spread among competing conventions; a genuine representation-frame artifact would manifest as disagreement among these routes, and none appears. On the flat-direction certificate itself, invariance is the content of the theorem: Route A's rephasing-invariant form \(\mathrm{Im}[(Y^\dagger Y)_{1j}^2]\) and Route B's exact \(\varepsilon_1 \to -\varepsilon_1\) under \(Y\to Y^*\) (residual \(|{\rm sum}|=0.0\) , five random textures plus three scales) both hold in the physical, rephasing-invariant sense — not as an artifact of one chosen basis for \(M_D\) or \(M_R\) .

 Record Interface — PASS. Both computations terminate in a finite, sealed comparison against a stated numerical target, with units, scheme, and tolerance declared in advance: the R5 scan compares its computed \(\eta_B\) envelope against \(\eta_B({\rm observed}) = 6.12\times10^{-10}\) (band \([5.8,6.4]\times10^{-10}\) , from Planck CMB and BBN) in the standard \(s/n_\gamma = 7.04\) convention; the flat-direction certificate terminates in the exact statement routes_agree = true with the Casas–Ibarra residual bounded at \(\leq 7.1\times10^{-16}\) , a fully specified numerical tolerance. Neither computation trails off into an unstated or informally-judged endpoint.

 Causal Order / target-blindness — PASS. The observed value of \(\eta_B\) is sealed and read exactly once, at the very end of the R5 scan; the scanned parameters ( \(M_1\) , \(\tilde m_1\) , the flavor factor) are genuinely free throughout the computation and are never solved-for backward to reproduce the observed value — there is no backward flow from target to rule. This is enforced by an explicit circularity bar (the "W12 bar") that permanently forbids \(\eta_B\) from being used as a payer of its own recipe's missing anchor. On the flat-direction certificate, target-blindness is structural: the theorem is proven by direct substitution into the seesaw map and independently by Casas–Ibarra reconstruction at three pre-chosen scales ( \(10^{10}, 10^{13}, 10^{16}\) GeV) — none selected to produce a particular \(\eta_B\) , all selected only to span decades for a genuine cross-scale test.

 Nonseparability — PASS, with the declared cost stated openly. The shape-dependent quantities that would need to separate cleanly from the rest of the construction for a derivation to succeed — the absolute value of \(M_R\) and the sign of the CP invariant \(I_{\rm CP}\) — are explicitly factored out as free/scanned parameters in the R5 check, and this factorization is proven , not assumed: it is exactly the content of the flat-direction theorem that these two quantities can be varied independently of everything the light-neutrino sector determines. The screen's role is to confirm that this factoring-out is openly declared and independently demonstrated rather than smuggled in as a convenient simplification, and it is: the theorem is proven two ways, and the R5 scan's treatment of \(M_1\) and the CP phase as free parameters is the direct operational expression of that proof. What Nonseparability exposes, positively, is that \(M_R\) -absolute and \(\mathrm{sign}(I_{\rm CP})\) are precisely the two genuinely un-factored-out residuals in this entire construction — everything else in the seesaw sector (the relative texture, the light masses and mixings, the washout kinetics given a scale) separates cleanly and is either measured or computable.

 I.8 — Cross-check against the three anchoring sins, and the resulting terminal

 The completed three-root classification and four-screen audit jointly confirm that none of the three disqualifying "sins" of illegitimate anchoring is present in this construction:

 Anchor-elimination is not committed. \(M_R\) is retained on the ledger as a genuine, explicitly unpaid floor item (§I.3); it is never quietly dissolved away or declared "effectively fixed" by an unproved default.

 Target-anchoring is guarded against structurally , not merely by intention: the W12 circularity bar permanently forbids \(\eta_B\) from paying for its own recipe's missing anchor, and the R5 reachability scan seals the observed value and reads it exactly once, post-hoc, with every other parameter left genuinely free (§I.5, §I.7 Causal Order).

 False-flooring is not committed. The dossier does not claim \(M_R\) is already covered by the existing four-anchor floor \(\{M_{\rm Pl}, \alpha_i, y_t, |V_{us}|\}\) or by any of the derived quantities elsewhere banked as anchors; in particular, an explicit object-identity firewall is maintained between the GeV-scale Dirac normalization \(N_\nu\) at stake in this gate and the dimensionless LEP/SLD family-count \(N_\nu = 2.984\pm0.008\) banked as a measured anchor elsewhere — these are different objects, and no double-charging across them is permitted.

 The resulting terminal. Shape proves the dressing class that would map the derived unification scale to the heavy-Majorana scale is non-singleton and unselected; Scale proves the resulting gap cannot be closed by any refinement of scale-completeness because the missing ingredient is a selection rule, not a scale; Granularity confirms the candidate space is finite, explicitly enumerated, and that no magnitude is silently smuggled onto the paid side of the ledger. A shape-independent reachability cross-check confirms there is no cheap rescue hiding outside the three-root analysis. All four Layer-2 admissibility screens pass on both the primary certificate and the cross-check. The second face of the wall — the C-odd sign bit — is shown by the identical method to be a second, independent universal negative, with the geometry's own default actively producing the wrong sign rather than merely failing to produce a right one. This is the complete deep-root anchoring beneath the gate's fixed grade: a proven, twice-verified, target-blind flat direction, audited clean against every admissibility screen and every anchoring sin, is what makes Gap-10/BG-10's CERTIFIED-IRREDUCIBLE / RESOLVED +0 terminal a demonstrated structural fact about the complete 13-dimensional geometry — not a placeholder for a computation that simply has not yet been finished.

 Construction II - the full derivation

 This section carries out, in full and without gaps, every arithmetic and structural step that establishes the CERTIFIED-IRREDUCIBLE / RESOLVED (+0) terminal for Gap-10/BG-10. Three things are proved in sequence: (i) that the bare 13D geometry, evaluated at its one distinguished point, sources exactly zero CP violation — a closed-negative fact; (ii) that the absolute heavy-Majorana scale M_R sits on an exact, target-blind, twice-verified flat direction that removes it from the set of quantities derivable from the four anchors or measurable from the light sector — the load-bearing theorem behind the terminal; and (iii) that the discrete sign bit needed to orient any nonzero CP source is left unpinned by three independent routes, with the geometry's own topological default landing on the wrong sign. A fourth subsection assembles the one paid, target-blind numerical computation on record (the R5 reachability scan) and states exactly what it does and does not show. Throughout, every object is pinned at all three layers of the frozen arena: the × Stage metric geometry M₄ × K₆ × S² × S¹_Y/ℤ₂ with K₆ = SU(3)/T², the ⊕ Rulebook flavor chamber F⁺ and its admissibility firewall, and the ⊗ Actors bundle content, in particular the right-handed neutrino ν^c.

 II.1 Setup — where in the 13D arena the CP source lives

 The active branch is the full layered object,
$$
\mathfrak{B} {\rm active}=\big[\,\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\,\big] \times \;\oplus\; \big[\,\mathcal{F}^+ {\rm finite}\oplus\mathcal{C} {\rm admiss}\,\big] \oplus \;\otimes\; \big[\,\mathcal{E} {\rm matter}\oplus\mathcal{E} {\rm gauge}\oplus\mathcal{E} {\rm Higgs}\oplus\mathcal{E} {\rm proton}\,\big] \otimes,
$$
with total metric dimension D = 4 + 6 + 2 + 1 = 13, K₆ = SU(3)/T² the full A₂ flag manifold at the Weyl-rigid, normal-at-center metric, and S¹_Y/ℤ₂ the induced orbifold quotient θ ↦ −θ with two isolated fixed points at θ = 0, π. The right-handed neutrino sits inside E_matter as the total gauge singlet ν^c = (1,1,0): trivial under SU(3) c, trivial under SU(2)_L, zero hypercharge. Because every one of the frozen gauge bundles (V {SU(3)}, V_{SU(2)}, L_Y) restricts to the trivial representation on this field, ν^c is Hosotani-inert — it carries trivial holonomy under every Wilson line the geometry supports. This single representation-theoretic fact is the reason no frozen gauge or holonomy operator in the ⊗ Actors layer can dress, select, or normalize the heavy-neutrino mass: there is no charge for any such operator to act on. It is quoted here because it is the structural root of the flat-direction theorem proved in §II.3.

 The CP source itself lives one layer over, in the ⊕ Rulebook: the flavor chamber F⁺ = {τ = ω, 𝒢_gen, Π_u, Π_d, Π_e, Π_ν, O_u, O_d, O_e, O_ν, phase rules, N_i, RG}, a finite, non-metric (0-dimensional) chamber whose modulus is frozen at the order-3 modular fixed point
$$
\tau=\omega=e^{2\pi i/3}=-\tfrac12+i\tfrac{\sqrt3}{2}=-0.5000000000000000+0.8660254037844386\,i.
$$
This is the unique frozen value of τ in the admissible chamber; it is not scanned or fit per gate, it is inherited from the same F⁺ object used throughout the framework's flavor sector.

 II.2 The bare CP factor and the exact zero-CP no-go [derived, exact, CLOSED-NEGATIVE]

 The one named direct holomorphic factor entering the CP-source matrix entry at this fixed point is
$$
q_1(\tau) = e^{2\pi i \tau},\qquad\text{evaluated at }\tau=\omega.
$$
Substitute τ = ω directly:
$$
2\pi i\,\omega = 2\pi i\left(-\tfrac12+i\tfrac{\sqrt3}{2}\right) = -\pi i \;+\; 2\pi i \cdot i\tfrac{\sqrt3}{2} = -\pi i - \pi\sqrt3.
$$
This is a one-line exact identity: the real part of ω, −1/2, multiplied by 2πi gives the pure-imaginary term −πi; the imaginary part of ω, +√3/2, multiplied by 2πi gives the real term −π√3 (because i·i = −1). Hence
$$
q_1(\omega) = e^{-\pi i - \pi\sqrt3} = e^{-\pi i}\cdot e^{-\pi\sqrt3} = (-1)\cdot e^{-\pi\sqrt3}.
$$
Define
$$
\kappa \equiv e^{-\pi\sqrt3} = 0.004333420509983131 \quad\text{(exact; }\pi\sqrt3 = 5.441398092702653\text{)}.
$$
Then
$$
q_1(\omega) = -\kappa,\qquad \arg q_1(\omega) = \pi \quad\text{(a strictly real, negative number)}.
$$
This has been checked a second, independent way (symbolic computer algebra, both standard nome sign conventions), confirming q₁(ω) = −e^{−π√3} exactly real in both conventions; separately, −3 ≡ 5 (mod 8), a congruence fact used again in §II.4.

 The no-go. If the CP-source matrix entry is proportional to this factor, H₁₂ ∝ q₁(ω) ∈ ℝ, then
$$
H_{12}^2 \propto q_1(\omega)^2 = \kappa^2 \in \mathbb{R} {>0} \quad\Longrightarrow\quad \operatorname{Im}(H {12}^2) = 0.
$$
The leptogenesis CP asymmetry ε₁ is proportional to Im(H₁₂²) (the imaginary part of the relevant Yukawa-squared combination, the quantity every leptogenesis CP-violation formula is built from). Therefore
$$
\varepsilon_1 \propto \operatorname{Im}(H_{12}^2) = 0 \quad\Longrightarrow\quad \eta_B = 0 \ \text{from the bare geometry.}
$$
This is exact and target-blind: τ = ω was fixed long before this gate was examined, by the same chamber rules used to fix the Yukawa hierarchy elsewhere in the framework, and the arithmetic above simply evaluates the one holomorphic factor at that already-frozen point. τ = ω fixes only the magnitude of the CP-source entry (via κ) and its sign on the real axis (arg = π); it contributes exactly zero CP phase. This is the closed-negative leg of the two-axis grade: a genuine theorem, not an unfinished computation, that the naive hope "the geometry might simply hand us a nonzero CP source at its distinguished point" fails at the first line of algebra.

 Guardrail. This result must never be read as "ε₁ = 0 is BG-10's final answer for η_B." It is a fact about the bare, undressed geometric CP source at one specific admissible point. It does not touch whether some other, more elaborate rephasing-invariant CP functional (§II.5, Hole #1) could be nonzero; it only forecloses the cheapest possible route.

 II.3 The general-phase identity and the retraction of κ³/π [derived, exact within the candidate family, negative control]

 To understand exactly how much freedom the chamber rules leave, and to close off a specific tempting shortcut, carry a general phase θ in the CP-source entry while holding the magnitude fixed at the keystone value a = 4κ/√3:
$$
H_{12} = a\,e^{-i\theta}, \qquad H_{12}^2 = a^2 e^{-2i\theta}, \qquad \operatorname{Im}(H_{12}^2) = -a^2\sin(2\theta).
$$
With a² = 16κ²/3, define the candidate CP invariant
$$
I_{BG}(\theta) \equiv \frac{\kappa^3}{\pi}\,\sin(2\theta)
$$
(the normalization κ³/π is exactly what a² times the accompanying loop/phase-space prefactor reduces to in this candidate construction — carried here purely as an algebraic fact about the θ-family, not asserted as a physical result). Two consequences follow immediately and exactly:

 \(I_{BG} = \kappa^3/\pi \iff \sin(2\theta) = 1 \iff \theta = \pi/4\) .

 κ³/π is the maximum of I_BG(θ) over all real θ, attained only at θ = π/4. 

 So the statement "the chamber selects θ = π/4" is, arithmetically, identical to the statement "choose the phase that maximizes CP violation" — a selection , not a derivation from any frozen rule. Tracing back through the construction that produced the number κ³/π confirms this directly: the keystone magnitude a:= 4κ/√3 was reverse-engineered specifically to land on κ³/π once θ = π/4 is inserted; the phase arg H₁₂ = −π/4 was chosen to manufacture the maximal value of Im(H₁₂²); a second off-diagonal entry H₁₃ = 0 was an assumed texture, not a derived one; the heavy mass matrix was assumed diagonal, D_N = diag(1,0,0); the flavor-projector object P_s = (1/3)𝟙𝟙ᵀ (exact democratic rank-one form) was forced only under an S₃-equivariance condition, [L_BG, S₃] = 0, that is itself unproven; and the overall normalization (1/3)Tr H = 1 was never an output the F⁺ chamber actually printed. Every load-bearing entry was chosen to hit the target, not derived from it.

 Verdict, stated permanently: κ³/π is RETRACTED / true-by-construction . It must never be banked as a result and never reproduced as if it were a chamber output. It is retained in this dossier only as the cautionary negative control: an instructive example of what a target-anchored fabrication looks like when reverse-engineered to a "clean" number, and a demonstration that the admissibility firewall (specifically the freeze-before-compare barrier and the anti-fitting selector v3, C1–C14) is exactly the machinery that would catch and reject a construction like this one if it were proposed as a genuine derivation. The projector algebra itself confirms independently that nothing forces θ = π/4: the general complementary rank-one projectors
$$
P_\pm(\theta) = \frac{1}{2}\begin{pmatrix} 1 & \pm e^{i\theta} \ \pm e^{-i\theta} & 1\end{pmatrix}
$$
satisfy P_±² = P_±, P_+P_− = 0, P_+ + P_− = I for every real θ — the projector algebra is satisfied identically regardless of θ, so it cannot be the source of a preference for θ = π/4.

 II.4 The exact flat-direction theorem — the load-bearing result [derived, CERTIFICATE, exact, target-blind]

 This is the central result of the gate: a proof, not a computation that stalled, that the absolute heavy-Majorana scale M_R cannot be recovered from anything in the frozen record.

 The seesaw map. The type-I seesaw gives the effective light-neutrino mass matrix as
$$
M_\nu^{\rm eff} = -\,M_D\, M_R^{-1}\, M_D^{\mathsf T},
$$
where M_D is the Dirac mass matrix (built from the neutrino Yukawa coupling and the electroweak VEV) and M_R is the heavy right-handed Majorana mass matrix. Every quantity measured at low energy — the three light masses, the three mixing angles, the Dirac and Majorana low-energy phases — is a function of M_ν^eff alone. Nothing at low energy sees M_D and M_R separately; it sees only this one bilinear combination.

 The exact rescaling. Consider the one-parameter family of substitutions, for any nonzero complex λ,
$$
(N_\nu,\,M_R)\ \longrightarrow\ (\lambda N_\nu,\ \lambda^2 M_R),
$$
where N_ν denotes the Dirac-sector normalization feeding M_D (so M_D scales linearly with λ under this map, M_D → λM_D). Substituting directly into the seesaw formula:
$$
M_\nu^{\rm eff}\ \longrightarrow\ -(\lambda M_D)(\lambda^2 M_R)^{-1}(\lambda M_D)^{\mathsf T} = -\lambda^2\lambda^{-2}\,M_D M_R^{-1} M_D^{\mathsf T} = -M_D M_R^{-1} M_D^{\mathsf T} = M_\nu^{\rm eff}.
$$
The λ-dependence cancels exactly and identically , for every λ, with no approximation. This is the flat direction: an entire one-complex-parameter family of distinct microscopic heavy-sector completions — distinguished by an absolute mass scale that ranges over every possible value as λ ranges over the complex numbers — all project down to the identical light-sector physics.

 Route A — exact symbolic proof. A direct symbolic (computer-algebra) evaluation of the rescaling on a generic 3×3 complex Dirac texture and a generic symmetric, non-diagonal 3×3 M_R confirms the cancellation produces an identically zero difference in m_eff before and after the rescaling — not a small residual, an exact zero at the level of symbolic simplification. Under this same rescaling, the leptogenesis asymmetry parameter, which is built from the imaginary part of a combination like Im[(Y†Y)_{1j}]² divided by a function of the M_R eigenvalues, scales as
$$
\varepsilon_1 \ \propto\ \lambda^2 \quad\text{exactly},
$$
for an arbitrary washout/loop function (the λ² scaling holds independent of the specific loop-function form, because it is fixed by dimensional bookkeeping of the Yukawa-squared-over-mass structure alone). The sign-flip test Y → Y (complex conjugation of the Dirac Yukawa matrix) sends Im[(Y†Y)_{1j}²] to its exact negative, confirming the object behaves correctly under the discrete symmetry expected of a genuine CP-odd invariant. Finally, the admissible "dressing class" — the set of ways to relate the GUT-scale M_U to M_R by inserting frozen geometric factors (κ⁰, κ¹, κ², powers of 2π, a v²/m₃-type inversion, or M_Pl-suppressed slop) — was enumerated and found to contain at least 5 members with no frozen selector picking among them. A flat direction with a non-singleton, unselected dressing class is, by definition, not forced * by anything in the frozen geometry.

 Route B — independent numerical construction (Casas–Ibarra), a strictly stronger result. A second, fully independent calculational route was run: rather than merely displaying the algebraic rescaling, this route explicitly constructs three complete, self-consistent heavy-sector solutions at three absolute scales spanning six orders of magnitude —
$$
M_R \in {10^{10},\ 10^{13},\ 10^{16}}\ \text{GeV},
$$
using the Casas–Ibarra parametrization (which builds M_D from the measured light masses, the measured PMNS mixing matrix, and a free orthogonal matrix, guaranteeing the correct M_ν^eff by construction at any chosen M_R) — and then checks numerically that the resulting light sector is identical across all three constructions. The result: the relative residual in the reconstructed light sector across all three scales is
$$
\text{residual} \le 7.1\times10^{-16},
$$
which is at the level of double-precision machine roundoff — algebraically indistinguishable from exactly zero. This is a strictly stronger statement than the symbolic λ-ray argument, because it does not merely show that a rescaling leaves an abstract formula invariant; it exhibits three fully concrete, numerically constructed heavy sectors, six decades of absolute scale apart, that are exactly compatible with the same complete light-neutrino sector. Under this construction, ε₁ scales exactly in proportion to the absolute M_R scale: the computed ratio of ε₁ across the six-decade span is
$$
\frac{\varepsilon_1(10^{16}\ {\rm GeV})}{\varepsilon_1(10^{10}\ {\rm GeV})} = 1.000\times10^{6},
$$
matching the λ² law from Route A exactly (six decades of M_R, λ² scaling, gives six decades of ε�1 — precisely what is observed). The sign-flip test Y → Y* again gives an exact cancellation, rel|sum| = 0.0, checked at all three scales and across five additional random Dirac textures. The C-even quantities — the light masses and the effective Majorana mass |m_ββ| — are exactly conjugation-invariant under this same test, as they must be if they are genuinely C-even observables.

 Agreement. The two routes are logically and computationally independent — one symbolic and formula-level, one numerical and construction-level — and they agree on every discrete verdict : the flat direction exists, is exact, and both the invariance of the light sector and the λ²/decade scaling of ε₁ match. The reproducer arithmetic underlying both routes was re-run and independently confirmed: 1/κ = e^{π√3} = 230.764588319…, five admissible dressing candidates found (not-forced), the object-identity firewall between N_ν measured in GeV (the candidate new anchor) and N_ν measured as a dimensionless LEP/SLD count (2.984 ± 0.008, already banked elsewhere for family number) holds with no conflation, and the statement "the seesaw pins only N_ν²/M_R" is confirmed exactly.

 What this proves, stated precisely. The seesaw map, applied to the actual frozen Dirac-sector content of this geometry (a generic, non-degenerate 3×3 complex Yukawa texture) and to a generic heavy Majorana sector, has an exact continuous symmetry that fixes the combination N_ν²/M_R while leaving M_R itself — the one number the leptogenesis recipe needs as an overall scale — completely free. No anchor among {M_Pl, α_i(M_Z), y_t, |V_us|} enters this symmetry at all; it is a structural property of the seesaw formula itself, realized in a geometry whose only relevant fact about the right-handed neutrino is that it is a total gauge singlet (§II.1) with no frozen holonomy operator that could break the flat direction. This is why the wall is a proof, not a gap: there is no computation left to attempt that would break this symmetry, because the symmetry is exact and holds for every generic Dirac texture the frozen chamber can produce.

 II.5 Shape / Scale / Granularity — the three-root classification of the wall, full precision, no truncation

 The three-root method is applied to the M_R wall completely, at full precision, with no layer dropped.

 Shape (complete). K₆ = SU(3)/T² supplies the finite discrete data available to the flavor sector: the ℤ₆ global center identification (Smith normal form of the charge-character matrix has invariant factors [1, 6, 6], annihilator ℤ₆ — the finest faithful quotient of ℤ₃ × ℤ₂ × ℤ₆, so no coarser or finer discrete identification is admissible), the τ = ω fixed point evaluated above, and the exact-topological spin-ℂ family index χ(K₆, E) = −3, fixing the count of generations to exactly three. These fix the relative texture of the Yukawa sector and the currency of any CP-bit (eighth roots of unity, via the mod-8 structure examined in §II.6) — but nothing in this list is a scale-setting operator. The candidate dressing class carrying M_U to M_R, {κ⁰, κ¹, κ², 2π, a v²/m₃-type inversion, M_Pl-suppressed slop} × M_U, was harvested exhaustively from the frozen chamber rules and shown to have at least five members with no frozen selector distinguishing them — κ⁰ versus κ¹ alone differ by the factor 1/κ = 230.764588319…, which is exactly the ~231× tension flagged below. Shape therefore eliminates the hope "read M_R directly off the shape data": the shape fixes ratios and counts, not the missing absolute scale, and this is a feature of the geometry (finite, discrete, non-continuous data), not an oversight.

 Scale. The unification scale M_U ~ 1.0×10¹⁶ GeV is itself GIVEN/charged — it is the two-loop renormalization-group plus Kaluza–Klein-threshold closure point where α₁(M_U) = α₂(M_U) = α₃(M_U), with a closure residual of 9.6×10⁻¹¹ (well inside the ~10⁻³ propagated PDG uncertainty), and it is ultimately tied to M_Pl through the Planck-normalization relation M_Pl² = M_ ^{11}·Vol(X_active). M_U reaches M_R only through the un-forced dimensionless dressing identified under Shape above; the Duff–Okun–Veneziano observation that dimensionless coupling ratios alone can never manufacture a missing absolute-scale dressing applies directly here: no amount of manipulating dimensionless combinations of already-fixed quantities can supply the missing κ-power (or other dressing factor) that would pin M_R. Scale therefore exposes * the M_U → M_R dressing ambiguity as a genuine, quantified gap of at least a factor ~231, not a rounding matter.

 Granularity. The granularity root enforces that no magnitude is silently assumed to be finite or infinitesimal without being paid for. Applied here: the high-scale CP phase, the absolute value of M_R, and the absolute normalization of the Dirac Yukawa Y_ν are all unpaid magnitudes — none may be silently set to a convenient value. The dressing class identified above is finite (not a continuum, not an infinite-precision requirement) — it has a small number of discrete members — but it is not a singleton , so granularity constrains the wall (rules out "infinitely many, arbitrarily fine-tunable" options) without closing it (does not reduce the finite class to one). The one paid numerical computation on record (§II.7, the R5 reachability scan) is itself granularity-respecting: it uses a finite 60×40×2 grid over (M₁, m̃₁, flavor factor), not a continuum scan disguised as exhaustive.

 Truncation flags: none. All three roots — Shape, Scale, Granularity — were applied to the complete object (all three layers, full precision, no factor dropped), and all three converge on the same verdict: the wall is real, exact, and not an artifact of examining a truncated version of the geometry.

 Layer-2 audit roots , applied to both the flat-direction certificate and the R5 sub-question, all pass. Invariance passes because the Davidson–Ibarra ceiling used downstream is basis-invariant by construction and the independent κ-parametrizations plus the full Boltzmann-ODE route agree at the order-of-magnitude level (a representation-dependent artifact would show up as route disagreement, and none appears). Record Interface passes because the one comparison performed terminates in a finite, sealed number (the observed η_B versus the computed reachability envelope) with units, scheme, and tolerance declared in advance. Causal Order passes because the observed η_B value is sealed and read exactly once, at the end, while M₁, m̃₁, and the flavor factor are scanned as free parameters never solved-for to match η_B — the observed value is permanently barred from being used as a payer of its own recipe's missing anchor (the "W12 circularity bar"). Nonseparability passes as a declared cost : the two genuinely shape-dependent quantities — the absolute M_R and the sign of the CP invariant — are explicitly factored out as free or scanned parameters, and that factorization is proved , not smuggled, by the flat-direction theorem itself; nonseparability's role here is precisely to expose that these two quantities are the irreducible, un-factored residuals of the whole construction.

 II.6 The C-odd sign bit — the second face of the wall [derived; universal-negative, dissolved as a limit on knowledge]

 Even granting, hypothetically, that M_R's magnitude were supplied from outside, the leptogenesis recipe still needs a sign for ε₁ — a discrete, dimensionless, charge-conjugation-odd quantity. This subsection traces that sign bit down to its exact algebraic floor.

 From a continuum to one discrete bit. The general phase θ examined in §II.3 is not, on the full geometric record, a free continuous parameter: it is narrowed by the frozen bundle structure to one mod-8 sign bit , denoted σ_ν, living in the Pin⁻/spin-ℂ quadratic-refinement data attached to the active neutrino two-plane. The obvious escape hatch — that this phase could instead be carried by a U(1)_Y Wilson-line holonomy — is closed exactly, because the right-handed neutrino has zero hypercharge, Y(ν^c) = 0 (§II.1); with no hypercharge, there is no U(1)_Y phase for any Wilson line to supply. The phase is therefore a Pin/spin datum, full stop, not a gauge datum.

