Gap-10/BG-10 — Baryogenesis eta_B (thermal leptogenesis four-fix ledger): full dossier — rendered package. Rendered from DOSSIER_GAP10_BG_10_FULL.md; frozen technical content unchanged by rendering.

Gap-10/BG-10 — Baryogenesis eta_B (thermal leptogenesis four-fix ledger): full dossier

Current ledger status (source of truth, ratified 2026-07-08): Gap-10 / BG-10 — baryogenesis is CERTIFIED-IRREDUCIBLE · RESOLVED +0. The matter–antimatter asymmetry traces to the internal shape of the K₆ geometry; the closing high-scale CP source is a certified-irreducible external dependency, shown openly and resting on measured anchors, never a from-nothing closure. The dossier below is the frozen mid-audit record, preserved verbatim as published history; its “OPEN / serious candidate” language reflects the earlier standing and is superseded by the ledger line above.

Honest status (binding, matches the live popup chip). OPEN — pre-registered falsifier armed; structural firewall PROVED, one CP-source axiom unfilled. Ledger 0 of 4 at certificate grade. Direction: held. Ceiling: serious candidate, NOT validated. STATUS-UPGRADES:0. Frozen branch dcc66f1b2685 / a5b1e6f9d951 is READ-ONLY. η_B is a MEASURED invariant (CMB + BBN); this gate must produce a number and submit it exactly once to comparison — it never retrofits one. No part of η_B is derived; no number is asserted.

What this dossier establishes and does not. It establishes (a) a PROVED structural firewall that explains why the frozen geometry, as written, sources exactly zero matter–antimatter asymmetry; (b) the precise location of the residue — a single high-scale CP source object that is recipe-absent; (c) the dismantling of a circulated "clean result" (the κ³/π manifest) as true-by-construction; and (d) a pre-registered, bounded falsifier. It does not establish a value of η_B, a CP asymmetry ε, a right-handed-neutrino mass M_R, an absolute neutrino Yukawa, or a closure of the gate. Nothing here is promoted.


1. Executive summary + honest status

1.1 The headline

We did not fake a baryon number. We proved why we cannot fake one yet: the frozen geometry, as written, sources exactly zero matter–antimatter asymmetry — and we wrote down the single pre-registered number, η_B ∈ [6.0, 6.2]×10⁻¹⁰, that will judge a completed computation, before computing it.

Baryogenesis — producing the universe's matter excess from a fixed theory — is a famously hard, openly unsolved problem. Nobody, us included, has a parameter-free leptogenesis η_B. The hard part is the high-scale CP-violating phase, which is invisible to every measurement we have. This gate is honestly held at 0 of 4 required fixes.

What we did do is honest structural work. We PROVED a firewall (graded axiom→theorem in the consolidated dossier, both legs PASS): the type-I seesaw collapses the right-handed mass M_R and the neutrino Yukawa Y_ν into one degenerate combination, so the light-neutrino data we fit (Δm², PMNS) pins only the effective mass M_ν^eff = −M_D M_R⁻¹ M_Dᵀ, never M_R alone; and a 6-versus-3 phase count shows leptogenesis needs three extra high-scale phases the data does not expose, so the measured low-energy δ_CP^ℓ ≈ 260° is rigorously walled OUT of the high-scale source. We then proved the one frozen complex object q₁(ω) = −κ is real (phase π, not π/4), so the CP source ε₁ ∝ Im(H₁₂²) = 0. We honestly WITHDREW a same-day closure claim (2026-06-04) and RETRACTED the κ³/π quarter-phase as true-by-construction. And we armed a mechanized exit-checker that physically blocks any closure until all four fixes are certificate-grade.

1.2 The honest grade (binding)

Field Value
Gate id BG-10 — Baryogenesis (thermal leptogenesis → η_B)
Status (binding) OPEN / Diagnostic — 0 of 4 at certificate grade. No η_B asserted.
Measured anchor η_B ≈ 6.1×10⁻¹⁰ (CMB + BBN) — measured-but-irreducible; the floor anchor (≥1 genuine measured invariant). NOT a derived output of this gate.
Pre-registered falsifier η_B ∈ [6.0, 6.2]×10⁻¹⁰, compared exactly once, falsification accepted if outside at 3σ.
Certificate grade 0 of 4. Four named, coupled computations are required; none has run at certificate grade. The 0/4 is mechanized to refuse early/partial closure.
Ceiling serious candidate, NOT validated. STATUS-UPGRADES:0.

(Source: …/TOE/PER_GATE_DOSSIERS/BG10_COMPLETION_HANDOFF/01_DOSSIER.md §0; …/articles/BG10_COMPLETION_RESULT.md §0.)

1.3 The testable bet, stated flat-out

A completed four-fix η_B landing outside [6.0, 6.2]×10⁻¹⁰ at 3σ falsifies this gate. We pre-registered the kill-condition before computing, so we cannot move the goalposts. And we are honest about the default direction: the frozen spin-C structure compresses the high-scale CP phase to a single mod-8 spin-c/Pin bit whose default value, fixed by the global index χ = −3 (≡ 5 mod 8), gives the WRONG sign for the asymmetry (−e^{iπ/4} / arg −135°, not +e^{iπ/4}). Right now the bare geometry points the wrong way. That is a real obstruction we name, not hide.


2. The community gap

2.1 The precise open problem

The observed universe is made of matter, not antimatter. Quantitatively, the baryon-to-photon ratio is

η_B ≡ n_B / n_γ ≈ 6.1×10⁻¹⁰      (CMB + BBN; the measured anchor of this gate)

A successful theory must produce this number from its own structure, satisfying the three Sakharov conditions: baryon-number (here B−L) violation, C and CP violation, and a departure from thermal equilibrium. Thermal leptogenesis is the mechanism this theory uses: heavy right-handed Majorana neutrinos N₁, N₂, N₃ decay out of equilibrium with a CP-violating asymmetry ε; a lepton asymmetry is generated, partially washed out by ΔL = 2 and flavor-resolved processes, and the surviving B−L is converted to a baryon asymmetry by electroweak sphalerons.

The community gap is that no theory has a parameter-free leptogenesis η_B. The decisive obstruction is universal: the high-scale CP-violating phase that sources ε is invisible to every low-energy measurement — neutrino oscillation experiments measure δ_CP^ℓ at low energy, but leptogenesis depends on additional high-scale phases that low-energy data does not constrain. This is not a defect of one theory; it is a structural feature of the type-I seesaw.

2.2 State of the art and best bounds

The textbook machinery is mature and citable (Davidson, Nardi, Nir, Phys. Rep. 466 (2008); Buchmüller, Di Bari, Plümacher; Blanchet, Di Bari). The key relations the field uses — and that we assemble correctly below — are:

(All of these constants and formulas are correctly assembled in …/TOE/bg10_fourfix_scaffold/BG10_ETAB_TEXTBOOK_ESTIMATE.md §2 and re-verified in …/BG10_HIGHSCALE_EXTRACTION_AND_ETAB.md §3.)

The state of the art's best bound — Davidson–Ibarra — gives the field's standard cap on ε₁; but even the bound is unstatable here without the lightest heavy mass M₁, which the geometry does not yet supply (see §3).

2.3 Prior attempts and why each falls short

Prior attempt What it tried Why it falls short
The κ³/π manifest (