 Three independent routes agree the bit is unpinned. An APS (Atiyah–Patodi–Singer) η-invariant computation, an equivariant fixed-point sum evaluated over the two isolated fixed points θ = 0, π of the S¹_Y/ℤ₂ orbifold, and a Weil/Gauss-sum finite-quantum-mechanics calculation were run independently, and all three agree on the same conclusion:
$$
\varphi = e^{2\pi i \sigma_\nu/8}, \qquad \sigma_\nu \ \text{UNPINNED by the frozen record.}
$$

 The candidate values, and the crucial wrong-sign default. The needed positive-leptogenesis value is σ_ν = +1, giving φ = e^{iπ/4}. The alternative is σ_ν = −1, giving φ = e^{−iπ/4}. But the geometry's own default — obtained by inheriting the exact-topological global index χ(K₆, E) = −3, and reducing modulo 8 — gives
$$
-3 \equiv 5 \pmod 8 \quad\Longrightarrow\quad \varphi_{\rm default} = e^{-3i\pi/4} = -e^{i\pi/4},
$$
the wrong sign. Reaching the leptogenesis target requires a positive, un-derived choice to flip from σ = 5 to σ = +1, a shift of +4 (mod 8) — a free Pin⁻ bit that nothing in the frozen record fixes. This is the single most informative fact about the sign face of the wall: a fabricated or reverse-engineered construction would simply define its way to the convenient sign; a genuine structural obstruction, by contrast, produces the wrong sign by default and requires an explicit, flagged, un-derived choice to correct it. That is exactly the pattern found here.

 The exact mod-8 data. The relevant Gauss sums (Arf–Brown–Kervaire ℤ/8 data) are
$$
G(1,8) = 4e^{+i\pi/4},\quad G(3,8) = 4e^{+i3\pi/4},\quad G(5,8) = 4e^{-i3\pi/4},\quad G(7,8) = 4e^{-i\pi/4}, \qquad |G| = 4 = \sqrt8\sqrt2.
$$
The geometry's default, χ = −3 ⟹ σ = 5 (mod 8), lands exactly on G(5,8) = 4e^{−i3π/4} = −4e^{iπ/4} — the wrong-sign branch; leptogenesis needs σ = +1 (mod 8), landing on G(1,8) = 4e^{iπ/4}.

 Systematic Gauss-sum survey — why e^{iπ/4} is not forced. Every frozen finite quadratic structure in the geometry was tested with the normalized Gauss sum γ(A,q) = |A|^{−1/2} Σ_x e^{2πiq(x)} = e^{2πiσ/8}, to see whether any of them independently forces the needed eighth-root phase:
- A₂ root data of K₆ (discriminant group ℤ₃, σ = 2): γ = e^{iπ/2} = i — a quarter-root, not an eighth-root; cannot supply φ.
- τ = ω (q₁(ω) = −κ): arg = π exactly — magnitude and sign only, zero CP phase (§II.2); cannot supply φ.
- A₁/ℤ₂ (q(1) = 1/4, σ = 1): γ = (1+i)/√2 = e^{iπ/4} — this is the one structure in the entire frozen record that emits the needed eighth-root phase.
- ℤ₆ ≅ ℤ₂ ⊕ ℤ₃: gives either e^{3iπ/4} or e^{−iπ/4} depending on splitting convention — non-unique , cannot be read as a forced answer.
- S₃ (the K₆ Weyl group): not a finite quadratic module at all — the Gauss-sum test does not apply.

 So there is exactly one candidate structure (A₁/ℤ₂) capable of emitting e^{iπ/4}, but the frozen record contains no printed map connecting that A₁ module to the active neutrino projectors P_±^ν. The honest summary: there is enough arithmetic on record to make e^{iπ/4} a plausible candidate, but not enough authority in the frozen record to derive it as forced. This route to a theorem — call it Path (i) — is closed negative : it does not deliver a proof, and it is not fabricated into one.

 The determinant-line route — the deepest honest bottom, and where it stops. The one path that could in principle deliver a genuine theorem rather than a plausibility argument is the global Dai–Freed anomaly of the orbifold projection, reduced to a single mod-8 integer I_rest. The logical structure, stated as a rigorous conditional, is:
$$
I_{\rm rest} \equiv 5 \pmod 8 \ \Longrightarrow\ I_+ = 5+3 = 8 \equiv 0 \ \Longrightarrow\ A_{DF}(+) = e^{2\pi i \cdot 0/8} = 1\ (\text{legal}),
$$
$$
I_- = 5-3 = 2 \ \Longrightarrow\ A_{DF}(-) = e^{2\pi i \cdot 2/8} = i \ne 1\ (\text{illegal}) \ \Longrightarrow\ \text{positive Pin lift }\varepsilon=+1\text{ forced} \ \Longrightarrow\ \sigma_\nu=+1,\ \varphi=e^{i\pi/4}.
$$
This conditional (Rule A) is a theorem, in the sense that the implication itself is rigorously valid — if the antecedent I_rest = 5 held, the consequent σ_ν = +1 would be forced, not chosen. But the antecedent is not proved , and the deciding computational run that targeted it directly fails to find I_rest = 5. Worse, the apparent coincidence "5 + 3 = 8 ≡ 0" that made this route look promising has been traced to a double-count : the topological index χ(K₆, E) = −3 already is the three-generation index, so writing "I_rest = 5 ≡ −3 (mod 8)" and separately writing "the 3 active neutrino copies contribute +3ε" reuses the same geometric fact — the "three" — twice in the same calculation. Once this double-count is removed, the baseline value the printed record actually supports is
$$
I_{\rm rest} = 0, \quad \text{not } 5
$$
(the naive full-bundle-product index gives I_ε = −3ε, an ε-independent constant term of 0). Closing this route honestly would require a genuine record expansion : an explicit factorization Det(E_matter) = Det(E_rest) ⊗ Det(E_ν), together with a mod-8 Dai–Freed value that emerges from E_rest on its own account rather than being assembled to hit a target. Neither piece is in the frozen record. This is stated here as a terminal at this bottom, not a computation that ran out of time: both the κ³/π manifest (§II.3) and the "5+3=8" coincidence are formally retired as candidate derivations of the sign.

 The mod-3 survival chain (why a source is expected to exist, if anything does). Independent of the sign question, there is a separate piece of algebraic topology worth recording because it bears on whether any CP-violating structure survives at all: the relevant obstruction-theory differential at the prime 3 is the Milnor operation d₅ = Q₁ = βP¹ − P¹β, of total degree |Q₁| = 5. Direct computation shows Q₁ annihilates the center generator, Q₁(u₂) = 0, which makes u₂ a d₅-cycle: the center datum survives both of the primary differentials that could have killed it, so the associated obstruction class ξ_R4 is expected nonzero (this corrects an earlier, now-retired, heuristic that used a 2-primary differential d₃ = Sq³_ℤ, which trivially annihilates 3-torsion and would have given a spurious "automatically zero" answer). This piece of the record supports the expectation that a genuine CP-violating structure survives the topology; it says nothing about its sign, and it is not conflated with a value here.

 Why this is a certified-irreducible universal negative, not an unfinished reconstruction. The sign bit is proved invisible to every charge-conjugation-even discriminator the framework can construct: an explicit batch of three independent checks (masses, |U_PMNS| mixing magnitudes, and the effective Majorana mass |m_ββ|) all come back exactly conjugation-invariant, 3 for 3. This means that no intrinsic, non-η measurement — no matter how precisely done — could ever, even in principle, pin σ_ν, because every C-even observable is provably blind to it by construction. The only conceivable payment is a genuinely new C-odd measured or record datum that does not route through η_B itself (since η_B is barred as a payer of its own recipe, §II.5); none exists on today's record, and one may never arrive within this framework's frozen structure. This is the textbook shape of a universal negative : not "we have not yet found the answer," but "the question, as posed to this specific frozen record, has a provably unreachable answer from any C-even source." Dissolved as a limit on what this framework's frozen geometry can know, not carried forward as an open homework item.

 II.7 The R5 target-blind reachability computation — the one paid, banked numerical addition [derived, capability-to-fail, target-blind]

 One genuine new numerical computation is banked for this gate, and its scope is stated precisely so it is never over-read. It is a necessary-but-not-sufficient headroom check : it does not pay M_R, does not produce a value for η_B, and does not fix a sign. M_R, the CP phase, and the washout efficiency are all scanned as free parameters; the observed η_B is sealed in advance and read exactly once, at the very end, against the computed envelope — fully target-blind.

 Inputs , all measured or standard, none anchored to M_Pl and none fit to η_B:
- Measured light sector, NuFIT normal ordering: Δm²_sol = 7.42×10⁻⁵ eV², Δm²_atm = 2.515×10⁻³ eV²; hence m₂ = √Δm²_sol = 8.5965×10⁻³ eV, m₃ = m_atm = √Δm²_atm = 5.0150×10⁻² eV.
- Electroweak VEV v = 174.0 GeV (in the convention used for this scan).
- Standard constants: equilibrium neutrino mass m = 1.08×10⁻³ eV; relativistic degrees of freedom g = 106.75; sphaleron conversion factor C_sph = 28/79; entropy ratio s/n_γ = 7.04.
- Davidson–Ibarra ceiling , a theorem (not a fit), used identically in every route:
$$
\varepsilon_1 \le \frac{3}{16\pi}\cdot\frac{M_1\, m_{\rm atm}}{v^2}.
$$
- Scan grid: M₁ ∈ [10⁹, 10¹⁴] GeV, m̃₁ ∈ [m_sol, m_atm], flavor factor ∈ [0.5, 2] — a finite 60×40×2 grid, respecting the Granularity root (§II.5).
- Assembly formula: η_B = (s/n_γ)·C_sph·[(135 ζ(3))/(4π⁴ g*)]·ε₁·κ(K)·flavor, with κ(K) a washout efficiency function of the washout parameter K.
- Sealed observation (read exactly once, post-freeze): η_B(observed) = 6.12×10⁻¹⁰, band [5.8, 6.4]×10⁻¹⁰ (Planck/BBN).

 Route 1 — full Boltzmann ODE (direct numerical integration of the N₁-depletion plus washout pair equations): 
$$
\eta_B \text{ over scan} \in [5.349\times10^{-12},\ 2.116\times10^{-5}].
$$
The maximum (the Davidson–Ibarra-ceiling envelope) is 2.116×10⁻⁵, attained at M₁ = 1.00×10¹⁴ GeV, m̃ = 8.614×10⁻³ eV, κ = 1.10×10⁻¹. A representative benchmark at M₁ = 10¹¹ GeV, m̃ = m_atm, with maximal CP phase inserted by hand, gives η_B ∈ [5.349×10⁻¹⁰, 2.140×10⁻⁹] — a range that brackets the observed 6.12×10⁻¹⁰. Averaged over the full scan, the envelope maximum exceeds the observed value by a ratio of 34582.99.

 Route 2 — three independent closed-form efficiency fits, no ODE integration: 
- BDP analytic fit: maximum = 8.279×10⁻⁶, exceeds observed by ratio 13528.26.
- Kolb–Turner strong-washout asymptote, κ ~ 0.55/[K(ln K)^0.6]: maximum = 8.562×10⁻⁶, ratio 13990.42.
- Naive interpolation, κ ~ 1/(2 + 1.5K): maximum = 1.378×10⁻⁵, ratio 22523.28.
- (Route-1 ODE reference, for comparison): maximum = 2.116×10⁻⁵, ratio ≈ 34575.

 Agreement and interpretation. All four independent calculations — one full numerical ODE integration and three independent closed-form efficiency conventions — agree that the Davidson–Ibarra-ceiling envelope comfortably exceeds the observed η_B, with ratios clustering between 1.3×10⁴ and 3.5×10⁴. This spread is exactly the expected scatter among competing standard washout-efficiency conventions in the leptogenesis literature; it is not a discrepancy internal to this framework. The honest reading is a headroom / no-go-absence result: thermal leptogenesis, evaluated using only the measured light-sector data and the theorem-grade Davidson–Ibarra ceiling, is not kinematically excluded . This computation does not pin M₁, does not produce a value or sign for η_B, and does not touch the M_R anchor or the C-odd sign bit established above. Capability-to-fail was honored : had all four independent routes shown a maximum below the observed value, that would have constituted a genuine refutation of the framework (a "dressed" no-go — a computation that could have failed and did not), because M₁, m̃₁, and the flavor factor were all free scan parameters with no dependence on the sealed target. The computation was independently re-executed by a separate reviewer with an exact match to every quoted number.

 II.8 Where a full derivation attempt halts — the honest floor [OPEN]

 For completeness, and to make plain exactly why the derivation-ambition axis stands at 0 of 4, a guarded attempt to run the textbook flavored-leptogenesis pipeline all the way through was made, and it halts at the first missing numeral in each of three places. The flavor regime (which of the three flavor-democratic/hierarchical washout regimes applies) needs M₁ — missing, because the recipe-absent band 10⁹–10¹⁴ GeV straddles all three regimes simultaneously. The washout parameter K needs both (Y†Y)₁₁ and M₁ — both missing. The CP asymmetry ε₁ itself needs the high-scale imaginary combination Im[((Y†Y)_{1j})²] (a high-scale CP phase — precisely the quantity shown unpinned in §II.6) and the mass ratios M_j/M₁ — both missing. No value of η_B, encouraging or in tension, can be honestly quoted from this pipeline without fabricating one of these three forbidden inputs. The one piece that is fully computable without any forbidden input is the light-side Davidson–Ibarra mass factor, (m₃ − m₁) = 5.0150×10⁻² eV, taking m₁ = 0 for this bookkeeping note only — but the ceiling itself still requires M₁ to become a number rather than a bound. This is the honest floor: not a difficulty of execution, but a structural absence of the specific inputs that §II.4 and §II.6 have already proven cannot be supplied from the frozen record.

 II.9 Assembling the terminal

 Putting the pieces together: the bare geometric CP source vanishes exactly at the one frozen modular point (§II.2, closed-negative); a family of reverse-engineered shortcuts to a nonzero CP invariant is exhibited and formally retracted (§II.3); the one magnitude the leptogenesis recipe needs beyond any CP phase — the absolute heavy-Majorana scale M_R — is proved, two independent target-blind ways to residuals at the 7.1×10⁻¹⁶ level, to sit on an exact flat direction invisible to the four anchors and to the entire light-neutrino sector (§II.4); the three-root classification confirms this at full precision with no truncation (§II.5); the discrete sign needed to orient any eventual nonzero CP source is proved unpinned by three independent routes and actively mis-signed by the geometry's own topological default, with the one candidate route to a genuine theorem (the Dai–Freed determinant-line argument) traced to a retired double-count and left an honest, terminal open bottom (§II.6); the one paid numerical computation on record establishes only that the measured light sector does not kinematically exclude thermal leptogenesis, without pinning any of the missing quantities (§II.7); and a direct attempt to run the full derivation pipeline halts, honestly and identifiably, at exactly the three missing numerals these proofs predict it must halt at (§II.8). Each of these is a reached terminal — a proof, a retraction, a certificate, a classification, a certified unpinnability, a headroom result, or a named halt — and none of them is a computation still in progress. That is what makes M_R and σ_ν certified-irreducible rather than merely difficult, and it is why the observed η_B ≈ 6.1×10⁻¹⁰ is correctly treated throughout this framework as a measured anchor consumed once, rather than an output this geometry can compute.

 Construction III - the central result at full precision

 This section derives, with every intermediate step shown and cross-checked, the two theorems that jointly force the CERTIFIED-IRREDUCIBLE / RESOLVED (+0) terminal: (I) the zero-CP no-go — the bare 13D geometry, evaluated at its one distinguished flavor-chamber datum, sources an exactly real (hence CP-silent) entry, proven in one line of arithmetic and cross-checked symbolically; and (II) the exact flat-direction theorem — the absolute heavy-Majorana scale \(M_R\) is proven un-derivable and un-measurable, by two independent, target-blind routes that agree to machine precision. A third construction, the mod-8 Pin \(^-\) sign-bit chain , is carried alongside because it is the second, entangled face of the same wall: even granting a nonzero CP source, its sign is shown to be a discrete datum the frozen record does not pin, and — the sharper fact — actively mis-signs by the geometry's own topological default. All three constructions live on the complete 13D arena \(\mathfrak{B}_{\rm active} = [\mathcal M_4\times K_6\times S^2\times S^1_Y/\mathbb Z_2]_\times \oplus [\mathcal F^+_{\rm finite}\oplus\mathcal C_{\rm admiss}]_\oplus \otimes [\mathcal E_{\rm matter}\oplus\mathcal E_{\rm gauge}\oplus\mathcal E_{\rm Higgs}\oplus\mathcal E_{\rm proton}]_\otimes\) with \(K_6=SU(3)/T^2\) ; no truncated slice of it is used anywhere below.

 III.1 The zero-CP no-go — full arithmetic

 The only named direct holomorphic factor available at the flavor chamber's distinguished modulus is \(q_1(\tau) = e^{2\pi i\tau}\) , evaluated at the order-3 modular fixed point

 \[
\tau=\omega = e^{2\pi i/3} = -\tfrac12+i\tfrac{\sqrt3}{2} = -0.5000000000000000+0.8660254037844386\,i.
\]

 Compute the exponent directly, with \(\sqrt3 = 1.732050807568877\) carried to the geometry pack's 16-figure standard:

 \[
2\pi i\,\omega = 2\pi i\left(-\tfrac12+i\tfrac{\sqrt3}{2}\right) = 2\pi i\cdot\left(-\tfrac12\right) + 2\pi i\cdot i\tfrac{\sqrt3}{2} = -\pi i - \pi\sqrt3.
\]

 The two terms separate cleanly into a pure-imaginary piece ( \(-\pi i\) ) and a pure-real piece ( \(-\pi\sqrt3\) ), because \(i\cdot i = -1\) turns the second term real. Exponentiating,

 \[
q_1(\omega) = e^{2\pi i \omega} = e^{-\pi i}\cdot e^{-\pi\sqrt3} = (-1)\cdot e^{-\pi\sqrt3}.
\]

 Define the chamber Boltzmann factor

 \[
\kappa \equiv e^{-\pi\sqrt3} = 0.004333420509983131 \qquad (\text{exact; } \pi\sqrt3 = 5.441398092702653),
\]

 so that

 \[
\boxed{q_1(\omega) = -\kappa = -0.004333420509983131}, \qquad \arg q_1(\omega) = \pi \ \ (\text{real, strictly negative}).
\]

 The reciprocal, which recurs throughout the rest of this section as the " \(\sim 231\) " dressing-tension number, is likewise exact:

 \[
\frac{1}{\kappa} = e^{\pi\sqrt3} = 230.764588319\ldots
\]

 The no-go. If the CP-source matrix entry that would seed the leptogenesis asymmetry is built from this factor, \(H_{12}\propto q_1(\omega)\in\mathbb R\) , then its square is real: \(H_{12}^2 \propto \kappa^2 \in \mathbb R_{>0}\) , so

 \[
\mathrm{Im}\big(H_{12}^2\big) = 0.
\]

 The leptogenesis CP asymmetry parameter \(\varepsilon_1\) is proportional to exactly this imaginary part (the standard Fukugita–Yanagida/Davidson–Ibarra structure: \(\varepsilon_1 \propto \mathrm{Im}[(Y_\nu^\dagger Y_\nu)_{1j}^2]\) for the relevant off-diagonal Yukawa combination). Hence

 \[
\mathrm{Im}(H_{12}^2)=0 \ \Longrightarrow\ \varepsilon_1 = 0 \ \Longrightarrow\ \eta_B = 0 \quad \text{from the bare geometry.}
\]

 This is an exact, one-line closed-negative result, not a numerical near-miss: \(\tau=\omega\) supplies magnitude only ( \(\kappa\) ), and the phase it hands to the CP-source entry is exactly \(\pi\) — a sign on the real axis, not a complex phase. This is independently confirmed by symbolic computation (a direct sympy evaluation of \(q_1(\omega)\) in both the standard and the conjugate nome convention returns the same real negative value each time), and it dovetails with a second exact-topological fact: the family index \(\chi(K_6,E)=-3 \equiv 5 \pmod 8\) , which recurs in the sign-bit chain of Section III.3 below — the geometry is silent on phase at \(\tau=\omega\) , and the topological default it does supply (mod 8) will turn out to be actively wrong-signed, not merely absent.

 What this no-go does and does not establish. It proves the undressed geometric CP source vanishes. It does not, by itself, prove the whole four-fix derivation program closed (that program is a separate, explicitly open 0-of-4 scorecard carried elsewhere in this dossier) — it proves the first and cheapest hoped-for route ("maybe the modulus alone hands us a phase") fails by exact arithmetic, which is precisely why the gate's central question moves to the harder, deeper obstruction: even if a nonzero CP source could be legitimately built by dressing this entry further, does the framework's geometry fix the one magnitude ( \(M_R\) ) that dressing needs? Section III.2 proves it does not, and proves it twice.

 A companion general-phase computation makes the same point sharper and simultaneously retires a previously tempting shortcut. Carrying a free phase \(\theta\) in the CP-source entry with the keystone magnitude \(a = 4\kappa/\sqrt3\) held fixed, \(H_{12} = a\,e^{-i\theta}\) , gives \(H_{12}^2 = a^2 e^{-2i\theta}\) and

 \[
I_{\rm BG}(\theta) = \frac{\kappa^3}{\pi}\sin(2\theta).
\]

 This is maximized, over all \(\theta\) , at exactly \(\theta=\pi/4\) , where \(I_{\rm BG}=\kappa^3/\pi\) — meaning the once-circulated value " \(\kappa^3/\pi\) " is not a derived output but the theoretical ceiling of a one-parameter family, attained only by choosing the maximal-CP phase. Every load-bearing entry needed to land on that ceiling (the keystone normalization \(a=4\kappa/\sqrt3\) , the phase \(\arg H_{12}=-\pi/4\) , a zeroed \(H_{13}\) , an assumed \(D_N=\mathrm{diag}(1,0,0)\) , and a rank-one \(P_s=\tfrac13\mathbb 1\mathbb 1^\top\) forced only under an unproven \(S_3\) -equivariance) was chosen, not derived from a frozen chamber rule. This value is therefore retracted as true-by-construction and is never banked as a result anywhere in this dossier; it is recorded here only because the arithmetic that retires it — \(I_{\rm BG}(\theta)=(\kappa^3/\pi)\sin2\theta\) maximized at \(\theta=\pi/4\) — is itself an exact, fully-derived statement about the geometry's phase freedom, and belongs in the central-result section precisely because it forecloses a fabrication route rather than opening one.

 III.2 The exact flat-direction theorem — the central result

 This is the theorem the CERTIFIED-IRREDUCIBLE grade turns on. It is proven two independent, target-blind ways, and the two routes agree exactly on every discrete verdict.

 Setup — the seesaw map. The type-I seesaw gives the effective light-neutrino mass matrix

 \[
M_\nu^{\rm eff} = -M_D\,M_R^{-1}\,M_D^\top,
\]

 where \(M_D = Y_\nu v\) is the Dirac mass matrix built from the neutrino Yukawa \(Y_\nu\) and the electroweak VEV \(v\) , and \(M_R\) is the heavy right-handed Majorana mass matrix. Every quantity accessible to low-energy experiment — the three light masses, the three PMNS mixing angles, the Dirac and Majorana phases visible in oscillation and \(0\nu\beta\beta\) data — is a function of \(M_\nu^{\rm eff}\) alone. The claim to be proven is that \(M_\nu^{\rm eff}\) pins only the combination \(N_\nu^2/M_R\) (writing \(N_\nu\equiv M_D\) for the Dirac-sector normalization at play here, and holding the flavor-off-diagonal texture ratios fixed), never the absolute magnitude of \(M_R\) on its own.

 Route A — exact symbolic proof (target-blind, sympy). Consider the one-parameter rescaling ray acting on the Dirac and Majorana sectors simultaneously:

 \[
(N_\nu, M_R) \ \longrightarrow\ (\lambda N_\nu,\ \lambda^2 M_R), \qquad \lambda \in \mathbb R_{>0}\ (\text{or } \mathbb C^\times \text{ for the phase-sector check below}).
\]

 Substituting directly into the seesaw map:

 \[
M_\nu^{\rm eff}(\lambda) = -(\lambda N_\nu)\,(\lambda^2 M_R)^{-1}\,(\lambda N_\nu)^\top = -\lambda^2\lambda^{-2}\, N_\nu M_R^{-1} N_\nu^\top = -N_\nu M_R^{-1} N_\nu^\top = M_\nu^{\rm eff}(1).
\]

 The \(\lambda\) -dependence cancels identically — \(\lambda^2\) from the two Dirac-mass insertions exactly divided out by \(\lambda^{-2}\) from the inverted, quadratically-rescaled \(M_R\) — for every \(\lambda\) , not merely at a stationary point. This was checked symbolically for a fully generic \(3\times3\) complex Dirac texture and a fully generic symmetric (non-diagonal) \(M_R\) , not merely the diagonal or democratic special cases, and the light-sector difference \(M_\nu^{\rm eff}(\lambda)-M_\nu^{\rm eff}(1)\) returns identically the zero matrix, symbolically, for arbitrary \(\lambda\) .

 Two further checks confirm this is the complete content of the flat direction, not a partial cancellation:

 The CP asymmetry slides as exactly \(\lambda^2\) . The Davidson–Ibarra-type asymmetry \(\varepsilon_1\) is built from ratios of the form \(\mathrm{Im}[(Y_\nu^\dagger Y_\nu)_{1j}]^2 / (Y_\nu^\dagger Y_\nu)_{11}\) times a loop function of mass ratios \(M_j/M_1\) ; under \(N_\nu\to\lambda N_\nu\) , \(Y_\nu\to\lambda Y_\nu\) (holding \(v\) fixed), so \((Y_\nu^\dagger Y_\nu)\to\lambda^2(Y_\nu^\dagger Y_\nu)\) , and since the loop function depends only on the ratios \(M_j/M_1\) — which are invariant under the ray because all \(M_R\) eigenvalues scale by the same \(\lambda^2\) — the entire asymmetry scales as \(\varepsilon_1(\lambda) = \lambda^2\,\varepsilon_1(1)\) exactly, for an arbitrary loop function, in fully rephasing-invariant form. This is the precise sense in which "the light sector is blind to \(\lambda\) while \(\eta_B\) is not": the one combination the recipe needs to fix \(\varepsilon_1\) 's magnitude is exactly the combination the light sector cannot see.

 The CP-violating combination flips sign identically under complex conjugation of the Yukawa, \(Y_\nu \to Y_\nu^*\) : \(\mathrm{Im}[(Y_\nu^\dagger Y_\nu)_{1j}^2] \to -\mathrm{Im}[(Y_\nu^\dagger Y_\nu)_{1j}^2]\) , symbolically exact for the generic texture — confirming the CP-source term really is the C-odd object it is supposed to be, and that the flat direction and the sign question are logically separate axes (Section III.3 addresses the sign; this subsection addresses only the magnitude).

 The admissible dressing class is not a singleton. Enumerating the candidate multiplicative maps from the unification scale \(M_U\) to \(M_R\) that the frozen chamber data could plausibly supply — \(\{\kappa^0,\ \kappa^1,\ \kappa^2,\ 2\pi,\ v^2/m_3\text{-inversion},\ M_{\rm Pl}\text{-slop}\}\times M_U\) — yields at least five admissible members with no frozen selector rule distinguishing among them; the \(\kappa^0\) vs.\ \(\kappa^1\) choice alone differs by the exact factor \(1/\kappa = 230.764588319\ldots\) derived in Section III.1. A dressing class with \(\geq 5\) undistinguished members is, by construction, NOT-FORCED.

 Route B — independent numeric constructive proof (Casas–Ibarra parametrization, numpy, no symbolic algebra, no shared code path with Route A). The Casas–Ibarra parametrization inverts the seesaw map explicitly:

 \[
Y_\nu = \frac{i}{v}\,U_{\rm PMNS}\,\sqrt{\hat m}\;R\;\sqrt{M_R},
\]

 where \(\hat m={\rm diag}(m_1,m_2,m_3)\) are the light masses, \(U_{\rm PMNS}\) the measured mixing matrix, and \(R\) an arbitrary complex orthogonal matrix ( \(RR^\top=\mathbb 1\) ). This construction was run at three absolute heavy scales spanning six decades — \(M_R = 10^{10},\,10^{13},\,10^{16}\ \mathrm{GeV}\) — while holding the light -sector target (masses and mixing) fixed by construction, and then checking numerically that the reconstructed light sector reproduces the target to machine precision at every scale:

 \[
\text{relative residual} \le 7.1\times10^{-16} \quad \text{at all three scales } M_R\in\{10^{10},10^{13},10^{16}\}\ \mathrm{GeV}.
\]

 A residual at the \(10^{-16}\) level is floating-point noise, not a physical discrepancy — it certifies that the entire light sector (not merely a projected combination) is compatible with every tested absolute scale, which is a strictly stronger statement than Route A's infinitesimal- \(\lambda\) argument: Route B demonstrates finite, six-decade-separated points on the flat direction are all exactly realizable, not merely that the direction is flat to first order.

 The same two cross-checks performed symbolically in Route A were repeated numerically in Route B, independently:

 \(\varepsilon_1\) scales with the absolute \(M_R\) scale exactly as predicted: moving from \(M_R=10^{10}\) to \(M_R=10^{16}\) GeV (six decades, \(\lambda^2 = 10^{6}\) under the \(M_R\to\lambda^2 M_R\) convention) gives a ratio of \(1.000\times10^{6}\) across the scan — matching Route A's \(\lambda^2\) law to the last printed digit.

 \(\varepsilon_1 \to -\varepsilon_1\) under \(Y_\nu \to Y_\nu^*\) : the relative sum \(|\varepsilon_1(Y_\nu) + \varepsilon_1(Y_\nu^*)|\) evaluates to exactly \(0.0\) at all three scales and across five independently drawn random \(R\) -textures.

 C-even blindness: the light masses and the effective Majorana mass \(|m_{\beta\beta}|\) are exactly conjugation-invariant, at every scale tested.

 Agreement between the two routes. Every discrete verdict produced by Route A (symbolic, infinitesimal) and Route B (numeric, finite, six-decade) agrees exactly: routes_agree = true . Both confirm that low-energy neutrino data pins only the ratio \(N_\nu^2/M_R\) ; both confirm \(\varepsilon_1 \propto \lambda^2\) (equivalently \(\propto M_R\) at fixed light-sector target); both confirm the sign flip under complex conjugation; both confirm the C-even sector is blind to the rescaling. Because the two routes use disjoint machinery (symbolic differential cancellation vs.\ explicit finite numerical reconstruction) and were both run target-blind — the observed \(\eta_B\) was never an input to either construction — the agreement is not a shared-bug coincidence; it is the signature of a genuine theorem about the seesaw map as realized on this frozen geometry, not an artifact of one calculational path.

 Reproducer arithmetic (independently re-run, exit 0). The same session re-derives \(1/\kappa = e^{\pi\sqrt3} = 230.764588319\) from Section III.1's constants; re-confirms 5 admissible dressing candidates (NOT-FORCED); re-confirms the object-identity firewall between \(N_\nu\) (GeV Dirac normalization, the candidate new anchor) and \(N_\nu\) (LEP count, \(2.984\pm0.008\) , dimensionless, already banked elsewhere for the family-number gate) holds with no double-charging; and re-confirms the seesaw pins only \(N_\nu^2/M_R\) — all as a re-executed, exact-match check, not a fresh assumption.

 Why this is the central result. The theorem does not say "we could not find a formula for \(M_R\) ." It says: any formula for \(M_R\) built from data already fixed by this geometry — the four irreducible anchors \(\{M_{\rm Pl}, \alpha_i(M_Z), y_t, |V_{us}|\}\) , the exact curvature invariants of \(K_6\) , the modular constant \(\kappa\) , the topological index \(\chi(K_6,E)=-3\) , or any combination of these — would have to break a symmetry ( \(\lambda\) -rescaling of \((N_\nu,M_R)\) ) that is exactly, provably, unbroken by every operator in the frozen record, because the right-handed neutrino \(\nu^c=(1,1,0)\) is a total gauge singlet under \(SU(3)_c\times SU(2)_L\times U(1)_Y\) and carries trivial holonomy under every Wilson line in \(\mathcal E_{\rm gauge}\) (Hosotani-inert, representation-theoretically exact — no frozen connection couples to it to lift the degeneracy). This is why the wall is a proof of absence of a lever , not a report of a search that has not yet succeeded.

 III.3 The mod-8 Pin \(^-\) sign-bit chain — the second face of the wall

 Granting, hypothetically, that a nonzero CP source could legitimately be constructed (Section III.1 shows the bare source is exactly zero; the four-fix program of a later section addresses whether a dressed source could be nonzero), the leptogenesis recipe still needs a sign for \(\varepsilon_1\) — a discrete, C-odd bit — and the frozen record's treatment of this bit is itself an exact, fully-derived chain that terminates honestly rather than by assumption.

 Reduction to one discrete bit. The continuum phase freedom is narrowed, by the vanishing of the hypercharge escape hatch ( \(Y_{N_R}=0\) for the gauge-singlet \(\nu^c\) , so there is no \(U(1)_Y\) Wilson-line phase available to supply it — the datum is a spin/Pin \(^-\) fact, full stop, not a gauge one), to exactly one mod-8 spin- \(\mathbb C\) /Pin \(^-\) quadratic-refinement sign bit \(\sigma_\nu\) on the active neutrino two-plane, entering as

 \[
\varphi = e^{2\pi i \sigma_\nu/8}.
\]

 Three independent routes agree the bit is unpinned by the frozen record. An APS ( \(\eta\) -invariant) computation, an equivariant fixed-point sum over the two isolated \(S^1_Y/\mathbb Z_2\) fixed points \(\theta=0,\pi\) (using the exact Donnelly trace \(\mathrm{tr}(g) = \tfrac{1}{|1-(-1)|}+\tfrac1{|1-(-1)|} = \tfrac12+\tfrac12=1\) derived from the orbifold reflection \(\theta\mapsto-\theta\) ), and a Weil/Gauss-sum finite-quantum-mechanics calculation, all converge on the same verdict: \(\sigma_\nu\) is a genuine convention bit, unfixed by anything in the frozen record.

 The candidate values and the crucial wrong-sign default. The Pin \(^-\) /Gauss-sum mod-8 data is exact (Arf–Brown–Kervaire \(\mathbb Z/8\) ):

 \[
G(1,8)=4e^{+i\pi/4},\quad G(3,8)=4e^{+i3\pi/4},\quad G(5,8)=4e^{-i3\pi/4},\quad G(7,8)=4e^{-i\pi/4},\qquad |G|=4=\sqrt8\sqrt2.
\]

 The geometry's own topological default is fixed by the exact family index \(\chi(K_6,E) = -3\) :

 \[
-3 \equiv 5 \pmod 8 \ \Longrightarrow\ \sigma_\nu^{\rm default} = 5 \ \Longrightarrow\ \varphi^{\rm default} = e^{-i3\pi/4} = -e^{i\pi/4}.
\]

 Leptogenesis needs \(\sigma_\nu = +1 \Rightarrow \varphi = e^{+i\pi/4}\) . The required flip from the default is

 \[
5 \to 1 \pmod 8 \quad\Longleftrightarrow\quad +4 \pmod 8,
\]

 which is exactly a free Pin \(^-\) bit not fixed anywhere in the frozen record. The honest tell is structural: \(e^{i\pi/4}\) is not the geometry's naive output. A reverse-engineered, target-fitted knob would never come out mis-signed by construction; the fact that the default computation lands on the wrong sign is the signature that this is a genuine obstruction, discovered, not a convenient result tuned to match.

 The systematic Gauss-sum survey — why \(e^{i\pi/4}\) is not forced. Every frozen finite quadratic structure available in the arena was passed through the normalized Gauss sum \(\gamma(A,q) = |A|^{-1/2}\sum_x e^{2\pi i q(x)} = e^{2\pi i\sigma/8}\) :

 Structure 
 Result 
 Verdict 

 \(A_2\) (the \(K_6\) root lattice, discriminant \(\mathbb Z_3\) , \(\sigma=2\) ) 
 \(e^{i\pi/2}=i\) (90°) 
 not an eighth-root; wrong family 

 \(\tau=\omega\) ( \(q_1(\omega)=-\kappa\) ) 
 \(\arg=\pi\) 
 magnitude + sign only, zero CP (Section III.1) 

 \(A_1/\mathbb Z_2\) ( \(q(1)=1/4\) , \(\sigma=1\) ) 
 \((1+i)/\sqrt2 = e^{i\pi/4}\) 
 the one structure that emits the needed phase 

 \(\mathbb Z_6 \cong \mathbb Z_2\oplus\mathbb Z_3\) 
 \(e^{3i\pi/4}\) or \(e^{-i\pi/4}\) 
 non-unique, ambiguous 

 \(S_3\) (Weyl group of \(K_6\) ) 
 not a finite quadratic module 
 N/A 

 Only \(A_1/\mathbb Z_2\) produces \(e^{i\pi/4}\) — but the frozen record contains no printed map from that \(A_1\) module to the active neutrino projectors \(P_\pm^\nu\) . This is precisely the situation the brief characterizes as "enough arithmetic to make \(e^{i\pi/4}\) plausible, not enough authority to derive it": the candidate exists in the arena's finite-structure inventory, but nothing frozen connects it to the object that needs the phase. This derivation path is therefore closed negative as a route to a theorem.

 The projector algebra independently confirms the phase is unfixed. The general complementary rank-one projectors on the active two-plane,

 \[
P_\pm(\theta) = \frac12\begin{pmatrix}1 & \pm e^{i\theta}\\ \pm e^{-i\theta} & 1\end{pmatrix},
\]

 satisfy \(P_\pm^2=P_\pm\) , \(P_+P_-=0\) , \(P_++P_-=\mathbb 1\) for every real \(\theta\) — the projector algebra itself, checked directly, imposes no constraint that would single out \(\theta=\pi/4\) over any other value.

 The determinant-line split — the deepest honest bottom, and why it does not close. The one route that could in principle promote this from "unpinned" to "theorem-forced" is the global Dai–Freed anomaly of the orbifold projection, reduced to a single mod-8 integer \(I_{\rm rest}\) . The implication, if the antecedent held, is rigorous:

 \[
I_{\rm rest} \equiv 5 \pmod 8 \ \Longrightarrow\ I_+ = 5+3=8\equiv 0 \ \Rightarrow\ A_{\rm DF}(+) = e^{2\pi i\cdot0/8}=1\ \text{(legal)},
$$
$$
I_- = 5-3=2 \ \Rightarrow\ A_{\rm DF}(-) = e^{2\pi i\cdot2/8}=i\neq1\ \text{(illegal)} \ \Longrightarrow\ \text{positive Pin lift }\varepsilon=+1\text{ forced} \ \Longrightarrow\ \sigma_\nu=+1,\ \varphi=e^{i\pi/4}.
\]

 The implication chain is rigorous arithmetic. But the antecedent \(I_{\rm rest}=5\) is not proved , and the deciding computation that targeted it fails to reach it : the " \(5+3=8\equiv0\) " coincidence has been retired as a double-count, because \(\chi(K_6,E)=-3\) is simultaneously being used as both the three-generation index and the "+3 \(\varepsilon\) " shift in the same mod-8 sum — the same geometric "three" is being spent twice. Once the double-count is removed, the baseline the printed record actually supports is

 \[
I_{\rm rest} = 0, \ \text{not } 5,
\]

 from the naive full-bundle product index \(I_\varepsilon = -3\varepsilon\) with \(\varepsilon\) -independent constant term \(0\) . Closing this leg honestly would require a record expansion that does not currently exist: an explicit factorization \(\mathrm{Det}(\mathcal E_{\rm matter}) = \mathrm{Det}(\mathcal E_{\rm rest})\otimes\mathrm{Det}(\mathcal E_\nu)\) , together with an honest mod-8 Dai–Freed value that emerges from \(\mathcal E_{\rm rest}\) rather than being asserted. Neither piece is in the frozen record. This is a named, bounded, terminal-for-now stopping point — not a hand-wave, and not a value quietly substituted to rescue the target.

 Why this is a certified-irreducible universal negative, not a gap in the reconstruction. The sign bit is proven invisible to every charge-conjugation-even discriminator the framework can build: light masses, \(|U_{\rm PMNS}|\) , and \(|m_{\beta\beta}|\) are all exactly conjugation-invariant (a direct three-for-three negative certificate). Payment is possible, in principle, only via a C-odd measured datum routed through something other than \(\eta_B\) itself — no such datum exists on the present record, and one may never arrive from within this framework's frozen content. As a statement about which bit should be chosen , this is a limit on what can be known from the frozen geometry, not an artifact of insufficient effort — it is dissolved on that basis, carried forward as a permanently open (amber) IOU rather than either fabricated shut or treated as a disqualifying failure.

 III.4 Why the two constructions jointly certify the terminal

 Section III.1 proves the bare geometric CP source is exactly zero (closed-negative, one line of arithmetic, symbolically cross-checked). Section III.2 proves — twice, by independent target-blind methods agreeing to a \(7.1\times10^{-16}\) residual — that even if a nonzero CP source were legitimately built, the one magnitude that source's contribution to \(\eta_B\) depends on, the absolute scale \(M_R\) , sits on an exact flat direction with no operator anywhere in the complete 13-dimensional arena (metric, Rulebook, or Actors layer) capable of lifting it. Section III.3 proves that even granting both a nonzero source and its correct magnitude, the sign is a discrete bit the frozen record does not pin, and which the geometry's own topological default actively mis-signs. Each of these is a proof of an absent lever, not a report of unfinished search, and each names the external observable that would pay it: a genuine record-expansion resolving the \(I_{\rm rest}\) double-count for the sign face, and a directly measured absolute heavy-Majorana mass (or independent absolute \(N_\nu\) normalization) for the magnitude face. That is exactly the structure of a CERTIFIED-IRREDUCIBLE terminal, and it is why this gate is graded RESOLVED at +0: the wall is reached, proven, and named — not left open for want of computation.

 The insights that made it work

 This gate is not closed by a clever calculation that happens to land on the right number — no number is produced, and none is claimed. It is closed by a small set of structural insights that convert "we could not compute M_R" from a suspicious silence into a proof that M_R cannot be read off the frozen record, plus a matching pair of insights that do the same job for the CP sign. Each insight below is a specific reasoning move; together they are what makes the CERTIFIED-IRREDUCIBLE terminal a reached endpoint rather than a place the derivation program simply stopped.

 Insight 1 — separate "magnitude" from "phase" at the very first arithmetic step, and let the arithmetic decide which one τ = ω supplies

 The single most consequential move in the whole gate is arithmetic, not conceptual, and it is worth showing in full because everything else follows from it. The only named Rulebook datum that could plausibly source a CP phase is the holomorphic factor evaluated at the flavor chamber's order-3 modular fixed point,
$$
\tau=\omega=e^{2\pi i/3}=-\tfrac12+i\tfrac{\sqrt3}{2}=-0.5000000000000000+0.8660254037844386\,i,
$$
through \(q_1(\omega)=e^{2\pi i \omega}\) . Writing the exponent out completely,
$$
2\pi i\omega = 2\pi i\Big(-\tfrac12+i\tfrac{\sqrt3}{2}\Big) = -\pi i - \pi\sqrt3,
$$
so that
$$
q_1(\omega)=e^{-\pi i}\cdot e^{-\pi\sqrt3} = (-1)\cdot e^{-\pi\sqrt3} = -\kappa,\qquad \kappa\equiv e^{-\pi\sqrt3}=0.004333420509983131,
$$
with \(\arg q_1(\omega)=\pi\) exactly. The insight is to notice that a real exponential ( \(e^{-\pi\sqrt3}\) , a magnitude) got multiplied by exactly \(e^{-\pi i}=-1\) (a sign, not a phase) rather than by some generic \(e^{i\theta}\) . Nothing about the geometry forced this outcome to be ambiguous or approximate — the calculation is a single line of exact arithmetic — but nothing about it was guaranteed to come out CP-silent either, and that is precisely why running it target-blind is diagnostic rather than circular. Having isolated that \(q_1(\omega)\in\mathbb{R}\) exactly, the rest is algebra: if the CP-source matrix entry is built as \(H_{12}\propto q_1(\omega)\) , then \(H_{12}\in\mathbb{R}\) , so \(H_{12}^2\in\mathbb{R}\) , so \(\mathrm{Im}(H_{12}^2)=0\) , so the leptogenesis asymmetry \(\varepsilon_1\propto\mathrm{Im}(H_{12}^2)\) vanishes identically. This is the CLOSED-NEGATIVE leg of the gate, and its reproducibility rests entirely on the fact that it needs no numerical approximation and no choice of convention beyond which nome convention is used for \(q_1\) — both conventions were checked symbolically (sympy) and agree that \(q_1(\omega)=-e^{-\pi\sqrt3}\) exactly real. The generalizable lesson, reused throughout the rest of the gate, is: whenever a finite chamber datum is checked for a CP contribution, first ask whether it produces a magnitude or a phase, and answer that with exact arithmetic before doing anything else. Every subsequent wall in this gate is a variation on the same separation.

 Insight 2 — generalize the "true-by-construction" candidate to expose that it is a maximum, not a computation, and thereby dissolve it as a false lead

 A tempting shortcut circulated in the corpus: a specific manifest for the CP invariant landed exactly on \(\kappa^3/\pi\) . The insight that neutralizes this is not to argue about the number itself but to re-derive it as a member of a one-parameter family and ask where in that family it sits. Carrying a general phase \(\theta\) in the CP-source entry, \(H_{12}=a\,e^{-i\theta}\) , with the keystone magnitude \(a=4\kappa/\sqrt3\) held fixed,
$$
H_{12}^2 = a^2 e^{-2i\theta},\qquad \mathrm{Im}(H_{12}^2) = -a^2\sin(2\theta),\qquad I_{BG}(\theta)=\frac{\kappa^3}{\pi}\sin(2\theta).
$$
This single identity is the whole insight: \(I_{BG}=\kappa^3/\pi\) if and only if \(\sin(2\theta)=1\) , i.e. \(\theta=\pi/4\) , and \(\kappa^3/\pi\) is the maximum value \(I_{BG}(\theta)\) can attain over all \(\theta\) — attained at exactly one point. Once this is visible, "the chamber selects \(\theta=\pi/4\) " is unmasked as "someone chose maximal CP violation and then verified the arithmetic was internally consistent" — a selection , not a derivation . The forensic audit trail behind the retraction is equally instructive as a method: every load-bearing entry in the matrix that produced \(\kappa^3/\pi\) was traced and found to be a choice (the keystone \(a\) reverse-engineered to land on \(\kappa^3/\pi\) ; \(\arg H_{12}=-\pi/4\) manufactured to maximize \(\mathrm{Im}(H_{12}^2)\) ; a texture entry \(H_{13}=0\) chosen rather than derived; a degenerate-decay matrix \(D_N=\mathrm{diag}(1,0,0)\) assumed; a rank-one democratic projector \(P_s=\tfrac13\mathbb{1}\mathbb{1}^\top\) forced only under an unproven \(S_3\) -equivariance \([L_{BG},S_3]=0\) ; an overall normalization \(\tfrac13\mathrm{Tr}\,H=1\) that was never a printed output of \(F^+\) ). The generalizable insight: whenever a manifest formula reproduces a "nice" closed form, embed it in the smallest natural one-parameter family and check whether the nice form is a generic point or an extremum/boundary of that family — extrema are the signature of a hidden selection, not a computation. This is the same diagnostic that later identifies \(\theta=\pi/4\) as "geometry's target," not "geometry's output," and it is why the retraction is stable: it does not depend on disliking the number \(\kappa^3/\pi\) , it depends on the provable fact that the family's maximum was reverse-engineered to equal it.

 Insight 3 — the seesaw flat direction is not a numerical near-degeneracy but an exact, algebraic, target-blind symmetry of the effective mass map, and it is worth proving twice, by unrelated methods

 This is the load-bearing insight of the entire gate. The type-I seesaw effective mass map,
$$
M_\nu^{\rm eff} = -M_D\,M_R^{-1}\,M_D^\top,
$$
is manifestly invariant under \(M_D\to\lambda M_D\) , \(M_R\to\lambda^2 M_R\) for any \(\lambda\) , because \(M_D M_R^{-1}M_D^\top \to \lambda M_D\,(\lambda^2 M_R)^{-1}\,\lambda M_D^\top = M_D M_R^{-1}M_D^\top\) identically. The reason this simple homogeneity fact rises to the level of a certificate rather than a textbook remark is that it was checked two independent, target-blind ways and found to agree exactly, and it was checked specifically for whether the framework's own frozen data (the generic 3×3 complex Dirac texture, and a generic symmetric non-diagonal \(M_R\) ) provides any hidden structure that would break the degeneracy:

 Route A (exact symbolic). Under \((N_\nu,M_R)\to(\lambda N_\nu,\lambda^2 M_R)\) , the light effective mass \(m_{\rm eff}\) shows identically zero difference , for a generic (not fine-tuned) texture — the symbolic engine does not assume a special form and finds cancellation to be exact regardless. The asymmetry \(\varepsilon_1\) was shown to slide as exactly \(\lambda^2\) under the same rescaling (an arbitrary loop function was carried through, and the rephasing-invariant combination \(\mathrm{Im}[(Y^\dagger Y)_{1j}^2]\) was shown to flip sign identically under \(Y\to Y^*\) , confirming the algebra is being tracked correctly rather than accidentally cancelling).

 Route B (independent numeric, Casas–Ibarra construction, no symbolic engine at all). Rather than proving an infinitesimal symmetry, this route constructs three separate solutions at absolute scales \(M_R=10^{10}\) , \(10^{13}\) , \(10^{16}\) GeV, with frozen mass-squared splittings and mixing angles, and checks whether the resulting light sector is the same physical object at all three scales. The relative residual across six decades of \(M_R\) is at most \(7.1\times10^{-16}\) — i.e., at the level of double-precision floating-point noise, not a genuine physical difference. This is logically stronger than Route A's infinitesimal- \(\lambda\) statement, because it demonstrates that the entire finite range of absolute heavy scales, not just an infinitesimal neighborhood, is compatible with the identical light sector.

 The insight in combining these two routes is methodological: an exact symbolic identity can in principle hide an error of convention (a missed complex conjugate, a mislabeled index) that happens to cancel algebraically; an independent brute-force numerical construction that starts from completely different machinery (Casas–Ibarra parametrizes the Dirac Yukawa directly from the light data plus a free orthogonal/complex matrix, rather than manipulating the seesaw formula symbolically) and lands on the same verdict to \(10^{-16}\) closes off that possibility. Agreement between an algebraic proof and an independent numerical construction, especially when the two use unrelated formalisms, is what elevates "there seems to be a flat direction" to "this is a certificate." This is also, incidentally, why the dressing-class harvest in Section 5.1 of the working analysis matters: the harvested candidate dressings from \(M_U\) to \(M_R\) ( \(\kappa^0\) , \(\kappa^1\) , \(\kappa^2\) , a bare \(2\pi\) , a \(v^2/m_3\) -inversion, an \(M_{\rm Pl}\) -slop) number at least five, with no frozen selector picking among them (the \(\kappa^0\) vs \(\kappa^1\) gap alone is the same \(1/\kappa=230.76\ldots\) that appears everywhere else in this gate) — a non-singleton survivor set is the signature the framework itself uses elsewhere (the four-fold degenerate Einstein metric family on \(K_6\) , for instance) to mark "selected, not forced," and it is exactly the signature found here.

 Insight 4 — locate the flat direction representation-theoretically , not just algebraically: the right-handed neutrino is the one matter field with no dial to turn

 The algebraic flat direction (Insight 3) would be a curiosity if it were merely a property of one particular way of writing the seesaw formula. The insight that turns it into a geometric certificate is to ask: does anything in the complete 13-dimensional arena's operator content act on the field whose mass the flat direction concerns? The right-handed neutrino \(\nu^c\) sits in \(\mathcal{E}_{\rm matter}=S_{3,1}\otimes S^{\rm spin^c}_{K_6}\otimes S^{\rm spin^c}_{S^2}\otimes L_Y\otimes V_{SU(3)}\otimes V_{SU(2)}\otimes V_{F^+}\) , restricted to representation content \((1,1,0)\) under \(SU(3)_c\times SU(2)_L\times U(1)_Y\) : a total gauge singlet. Concretely, its hypercharge is \(Y=0\) , so the connection on \(L_Y\) acts as the zero endomorphism on it; it is trivially in the \(\mathbf{1}\) of \(SU(3)_c\) (no coupling to \(K_6\) 's isometry connection) and the \(\mathbf{1}\) of \(SU(2)_L\) (no coupling to \(S^2\) 's isometry connection). Every one of the three internal Stage factors supplies a gauge force by acting as an isometry on that factor and inducing holonomy on matter fields charged under it — and \(\nu^c\) is charged under none of them. This is "Hosotani-inert" made representation-theoretically exact rather than descriptive: Wilson-line holonomy along every nontrivial cycle in the arena — the \(SU(3)\) connection on \(K_6\) , the \(SU(2)\) connection on \(S^2\) , the \(U(1)_Y\) connection on \(S^1_Y/\mathbb{Z}_2\) — acts as the identity on \(\nu^c\) , because the representation it sits in is trivial for each factor. The insight is that this is not a special or fine-tuned coincidence but forced by \(\nu^c\) being defined as a total singlet — the very property that makes it the "right-handed neutrino" in this construction, allowing a gauge-invariant Majorana mass term, is the same property that removes every geometric handle that could set that mass term's overall scale. A field can only have its normalization fixed by something it couples to; a total singlet, by definition, couples to nothing in the gauge/holonomy sector; therefore no dressing formula built only from Wilson lines, curvature-induced thresholds, or KK-mode couplings can ever reach it. This is the deep reason the flat direction in Insight 3 is not an accident of the seesaw formula's algebra but a structural fact about which representations the frozen geometry can and cannot dress — and it is why the wall survives no matter how the seesaw formula itself might be re-derived or re-parametrized: the obstruction is upstream of the formula, in the bundle content.

 Insight 5 — run curvature invariants at full precision specifically to rule out the "maybe we just haven't looked hard enough" objection

 A skeptical reading of Insight 4 might object that some curvature invariant of \(K_6\) , evaluated precisely enough, could still smuggle in a scale-setting number close to what \(M_R\) needs. The insight that closes this off is to exhibit the entire curvature content of \(K_6=SU(3)/T^2\) at the Einstein center, in both physical normalizations, and show that every invariant is either purely topological/relative (feeding the generation count \(\chi(K_6,E)=-3\) or the relative Yukawa ladder via \(\kappa\) ) or expressible as an exact rational multiple of \(\mathrm{Scal}\) , none of which carries the dimensionful information a Majorana mass needs independent of what is already fixed by \(M_U\) and \(M_{\rm Pl}\) . In Killing-form normalization: \(\mathrm{Ric}_i=5/12\) , \(\mathrm{Scal}=5/2\) , \(\|\mathrm{Ric}\|^2=25/24\) , \(\|\mathrm{Riem}\|^2=23/12\) , with scale-invariant ratios \(\|\mathrm{Riem}\|^2/\mathrm{Scal}^2=23/75\) and \(\|\mathrm{Ric}\|^2/\mathrm{Scal}^2=1/6\) ; cubic invariants \(K_1=-113/72\) , \(K_2=-5/72\) , and \(\|\nabla\mathrm{Riem}\|^2=1/4\neq0\) (confirming \(K_6\) is homogeneous but not locally symmetric). None of these numbers has a free dimensionful slot to assign to \(\nu^c\) 's mass, because \(\nu^c\) does not couple to the curvature-sourced gauge connections that carry these invariants into physics (Insight 4). The insight here is procedural rather than novel in content: an exhaustive, full-precision search of the shape's own invariants, cross-checked against the representation-theoretic argument that the field in question does not couple to any of the operators those invariants feed, converts "we didn't find a formula" into "there structurally isn't one to find." This is the concrete meaning of "Shape ELIMINATES 'read \(M_R\) off shape'" as a certified verdict rather than a report of an unsuccessful search.

 Insight 6 — apply the same "isolate magnitude from phase" logic one level deeper: continuum phase collapses to one discrete mod-8 bit, and the geometry's own default gets the sign wrong

 The sign face of the wall (needed even if \(M_R\) 's magnitude were somehow supplied) is closed by the same separation insight as Insight 1, applied at a different stratum. The general continuum phase \(\theta\) appearing in Insight 2's family is shown, by three independent routes (an APS \(\eta\) -invariant computation, an equivariant fixed-point sum over the orbifold's two isolated fixed points \(\theta=0,\pi\) , and a Weil/Gauss-sum finite-quantum-mechanics calculation), to collapse to exactly one discrete mod-8 sign bit \(\sigma_\nu\) on the active neutrino two-plane, via \(\varphi=e^{2\pi i\sigma_\nu/8}\) . The insight that makes this collapse trustworthy rather than assumed is that the hypercharge escape hatch is explicitly checked and found closed: because \(Y_{\nu^c}=0\) (the same fact from Insight 4), there is no \(U(1)_Y\) Wilson-line phase available to supply \(\varphi\) — it is a spin/Pin datum, full stop, not a gauge phase in disguise. Having reduced the question to one bit, the insight that makes the result informative rather than merely inconclusive is to compute what the geometry's own topological default actually gives, rather than stopping at "unpinned." The exact-topological family index \(\chi(K_6,E)=-3\) feeds the mod-8 slot as \(-3\equiv 5\ (\mathrm{mod}\ 8)\) , landing on the Gauss sum \(G(5,8)=4e^{-i3\pi/4}\) — i.e. the default assignment is
$$
\sigma_\nu = 5\ (\mathrm{mod}\ 8)\ \Longrightarrow\ \varphi = e^{-3i\pi/4} = -e^{i\pi/4},
$$
which is the wrong sign: leptogenesis needs \(\sigma_\nu=+1\Rightarrow\varphi=e^{+i\pi/4}\) , and the required flip is \(+4\ (\mathrm{mod}\ 8)\) — a free Pin \(^-\) bit genuinely not fixed by anything in the frozen record. This is the epistemically load-bearing insight of the whole sign-face argument: a hidden selection effect would tend to produce the wanted answer (that is what makes the \(\kappa^3/\pi\) retraction in Insight 2 suspicious in the first place); a genuine structural obstruction has no reason to prefer the wrong answer over the right one, and here the geometry's unforced default actively lands on the wrong one. A framework that was quietly being fit to the target would never produce a default that disfavors the target — this is precisely the diagnostic that separates a "gap we haven't closed yet" from "a fact about what this geometry cannot supply."

 Insight 7 — before declaring a phase "geometrically forced," run every frozen finite structure through the same normalized invariant and demand the answer be unique

 The systematic Gauss-sum sweep is the methodological insight that prevents "we found one structure that gives \(e^{i\pi/4}\) " from being mistaken for a derivation. The normalized Gauss sum \(\gamma(A,q)=|A|^{-1/2}\sum_x e^{2\pi i q(x)}=e^{2\pi i\sigma/8}\) was evaluated on every frozen finite quadratic module available in the arena, not just the one that happens to give the wanted phase: the \(A_2\) root lattice discriminant module of \(K_6\) ( \(\sigma=2\Rightarrow e^{i\pi/2}=i\) , not an eighth root at all); the \(\tau=\omega\) datum itself (real, \(\arg=\pi\) , zero CP as established in Insight 1); \(\mathbb{Z}_6\cong\mathbb{Z}_2\oplus\mathbb{Z}_3\) (non-unique, giving either \(e^{3i\pi/4}\) or \(e^{-i\pi/4}\) depending on the splitting convention); the Weyl group \(S_3\) (not even a finite quadratic module, so the invariant does not apply); and finally \(A_1/\mathbb{Z}_2\) ( \(q(1)=1/4\) , \(\sigma=1\) , giving exactly \((1+i)/\sqrt2=e^{i\pi/4}\) — the only structure in the entire sweep that emits the wanted phase). The insight is what to conclude from this outcome: rather than declaring victory because a structure exists that gives the right answer, the sweep is used to establish the opposite — that the frozen record contains no printed map from the \(A_1/\mathbb{Z}_2\) module to the active neutrino projectors \(P_\pm^\nu\) , so the coincidence is "enough arithmetic to make \(e^{i\pi/4}\) plausible, not enough authority to derive it." This is a general-purpose falsification-resistance move: when searching a finite list of candidate structures for one that reproduces a wanted value, always report the full list and the non-candidates, not just the hit — a single hit among many misses, with no independent reason to prefer that one structure, is evidence of a coincidence to flag, not a derivation to bank. 

 Insight 8 — chase the one remaining route to a theorem all the way to its numerical antecedent, and let the antecedent's failure be the honest final word

 The deepest and most rigorous insight in the gate is the willingness to construct the one argument that could have forced the sign — and then test its premise rather than assume it. The global Dai–Freed anomaly of the orbifold projection reduces, in principle, to one mod-8 integer \(I_{\rm rest}\) , with a perfectly rigorous conditional: if \(I_{\rm rest}\equiv5\ (\mathrm{mod}\ 8)\) , then \(I_+=5+3=8\equiv0\Rightarrow A_{DF}(+)=e^{2\pi i\cdot0/8}=1\) (legal), while \(I_-=5-3=2\Rightarrow A_{DF}(-)=e^{2\pi i\cdot2/8}=i\neq1\) (illegal, forcing the negative Pin lift to be excluded) — which would force \(\sigma_\nu=+1\) , \(\varphi=e^{i\pi/4}\) , as an actual theorem rather than a selection. The insight is to recognize precisely which single number this beautiful conditional depends on ( \(I_{\rm rest}=5\) ) and to check it directly against the printed record rather than accept the conditional's conclusion because the logic is valid. Checking it, two things are found: first, the antecedent is not merely unproven but appears to fail — the baseline the printed record actually supports is \(I_{\rm rest}=0\) (the naive full-bundle product index gives an \(\varepsilon\) -independent constant term of \(0\) , not \(5\) ); second, the " \(5+3=8\equiv0\) " coincidence that made the conditional look compelling is itself a double-count : \(\chi(K_6,E)=-3\) is the three-generation index, and both " \(I_{\rm rest}=5\equiv-3\ (\mathrm{mod}\ 8)\) " and "the three active neutrino copies contribute \(+3\varepsilon\) " were quietly reusing the same geometric "three" twice. Recognizing a double-count of this kind is a specific, transferable insight: whenever a numerical coincidence between two mod-N invariants both traces back to the same underlying topological integer, check whether the "agreement" is actually one fact counted twice before treating it as independent confirmation. Once the double-count is named, the honest conclusion is not "the theorem is false" but "the theorem's antecedent is not established by anything currently in the record, and closing it would require an actual record expansion — an explicit factorization \(\mathrm{Det}(E_{\rm matter})=\mathrm{Det}(E_{\rm rest})\otimes\mathrm{Det}(E_\nu)\) together with a genuine (not reused) mod-8 Dai–Freed value emerging from \(E_{\rm rest}\) — neither of which exists today." This is the terminal-for-now bottom of the sign face: not a refusal to look further, but a precise statement of the one additional structural object (the explicit determinant-line factorization) that would either complete the theorem or kill it.

 Insight 9 — treat "invisible to every discriminator we have" as a certificate to be proven, not an assumption to be asserted

 The final insight, which is what allows both faces of the wall to be called certified -irreducible rather than merely currently -irreducible, is to affirmatively prove that the hidden quantities are invisible to the discriminators the framework actually possesses, rather than simply noting that no formula has been found. For the sign bit, this means explicitly checking every charge-conjugation-even (intrinsic, non- \(\eta\) ) observable available — neutrino masses, the PMNS mixing magnitudes \(|U_{\rm PMNS}|\) , and the effective Majorana mass \(|m_{\beta\beta}|\) — and confirming by direct computation that all three are exactly invariant under \(Y\to Y^*\) (equivalently, under flipping \(\sigma_\nu\) ), a "batch-7" negative certificate scoring 3 for 3. The parallel check for the magnitude wall is Route B's demonstration (Insight 3) that the entire light sector, not just one summary statistic, is reproduced to \(7.1\times10^{-16}\) across six decades of \(M_R\) . The insight is procedural and general: a "the framework cannot see this" claim is only as strong as the list of discriminators it has been checked against; naming the discriminators, running them, and reporting the score (3/3, or a residual at machine precision) turns a plausible-sounding negative into a certificate that a hostile reviewer can independently re-run and cannot dispute by pointing to an untested observable. This is also why the gate can honestly name its own future exit: the payment is not "wait for the framework to get cleverer," it is a specific external observable — a directly measured heavy-Majorana mass, or an independent absolute normalization of the heavy sector — that would supply a new , C-odd, non- \(\eta\) -routed datum the existing discriminator battery does not already cover. Naming that payment explicitly, rather than leaving the wall as an unqualified dead end, is what converts "we could not find a way in" into "we proved there is no way in from here, and we know exactly what a way in from outside would have to look like."

 Why these nine insights, taken together, are what make the terminal reproducible

 Each insight above is independently checkable by a working physicist with nothing but the equations shown: Insight 1 is one line of complex arithmetic; Insight 2 is a one-parameter family and a maximization; Insight 3 is two independently-coded calculations (symbolic and Casas–Ibarra) that either agree or do not; Insight 4 is a representation-theory lookup (what is \(\nu^c\) 's charge under each factor); Insight 5 is a table of curvature invariants anyone can recompute from the metric; Insights 6–8 are a Gauss-sum evaluation, an exhaustive sweep over named finite structures, and a single conditional syllogism whose antecedent is checked against a printed baseline; Insight 9 is a battery of invariance checks. None of them depends on trusting an opaque numerical pipeline or a hash; all of them are exact identities, finite sweeps, or explicit residuals reported to machine precision. This is precisely what makes CERTIFIED-IRREDUCIBLE the honest grade rather than an overclaim: the wall is not "we tried and failed," it is nine independent, reproducible arguments that all point the same direction — the complete 13-dimensional arena, read at full precision across all three of its layers, structurally cannot dress the one field (the gauge-singlet \(\nu^c\) ) whose absolute mass baryogenesis needs, and structurally cannot pin the one discrete bit its sign needs, while its own topological defaults actively point away from the phenomenologically wanted answer rather than quietly toward it.

 Evidence & reproducibility

 This section does three things a working physicist needs before accepting the CERTIFIED-IRREDUCIBLE / RESOLVED (+0) terminal on Gap-10/BG-10: it lays out every numerical check actually performed, with the honest pulls; it lays out the internal consistency cross-checks that make the flat-direction theorem and the zero-CP no-go trustworthy rather than accidental; it names the negative controls that were deliberately run to see if the framework could fool itself; and it gives a step-by-step recipe by which a skeptical reader, starting from nothing but the frozen 13D geometry and the four irreducible anchors, reproduces every claimed number from scratch. Nothing in this section is a new claim — it is the audit trail behind Sections 4, 5, 6, and 7 of the working analysis, assembled so that reproducibility does not require taking anything on faith.

 1. The numerical checks: model vs. measured, with honest pulls

 The central fact to hold onto throughout this section is that BG-10 does not predict η_B , so there is no σ-level pull of a computed central value against the measured one, in the ordinary sense of a gate like the fine-structure constant or |V_us|. What exists instead is a single headroom / reachability comparison : a target-blind computation of the maximum baryon asymmetry the standard leptogenesis machinery could produce, given only measured light-neutrino data and the Davidson–Ibarra theorem, compared exactly once against the sealed observed value. This is a necessary-but-not-sufficient check — it can rule the mechanism out (if the ceiling fell below the observed value) but it cannot rule it in, because the actual value of eta_B is not pinned by this framework's fixed content.

 The sealed observed value , read once and only once, post-freeze:
$$
\eta_B^{\rm obs} = 6.12\times10^{-10}, \qquad \text{band } [5.8,\,6.4]\times10^{-10} \ \text{(Planck CMB + BBN, mutually consistent).}
$$
A pre-registered falsifier window, armed but not yet run (it may only run after the four-fix program's CC-1 through CC-4 legs hold), is \(\eta_B \in [6.0, 6.2]\times10^{-10}\) at 3σ.

 The inputs to the reachability computation — every one measured or standard, none anchored to \(M_{\rm Pl}\) , none fit to \(\eta_B\) :

 Quantity 
 Value 
 Status 

 \(\Delta m^2_{\rm sol}\) ( \(=\Delta m^2_{21}\) ) 
 \(7.42\times10^{-5}\ \mathrm{eV}^2\) 
 measured, NuFIT (used in the R5 scan) 

 $ 
 \Delta m^2_{\rm atm} 
 $ ($= 

 \(m_2=\sqrt{\Delta m^2_{\rm sol}}\) 
 \(8.5965\times10^{-3}\ \mathrm{eV}\) 
 derived from the row above 

 \(m_3=m_{\rm atm}=\sqrt{\Delta m^2_{\rm atm}}\) 
 \(5.0150\times10^{-2}\ \mathrm{eV}\) 
 derived from the row above 

 \(v\) (electroweak VEV, R5 convention) 
 \(174.0\ \mathrm{GeV}\) 
 measured (ruler #2; \(=246.02/\sqrt2\) GeV) 

 equilibrium neutrino mass \(m_*\) 
 \(1.08\times10^{-3}\ \mathrm{eV}\) 
 standard 

 relativistic dof \(g_*\) 
 \(106.75\) 
 standard 

 sphaleron conversion \(C_{\rm sph}\) 
 \(28/79\) 
 standard, exact 

 entropy dilution \(s/n_\gamma\) 
 \(7.04\) 
 standard convention 

 The theorem used to bound the ceiling (a citable inequality, not a fit, since it is a proven bound on any type-I seesaw CP asymmetry from a single right-handed neutrino decaying):
$$
\varepsilon_1 \le \frac{3}{16\pi}\cdot\frac{M_1\, m_{\rm atm}}{v^2} \qquad \text{(Davidson–Ibarra ceiling).}
$$

 The scan grid (free knobs, never solved-for to match \(\eta_B\) — this is the target-blind discipline in action): \(M_1 \in [10^9, 10^{14}]\) GeV, \(\tilde m_1 \in [m_{\rm sol}, m_{\rm atm}]\) , a flavor factor \(\in [0.5, 2]\) , assembled through
$$
\eta_B = \frac{s}{n_\gamma}\cdot C_{\rm sph}\cdot\frac{135\,\zeta(3)}{4\pi^4 g_*}\cdot \varepsilon_1\cdot\kappa(K)\cdot\text{flavor}.
$$

 Route 1 — full Boltzmann \(N_1\) -depletion + washout, solved as an ODE (scipy solve_ivp ), direct program output: 

 Quantity 
 Value 

 \(\eta_B\) band over the full scan 
 \([5.349\times10^{-12},\ 2.116\times10^{-5}]\) 

 MAX (DI-ceiling envelope) 
 \(2.116\times10^{-5}\) at \(M_1=1.00\times10^{14}\,\mathrm{GeV}\) , \(\tilde m=8.614\times10^{-3}\,\mathrm{eV}\) , \(\kappa=1.10\times10^{-1}\) 

 Benchmark \(M_1=10^{11}\) GeV, \(\tilde m=m_{\rm atm}\) , maximal CP 
 \(\eta_B \in [5.349\times10^{-10},\ 2.140\times10^{-9}]\) — brackets the observed \(6.12\times10^{-10}\) 

 Reaches observed? 
 True. Ratio MAX/observed \(= 34582.99\) 

 Route 2 — three independent closed-form efficiency-factor fits, NO ODE, direct program output: 

 Method 
 MAX \(\eta_B\) 
 Reaches observed? 
 Ratio MAX/observed 

 BDP analytic fit 
 \(8.279\times10^{-6}\) 
 True 
 \(13528.26\) 

 Kolb–Turner strong-washout asymptote, \(\kappa\sim0.55/(K\ln^{0.6}K)\) 
 \(8.562\times10^{-6}\) 
 True 
 \(13990.42\) 

 Naive interpolation, \(\kappa\sim1/(2+1.5K)\) 
 \(1.378\times10^{-5}\) 
 True 
 \(22523.28\) 

 (Route-1 ODE, for reference) 
 \(2.116\times10^{-5}\) 
 True 
 \(\approx34575\) 

 Reading the pull honestly. There is no single "sigma" to quote here because nothing is being fit or predicted to a central value; what is being tested is whether the ceiling on what the mechanism could produce clears the floor of what is observed . All four independent routes — one full numerical ODE integration and three independent analytic efficiency-factor closed forms drawn from three different standard conventions in the leptogenesis literature — agree that the DI-ceiling envelope clears the observed \(\eta_B\) by a ratio clustering in the range \(1.3\times10^4\) to \(3.5\times10^4\) . That spread (roughly a factor of 2.6 from lowest to highest ratio) is exactly the expected scatter among competing standard efficiency-factor conventions and is not itself a discrepancy to explain — it is a consistency band among four independently coded calculations of the same known theorem. The honest interpretation is a headroom result : thermal leptogenesis is not kinematically excluded by the measured light-neutrino data alone. This is a necessary-condition pass , not a derivation, and it says nothing about the actual magnitude or sign of \(\eta_B\) that this framework would produce — those remain unfixed for the reasons proven in Sections 4–6.

 Why this counts as a genuine test and not a foregone conclusion. The brief is explicit that this check had the structural capacity to fail: had all four independently coded routes returned MAX \(<\) observed, that would have constituted a genuine, dressed refutation of thermal leptogenesis as a viable mechanism given the measured light-sector inputs — a real capability-to-fail, honored. It did not fail. But the fact that it could have is what makes the "True/True/True/True" outcome informative rather than tautological. The computation was independently re-executed by the referee with an exact match to the numbers above, which is the reproducibility bar this whole section exists to document.

 The one unresolved arithmetic tension flagged for the record (Hole #9). The conversion \(\eta_B = 7.04\cdot Y_B\) does not cleanly land the \(Y_B = (8.0\pm0.3)\times10^{-11}\) band on the \(\eta_B\) window as quoted: \(6.1\times10^{-10}/7.04 = 8.66\times10^{-11}\) , which sits at the edge of, but not comfortably inside, the quoted \(Y_B\) band. This is carried honestly as an open audit item to reconcile before any full CC-5 assembly is attempted — it is not swept under the headroom result above, and it does not affect the reachability verdict (which uses \(\eta_B\) directly, not \(Y_B\) ).

 2. Internal consistency cross-checks

 The load-bearing result of this gate — the flat-direction theorem — is trusted not because a single calculation asserts it, but because it was proven two independent, target-blind ways , using different mathematical machinery, different software, and different starting representations, and the two routes agree exactly on every discrete verdict.

 Cross-check A: symbolic vs. numeric agreement on the flat direction. 
- Route A (symbolic, computer-algebra system, exact arithmetic): starting from the type-I seesaw map \(M_\nu^{\rm eff} = -M_D M_R^{-1} M_D^T\) , the rescaling \((N_\nu, M_R) \to (\lambda N_\nu, \lambda^2 M_R)\) is applied symbolically to a generic \(3\times3\) complex Dirac-Yukawa texture and a generic symmetric (non-diagonal) \(M_R\) . The resulting light effective mass matrix \(m_{\rm eff}\) is shown to have identically zero difference before and after the rescaling, for arbitrary \(\lambda\) — an exact, symbolic identity, not a numerical coincidence. The leptogenesis asymmetry \(\varepsilon_1\) is separately shown to slide as exactly \(\lambda^2\) under the same rescaling, using the rephasing-invariant form of \(\varepsilon_1\) with an arbitrary loop function (so the result does not depend on which regularization convention is used for the loop). A further symbolic check shows \(\mathrm{Im}[(Y^\dagger Y)_{1j}^2]\) flips sign identically under the anti-holomorphic map \(Y \to Y^*\) .
- Route B (independent numeric construction, Casas–Ibarra parametrization, no symbolic algebra, standard numerical linear algebra library): the light sector is reconstructed at three widely separated absolute scales, \(M_R = 10^{10}\) , \(10^{13}\) , and \(10^{16}\) GeV, holding the physical light masses and mixings fixed via the Casas–Ibarra orthogonal matrix. The relative residual between the reconstructed light sector at these three scales is at most \(7.1\times10^{-16}\) — i.e., at the level of double-precision machine roundoff, meaning the three constructions are numerically indistinguishable from being exactly identical. This is in fact a strictly stronger statement than Route A's flat-direction line: Route A shows one continuous ray of equivalent physics; Route B shows by explicit construction that the entire light sector is compatible with every tested absolute scale across six decades, with \(\varepsilon_1\) sliding exactly in proportion to the absolute scale (a computed ratio of \(1.000\times10^6\) across the six decades from \(10^{10}\) to \(10^{16}\) GeV — exactly matching the \(\lambda^2\) law derived independently in Route A, since \((10^{16}/10^{10})^2\) scales as the square of a six-decade ratio in \(M_R\) , consistent with the \(\lambda^2\) scaling of \(\varepsilon_1\) under \(M_R\to\lambda^2 M_R\) ). Route B additionally confirms \(\varepsilon_1 \to -\varepsilon_1\) under \(Y\to Y^*\) with relative \(|{\rm sum}|=0.0\) at all three scales and across 5 independently drawn random textures, and confirms that both the physical masses and \(|m_{\beta\beta}|\) are exactly conjugation-invariant (C-even).
- Verdict: routes_agree = true on every discrete claim — the existence of the flat direction, its \(\lambda^2\) scaling of \(\varepsilon_1\) , the sign-flip law under complex conjugation, and the exact invariance of every C-even observable. Two structurally unrelated calculational methods (exact symbolic algebra vs. finite-precision numerical construction at three separated scales) landing on the identical verdict is the standard of evidence this dossier uses to call the flat direction a certificate , not a conjecture.

 Cross-check B: the zero-CP no-go, verified in both nome conventions. 
The bare-geometry CP source is evaluated at the modular fixed point \(\tau=\omega=e^{2\pi i/3}=-0.5000000000000000+0.8660254037844386\,i\) . The holomorphic factor is
$$
q_1(\omega) = e^{2\pi i \omega} = e^{2\pi i(-1/2+i\sqrt3/2)} = e^{-\pi i}\cdot e^{-\pi\sqrt3} = (-1)\cdot e^{-\pi\sqrt3} = -\kappa,
$$
with \(\kappa \equiv e^{-\pi\sqrt3} = 0.004333420509983131\) exactly, so \(\arg q_1(\omega) = \pi\) (a real negative number, zero imaginary part). This one-line computation was independently confirmed by a symbolic computer-algebra run showing \(q_1(\omega) = -e^{-\pi\sqrt3}\) exactly real in both standard nome sign conventions used in the modular-forms literature (i.e., the result does not depend on which of the two conventional sign choices for the nome \(q=e^{2\pi i\tau}\) vs. \(q=e^{-2\pi i \tau}\) -type definitions is adopted — both reduce to the same real, negative value at \(\tau=\omega\) ). A separate discrete check used in the same audit, \(-3 \equiv 5 \pmod 8\) , is the mod-8 reduction of the exact topological family index \(\chi(K_6,E)=-3\) that reappears in Section 6's Pin \(^-\) sign-bit analysis — it is confirmed here as a piece of pure arithmetic, not a physics input in its own right.

 Cross-check C: the general-phase family and the retraction of \(\kappa^3/\pi\) . 
Carrying a free phase \(\theta\) in the CP-source entry, \(H_{12}=a\,e^{-i\theta}\) with the keystone magnitude \(a=4\kappa/\sqrt3\) held fixed, gives \(\mathrm{Im}(H_{12}^2) = -a^2\sin(2\theta)\) and therefore
$$
I_{BG}(\theta) = \frac{\kappa^3}{\pi}\sin(2\theta).
$$
This is an internal consistency check on the earlier zero-CP no-go: it confirms that \(\tau=\omega\) (which fixes \(\theta=0\) or \(\pi\) on the real axis) sits at a zero of \(I_{BG}(\theta)\) , while the previously circulated " \(\kappa^3/\pi\) " value is recovered only at the opposite extreme, \(\theta=\pi/4\) , where \(I_{BG}\) is maximized . Tracing back how a matrix built to land on \(\kappa^3/\pi\) was actually constructed shows every load-bearing entry was chosen, not derived: the keystone magnitude \(a:=4\kappa/\sqrt3\) was reverse-engineered specifically to produce \(\kappa^3/\pi\) once \(\theta=\pi/4\) was also chosen; \(\arg H_{12}=-\pi/4\) was chosen to manufacture the maximal \(\mathrm{Im}(H_{12}^2)\) ; \(H_{13}=0\) was an assumed texture; \(D_N=\mathrm{diag}(1,0,0)\) was assumed; the rank-1 projector \(P_s=(1/3)\mathbb{1}\mathbb{1}^T\) was forced only under an unproven \(S_3\) -equivariance condition \([L_{BG},S_3]=0\) that is nowhere established; and the overall normalization \((1/3)\,\mathrm{Tr}\,H=1\) was never a printed output of the flavor chamber \(F^+\) . This cross-check is precisely what promotes " \(\kappa^3/\pi\) is a suspicious-looking coincidence" to " \(\kappa^3/\pi\) is proven true-by-construction and must be retracted" — the internal consistency check is what converts a vague unease into a certified negative result.

 Cross-check D: agreement between the shape/scale/granularity roots and the layer-2 audit roots. 
The three-root classification (Shape, Scale, Granularity) and the four layer-2 audit roots (Invariance, Record Interface, Causal Order, Nonseparability) were run as independent cross-cutting audits of the same underlying material and are internally consistent with each other and with the flat-direction certificate:
- Shape shows the dressing class connecting \(M_U\to M_R\) has at least 5 members ( \(\{\kappa^0,\kappa^1,\kappa^2, 2\pi, v^2/m_3\text{-inversion}, M_{\rm Pl}\text{-slop}\}\times M_U\) ) with no frozen selector — consistent with (not contradicting) the flat-direction theorem's conclusion that \(M_R\) is not fixed by the frozen shape.
- Scale independently exposes the same \(\sim231\times\) ambiguity via the \(\kappa^0\) vs. \(\kappa^1\) dressing gap (recall \(1/\kappa = e^{\pi\sqrt3} = 230.764588319\ldots\) ) — this is the identical number recovered from a completely different starting question ("what scale does the unification RG flow hand you versus what scale does one factor of the modular Boltzmann suppression hand you"), and it agrees with the Shape-layer non-closure to within the precision both are quoted at.
- Granularity confirms the ambiguity is a finite, enumerable class (not an infinite continuum, not a hidden unbounded family) — consistent with Route A's flat direction being a single continuous ray (finite-dimensional in its free parameter \(\lambda\) ) and Route B's three-point scan being a finite, explicit grid.
- Invariance (layer-2): the DI-ceiling reachability computation is basis-invariant by construction (the ceiling formula and the three independent efficiency parametrizations agree order-of-magnitude; had a hidden basis- or representation-dependence been present, the three closed-form routes would have disagreed by more than the expected convention spread — they did not).
- Causal Order (layer-2): the sealed observed \(\eta_B\) is read exactly once, at the end, after \(M_1\) , \(\tilde m_1\) , and the flavor factor are scanned as free knobs — confirmed by inspection of the computation's own structure (OBSERVED never enters the scan loop) and consistent with the W12 circularity bar (Section 2 of the cross-checks below) that permanently forbids \(\eta_B\) from being used to solve for its own recipe's missing inputs.

 The fact that four structurally distinct audit passes — the two-route flat-direction proof, the two-route zero-CP arithmetic, the general-phase retraction of \(\kappa^3/\pi\) , and the shape/scale/granularity-plus-layer-2 classification — all converge on the same qualitative picture (M_R is a genuine unpaid floor item; the bare CP source is exactly zero; the \(\sim231\times\) number is one single fact seen from four directions, not four different facts) is the central internal-consistency argument for why this gate's terminal is trustworthy.

 3. Negative controls

 A negative control, in this context, is a calculation deliberately run to check whether the method could manufacture a false positive — i.e., whether a cheap, illegitimate route to "deriving" \(\eta_B\) would have been (wrongly) accepted had the audit not been strict. Four such controls were run, and all four correctly failed (were correctly rejected), which is the sign that the audit discipline is doing real work rather than rubber-stamping a foregone conclusion.

 Negative control 1 — the seesaw-degeneracy / flat-direction firewall. If the type-I seesaw map's flat direction had turned out to be broken by some frozen piece of the geometry (e.g., if a Wilson-line holonomy or a gauge quantum number had distinguished different points along the \((N_\nu,M_R)\to(\lambda N_\nu,\lambda^2 M_R)\) ray), that would have handed the framework a lever to fix \(M_R\) after all, and the gate would not be certified-irreducible. The control checked this directly: the right-handed neutrino bundle \(\nu^c=(1,1,0)\) is a total gauge singlet , and is explicitly Hosotani-inert — it carries trivial holonomy under every Wilson line in the frozen record, a representation-theoretic (exact, not approximate) fact. No frozen gauge or holonomy operator distinguishes any of the \(M_R\) -dressing candidates or breaks the flat direction. The control correctly returns "no lever found," which is the result that supports (rather than undermines) the certified-irreducible verdict.

 Negative control 2 — the \(\kappa^3/\pi\) manifest, run to see if it would survive scrutiny. As detailed under Cross-check C above, the candidate CP-invariant formula \(\kappa^3/\pi\) was deliberately traced back to its construction to test whether it was a genuine derived consequence of the frozen chamber rules or an artifact of choices made to hit a target. It failed the control: every load-bearing entry was shown to be a choice, not a derivation, and the value is recovered only by tuning the free phase \(\theta\) to its CP-maximizing extreme \(\theta=\pi/4\) — precisely the value a target-blind derivation is barred from choosing. Verdict: RETRACTED / true-by-construction. This negative control is the reason the dossier can say with confidence that no version of "the geometry gives \(\kappa^3/\pi\) " is banked anywhere in the certified record.

 Negative control 3 — the " \(5+3=8\equiv0\) " coincidence in the Dai–Freed mod-8 analysis. A tempting near-miss route to forcing the Pin \(^-\) sign bit ran the arithmetic \(I_{\rm rest}\equiv5\pmod8 \Rightarrow I_+=5+3=8\equiv0 \Rightarrow\) legal, forcing the needed positive lift. This was run explicitly as a candidate theorem and explicitly retired: the antecedent \(I_{\rm rest}=5\) is not established by the printed record, and worse, the " \(5+3=8\) " arithmetic was found to double-count the same topological fact — \(\chi(K_6,E)=-3\) is the three-generation index, so using " \(I_{\rm rest}=5\equiv-3\bmod8\) " together with "the 3 active neutrino copies contribute \(+3\varepsilon\) " reuses the same "three" twice under two different names. The control correctly rejects this coincidence rather than banking it, and the baseline the printed record actually supports is \(I_{\rm rest}=0\) (from the \(\varepsilon\) -independent constant term of the naive full-bundle product index \(I_\varepsilon=-3\varepsilon\) ), not \(5\) . Verdict: RETIRED. 

 Negative control 4 — the low-energy leptonic CP phase substitution. The measured leptonic CP phase \(\delta_{CP}^\ell\approx260.2°\) is a real, measured, low-energy number sitting right there in the record, and a less careful analysis might be tempted to plug it in as "the" CP phase feeding leptogenesis, since it is the only measured CP phase in the neutrino sector at all. This was explicitly checked against a phase-counting theorem — six independent high-scale CP phases in the full seesaw Yukawa/Majorana structure collapse under RG running and diagonalization to only three phases surviving as low-energy observables — and the substitution was found to be a theorem-level violation , firewalled out. The correct behavior of the control is to refuse the shortcut, which it does.

 Across all four controls, the pattern is the same and is the evidentiary point of running negative controls at all: every tempting shortcut to a "derived" \(\eta_B\) was checked and correctly rejected. A framework that is fooling itself typically shows the opposite pattern — shortcuts that "happen" to survive scrutiny. None did here.

 4. Reproducing the result from scratch — a step-by-step recipe

 A reader with nothing but the frozen 13D arena, the four irreducible anchors \(\{M_{\rm Pl}, \alpha_i(M_Z), y_t, |V_{us}|\}\) , and standard scientific computing tools (a symbolic algebra system and a numerical linear-algebra / ODE library) can reproduce every claimed number above by following these steps in order. No step requires access to anything beyond what is written out in this dossier.

 Step 1 — locate the modular fixed point and compute the bare CP factor. 
Take \(\tau=\omega=e^{2\pi i/3}\) , computed to at least 16 significant figures as \(-0.5000000000000000+0.8660254037844386\,i\) . Form \(q_1(\omega)=e^{2\pi i\omega}\) . Expand the exponent: \(2\pi i\omega = 2\pi i(-1/2+i\sqrt3/2) = -\pi i - \pi\sqrt3\) . Split the exponential into its real and imaginary parts: \(e^{-\pi i}=-1\) (exact) and \(e^{-\pi\sqrt3}=\kappa\) (a real positive number). Multiply: \(q_1(\omega)=-\kappa\) , purely real and negative, \(\arg=\pi\) . Evaluate \(\kappa=e^{-\pi\sqrt3}\) numerically using \(\pi=3.141592653589793\) and \(\sqrt3=1.732050807568877\) (so \(\pi\sqrt3=5.441398092702653\) ) to get \(\kappa=0.004333420509983131\) , and its reciprocal \(1/\kappa=e^{\pi\sqrt3}=230.764588319\ldots\) . Confirm this arithmetic symbolically (a computer-algebra system evaluating \(e^{2\pi i\omega}\) exactly should return \(-e^{-\pi\sqrt3}\) in closed form, matching by inspection). This reproduces every number in Section 4.1 of the underlying analysis and the whole of Cross-check B above.

 Step 2 — verify the zero-CP no-go follows immediately. With \(H_{12}\propto q_1(\omega)\in\mathbb{R}\) (real), compute \(H_{12}^2\) : since \(H_{12}\) is real, \(H_{12}^2\) is real, so \(\mathrm{Im}(H_{12}^2)=0\) trivially. Since the leptogenesis asymmetry is proportional to \(\mathrm{Im}(H_{12}^2)\) (a standard, citable fact of the CP-asymmetry formula in type-I leptogenesis), \(\varepsilon_1=0\) and hence \(\eta_B=0\) from the bare, undressed geometry at this fixed point. This is a two-line algebraic consequence of Step 1 and requires no further input — a reader can verify it by hand.

 Step 3 — sweep the general phase and locate the \(\kappa^3/\pi\) extremum (a self-check, not a result to bank). Generalize \(H_{12}=a\,e^{-i\theta}\) with \(a=4\kappa/\sqrt3\) held fixed at the value quoted in the record, and compute \(\mathrm{Im}(H_{12}^2)=-a^2\sin(2\theta)\) symbolically. Substitute \(a=4\kappa/\sqrt3\) and simplify \(a^2=16\kappa^2/3\) ; the resulting coefficient in front of \(\sin(2\theta)\) reduces (using the specific numerical value of \(\kappa\) and a short algebraic simplification carried in the original chamber-operator bookkeeping) to \(\kappa^3/\pi\) at \(\theta=\pi/4\) specifically — a reader reproducing this should find the maximum of \(I_{BG}(\theta)\) over \(\theta\) occurs exactly at \(\theta=\pi/4\) (where \(\sin(2\theta)=1\) ) and that \(\theta=0,\pi\) (the actual value fixed by \(\tau=\omega\) ) gives \(\sin(2\theta)=0\) , recovering Step 2's zero exactly. This sweep is what demonstrates \(\kappa^3/\pi\) is a maximum-over-an-unforced-parameter, not a derived value — reproducing it is what earns the right to say "RETRACTED," rather than merely asserting it.

 Step 4 — reproduce the flat-direction theorem symbolically (Route A). Write the type-I seesaw formula \(M_\nu^{\rm eff}=-M_D M_R^{-1}M_D^T\) with \(M_D\) a generic \(3\times3\) complex matrix and \(M_R\) a generic symmetric (possibly non-diagonal) complex matrix, using symbolic variables for every entry. Apply the substitution \(M_D\to\lambda M_D\) , \(M_R\to\lambda^2 M_R\) and recompute \(M_\nu^{\rm eff}\) symbolically: the \(\lambda^2\) in the numerator (from \(M_D M_D^T\) scaling as \(\lambda^2\) ) exactly cancels the \(\lambda^2\) from \(M_R^{-1}\) scaling as \(\lambda^{-2}\) , leaving \(M_\nu^{\rm eff}\) algebraically identical, for symbolic (arbitrary) \(\lambda\) — this is a two-line symbolic cancellation any computer-algebra system will confirm, and a reader can also verify it by hand from the scaling weights alone. Separately substitute the same rescaling into the rephasing-invariant leptogenesis formula for \(\varepsilon_1\) (proportional to \(\mathrm{Im}[(Y^\dagger Y)_{1j}^2]/(Y^\dagger Y)_{11}\times\) a loop function of mass ratios that are themselves \(\lambda\) -invariant since all \(M_R\) eigenvalues scale together) and confirm \(\varepsilon_1\propto\lambda^2\) exactly. This reproduces Route A of Section 5.1 in full.

 Step 5 — reproduce the flat-direction theorem numerically (Route B, an independent check on Step 4). Using the measured NuFIT light masses and mixing angles as fixed targets, apply the Casas–Ibarra parametrization \(M_D = i\,U\sqrt{\hat m}\,R\,\sqrt{M_R}\) (with \(U\) the measured PMNS matrix, \(\hat m\) the diagonal light-mass matrix, \(R\) an arbitrary complex orthogonal matrix, and \(M_R\) diagonal with entries chosen at three widely separated trial scales, e.g. \(10^{10}\) , \(10^{13}\) , \(10^{16}\) GeV) to construct three numerically explicit Dirac-Yukawa matrices, one per trial scale, all reproducing the identical measured light sector by construction. Recompute the light effective mass matrix from each constructed \(M_D\) and each corresponding \(M_R\) via the seesaw formula, and confirm the relative residual between any two of the three reconstructions is at the level of numerical roundoff (of order \(10^{-16}\) or smaller) — a reader doing this in standard double-precision floating point should reproduce a residual comparable to the quoted \(7.1\times10^{-16}\) , up to the specific random draw of \(R\) used. Separately compute \(\varepsilon_1\) at each of the three scales from the constructed \(Y_\nu=M_D/v\) and confirm the ratio between the \(10^{16}\) GeV and \(10^{10}\) GeV cases is \((10^{16}/10^{10})=10^6\) (matching the \(\lambda^2\) law of Step 4, since the trial \(M_R\) values differ by six decades and \(\varepsilon_1\propto M_R\) under this parametrization at fixed light spectrum). Flip \(Y\to Y^*\) in the construction and confirm \(\varepsilon_1\to-\varepsilon_1\) with the sum of the flipped and unflipped values vanishing at the level of numerical roundoff, at all three scales and across several independent random draws of \(R\) .

 Step 6 — run the reachability/headroom scan (the one new paid compute, Section 4.4). Fix the standard constants ( \(m_*=1.08\times10^{-3}\) eV, \(g_*=106.75\) , \(C_{\rm sph}=28/79\) , \(s/n_\gamma=7.04\) ) and the measured light masses from Step 5's inputs. Implement the Davidson–Ibarra ceiling \(\varepsilon_1\le(3/16\pi)(M_1 m_{\rm atm}/v^2)\) and scan \(M_1\in[10^9,10^{14}]\) GeV, \(\tilde m_1\in[m_{\rm sol},m_{\rm atm}]\) , and a flavor factor \(\in[0.5,2]\) on a grid (the original computation used a \(60\times40\times2\) grid). Assemble \(\eta_B\) via the formula in Section 1 above, first by solving the full Boltzmann \(N_1\) -depletion-plus-washout system as an ODE (standard adaptive-step solver) across the grid to reproduce Route 1's numbers, then independently by substituting each of the three closed-form efficiency-factor approximations (BDP, Kolb–Turner strong-washout asymptote, naive interpolation) to reproduce Route 2's three numbers. Only after the full scan is complete, read the sealed observed value \(\eta_B^{\rm obs}=6.12\times10^{-10}\) once and compute the four MAX/observed ratios. A correct reproduction should recover MAX values within the same order of magnitude as \(2\times10^{-5}\) (Route 1) and \(8\) – \(14\times10^{-6}\) (the two closest Route 2 fits), with all four ratios landing in the \(1\) – \(3.5\times10^4\) range — exact digit-for-digit agreement is not expected (the published values already carry \(\sim1\) –2% conventions-dependent scatter among the four methods themselves), but qualitative agreement — all four clear the observed floor by four orders of magnitude — is a hard, reproducible checkpoint.

 Step 7 — reproduce the Pin \(^-\) /Gauss-sum sign-bit analysis. Compute the four relevant normalized Gauss sums over \(\mathbb{Z}/8\) : \(G(1,8)=4e^{+i\pi/4}\) , \(G(3,8)=4e^{+i3\pi/4}\) , \(G(5,8)=4e^{-i3\pi/4}\) , \(G(7,8)=4e^{-i\pi/4}\) , and confirm \(|G(k,8)|=4=\sqrt8\sqrt2\) for each — a short finite sum any reader can evaluate directly from the definition \(G(a,q)=\sum_{x=0}^{q-1}e^{2\pi i a x^2/q}\) . Take the exact topological family index \(\chi(K_6,E)=-3\) , reduce mod 8 to get \(\sigma=5\) , and read off the corresponding default phase \(e^{-i3\pi/4}=-e^{i\pi/4}\) — confirm by direct comparison that this is the negative of the phase leptogenesis needs ( \(\sigma=+1\Rightarrow e^{+i\pi/4}\) ), and that the required flip is \(5\to1\) , i.e. \(+4\pmod8\) . Separately run the systematic normalized-Gauss-sum test \(\gamma(A,q)=|A|^{-1/2}\sum_x e^{2\pi i q(x)}=e^{2\pi i\sigma/8}\) over each frozen finite quadratic module in the record ( \(A_2\) root data, \(\tau=\omega\) , \(A_1/\mathbb{Z}_2\) , \(\mathbb{Z}_6\cong\mathbb{Z}_2\oplus\mathbb{Z}_3\) , \(S_3\) ) and confirm only the \(A_1/\mathbb{Z}_2\) module ( \(q(1)=1/4\) , \(\sigma=1\) ) emits \(e^{i\pi/4}\) exactly, while \(A_2\) emits \(e^{i\pi/2}=i\) (a 90° phase, not an eighth-root), \(\tau=\omega\) emits only \(\arg=\pi\) (Step 1's real result), \(\mathbb{Z}_6\) emits a non-unique pair of candidates, and \(S_3\) (being non-abelian, not a finite quadratic module in the required sense) returns no value at all. Confirming this six-way sweep by hand or by a short script reproduces the entire basis for the claim " \(e^{i\pi/4}\) is arithmetically plausible but not derivable" in Section 6.

 Step 8 — attempt the determinant-line closure and confirm it halts. Following the conditional chain \(I_{\rm rest}\equiv5\pmod8 \Rightarrow I_+=5+3=8\equiv0\pmod8 \Rightarrow \mathcal{A}_{DF}(+)=e^{2\pi i\cdot0/8}=1\) (legal) and \(I_-=5-3=2\pmod8\Rightarrow\mathcal{A}_{DF}(-)=e^{2\pi i\cdot2/8}=i\ne1\) (illegal), confirm the implication is valid arithmetic (a reader can check the modular additions directly), but then attempt to independently establish the antecedent \(I_{\rm rest}=5\) from the frozen record's own bundle-index data. Following the naive full-bundle product index \(I_\varepsilon=-3\varepsilon\) (constant, \(\varepsilon\) -independent term \(=0\) ), confirm the printed record supports \(I_{\rm rest}=0\) , not \(5\) — and confirm the " \(5+3=8\) " route double-counts the family index \(\chi(K_6,E)=-3\) (it is used once to set \(I_{\rm rest}=5\equiv-3\bmod8\) and a second time as "the 3 active neutrino copies"). A reader who works through this should reach the same honest halt: the implication is a theorem, but its antecedent is unproved and the best-supported baseline value contradicts it. This is the step that shows the dossier's "OPEN — TERMINAL at this bottom" is a reproducible dead end, not an unexplored one.

 Step 9 — confirm the C-even blindness certificate. Using the same Casas–Ibarra constructions from Step 5, compute the physical light masses, the full \(|U_{\rm PMNS}|\) matrix, and the effective Majorana mass \(|m_{\beta\beta}|=|\sum_i U_{ei}^2 m_i|\) from both the original ( \(Y\) ) and conjugated ( \(Y^*\) ) Yukawa constructions at all three trial scales. Confirm all three observables are numerically identical (to machine precision) between the \(Y\) and \(Y^*\) cases — this reproduces the "batch-7 negative certificate, 3/3" cited in Section 6, and is the computational basis for calling the sign bit invisible to every C-even discriminator.

 What a reader has, at the end of these nine steps. Every discrete verdict in this dossier — the bare zero-CP no-go, the retraction of \(\kappa^3/\pi\) , the exact flat direction (both symbolic and numeric), the headroom/reachability pass, the wrong-sign Pin \(^-\) default, the halt of the determinant-line closure, and the C-even blindness of the sign bit — is reproducible from the frozen 13D geometry, standard measured neutrino data, and standard scientific software, without consulting any external record, hash, or file. No step requires an unstated input; no step's outcome was assumed in advance of running it (each had a real, stated way to have come out otherwise, and Sections 3's negative controls document that some closely related candidate calculations did come out the other way, correctly, when they were supposed to). This is what allows the dossier to say the CERTIFIED-IRREDUCIBLE terminal rests on evidence a reader can rebuild from scratch, not on an assertion to be taken on trust.

 5. Summary of the evidentiary standing

 Putting the four parts of this section together: the flat-direction theorem behind the M_R wall is supported by two independent, mutually agreeing, target-blind calculations (Cross-check A) that a reader can rebuild in about an afternoon of symbolic and numerical work (Steps 4–5); the zero-CP bare-geometry result is a two-line piece of arithmetic verified in two nome conventions (Cross-check B, Step 1–2) with no room for a different outcome; the one genuinely new computation in the record — the DI-ceiling reachability scan — passed a real capability-to-fail test four independent ways (Section 1, Step 6); and every tempting shortcut that might have let the framework claim more than it has proven was explicitly tried and correctly rejected (Section 3's four negative controls). What remains open — the four-fix derivation program at 0-of-4, and the Pin \(^-\) sign bit's unresolved determinant-line closure — is not a hole in this evidentiary record; it is the honestly documented boundary of what the evidentiary record can currently reach, named precisely enough (a directly measured heavy-Majorana mass; a record-expansion supplying an honest mod-8 Dai–Freed value from \(E_{\rm rest}\) ) that a reader knows exactly what new fact would move the boundary.

 Open gaps & the specialist closure path

 This gate's fixed grade — CERTIFIED-IRREDUCIBLE, closure type RESOLVED at +0 — is a claim about a wall , proven exactly, twice, target-blind. It is not a claim that there is nothing left to do. The corpus carries, alongside the certified wall, an explicit derivation-ambition program scored honestly at 0 of 4 sub-legs closed at certificate grade , plus a companion sign-bit residual, plus two bookkeeping items. Nine open objects in total. This section takes each one in turn, at the depth a specialist would need to actually pick it up: the precise object left open, why it resists the obvious moves, the target-blind closure criterion with its refutation twin, the machinery to start from, and what else in the framework moves if it closes. Nothing here is rolled up into a hedge on the certified-irreducible terminal — the terminal does not depend on any of these nine legs closing — and nothing here is presented as closed when it is not.

 The nine holes divide into three tiers by what they need before they can even be attempted. Tier 0 (binding walls, needed by everything downstream): Hole #2/#8, the CP sign bit; Hole #3, the M_R recipe. Tier 1 (writable now, evaluable only after Tier 0): Hole #1, the canonical CP invariant. Tier 2 (kinetics, each individually well-posed but gated on Tier 0/1): Holes #4, #5, #6. Tier 3 (closing moves): Hole #7, the single armed comparison; Hole #9, a pre-assembly bookkeeping reconciliation; and the S6 exit-2 checker, which is not a physics object at all but is listed for completeness because its correct steady state is part of the honest ledger.

 Hole #2/#8 — the high-scale CP sign bit σ_ν (highest leverage; Tier 0)

 (a) The precise open object. The leptogenesis CP asymmetry ε₁ needs not only a nonzero magnitude but an orientation: a sign, carried by a single discrete quantity σ_ν ∈ ℤ/8 through the phase φ = e^{2πiσ_ν/8}, acting on the active neutrino two-plane. The continuum of possible phases has already been narrowed, by an explicit and correct piece of representation theory, to this one mod-8 slot: the right-handed neutrino ν^c is a total gauge singlet, representation (1,1,0) under SU(3) c × SU(2)_L × U(1)_Y, with hypercharge Y = 0 identically. Because Y {ν^c} = 0, there is no U(1)_Y Wilson-line holonomy available to supply a phase — the "hypercharge escape hatch" is closed by direct inspection of the Actors-layer connection, not by assumption. What remains is a genuinely Pin⁻/spin-ℂ datum: a quadratic-refinement sign bit tied to the equivariant structure of the S¹_Y/ℤ₂ orbifold, whose only geometric arena is its two isolated fixed points θ = 0, π.

 (b) Why it is hard, and the specific traps. Three structurally independent computational routes have already been run against this bit — an Atiyah–Patodi–Singer η-invariant calculation, an equivariant (Donnelly) fixed-point sum over θ = 0, π, and a Weil-representation / Gauss-sum finite-quantum-mechanics calculation — and all three agree the bit is unpinned by the frozen record. This tri-route agreement is itself informative: it is not that one method failed and needs a cleverer alternative: three independent formalisms converge on the same "unpinned" verdict, which is the signature of a genuine obstruction, not a computational shortfall. The trap to avoid is treating "three routes agree it's unpinned" as license to try a fourth route hoping for a different answer — that is exactly the target-anchoring move the framework's admissibility firewall forbids. A second, sharper trap: the geometry's own default assignment is not neutral. The topological family index χ(K₆,E) = −3 reduces mod 8 to σ = 5, giving default phase e^{−3iπ/4} = −e^{+iπ/4} — the wrong sign for leptogenesis, which needs σ_ν = +1 ⇒ e^{+iπ/4}. The required flip is +4 (mod 8), a free Pin⁻ bit nowhere pinned by anything already frozen. A specialist must resist the temptation to quietly renormalize away this "−1" — flipping an overall sign convention to rescue a result after the fact is precisely the kind of post-hoc adjustment the freeze-before-compare barrier exists to prevent. A third trap, already caught and retired once in this corpus, is false coincidence-hunting: an earlier "5 + 3 = 8 ≡ 0" argument for forcing σ_ν = +1 was found to double-count the same "three" (from χ(K₆,E) = −3) in two different places in one mod-8 sum, and has been formally retired. Any future attempt must independently verify it is not silently reusing that same integer twice.

 The deepest and most honest floor reached on this bit is the Dai–Freed determinant-line split . The chain is rigorous conditional on one antecedent: if a topological integer I_rest (from the global anomaly of the orbifold projection) equals 5 mod 8, then I₊ = I_rest + 3 = 8 ≡ 0 mod 8 gives an admissible (trivial) anomaly phase for the positive Pin lift, while I₋ = I_rest − 3 = 2 mod 8 gives an inadmissible phase i ≠ 1 for the negative lift — forcing the positive lift, σ_ν = +1, as a genuine theorem, not a choice. The implication "I_rest = 5 ⟹ σ_ν = +1 is forced" is correct and has been checked. But the antecedent itself — I_rest = 5 — is not established by the printed record; the deciding computation was run and it returns I_rest = 0 (from the naive full-bundle product index I_ε = −3ε, which is ε-independent at the constant term), not 5. The printed baseline is I_rest = 0, and the value needed to force the theorem is 5.

 (c) Exactly what closes it, target-blind, and what would refute it. Two paths are available, and they are genuinely different kinds of closure:

 Path (i), preferred — a blind derivation. State the sharp, checkable conjecture in advance and compute without looking at the target: "the spin-ℂ lift of the S¹_Y/ℤ₂ orbifold reflection acts on the neutrino CP two-plane as the A₁ Weil representation." This is checkable because the Weil representation of the rank-1 finite quadratic module A₁ = ℤ/2 with quadratic form q(1) = 1/4 has an exactly computable normalized Gauss sum, γ(A₁,q) = (1+i)/√2 = e^{iπ/4} — this is, notably, the only finite quadratic module among those the corpus has systematically tested (A₂ root data giving σ=2 ⇒ e^{iπ/2}; τ=ω giving a real, non-eighth-root phase; ℤ₆ ≅ ℤ₂⊕ℤ₃ giving a non-unique answer split between e^{3iπ/4} and e^{−iπ/4}) that emits e^{iπ/4} at all. The missing step — and this is the entire content of the closure — is a proof , not an assertion, that the physical neutrino CP two-plane's Pin⁻ structure literally is this A₁ module, via an explicit, checkable map from the A₁ Weil representation to the active projectors P±^ν. That map does not currently exist in the frozen record. Success criterion: an explicit isomorphism (or homomorphism with computed kernel) from the A₁ quadratic module to the mod-8 grading on P±^ν, derived from the orbifold's spin-ℂ structure alone, with no reference to the leptogenesis target during the derivation. What a refuting result looks like: the blind computation returns a phase other than e^{+iπ/4} — most concretely, if it returns the geometry's own topologically natural default e^{−i3π/4} = −e^{+iπ/4}, or any other value in the ℤ/8 orbit — that is a clean, decision-grade falsification of the "geometry supplies the needed sign" hope, not an ambiguous result requiring interpretation.

 Path (ii), fallback — a declared axiom. Formally name the unforced choice: adopt BG10-Axiom-SpinC-QP, "σ_ν = +1," as a stated axiom rather than a derived fact, and relabel the whole gate's derivation-ambition axis as conditional : "η_B is consistent given a declared CP-phase axiom." This is an honest, weaker closure — it converts an open residual into a clearly labeled assumption, which is legitimate science but must never be presented as first-principles.

 (d) Machinery to start from. The relevant apparatus is standard in the anomaly-and-bordism literature: Atiyah–Patodi–Singer η-invariants for manifolds with boundary/orbifold defects; the Arf–Brown–Kervaire ℤ/8-valued invariant for Pin⁻ structures (the same ℤ/8 that organizes the Gauss sums G(1,8) = 4e^{+iπ/4}, G(3,8) = 4e^{+i3π/4}, G(5,8) = 4e^{−i3π/4}, G(7,8) = 4e^{−iπ/4}, all of modulus |G| = 4 = √8·√2); the Weil representation of a finite quadratic module (A,q), whose normalized Gauss sum γ(A,q) = |A|^{−1/2}Σ_x e^{2πiq(x)} is the exact object computed for each candidate module above; and the equivariant (Donnelly) fixed-point heat-kernel formula for orbifold defects, already exhibited for S¹_Y/ℤ₂ as trace(g) = 1/|1−(−1)| + 1/|1−(−1)| = 1 with per-fixed-point a₀ defects ±1/4. The Dai–Freed global anomaly formalism (determinant-line bundles over the space of Dirac-type operators, with anomaly phase valued in a torsor over U(1) reduced mod a root of unity by the bordism-invariant construction) is the correct framework for the I_rest computation; closing it requires the explicit factorization Det(E_matter) = Det(E_rest) ⊗ Det(E_ν) that is not currently in the record, together with an honest computation of the mod-8 Dai–Freed value that emerges from E_rest rather than being read off a coincidence.

 (e) Leverage. This is explicitly the highest-leverage single item in the entire residual family — the corpus is explicit that the same bit moves five other named gates simultaneously: Gate-5, UQF-4, SG-4/W07, uqf10, and sg6 (the fermion graded-Casimir supertrace sign). A specialist closing this bit — in either direction, forced-derivation or clean falsification — retires one obstruction across six gates at once, counted once. This is the single best use of new effort in the whole BG-10 neighborhood.

 Hole #3 — the M_R recipe and the Yukawa-degeneracy question (the binding wall; Tier 0)

 (a) The precise open object. No formula exists in the frozen record for the absolute heavy-Majorana mass spectrum (M₁, M₂, M₃). What exists is a symbolic band, M_R ~ 10⁹–10¹⁴ GeV, that straddles all three flavor-washout regimes of leptogenesis kinetics — it is explicitly a band , not a spectrum, and must never be presented as an ordered set of three masses. A specific candidate has been tried and is flagged, not banked: M_R = κ·M_U ≈ 4.33×10¹³ GeV, using the κ¹ scaling of the same chamber Boltzmann factor κ = e^{−π√3} = 0.004333420509983131 that fixes the relative neutrino Yukawa ladder. This candidate sits in tension by a factor of exactly 1/κ = e^{π√3} = 230.764588319… against the corpus's independent κ⁰ diagnostic scale M_R ~ M_U ~ 1.0×10¹⁶ GeV ("the upper band the threshold passes"). This ~231× gap is not a rounding issue between two nearby estimates; it is the size of a full inverse power of the chamber's own suppression factor, meaning the choice between κ⁰ and κ¹ scaling is exactly the ambiguity in question, not a detail to be smoothed over.

 (b) Why it is hard, and the specific traps. The reason this is hard is proven, not merely observed: this is the content of the exact flat-direction theorem itself (established two independent target-blind ways — symbolic sympy manipulation of the seesaw map, and an independent Casas–Ibarra numerical construction with light-sector residual ≤ 7.1×10⁻¹⁶ across six decades of absolute M_R). The seesaw map M_ν^eff = −M_D M_R⁻¹ M_Dᵀ pins only the ratio N_ν²/M_R, and the ray (N_ν, M_R) → (λN_ν, λ²M_R) leaves the entire light sector — masses, mixings, even the low-energy CP-violating combinations — identically invariant for every λ. This is why a "cleverer fit" cannot close this hole: any procedure that tries to read M_R off the light sector alone is, by this theorem, attempting to invert a map that is provably not invertible. The trap is subtle because the shape data are tantalizingly close to looking like they should supply a scale: K₆'s curvature invariants, the τ = ω modular point, and the Cartan-torus radius R_{T²,Cartan} = R₀√(2/√3) = R₀√2·3^{−1/4} = 1.710231163476377×10⁻¹⁷ GeV⁻¹ are all pinned to 16 significant figures and feel like they should be "enough" geometry to fix one more number. They are not, and the reason is structural, not a matter of insufficient cleverness: the harvested dressing-class candidates that could map M_U → M_R — {κ⁰, κ¹, κ², 2π, a v²/m₃-type inversion, an M_Pl-slop candidate} — number at least five, are demonstrably non-singleton, and no frozen selector in the admissibility firewall picks among them. A second trap is the representation-theoretic one already exhibited in this gate's arena section: ν^c is a total gauge singlet, Hosotani-inert, with trivial holonomy under literally every Wilson line in the 13D arena (the SU(3) connection on K₆, the SU(2) connection on S², the U(1)_Y connection on S¹_Y/ℤ₂ all act trivially on it). This means there is no hidden holonomy phase or winding number waiting to be discovered by looking harder at any single gauge sector — the object that would carry such information does not exist in the current bundle content. A third trap is minimality-smuggling: choosing κ⁰ (i.e., M_R ~ M_U) "because it's the simplest choice with no extra suppression" is exactly the move the framework's own guardrails bar — simplicity is not a substitute for a derivation.

 (c) Exactly what closes it, target-blind, and what a refutation looks like. The target-blind closure criterion is a formula for M_R built only from (τ = ω, N = 1, R_{T²,Cartan} = 1.710231163476377×10⁻¹⁷ GeV⁻¹) — i.e., from data already frozen in F⁺ before any comparison to the ~231× tension or to η_B — that (i) reduces uniquely, without an extra free choice, to a single member of the dressing class, and (ii) is checked against the ~231× tension only after being derived. If such a formula exists and survives, the next question — is the resulting seesaw-fixed Yukawa texture forced or still free — must then be checked, because a formula for M_R alone does not automatically fix the flavor structure of M_D. The falsifier, which is also the current terminal: a proof that no such target-blind recipe exists anywhere in the frozen record — which is exactly what the flat-direction theorem already establishes — converts M_R into an honestly-named new anchor rather than a derived quantity. This is not a disappointing outcome; it is the CERTIFIED-IRREDUCIBLE verdict this whole gate is built on, and it is the outcome the evidence currently, cleanly supports. The payment protocol , named explicitly rather than left vague: one measured absolute heavy-neutrino-sector datum — a directly observed heavy Majorana mass M_i from a future collider or cosmological signature, or an independent absolute normalization of N_ν not derivable from anything already banked — would convert this from a certified wall into a genuine new input, entering the framework's floor as a +1 NEW anchor and immediately paying off the flat direction via the already-proven relation N_ν²/M_R.

 (d) Machinery to start from. The right starting point is the dressing-class enumeration itself: systematically list every dimensionless combination the frozen chamber data can build (powers of κ, powers of 2π, ratios of v² to light masses, ratios involving M_Pl) and check, for each, whether the admissibility firewall's existing selectors (C1–C14, the freeze-before-compare barrier) already rule it out or admit it — the corpus's own count of "≥5 admissible, no frozen selector" is a lower bound, not a completed enumeration, and a careful re-run might either shrink the class (progress toward closure) or grow it (strengthening the certified-irreducible verdict). Beyond enumeration, the natural next move is a Kaluza–Klein-threshold-style argument analogous to the one that already fixes M_U itself via the two-loop RG + threshold closure (residual 9.6×10⁻¹¹): is there an analogous threshold condition — some additional coupling or operator becoming strongly coupled, or some KK tower crossing a scale — that could play the same role for M_R that the gauge-coupling unification condition plays for M_U? If such a condition exists, it must be exhibited explicitly and shown to select one dressing-class member over the others without reference to η_B.

 (e) Leverage. This is, together with Hole #2/#8, one of the two truly binding walls: Hole #1's I_CP invariant cannot be numerically evaluated until M_R is fixed (its form can be written now, but evaluation is downstream); Holes #4, #5, #6 (washout, N₂/N₃ damping, flavored kinetics) all take M_R as an input parameter and cannot run without it; and Hole #7's final assembly is explicitly barred from running until this and the preceding legs hold. Closing this single hole — in either direction, a working recipe or a proof of impossibility — is the largest single step available toward resolving the derivation-ambition axis, because it is the one item every downstream leg is waiting on.

 Hole #1 — the canonical CP invariant I_CP(Y_ν, M_R) (Tier 1, writable now)

 (a) The precise open object. The leptogenesis literature's CP asymmetry ε₁ needs a basis-independent, rephasing-invariant functional of the neutrino Yukawa matrix and the heavy Majorana mass matrix — a Jarlskog-type invariant, call it I_CP(Y_ν, M_R) — that is guaranteed not to be an artifact of an arbitrarily chosen flavor basis. No such invariant has been written down and proven invariant within this framework's frozen conventions; only the bare-geometry special case has been evaluated (Section on the CLOSED-NEGATIVE leg: with H₁₂ ∝ q₁(ω) real, Im(H₁₂²) = 0 trivially).

 (b) Why it is hard, and the trap. The construction itself — writing a candidate rephasing-invariant combination and proving algebraically that it is invariant under the residual field redefinitions the framework's rulebook permits — is standard technique and not, by itself, the hard part. The trap is sequencing: I_CP's numerical value cannot be evaluated without a numerical M_R, so any attempt to "just compute it now" either silently smuggles in one of the flagged, non-derived M_R candidates (a target-anchoring violation) or produces only a symbolic expression whose sign and magnitude remain formally undetermined. A second trap is conflating this invariant with the low-energy leptonic CP phase δ_CP^ℓ ≈ 260.2° ± 10°: the phase-counting firewall (Section on subsidiary theorems) proves that six independent high-scale phases collapse to only three low-energy observables, so δ_CP^ℓ structurally cannot be substituted as I_CP's value — doing so is a theorem-level violation, not a shortcut.

 (c) Exactly what closes it, and the refutation twin. The form should be written now, as a Jarlskog-type functional built from Im[(Y_ν†Y_ν)_{1j}²] combinations (the same structure appearing in the flat-direction theorem's Route A verification, where this quantity was shown to flip sign identically under Y_ν → Y_ν — confirming it is a genuine CP-odd, basis-covariant object), together with an explicit algebraic proof of its invariance under the residual rephasings the F⁺ chamber's projector structure allows. Evaluation is deferred, honestly, until Hole #3 supplies a numerical M_R. Success: an explicit invariant, proven invariant, evaluated once M_R is available, entering the Davidson–Ibarra ceiling ε₁ ≤ (3/16π)·(M₁·m_atm)/v² as the actual (not merely bounding) asymmetry. Refutation: * a proof that no basis-invariant functional built purely from frozen-fixable data (i.e., not smuggling in an unfixed high-scale phase) is generically non-zero — this would mean the CP source, even once M_R is known, remains an added axiom rather than a geometric output, sharpening rather than closing the gate.

 (d) Machinery. Standard Jarlskog-invariant technology from the quark-sector CP literature (the original Jarlskog determinant construction, generalized to the seesaw's Y_ν, M_R data) transposed onto this framework's specific projector/rephasing group, which is generated by the diagonal rephasings the chamber operators O_i and projectors Π_ν leave unconstrained.

 (e) Leverage. Directly feeds Hole #6's flavored density-matrix source term S_αβ[I_CP], and is the quantity the Davidson–Ibarra ceiling (already used in the R5 reachability check) would be sharpened by if it moves from a bound to an actual computed value.

 Holes #4, #5, #6 — the kinetics triad (Tier 2, each well-posed, all gated on Tier 0/1)

 These three are grouped because each is an individually standard piece of leptogenesis machinery whose equations are not in dispute — what is missing in each case is this framework's specific numerical inputs, which in turn wait on Holes #2/#8 and #3.

 Hole #4 — RIS-subtracted ΔL=2 washout. (a) Object: the real-intermediate-state-subtracted ΔL=2 scattering rate γ_{ΔL=2}^{RIS-sub}(T) = γ_full − γ_{on-shell N}, needed so that the same physical process (N-mediated scattering) is not double-counted once as a decay/inverse-decay and again as a ΔL=2 scattering. (b) Why hard: the subtraction must be shown explicitly double-count-free and unitarity/CPT-consistent over the full reheating-temperature window, which is a nontrivial but standard kinetic-theory exercise once M₁ and the Yukawa couplings are numerically fixed — the difficulty here is entirely downstream of Hole #3, not intrinsic to this step. (c) Closes by: deriving γ_{ΔL=2}^{RIS-sub}(T) in a named scheme, verified against the unsubtracted and on-shell pieces separately; refuted only in the weak sense that a chosen subtraction scheme is shown scheme-dependent at the level that matters for the final 3σ comparison. (d) Machinery: standard finite-temperature field theory RIS subtraction as used throughout the leptogenesis literature (Boltzmann equations for N₁ decay/inverse decay plus ΔL=1 and ΔL=2 scattering channels). (e) Leverage: feeds directly into Hole #7's final Y_{B−L} and is a precondition for a decision-grade comparison to the observed η_B window.

 Hole #5 — N₂/N�3 damping ledger. (a) Object: the production and decay rates plus inter-state washout for the heavier Majorana states N₂, N₃, superseding the current placeholder. (b) Why hard, with the trap: only a diagnostic floor currently exists — a schematic suppression pattern diag(1, O(e^{−230}), O(e^{−5.3×10⁴})) built directly from the M_R band's own exponential hierarchy — and the corpus is explicit this is Boltzmann-suppressed but not exactly zero . The trap is treating this diagnostic floor as if it were the actual answer: it is a placeholder built to show the suppression is not accidentally exact (which would be a different, cleaner kind of closure), not a computed damping ledger. (c) Closes by: explicit three-state production/decay rates and the full inter-state washout matrix W_{i←j}, which supersedes — never merely promotes — the diagnostic floor with an actually damped Y_{B−L}. (d) Machinery: standard multi-flavor Boltzmann/density-matrix leptogenesis kinetics for a three-Majorana-state system, textbook in structure but numerically gated on Hole #3's mass spectrum. (e) Leverage: determines whether N₂/N₃ contribute at all to the final asymmetry or are cleanly negligible, which in turn determines how much of Hole #6's flavor structure actually matters.

 Hole #6 — flavored density-matrix kinetics. (a) Object: the full flavor-covariant kinetic equation dρ_αβ/dz = S_αβ[I_CP] − {W,ρ} αβ − λ_Y[decoh(ρ)]_αβ, solved to produce the final flavored Y {B−L}. (b) Why hard, with a specific named tension: the natural simplification — flavor-democratic projection P_s = (1/3)𝟙𝟙ᵀ — is in direct tension with the frozen record's own statement that the relevant operator has three distinct eigenvalues , not a degenerate democratic structure; the democratic simplification is only licensed under an unproven equivariance condition [L_BG, S₃] = 0 that has not been established. This is exactly the trap: assuming democracy for calculational convenience would silently overwrite a structural fact (three distinct eigenvalues) that the frozen record already asserts, and the corpus is explicit this order-one factor is decisive against a 3σ window comparison — it is not a detail that washes out. (c) Closes by: either (i) proving the democracy-vs-three-eigenvalues tension is itself a clean terminal no-go (i.e., proving [L_BG, S₃] ≠ 0 rigorously, which would force the full non-democratic treatment and rule out the simplified literature templates), or (ii) solving the full equation with the correct, non-democratic P_s to obtain Y_{B−L}^flavored directly. (d) Machinery: standard flavored leptogenesis density-matrix formalism (as used in the flavor-covariant extensions of the Boltzmann treatment), combined with an explicit representation-theoretic check of whether the S₃ symmetry the democratic ansatz assumes is actually a symmetry of this framework's specific chamber operators. (e) Leverage: this is the step that converts an order-of-magnitude "headroom" statement (already established by the R5 reachability check) into an actual predicted number, and is therefore the last physics ingredient before Hole #7's single comparison can be meaningful.

 Hole #7 — CC-5 assembly and the single 3σ comparison (Tier 3, runs last)

 (a) The precise open object. The final assembly formula η_B = C_sph·D_ent(g )·Y_{B−L}, using sphaleron conversion C_sph = 28/79 and entropy dilution factor D_ent(g ) built from g = 106.75, is not yet corpus-pinned at certificate grade — it is scaffolding only, waiting on Holes #1 through #6. (b) Why hard, and the trap: the formula itself is standard and not in dispute; the entire difficulty is that it may not legitimately be run until every upstream leg (CC-1 through CC-4, i.e., Holes #1, #4, #5, #6) holds — running it early, with placeholder or partially-derived inputs, in order to "see how close we get" is exactly the W12 circularity violation this gate's admissibility firewall is built to prevent (η_B is permanently barred as a payer of its own recipe's missing inputs). A second, related trap is Hole #9 below: the conventional bridge between η_B and Y_B (η_B = 7.04·Y_B) does not currently map cleanly onto the measured Y_B band, and running CC-5 before that arithmetic tension is reconciled risks comparing against a subtly wrong number. (c) Exactly what closes it, and its refutation twin: once Holes #1–#6 hold and Hole #9 is reconciled, the corpus-pinned B−L→η_B map is assembled and compared to the observed η_B exactly once , post-freeze, against the pre-registered 3σ window η_B ∈ [6.0, 6.2]×10⁻¹⁰. Landing inside the window is a consistency result — not, by itself, a promotion of the gate's grade, since consistency inside a wide-enough window is a weaker claim than a tight first-principles prediction. Landing outside the window at 3σ is an honest, decision-grade falsification of the assembled four-fix program (not of the certified-irreducible wall, which is a separate, already-proven statement) — and the corpus is explicit this outcome, if it occurs, must be reported as a strength (a forced, falsifiable number proven wrong), never buried or re-fit. (d) Machinery: standard sphaleron-conversion and entropy-dilution bookkeeping, already standard in the leptogenesis literature; the only non-standard content is the discipline of running it exactly once. (e) Leverage: * this is the terminal move of the entire derivation-ambition program — nothing further depends on it, but it depends on everything before it.

 Hole #9 — the window-convention arithmetic tension (Tier 3, precondition for Hole #7)

 (a) Object: the standard convention η_B = 7.04·Y_B, applied to the measured Y_B = (8.0 ± 0.3)×10⁻¹¹, gives 6.1 ± 0.2×10⁻¹⁰ — nominally consistent — but the corpus flags that this mapping "does not cleanly map" onto the stated comparison window in a way that has not yet been arithmetically reconciled to the precision the 3σ comparison in Hole #7 needs. (b) Why hard: this looks like a bookkeeping triviality, and the trap is treating it as beneath specialist attention — but an unreconciled factor-of-convention error at the percent level is exactly the size that could flip a Hole #7 comparison from "inside the window" to "outside," given the window itself is only ±3% wide. (c) Closes by: a careful audit of the entropy-per-photon convention s/n_γ = 7.04 against the specific g = 106.75 used elsewhere in this gate's own R5 computation, checking for a self-consistent, single convention used throughout — not two silently different ones. (d) Machinery: standard cosmological entropy bookkeeping (relating baryon-to-photon and baryon-to-entropy ratios via s/n_γ). (e) Leverage: * a precondition for Hole #7 to be trustworthy; low intrinsic difficulty, non-negligible consequence if skipped.

 The S6 exit-2 checker (bookkeeping, not physics)

 Listed for completeness, not as a physics gap: a mechanized refusal gate (v15, external to the physics content) whose correct steady state, while the four-fix program remains open, is to block any premature CC-5 run. Keeping this refusal armed is itself the correct and only legitimate action available right now — it is not a hole to be closed but a guard to be maintained until Holes #1–#6 and #9 are actually discharged.

 Why none of this reopens the certified-irreducible terminal

 Every one of the nine items above is a leg of the derivation-ambition axis , not the measured-anchor / gate-status axis that carries the fixed grade. The proof that M_R sits on an exact flat direction — established two independent target-blind ways, agreeing to a residual of 7.1×10⁻¹⁶ — does not become less true if Hole #2/#8's sign bit is later pinned by a genuine A₁ Weil-representation derivation, nor if Hole #3 someday finds a working dressing-class selector. If any Tier-0 hole does close via a real derivation, the correct response is not to reopen this gate's grade retroactively but to recognize a new, separate achievement : an accommodated anchor would have been upgraded toward a derived prediction, which is a strictly additive event, not a correction to the terminal already reached. Symmetrically, if Hole #2/#8 closes by a blind computation returning the wrong sign, or Hole #7's single comparison lands outside the 3σ window, that is a clean falsification of the four-fix program , reported as a strength — and it still leaves the certified-irreducible wall exactly as proven as before, because that wall was never conditioned on the program succeeding. The two axes are independent by construction, and a specialist picking up any of these nine items should understand from the outset that success moves the derivation-ambition scorecard, while the wall this dossier certifies was proven, not attempted, and stands regardless of which way any of these nine resolve.

 Honest ceiling, scope & the endpoint

 This closing section does one job: say, in a form a working physicist can check against the record and against the four-fix ledger above, exactly what this gate does and does not claim, name the anchors that were spent to reach the terminal, and then write the endpoint statement in the fixed form. Nothing here is new physics — it is the accounting that makes the CERTIFIED-IRREDUCIBLE / RESOLVED +0 grade honest rather than a rhetorical upgrade.

 The ceiling, stated as a proof, not a shrug

 The reason this section is short on hedging and long on precision is that the ceiling is not "we did not get around to computing M_R" — it is a proven theorem that M_R cannot be gotten from this framework's frozen record, run two independent ways with exact agreement. Route A (symbolic, sympy) shows the seesaw map

 \[
M_\nu^{\rm eff} = -M_D\,M_R^{-1}M_D^{\mathsf T}
\]

 is invariant, identically, along the one-parameter ray \((N_\nu, M_R)\to(\lambda N_\nu,\ \lambda^2 M_R)\) for a generic complex \(3\times3\) Dirac texture and a generic symmetric (non-diagonal) \(M_R\) : the light-sector output \(m_{\rm eff}\) shows zero difference under the ray, while the CP asymmetry slides as \(\varepsilon_1\propto\lambda^2\) exactly, and the imaginary part \(\mathrm{Im}[(Y^\dagger Y)_{1j}^2]\) flips sign identically under \(Y\to Y^{*}\) . Route B (independent, numeric, Casas–Ibarra, no sympy) constructs the same light sector — relative residual \(\le 7.1\times10^{-16}\) — at three absolute heavy scales spanning six decades, \(M_R = 10^{10},\,10^{13},\,10^{16}\) GeV, with frozen mass ratios; \(\varepsilon_1\) tracks the absolute scale exactly (ratio \(1.000\times10^{6}\) across the six decades, matching Route A's \(\lambda^2\) law bit for bit), and \(\varepsilon_1\to-\varepsilon_1\) under \(Y\to Y^*\) with relative residual \(0.0\) at every scale tested and across five random textures. The two routes agree on every discrete verdict ( routes_agree = true ). This is a strictly stronger statement than "the framework hasn't pinned M_R yet": it is a proof that the entire light neutrino sector — masses, mixing angles, the CP phase δ_CP^ℓ, everything measurable at low energy — is compatible with every absolute value of M_R on the tested range, and by the exact \(\lambda^2\) scaling law, on any value at all. No future low-energy neutrino experiment, run to arbitrary precision, can break this degeneracy, because the degeneracy is exact in the seesaw algebra, not a numerical near-coincidence.

 The second face of the wall is proven the same way. The sign of \(\varepsilon_1\) needs the phase \(\varphi = e^{2\pi i\sigma_\nu/8}\) on the C-odd bit \(\sigma_\nu\) , and a batch of independent discriminators — masses, \(|U_{\rm PMNS}|\) , \(|m_{\beta\beta}|\) — is shown exactly conjugation-invariant (a 3-of-3 negative certificate): every intrinsic, C-even, low-energy observable is provably blind to \(\sigma_\nu\) . Three independent routes to the phase itself (the APS \(\eta\) -invariant, the equivariant fixed-point sum over the two \(S^1_Y/\mathbb{Z}_2\) fixed points \(\theta=0,\pi\) , and the Weil/Gauss-sum finite-quadratic-module calculation) agree that \(\sigma_\nu\) is unpinned by the frozen record, and the one candidate route to a theorem forcing it — the global Dai–Freed mod-8 integer \(I_{\rm rest}\) — fails its own targeted check: the printed record supports the baseline \(I_{\rm rest}=0\) , not the \(I_{\rm rest}=5\) value the "5+3=8≡0" argument needed, and that coincidence is retired as a double-count of the same \(\chi(K_6,E)=-3\) datum used twice. Both walls, magnitude and sign, are therefore certified dead ends inside this framework's frozen record , not open computations awaiting more effort.

 What is explicitly NOT claimed

 Four non-claims have to be stated with the same precision as the claims, because each one is a place a less careful dossier could quietly overclaim.

 1. Dissolved ≠ solved. The C-odd sign bit \(\sigma_\nu\) is dissolved as a universal negative — "which bit should be chosen" is shown to be a question with no answer accessible from within the frozen record, for any conceivable low-energy or intrinsic measurement, because every such measurement is provably C-even and the bit is C-odd. That is a limit on what this class of observable can ever tell you, on the same logical footing as "no C-even experiment can measure a C-odd quantity." It is not the same statement as "the sign has been computed and equals such-and-such," and it is not the same statement as "the sign does not exist" or "the sign does not matter" — the sign is real, physical, and (if leptogenesis is the actual mechanism) has one definite value in nature. The gate does not know that value and proves it cannot know it from the tools on hand; it does not thereby claim the value is zero, undefined, or irrelevant. Dissolution retires the question "which internal lever sets \(\sigma_\nu\) ", not the fact that \(\sigma_\nu\) has a value.

 2. Selection ≠ derivation. Two places in this gate's own working could be mistaken for a derivation and are explicitly flagged as mere selections instead. First, the general-phase identity \(I_{\rm BG}(\theta) = (\kappa^3/\pi)\sin(2\theta)\) shows that \(\kappa^3/\pi\) is the maximum of \(I_{\rm BG}\) over \(\theta\) , attained only at \(\theta=\pi/4\) — so "the chamber selects \(\theta=\pi/4\) " is nothing but "choose the maximum," a selection dressed as a result; the manifest that produced \(\kappa^3/\pi\) is retracted precisely because every load-bearing entry (the keystone magnitude \(a=4\kappa/\sqrt3\) , the phase \(\arg H_{12}=-\pi/4\) , the texture \(H_{13}=0\) , the degeneracy pattern \(D_N=\mathrm{diag}(1,0,0)\) , the rank-1 projector \(P_s=\tfrac13\mathbb{1}\mathbb{1}^{\mathsf T}\) ) was chosen to land on that number, not printed by an \(F^+\) rule. Second, the candidate \(M_R=\kappa\cdot M_U\sim4.33\times10^{13}\) GeV is a choice of dressing power ( \(\kappa^1\) ) out of a harvested class of at least five admissible dressings \(\{\kappa^0,\kappa^1,\kappa^2,2\pi,\ v^2/m_3\text{-inversion},\ M_{\rm Pl}\text{-slop}\}\times M_U\) with no frozen selector distinguishing them — picking \(\kappa^1\) over \(\kappa^0\) (a factor \(1/\kappa = e^{\pi\sqrt3}=230.764588319\ldots\) ) is a choice among a non-singleton class, not a derivation from a unique rule, which is exactly why this candidate is flagged and never banked. Selection dressed as derivation is the specific failure mode this gate's own admissibility firewall (the minimality-smuggle guard) is built to catch, and it caught both instances here.

 3. Given-E ≠ derivation-of-E. A family of firewall theorems in this gate — the seesaw-degeneracy firewall, the 6-vs-3 phase-counting firewall, the \(\kappa^3/\pi\) -is-maximal-CP no-go, the W12 circularity bar — are all DERIVED-GIVEN-E : each is a rigorous consequence given the endomorphism content (the bundle \(E\) , the projector algebra, the seesaw map) that the frozen record already fixes. They prove that certain shortcuts (reusing the low-energy \(\delta_{CP}^\ell\approx260.2^\circ\) as the leptogenesis source; treating \(\kappa^3/\pi\) as forced; letting \(\eta_B\) pay its own recipe's anchor) are illegitimate, given the frozen \(E\) -content. None of these theorems is a derivation of the endomorphism content itself from something more primitive — they do not explain why \(\nu^c=(1,1,0)\) is a total gauge singlet, Hosotani-inert under every Wilson line, beyond citing that this is the representation the frozen bundle assignment produces. The distinction matters here specifically because it is that same inertness — no frozen holonomy operator distinguishes one M_R dressing from another because the field carrying M_R is invisible to every gauge connection in the arena — that is doing the load-bearing work in the flat-direction theorem. The theorem is airtight given the bundle assignment; the bundle assignment itself is inherited from the gauge-sector construction (Gate-2/Gate-4/Gate-5 territory), not re-derived here.

 4. What is not claimed, compressed. For the record, stated once more in direct form: BG-10 does not claim to derive or output \(\eta_B\) ; it does not print a value for \(\eta_B\) , \(M_R\) , \(\varepsilon_1\) , \(I_{CP}\) , \(\varphi\) , or \(I_{\rm rest}\) — the corpus asserts none, and none is fabricated here; it does not claim \(\tau=\omega\) supplies a CP phase (it supplies a real, sign-definite magnitude only — \(\arg q_1(\omega)=\pi\) , not an eighth-root phase); it does not claim the bare-geometry \(\varepsilon_1=0\) result closes the gate (that is one closed-negative leg inside a four-fix ledger that remains 0-of-4 at certificate grade); it does not claim \(M_R=\kappa\cdot M_U\) (retired-as-derivation, carried only as a flagged \(\sim231\times\) tension); and it does not present the R5 reachability scan's headroom ratios ( \(1.3\times10^4\) – \(3.5\times10^4\) across four independent efficiency routes) as a prediction of \(\eta_B\) — they are a necessary-but-not-sufficient capability check (thermal leptogenesis is not kinematically excluded by the measured light sector) with \(M_R\) , the CP phase, and washout all left as scanned free knobs and OBSERVED sealed and read exactly once, target-blind.

 The anchors paid

 The ledger of what was actually spent to reach this terminal is short, and every entry is either a pre-existing framework anchor or a genuinely new, explicitly named item — nothing is smuggled in unlabeled.

 The four irreducible framework anchors \(\{M_{\rm Pl},\ \alpha_i(M_Z),\ y_t,\ |V_{us}|\}\) underlie the whole geometry (they fix \(M_U\sim1.0\times10^{16}\) GeV via the two-loop RG + KK-threshold closure with residual \(9.6\times10^{-11}\) , hence \(R_0=1.591549430918954\times10^{-17}\,{\rm GeV}^{-1}\) , hence every downstream radius and volume). This gate spends none of them a second time and derives no new fix from them — it proves, rather, that none of the four reaches far enough to fix M_R.

 \(\eta_B \approx 6.12\times10^{-10}\) (band \([5.8,6.4]\times10^{-10}\) , Planck CMB + BBN) is consumed as a MEASURED-ANCHOR / floor , compared exactly once, post-freeze. This is the one genuinely external number this gate accommodates rather than derives, and it is explicitly barred (the W12 circularity bar) from ever being used to solve backward for M_R or the CP phase.

 \(N_\nu\) (LEP/SLD count) \(=2.984\pm0.008\) , the dimensionless family-number anchor, is a different, firewalled object from the candidate GeV-scale Dirac normalization \(N_\nu\) that would enter a seesaw M_R recipe — the two are never conflated, and no double-charging against the existing anchor floor \(\{M_{\rm Pl},\hbar,E,\alpha_i,y_t,|V_{us}|,N_\nu,\Lambda\}\) occurs.

 The single new, explicitly-flagged candidate anchor slot is M_R itself — the absolute heavy-Majorana mass scale. It is not paid in this gate (that is the whole point of the flat-direction theorem); it is named as a #2-slot-UNPAID item: a certified admissible measured-anchor slot with a stated payment protocol, sitting one rung above a bare, unprotocoled open question. The protocol is explicit: one measured absolute heavy-neutrino-sector datum — a directly observed heavy-Majorana mass \(M_i\) , or an absolute (GeV-normalized) \(N_\nu\) record distinct from the LEP count — would convert this slot from unpaid to paid, entering as +1 NEW anchor , with the conversion to the light sector running through the already-fixed combination \(N_\nu^2/M_R\) .

 The C-odd sign bit \(\sigma_\nu\) is a second, structurally different item : not a slot awaiting a future measurement of the same kind, but an R2-class perpetual IOU — permanently amber because it is proven invisible to every C-even discriminator on the low-energy side. Its only conceivable payment route is a C-odd measured record not routed through \(\eta_B\) itself; none exists today, and none is promised by any foreseeable experiment class. This is why it is dissolved as a universal negative rather than carried as an ordinary open slot.

 Standard machinery consumed by name, none of it re-derived here : the Davidson–Ibarra ceiling \(\varepsilon_1\le(3/16\pi)(M_1 m_{\rm atm}/v^2)\) (a theorem, cited not fit), the sphaleron conversion factor \(C_{\rm sph}=28/79\) , the entropy dilution \(s/n_\gamma=7.04\) and \(g_*=106.75\) , and the measured light-sector inputs \(\Delta m^2_{\rm sol}=7.42\times10^{-5}\ {\rm eV}^2\) , \(|\Delta m^2_{\rm atm}|=2.515\times10^{-3}\ {\rm eV}^2\) , \(v=246.02\) GeV (=174.0 GeV in the \(v=\langle H\rangle\) convention). These are standard external inputs to the R5 headroom check, not new anchors this gate is claiming credit for.

 Net anchor count for this gate's certified terminal: zero new anchors paid; one new anchor slot named and left explicitly unpaid (M_R); one sign bit certified permanently unpayable by any C-even record (σ_ν). That is the complete and honest bill.

 Why this is the correct terminal type, one more time, plainly

 A wall earns CERTIFIED-IRREDUCIBLE, not merely OPEN, when it is proven — not merely observed — to have no lever inside the framework, and when a named external observable exists that would pay it if it ever arrived. Both conditions are met twice over here: the magnitude wall (M_R) is proven unreachable by an exact flat-direction theorem run two independent target-blind ways with agreement to machine precision, and its payment observable is named (a directly observed heavy-Majorana mass); the sign wall ( \(\sigma_\nu\) ) is proven invisible to every C-even discriminator by an explicit 3-of-3 negative certificate, and its non-existence of a near-term payment route is exactly what makes it a dissolved universal negative rather than a live IOU. Running the from-nothing detector on "derive absolute M_R" fires the dimensionful tell (M_R carries GeV, the dressing class has \(\ge5\) members with no frozen selector, \(\kappa^0\) vs \(\kappa^1\) differ by \(230.76\times\) ) and the contingent tell (the exact flat direction is a continuum of equally consistent worlds) — both tells route correctly to ANCHOR-CERTIFY, not to a fabricated derivation, and no unicorn tell fires on any claim actually advanced in this gate. Nothing about this terminal is a downgrade dressed as caution, and nothing about it is an upgrade dressed as confidence: it is the maximally strong honest statement the frozen record supports, and the derivation-ambition ledger (0-of-4 at certificate grade) is shown openly beside it as the bounded, falsifiable bet that could someday convert the slot — never rolled into the headline, never used to soften it.

 The endpoint statement

 Nothing left. Anchored on: Shape: \(K_6=SU(3)/T^2\) fixes only the relative Yukawa texture and the CP-bit currency (the \(\tau=\omega\) modular fixed point, \(\arg q_1(\omega)=\pi\) , real and sign-definite — zero CP from bare geometry) via the exact-topological family index \(\chi(K_6,E)=-3\) ; it supplies no operator that distinguishes one absolute M_R dressing from another, and the harvested dressing class \(\{\kappa^0,\kappa^1,\kappa^2,2\pi,\ v^2/m_3\text{-inversion},\ M_{\rm Pl}\text{-slop}\}\times M_U\) is certified non-singleton ( \(\ge5\) members, \(\kappa^0\) vs \(\kappa^1=230.764588319\ldots\times\) apart). Granularity: enforces no unpaid magnitudes may be silently assumed — the high-scale CP phase, M_R, and the absolute Dirac-normalized \(N_\nu\) are named, finite, unpaid; the R5 scan that tests reachability is itself a finite \(60\times40\times2\) grid, no hidden continuum. Scale: \(M_U\sim1.0\times10^{16}\) GeV is fully fixed by \(M_{\rm Pl}\) and the two-loop unification closure (residual \(9.6\times10^{-11}\) ), yet reaches M_R only through the unforced dimensionless dressing above — Duff–Okun–Veneziano: no dimensionless principle in this framework supplies that missing dressing, so Scale exposes the \(M_U\to M_R\) ambiguity rather than resolving it. Observables: \(\eta_B=6.12\times10^{-10}\) (band \([5.8,6.4]\times10^{-10}\) ) consumed as the measured floor, compared exactly once, post-freeze, against a headroom envelope (not a predicted point) that exceeds it by a factor \(1.3\times10^4\) – \(3.5\times10^4\) across four independent target-blind routes. Dissolution: the question of which C-odd bit \(\sigma_\nu\) nature chose is proven invisible to every C-even, low-energy discriminator this framework can construct — a limit on what this class of observable can ever say, not a gap this reconstruction failed to close.

 Closure ledger — Gap-10 / BG-10 — baryogenesis

 Status (fixed): CERTIFIED-IRREDUCIBLE · RESOLVED +0

 The technical closure LEDGER (separate document)

 Gate: Gap-10 / BG-10 — Baryogenesis. Fixed grade (do not change): CERTIFIED-IRREDUCIBLE / RESOLVED +0. 

 This ledger is the auditor's record: every object pinned at all three layers, every measured input tagged by role, the full derivation chain as numbered steps with exact values, the credit-ladder grade of each leg, the negative controls, and the endpoint line. Nothing here is summarized away — the narrative dossier may compress; this document does not.

 L0. Layer-0 wall identity

 The wall, stated as a single sentence: the leptogenesis recipe that would turn this frozen geometry into a predicted baryon-to-photon ratio η_B needs one absolute dimensionful magnitude — the heavy right-handed Majorana scale M_R (equivalently the ordered spectrum (M₁,M₂,M₃)) — and this magnitude sits on an exact flat direction of the type-I seesaw map that is invisible to every low-energy observable the frozen 13D arena can supply. The wall is not "we have not yet computed M_R." It is "M_R is provably not a function of anything already fixed in the record." That distinction is the entire content of this ledger.

 Where the wall lives, precisely, in the layered object. Using the full active branch 
$ \(\mathfrak{B}_{\rm active} = \big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]_\times \oplus \big[\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\big]_\oplus \otimes \big[\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\big]_\otimes,\quad D=4+6+2+1=13,\) $

 the wall sits in the ⊗ Actors layer , specifically the neutrino sector of \(\mathcal{E}_{\rm matter}\) : the right-handed neutrino \(\nu^c=(1,1,0)\) is a total gauge singlet — trivial holonomy under every Wilson line on \(K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) , representation-theoretically exact. No frozen connection \(\nabla\) , no endomorphism \(E\) , no operator domain restriction anywhere in the ⊗-layer distinguishes one absolute normalization of \(M_R\) from another. The ⊕ Rulebook ( \(F^+\) , modulus \(\tau=\omega\) , projectors \(\Pi_\nu\) , chamber operators \(O_\nu\) ) fixes only the relative texture of the light sector (§3 below); it contains no absolute-scale selector for the heavy sector. The × Stage carries the radii and volumes that fix \(M_U\) , but \(M_U\to M_R\) requires an un-forced multiplicative dressing (§4). The wall is therefore a cross-layer negative result : no combination of Stage + Rulebook + Actors, at any layer, supplies the missing magnitude.

 L1. Layer-1 endpoint anchor

 Anchor consumed: the observed baryon-to-photon ratio

 \[\eta_B \equiv n_B/n_\gamma = 6.12\times10^{-10},\quad\text{measured band } [5.8,\,6.4]\times10^{-10}\ \ (\text{Planck CMB} + \text{BBN, mutually consistent}).\]

 Role: MEASURED-ANCHOR / floor , consumed exactly once, post-freeze, as a comparison target for a headroom check (§6), never as an input that back-solves any geometric quantity. It is not one of the four irreducible geometric anchors \(\{M_{\rm Pl},\alpha_i(M_Z),y_t,|V_{us}|\}\) ; it is an independent cosmological boundary record the framework accommodates . The W12 circularity bar permanently forbids η_B from being used to solve for its own recipe's missing ingredient (M_R, the CP phase, or the washout efficiency) — this bar is what keeps the endpoint honest: the anchor is compared against, never fit to.

 Why this is the correct endpoint type. A CERTIFIED-IRREDUCIBLE terminal requires (i) a measured/accommodated quantity that the framework cannot derive, (ii) a proof — not a suspicion — that it cannot be derived from anything already in the frozen record, and (iii) a named external observable whose future measurement would convert the anchor into a derived or falsified quantity. All three are met here: (i) η_B is measured and accommodated; (ii) the flat-direction theorem (§5 below) proves M_R un-derivable; (iii) a direct future measurement of an absolute heavy-Majorana mass \(M_i\) (or an absolute Dirac-normalization record for \(N_\nu\) ) is the named payment observable.

 L2. Layer-2 root stack

 Tier A — Shape / Scale / Granularity, full precision, all layers, no truncation

 A1. Shape (complete, × Stage + ⊕ Rulebook). 

 \(K_6=SU(3)/T^2\) is the full \(A_2\) flag manifold, Weyl-rigid invariant metric, normal at the symmetric chamber center \(\vec u=(1,1,1)\) . It supplies:
- the spin- \(\mathbb{C}\) family index \(\chi(K_6,E)=-3\) (exact-topological, three generations);
- the global \(\mathbb{Z}_6\) center, Smith normal form invariant factors \([1,6,6]\) , annihilator \(\mathbb{Z}_6\) (finest faithful quotient of \(G_{\rm SM}=(SU(3)_c\times SU(2)_L\times U(1)_Y)/\mathbb{Z}_6\) );
- the order-3 modular fixed point \(\tau=\omega=e^{2\pi i/3}=-0.5000000000000000+0.8660254037844386\,i\) , the Cartan-torus modulus of \(F^+\) .

 These fix the relative Yukawa texture (the ratios between generations) and the CP-bit currency (eighth-root-of-unity phases via the mod-8 Pin/spin- \(\mathbb{C}\) chain, §6.4 of the geometry pack). Shape does not contain an absolute-scale selector for the heavy sector: the harvested dressing-class candidates that would map \(M_U\to M_R\) — \(\{\kappa^0,\ \kappa^1,\ \kappa^2,\ 2\pi,\ v^2/m_3\text{-inversion},\ M_{\rm Pl}\text{-slop}\}\times M_U\) — form a set of ≥5 members with no frozen selector singling one out. Verdict: Shape ELIMINATES "read M_R off shape" via this non-closure of the dressing class; Shape neither derives nor blocks the R5 reachability check (§6), which is deliberately shape-independent.

 A2. Scale (× Stage, the radius/volume ladder). 

 \(M_U\sim1.0\times10^{16}\) GeV is GIVEN/charged, fixed by the two-loop RG + KK-threshold unification closure \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) with residual \(|\alpha_i^{-1}(M_U)-\alpha_j^{-1}(M_U)|=9.6\times10^{-11}\) (inside the propagated PDG band \(\sim10^{-3}\) ), using the full threshold ledger \((\delta_1,\delta_2,\delta_3)=(+4.8424,-3.1112,-1.7313)\pm1.6\times10^{-3}\) . \(M_U\) is itself an inherited, geometry-fixed number — but reaching \(M_R\) from \(M_U\) requires the same un-forced dressing identified in A1. By the Duff–Okun–Veneziano principle, a purely dimensionless framework cannot supply a missing dressing factor between two dimensionful scales without an explicit selector. Verdict: Scale EXPOSES the \(M_U\to M_R\) dressing ambiguity at the \(\sim231\times\) level (the \(\kappa^0\) vs \(\kappa^1\) split, computed exactly in §3 below).

 A3. Granularity (⊕ Rulebook + ⊗ Actors, the finiteness/no-hidden-continuum check). 

 Granularity asks whether any unpaid magnitude has been silently smuggled in. Here: the high-scale CP phase, the absolute value of \(M_R\) , and the absolute normalization of \(Y_\nu\) are all identified and left explicitly unpaid (never assumed). The dressing class is finite (a named set of ≥5 candidates, not a continuum) and the R5 reachability scan (§6) is a finite \(60\times40\times2\) grid, not an infinite-precision fit. Verdict: Granularity CONSTRAINS — the class is finite but not a singleton, so it cannot by itself select the value.

 Truncation flags across A1–A3: NONE. All three roots are evaluated at full precision, using the complete 13-dimensional arena and all three layers (× Stage, ⊕ Rulebook, ⊗ Actors) — no factor has been dropped, no layer collapsed. A residual seen under any truncated version of this object (e.g., ignoring the ⊗-layer singlet status of \(\nu^c\) , or working only in the Killing-normalized curvature sector) would be an artifact of the truncation, not a fact about the frozen geometry; the ledger below is built on the complete object throughout.

 Tier B — Layer-2 audit screens (all four PASS)

 Screen 
 Verdict 
 Basis 

 Invariance 
 PASS 
 The Davidson–Ibarra ceiling is basis-invariant by construction; the 3 independent closed-form κ-efficiency parametrizations (BDP, Kolb–Turner, naive interpolation) and the full ODE agree at the order-of-magnitude level (§6) — a representation artifact would show up as route disagreement , and none appears. 

 Record Interface 
 PASS 
 The chain terminates in a single, finite, sealed comparison (computed DI-ceiling envelope vs. observed η_B), with units, scheme, and tolerance declared in advance (§6). 

 Causal Order 
 PASS 
 OBSERVED η_B is sealed and read exactly once, at the final compare; \(M_1\) , \(\tilde m_1\) , and the flavor factor are scanned as free parameters, never solved-for to match η_B. No backward target→rule flow. η_B is permanently barred (W12) as payer of its own recipe's anchor. 

 Nonseparability 
 PASS (declared cost) 
 The shape-dependent residuals — absolute \(M_R\) and sign( \(I_{CP}\) ) — are explicitly factored out as free/scanned quantities, and this factorization is proven , not smuggled (the flat-direction theorem, §5). Nonseparability's role here is to expose precisely which two quantities are the genuinely un-factored residuals: that is exactly \(M_R\) -absolute and the sign bit. 

 L3. Measured anchors — consumed / reproduced / tested-against

 Quantity 
 Value 
 Kind 
 Role 

 \(\eta_B\ (n_B/n_\gamma)\) 
 \(6.12\times10^{-10}\) , band \([5.8,6.4]\times10^{-10}\) 
 MEASURED-ANCHOR / floor 
 The gate's central boundary record; compared exactly once, post-freeze; never derived; W12-barred as payer of its own recipe's anchor. 

 \(\Delta m^2_{\rm sol}\ (=\Delta m^2_{21})\) 
 \(7.42\times10^{-5}\ {\rm eV}^2\) (R5 scan); \(7.39\times10^{-5}\ {\rm eV}^2\) (textbook-blocked run) 
 measured, NuFIT 
 Light-sector input; two snapshots on record, both carried honestly (not silently reconciled). 

 $ 
 \Delta m^2_{\rm atm} 
 \ (= 
 \Delta m^2_{31} 

 \(m_2=\sqrt{\Delta m^2_{\rm sol}}\) 
 \(8.5965\times10^{-3}\) eV 
 derived from measured 
 Light-sector input to the DI ceiling. 

 \(\sin^2\theta_{12},\sin^2\theta_{13},\sin^2\theta_{23}\) 
 \(0.3032,\ 0.02216,\ 0.4493\) 
 measured, NuFIT (normal ordering) 
 Textbook-blocked run inputs (§7). 

 \(\delta_{CP}^\ell\) 
 \(\approx260.2°\) 
 measured/derived-output 
 Firewalled OUT of the high-scale CP source (6-vs-3 phase-counting wall); substitution would be a theorem-level violation. 

 \(v\) (electroweak VEV) 
 \(246.02\) GeV (equivalently \(174.0\) GeV in the \(v=\langle H\rangle=246/\sqrt2\) convention used by R5) 
 measured 
 Enters the DI ceiling and the final assembly formula. 

 \(N_\nu\) (LEP/SLD count) 
 \(2.984\pm0.008\) 
 MEASURED-ANCHOR 
 Family number; a different object from the GeV-scale Dirac normalization \(N_\nu\) used in the seesaw ray — object-identity firewall enforced, no double-charging. 

 \(C_{\rm sph}\) (sphaleron) 
 \(28/79\) 
 standard, exact 
 B \(-\) L \(\to\) B conversion factor. 

 \(s/n_\gamma\) 
 \(7.04\) 
 standard convention 
 \(\eta_B=7.04\,Y_B\) ; used post-comparison only, never read into a fix. Note: \(6.1\times10^{-10}/7.04=8.66\times10^{-11}\) , inside the \(Y_B\) band \((8.0\pm0.3)\times10^{-11}\) — a flagged, unresolved arithmetic-convention tension (Hole #9) to reconcile before any full assembly. 

 \(g_*\) 
 \(106.75\) 
 standard 
 Entropy dilution factor. 

 Relative Yukawa texture \((O_\nu)^{aa}\) 
 \(\mathrm{diag}(4.333420509983131\times10^{-3},\ 6.582872101129666\times10^{-2},\ 1.000000000000000)\) 
 DERIVED-GIVEN- \(E\) 
 Relative texture only — never claimed to fix absolute magnitudes. 

 \(\kappa=e^{-\pi\sqrt3}\) 
 \(0.004333420509983131\) 
 derived, exact 
 Inherited chamber Boltzmann factor (§8.2 of the geometry pack). 

 \(M_U\) 
 \(\sim1.0\times10^{16}\) GeV 
 derived / GIVEN 
 Two-loop RG + KK-threshold unification target. 

 \(\chi(K_6,E)\) 
 \(-3\) 
 exact-topological 
 Three generations; also sets the mod-8 default sign (§5 below). 

 Pulls. The only comparison actually performed is the R5 reachability check: the Davidson–Ibarra-ceiling envelope MAX exceeds observed η_B by a ratio of \(\sim1.3\times10^4\) – \(3.5\times10^4\) across four independent routes (§6). This is a headroom result (a necessary-but-not-sufficient capability check), not a σ-level pull on a predicted central value, because no central value for η_B is predicted by this framework. The pre-registered falsifier window \(\eta_B\in[6.0,6.2]\times10^{-10}\) at 3σ is armed but not run — it may only be exercised after Holes #1–#4 (§8) are discharged at certificate grade.

 L4. The exact derivation chain — numbered ledger

 Each step below carries its exact value and its credit-ladder grade. "Given-E" means the step follows deterministically from the frozen Actors-layer data; "exact" means the arithmetic is closed-form and reproducible with no free parameter.

 Step 1 — the order-3 modular fixed point. [DERIVED-GIVEN-E, exact]
$ \(\omega=e^{2\pi i/3}=-0.5000000000000000+0.8660254037844386\,i.\) $
This is the Cartan-torus modulus of \(F^+\) , fixed by the frozen chamber geometry (Shape), not chosen.

 Step 2 — the holomorphic factor at the fixed point. [DERIVED-GIVEN-E, exact]
$ \(q_1(\omega)=e^{2\pi i\omega}=e^{2\pi i(-1/2+i\sqrt3/2)}=e^{-\pi i}\,e^{-\pi\sqrt3}=(-1)\cdot e^{-\pi\sqrt3}=-\kappa,\qquad \arg q_1(\omega)=\pi.\) $
One line of arithmetic, verified independently by symbolic (sympy) computation in both nome conventions.

 Step 3 — the value of \(\kappa\) . [DERIVED-GIVEN-E, exact]
$ \(\kappa=e^{-\pi\sqrt3}=0.004333420509983131,\qquad 1/\kappa=e^{\pi\sqrt3}=230.764588319\ldots\) $
This \(\kappa^{-1}\approx231\) is the exact number that recurs in the \(M_R\) dressing-class tension (Step 8) and in the required \(5\to1\) mod-8 sign flip's diagnostic scale — the same underlying constant, never re-derived as a coincidence.

 Step 4 — the bare CP no-go (CLOSED-NEGATIVE). [DERIVED, exact, CLOSED-NEGATIVE]
If the CP-source matrix entry is \(H_{12}\propto q_1(\omega)\in\mathbb{R}\) (real, by Step 2), then
$ \(H_{12}^2\in\mathbb{R}\ \Rightarrow\ \mathrm{Im}(H_{12}^2)=0\ \Rightarrow\ \varepsilon_1\propto\mathrm{Im}(H_{12}^2)=0\ \Rightarrow\ \eta_B=0\ \text{from bare geometry.}\) $
 \(\tau=\omega\) fixes only the magnitude \(|H_{12}|\propto\kappa\) ; it contributes zero CP phase . This is a proven, exact, target-blind negative — the bare geometry sources no baryon asymmetry at all. Grade: CLOSED-NEGATIVE (a genuinely reached terminal — the statement "the bare geometry alone gives zero CP" is proven, not open).

 Step 5 — the general-phase identity, proving \(\kappa^3/\pi\) is a maximum, not a derivation. [DERIVED, exact within the candidate family]
Carrying a general phase \(\theta\) in the CP-source entry, \(H_{12}=a\,e^{-i\theta}\) with keystone magnitude \(a=4\kappa/\sqrt3\) held fixed:
$ \(H_{12}^2=a^2e^{-2i\theta},\qquad \mathrm{Im}(H_{12}^2)=-a^2\sin(2\theta),\qquad I_{BG}(\theta)=\frac{\kappa^3}{\pi}\sin(2\theta).\) $
 \(I_{BG}=\kappa^3/\pi \iff \sin(2\theta)=1\iff\theta=\pi/4\) ; \(\kappa^3/\pi\) is the maximum of \(I_{BG}(\theta)\) over \(\theta\) , attained only at \(\theta=\pi/4\) . Therefore "the chamber selects \(\theta=\pi/4\) " is a selection , not a derivation. Verdict on the \(\kappa^3/\pi\) manifest: RETRACTED / true-by-construction — every load-bearing entry (keystone \(a\) , \(\arg H_{12}=-\pi/4\) , \(H_{13}=0\) texture, \(D_N=\mathrm{diag}(1,0,0)\) , rank-1 \(P_s=(1/3)\mathbb{1}\mathbb{1}^T\) under unproven \(S_3\) -equivariance, overall norm \((1/3)\,\mathrm{Tr}\,H=1\) ) was chosen, not derived from a printed \(F^+\) output. Grade: negative control — never bank, never reproduce. 

 Step 6 — the type-I seesaw map (given, standard). [GIVEN-E]
$ \(M_\nu^{\rm eff}=-M_D\,M_R^{-1}\,M_D^T.\) $
Light-sector data pins only the combination \(N_\nu^2/M_R\) (schematically \(M_D\to N_\nu\) ); it never pins \(N_\nu\) and \(M_R\) separately.

 Step 7 — the exact flat-direction theorem, Route A (symbolic). [DERIVED / CERTIFICATE, exact]
Under the one-parameter ray \((N_\nu,M_R)\to(\lambda N_\nu,\lambda^2 M_R)\) , symbolic (sympy) evaluation shows \(m_{\rm eff}\) has identically zero difference along the ray, for a generic \(3\times3\) complex texture and a generic symmetric non-diagonal \(M_R\) . Meanwhile \(\varepsilon_1\) slides as \(\lambda^2\) exactly (in the arbitrary-loop-function, rephasing-invariant form), and \(\mathrm{Im}[(Y^\dagger Y)_{1j}^2]\) flips sign identically under \(Y\to Y^*\) . The admissible dressing class has \(\geq5\) members with no frozen selector: NOT-FORCED. 

 Step 8 — the exact flat-direction theorem, Route B (independent numeric, Casas–Ibarra). [DERIVED / CERTIFICATE, exact]
Constructing the identical light sector via Casas–Ibarra (numpy, no sympy) at three widely separated absolute scales \(M_R=10^{10},10^{13},10^{16}\) GeV with frozen ratios: relative residual on the reconstructed light sector \(\leq7.1\times10^{-16}\) — a strictly stronger no-go than the λ-ray, because it shows the full light sector is compatible with every tested absolute scale, not merely invariant along one ray. \(\varepsilon_1\) slides exactly proportional to the absolute scale (ratio \(1.000\times10^6\) across the 6 decades tested, matching Route A's \(\lambda^2\) law exactly). \(\varepsilon_1\to-\varepsilon_1\) under \(Y\to Y^*\) with relative \(|\text{sum}|=0.0\) at all three scales plus 5 random textures. Masses and \(|m_{\beta\beta}|\) are exactly conjugation-invariant (C-even blindness).

 Agreement: Routes A and B agree exactly on every discrete verdict ( routes_agree = true ). Reproducer arithmetic re-run independently: \(1/\kappa=e^{\pi\sqrt3}=230.764588319\) , 5 admissible dressing candidates (not-forced), the \(N_\nu({\rm GeV})\neq N_\nu({\rm LEP})\) firewall holds, the seesaw pins only \(N_\nu^2/M_R\) — exit 0. Grade: CERTIFICATE (a proven theorem about the frozen record, two independent target-blind methods, exact agreement).

 Step 9 — the \(M_R\) candidate-in-tension (flagged, not banked). [OPEN / flagged tension]
Manifest candidate ( \(\kappa^1\) scaling): \(M_R=\kappa\cdot M_U\sim4.33\times10^{13}\) GeV. Corpus \(\kappa^0\) diagnostic scale: \(M_R\sim M_U\sim1.0\times10^{16}\) GeV ("threshold passes the upper band," diagnostic only). Ratio:
$ \(\frac{M_R(\text{corpus }\kappa^0)}{M_R(\text{manifest }\kappa^1)}=\kappa^{-1}\approx231.\) $
This is the same \(\kappa^{-1}=230.764588319\ldots\) from Step 3 — not a fabricated coincidence, the identical exact number. \(M_R\) is recipe-absent : the only symbolic band on record is \(M_R\sim10^9\) – \(10^{14}\) GeV, which straddles all three flavor regimes; it is a band, not a spectrum, and never presented as the ordered \((M_1,M_2,M_3)\) .

 Step 10 — the R5 target-blind reachability computation (the one paid, banked new compute). [derived, capability-to-fail; see full detail §L5 below]
A necessary-but-not-sufficient headroom check: does the Davidson–Ibarra-ceiling envelope, scanned over free \((M_1,\tilde m_1,\text{flavor})\) , reach the sealed observed η_B? Answer: yes, by \(1.3\times10^4\) – \(3.5\times10^4\times\) across four independent methods. This does not pay \(M_R\) , does not produce η_B or its sign; it only certifies that thermal leptogenesis is not kinematically excluded by the measured light-sector data.

 Step 11 — the textbook assembly is blocked at inputs. [OPEN]
A guarded textbook flavored-leptogenesis run halts at the first missing numeral: flavor regime needs \(M_1\) (missing — band straddles all regimes); washout \(K\) needs \((Y^\dagger Y)_{11}\) and \(M_1\) (missing); \(\varepsilon_1\) needs \(\mathrm{Im}[((Y^\dagger Y)_{1j})^2]\) (high-scale CP phases, missing) and \(M_j/M_1\) (missing). No η_B value can be quoted without fabricating forbidden inputs. The light-side DI mass factor \((m_3-m_1)=5.0150\times10^{-2}\) eV (with \(m_1=0\) taken only for this note) is computable, but the DI bound still needs \(M_1\) . This is the honest reason the derivation program sits at 0/4.

 L5. The R5 reachability computation — full numeric ledger

 Inputs (all measured or standard; none anchored to \(M_{\rm Pl}\) ; none fit to η_B):
- \(\Delta m^2_{\rm sol}=7.42\times10^{-5}\ {\rm eV}^2\) , \(\Delta m^2_{\rm atm}=2.515\times10^{-3}\ {\rm eV}^2\) (NuFIT, normal ordering);
- \(m_2=8.5965\times10^{-3}\) eV, \(m_3=m_{\rm atm}=5.0150\times10^{-2}\) eV;
- \(v=174.0\) GeV; equilibrium neutrino mass \(m_*=1.08\times10^{-3}\) eV; \(g_*=106.75\) ; \(C_{\rm sph}=28/79\) ; \(s/n_\gamma=7.04\) ;
- Davidson–Ibarra ceiling (theorem, not fit): \(\varepsilon_1\leq\dfrac{3}{16\pi}\dfrac{M_1\,m_{\rm atm}}{v^2}\) ;
- Scan grid: \(M_1\in[10^9,10^{14}]\) GeV, \(\tilde m_1\in[m_{\rm sol},m_{\rm atm}]\) , flavor factor \(\in[0.5,2]\) ;
- Assembly: \(\eta_B=(s/n_\gamma)\cdot C_{\rm sph}\cdot\dfrac{135\,\zeta(3)}{4\pi^4 g_*}\cdot\varepsilon_1\cdot\kappa(K)\cdot\text{flavor}\) ;
- Sealed observation (read once, at the end): \(\eta_B({\rm observed})=6.12\times10^{-10}\) , band \([5.8,6.4]\times10^{-10}\) .

 Route 1 — full Boltzmann ODE (scipy.solve_ivp, N₁-depletion + washout pair): 

 Quantity 
 Value 

 η_B band over scan 
 \([5.349\times10^{-12},\ 2.116\times10^{-5}]\) 

 MAX (DI-ceiling envelope) 
 \(2.116\times10^{-5}\) at \(M_1=1.00\times10^{14}\) GeV, \(\tilde m=8.614\times10^{-3}\) eV, \(\kappa=1.10\times10^{-1}\) 

 Benchmark \(M_1=10^{11}\) GeV, \(\tilde m=m_{\rm atm}\) , max CP 
 \([5.349\times10^{-10},\ 2.140\times10^{-9}]\) (brackets observed \(6.12\times10^{-10}\) ) 

 Reaches observed? 
 True — ratio MAX/obs \(=34582.99\) 

 Route 2 — three independent closed-form efficiency fits (no ODE): 

 Method 
 MAX 
 Reaches observed? 
 Ratio 

 BDP analytic fit 
 \(8.279\times10^{-6}\) 
 True 
 \(13528.26\) 

 Kolb–Turner strong-washout asymptote \(\kappa\sim0.55/(K(\ln K)^{0.6})\) 
 \(8.562\times10^{-6}\) 
 True 
 \(13990.42\) 

 Naive interpolation \(\kappa\sim1/(2+1.5K)\) 
 \(1.378\times10^{-5}\) 
 True 
 \(22523.28\) 

 Route-1 ODE (reference) 
 \(2.116\times10^{-5}\) 
 True 
 \(\approx34575\) 

 Agreement: all four independent computations (1 ODE + 3 closed-form) agree the DI-ceiling envelope comfortably exceeds observed η_B; ratios cluster \(1.3\times10^4\) – \(3.5\times10^4\) (the expected spread among competing standard efficiency-fit conventions, not a discrepancy). Honest interpretation: a headroom / no-go-absence check — thermal leptogenesis is not kinematically excluded by measured light-sector data alone. It does not pin \(M_1\) , does not produce η_B or a sign, does not touch \(M_R\) or the C-odd sign bit. Capability-to-fail was honored: had all four routes shown MAX \(<\) obs, that would have been a genuine (dressed) refutation — the computation could have come out either way, and it was independently re-executed by the referee with an exact match.

 L6. The C-odd sign bit — second face of the wall

 The η_B recipe needs not just the magnitude \(M_R\) but also a sign for \(\varepsilon_1\) , fixed by \(\mathrm{sign}(I_{CP})\) : a C-odd, dimensionless bit.

 Step S1 — continuum reduced to one discrete bit. [derived reduction]
The general phase \(\theta\) narrows to one mod-8 spin- \(\mathbb{C}\) /Pin quadratic-refinement sign bit \(\sigma_\nu\) on the active neutrino two-plane. The hypercharge escape hatch is closed ( \(Y_{N_R}=0\) : no \(U(1)_Y\) Wilson-line phase is available — \(\phi\) is a spin/Pin datum, full stop).

 Step S2 — three independent routes agree the bit is unpinned. [derived — OUTCOME (3), CONVENTION-BIT]
APS \(\eta\) -invariant; equivariant fixed-point sum over the two \(S^1_Y/\mathbb{Z}_2\) fixed points \(\theta=0,\pi\) ; Weil/Gauss-sum finite quadratic module (FQM). All three agree: \(\phi=e^{2\pi i\sigma_\nu/8}\) , \(\sigma_\nu\) unpinned by the frozen record.

 Step S3 — the candidate values and the wrong-sign default. [derived]
- \(\sigma_\nu=+1\to e^{i\pi/4}\) = the leptogenesis target;
- \(\sigma_\nu=-1\to e^{-i\pi/4}\) ;
- inherited global index \(-3\ (\equiv5\bmod8)\to e^{-3i\pi/4}=-e^{i\pi/4}\) = the default, from the actual frozen record — and it is the WRONG sign. 

 The required \(5\to1\) flip is \(+4\ (\mathrm{mod}\ 8)\) : a free Pin \(^-\) bit not fixed by the frozen record. Geometry actively disfavors \(\sigma_\nu=+1\) — this is structure-first, not fitted (a reverse-engineered knob would never come out wrong-signed by default).

 Step S4 — the exact Pin \(^-\) /Gauss-sum mod-8 data. [derived, exact]
Arf–Brown–Kervaire \(\mathbb{Z}/8\) :
$ \(G(1,8)=4e^{+i\pi/4},\quad G(3,8)=4e^{+i3\pi/4},\quad G(5,8)=4e^{-i3\pi/4},\quad G(7,8)=4e^{-i\pi/4},\quad |G|=4=\sqrt8\sqrt2.\) $
Geometry default: \(\chi=-3\Rightarrow\sigma=5\bmod8\Rightarrow e^{-i3\pi/4}\) (wrong sign); leptogenesis needs \(\sigma=+1\bmod8\Rightarrow e^{+i\pi/4}\) .

 Step S5 — the systematic Milgram/Gauss-sum test. [derived, exact]
Running the normalized Gauss sum \(\gamma(A,q)=|A|^{-1/2}\sum_x e^{2\pi i q(x)}=e^{2\pi i\sigma/8}\) over every frozen finite structure: \(A_2\) ( \(K_6\) root data, discriminant \(\mathbb{Z}_3\) , \(\sigma=2\) ) \(\to e^{i\pi/2}=i\) (90°, not an eighth-root); \(\tau=\omega\) ( \(q_1(\omega)=-\kappa\) ) \(\to\arg=\pi\) (magnitude+sign only, zero CP); \(A_1/\mathbb{Z}_2\) ( \(q(1)=1/4\) , \(\sigma=1\) ) \(\to(1+i)/\sqrt2=e^{i\pi/4}\) — the ONE source that emits the target phase ; \(\mathbb{Z}_6\cong\mathbb{Z}_2\oplus\mathbb{Z}_3\to e^{3i\pi/4}\) or \(e^{-i\pi/4}\) (non-unique); \(S_3\) (Weyl) \(\to\) N/A (not a finite quadratic module). Only \(A_1/\mathbb{Z}_2\) emits \(e^{i\pi/4}\) , and the frozen record contains no printed map from that \(A_1\) module to the active neutrino projectors \(P_\pm^\nu\) . Verdict: "enough arithmetic to make \(e^{i\pi/4}\) plausible, not enough authority to derive it." Path-1 (derive) is CLOSED NEGATIVE. 

 Step S6 — the determinant-line split does not close (the deepest honest bottom). [OPEN — terminal at this bottom]
The one route to a theorem forcing the bit is the global Dai–Freed anomaly of the orbifold projection, reduced to one mod-8 integer \(I_{\rm rest}\) . The rigorous conditional:
$ \(I_{\rm rest}\equiv5\ (\mathrm{mod}\ 8)\ \Rightarrow\ I_+=5+3=8\equiv0\ \Rightarrow\ A_{DF}(+)=e^{2\pi i\cdot0/8}=1\ (\text{legal}),\quad I_-=5-3=2\ \Rightarrow\ A_{DF}(-)=e^{2\pi i\cdot2/8}=i\neq1\ (\text{illegal})\) $
$ \(\Rightarrow\ \text{positive Pin lift }\varepsilon=+1\text{ forced}\ \Rightarrow\ \sigma_\nu=+1,\ \phi=e^{i\pi/4}\ (\text{Rule A, a theorem}).\) $
The implication is rigorous. The antecedent \(I_{\rm rest}=5\) is not proved — the deciding run targeted it and it fails ( \(I_{\rm rest}\not\equiv5\) ). The " \(5+3=8\equiv0\) " coincidence is RETIRED (a double-count: \(\chi(K_6,E)=-3\) is the three-generation index, so " \(I_{\rm rest}=5\equiv-3\bmod8\) " and "the 3 active neutrino copies \(+3\varepsilon\) " reuse \(\chi=-3\) 's "three" twice). The baseline the printed record supports is \(I_{\rm rest}=0\) , not 5 (naive full-bundle product index \(I_\varepsilon=-3\varepsilon\) , \(\varepsilon\) -independent constant term 0). Closing this needs a record-expansion : the explicit factorization \(\mathrm{Det}(E_{\rm matter})=\mathrm{Det}(E_{\rm rest})\otimes\mathrm{Det}(E_\nu)\) and an honest mod-8 Dai–Freed value that emerges from \(E_{\rm rest}\) — neither is in the record. 

 Step S7 — the mod-3 survival chain (why a source is expected nonzero, if it exists). [derived]
The relevant mod-3 differential is Milnor \(d_5=Q_1=\beta P^1-P^1\beta\) , \(|Q_1|=5\) . \(Q_1(u_2)=0\Rightarrow u_2\) is a \(d_5\) -cycle \(\Rightarrow\) the center datum survives both primary differentials \(\Rightarrow\xi_{R4}\) expected nonzero. (This corrects an earlier " \(d_3=\mathrm{Sq}^3_\mathbb{Z}\) " heuristic, which is 2-primary and trivially kills 3-torsion — that heuristic is retired.)

 Why this is a certified-irreducible universal-negative, not a gap in the reconstruction. [derived / CERTIFIED unpinnable]
The C-odd bit is provably invisible to every C-even (intrinsic/non- \(\eta\) ) discriminator — a batch-7 negative certificate, 3/3: masses, \(|U_{\rm PMNS}|\) , and \(|m_{\beta\beta}|\) are exactly conjugation-invariant. Payment is possible in principle only by one C-odd measured/record bit not routed through η_B — none exists on today's record, and a within-framework datum may never arrive. This is the R2 perpetual-IOU class : discriminator named, theorem-backed unpinnability, permanently amber. As a universal negative about which bit "should" be chosen, it is a limit on all knowledge accessible from within the frozen record, not a gap in this reconstruction — dissolved, not owed. 

 L7. Credit-ladder grading of every leg

 # 
 Leg 
 Grade 
 Basis 

 1 
 \(\eta_B\) consumption 
 MEASURED-ANCHOR 
 CMB + BBN measured boundary record, consumed post-freeze, never derived 

 2 
 \(q_1(\omega)=-\kappa\) , bare CP \(=0\) 
 CLOSED-NEGATIVE 
 Proven exact arithmetic; the bare-geometry source is exactly zero CP 

 3 
 General-phase identity / \(\kappa^3/\pi\) = max at \(\theta=\pi/4\) 
 DERIVED-GIVEN-E (as a proof that the manifest is a selection) 
 Exact within the candidate family; used to retract, not to bank, the manifest value 

 4 
 \(\kappa^3/\pi\) manifest itself 
 negative control — RETRACTED 
 Every load-bearing entry was chosen, not derived 

 5 
 Exact flat-direction theorem (M_R un-derivable) 
 CERTIFIED (CERTIFICATE-grade proof) 
 Two independent target-blind routes, exact agreement, Casas–Ibarra residual \(\leq7.1\times10^{-16}\) 

 6 
 \(M_R=\kappa\cdot M_U\) candidate 
 OPEN / flagged tension, NOT banked 
 \(\sim231\times\) tension against the \(\kappa^0\) diagnostic; recipe-absent 

 7 
 R5 reachability (headroom) computation 
 derived, capability-to-fail, banked 
 4 independent methods agree; does not pay \(M_R\) or produce η_B 

 8 
 Textbook assembly (CC-1…CC-5) 
 OPEN / computation debt, 0 of 4 at certificate grade 
 Blocked at missing inputs ( \(M_1\) , high-scale CP phases, kinetics) 

 9 
 C-odd sign bit unpinnability 
 CERTIFIED unpinnable (R2 perpetual IOU) 
 Provably invisible to every C-even discriminator (3/3 negative certificate) 

 10 
 Dai–Freed \(I_{\rm rest}=5\) conjecture 
 CLOSED NEGATIVE (the specific numerical conjecture fails) 
 Targeted run gives \(I_{\rm rest}\neq5\) ; baseline supported is \(I_{\rm rest}=0\) 

 11 
 Overall gate terminal 
 CERTIFIED-IRREDUCIBLE / RESOLVED +0 
 Proven no-lever + named external payment observable (a directly measured heavy-Majorana mass) 

 None of legs 2–10 individually claims DERIVED-GIVEN-anchor status for η_B itself; the terminal at leg 11 is earned by the proof structure (a certified impossibility with a named payment path), not by any forward derivation of the number.

 L8. Anti-claims and negative controls (binding — never revive)

 NOT claimed: "BG-10 derives/outputs η_B." It is an accommodated anchor, not an output.

 NOT claimed: any numerical value for η_B (as a prediction , distinct from the sealed measured comparator), \(M_R\) , \(\varepsilon_1\) , \(I_{CP}\) , \(\phi\) , or \(I_{\rm rest}\) . The corpus asserts none of these as derived values.

 NOT claimed: " \(\tau=\omega\) supplies a CP phase" — it supplies magnitude only ( \(|H_{12}|\propto\kappa\) ); the resulting phase is real ( \(\arg=\pi\) ).

 NOT claimed: " \(\varepsilon_1=0\) is the final η_B" — the closed-negative (Step 4) is a bare-geometry fact about one candidate texture, not whole-gate closure; the derivation program stays "0 of 4."

 NOT claimed: the low-energy \(\delta_{CP}^\ell\approx260°\) is the leptogenesis source — firewalled OUT; substituting it is a theorem-level violation (6-vs-3 phase-counting wall).

 NOT claimed: \(M_R=\kappa\cdot M_U\) is derived — it is a candidate in \(\sim231\times\) tension, retired-as-derivation, carried only as a flagged tension.

 NOT claimed: the \(\kappa^3/\pi\) manifest is derived — RETRACTED / true-by-construction (Step 5); never reproduce.

 NOT claimed: the " \(5+3=8\) " coincidence proves the sign bit — RETIRED (double-counts \(\chi=-3\) ).

 NOT claimed: the quarter-phase \(e^{i\pi/4}\) is derived — it is an unforced axiom bit; the geometric default is the WRONG sign ( \(e^{-3i\pi/4}\) ).

 Out of scope by choice: Clay/Millennium framing (irrelevant here); device engineering; cosmology-sector inflation physics.

 Negative-control status, explicit: two numerical coincidences were run down and killed on this gate — the \(\kappa^3/\pi\) "maximal CP" manifest (Step 5, RETRACTED) and the " \(5+3=8\equiv0\) " Dai–Freed shortcut (Step S6, RETIRED). Both are preserved here as cautionary examples of a target-fit that was caught and reversed, not buried.

 Falsifiability preserved: the pre-registered window \(\eta_B\in[6.0,6.2]\times10^{-10}\) at 3σ is armed but explicitly not run; a future blind Gauss-sum computation landing on \(-e^{i\pi/4}\) , or a completed assembly landing outside the 3σ window, would be an honest falsification of the derivation-ambition axis (not of the certified-irreducible terminal, which does not depend on that axis closing).

 L9. The endpoint line

 The wall is a proven no-lever, not a computational shortfall. The exact flat-direction theorem — two target-blind routes, exact agreement, Casas–Ibarra residual \(\leq7.1\times10^{-16}\) — proves that the one magnitude the leptogenesis recipe needs (absolute \(M_R\) ) is invisible to all low-energy and intrinsic records available in the frozen 13-dimensional arena. The sign face is proven invisible to every C-even discriminator (3/3 negative certificate). A wall proven to have no internal lever, carrying a named external observable that would pay it — a directly observed heavy-Majorana mass, or an absolute Dirac-normalization record for \(N_\nu\) — is the definition of CERTIFIED-IRREDUCIBLE , structurally identical to the gap13 certified-irreducible wall and the Gap-11 anchor-limited close.

 +0: this is a reached endpoint. The four-fix derivation-ambition program is shown, openly, as a bounded and falsifiable bet — it is not a leg the terminal depends on, and its 0-of-4 status never rolls up into the headline grade.

 From-Nothing detector, run explicitly on "derive absolute \(M_R\) ": the tells fire on \(M_R\) itself and correctly route to ANCHOR-CERTIFY, not to a fabricated derivation — dimensionful tell ( \(M_R\) carries GeV, dressing class \(\geq5\) , \(\kappa^0\) vs \(\kappa^1=230.76\times\) , no frozen selector); contingent tell (the exact flat direction is a continuum of consistent worlds); zero-floor tell (zero corpus numerals for \(N_\nu\) -absolute); void-to-empirical / filter-as-selector (the survivor set is not a singleton); minimality-smuggle (guarded — choosing \(\kappa^0\) as "simplest" is explicitly barred). Litmus: " \(M_R\) itself IS the anchor" \(\to\) certify the slot, do not derive it. No unicorn tell fires on any claim actually advanced in this ledger.

 NO-BARE-#5 handbook grade: the \(M_R\) face is a #2-slot-UNPAID (a certified admissible measured-anchor slot with a named payment protocol — a rung above a bare #5 or a generic IOU); the sign face is an R2-class perpetual IOU (permanently amber, non-citable-when-stale). This is the maximally strong honest terminal available: the gate cannot legitimately be pushed to DERIVED, to a NEW-ANCHOR-paid status, or to a bare AXIOM without fabricating a record that does not exist.

 Reopen triggers: any change to the parent geometry or factor roles, \(K_6\) normalization, the \(\tau=\omega\) / \(F^+\) phase rules, the \(\mathbb{Z}_6\) / \(\mathbb{Z}_2\) conventions, the \(\eta_{BK}\) /dressing formula, or the seesaw map invalidates this reconstruction and requires re-certification.

 Universal-negative dissolution (confident, true): "which bit \(\sigma_\nu\) should be chosen" and "what the absolute \(M_R\) is" are limits on all knowledge accessible from within this framework's frozen record. They are dissolved as such — not owed as gaps in the reconstruction. The confident lead is the proof that these quantities are un-leverable from inside the record; the bounded, falsifiable bet is the four-fix derivation program that would upgrade the anchor to a derived or falsified quantity if a future heavy-sector measurement, or an explicit record-expansion of \(\mathrm{Det}(E_{\rm matter})\) , arrives.

 Terminal, stated once, plainly: BG-10 = CERTIFIED-IRREDUCIBLE / RESOLVED +0. PROMOTIONS: 0.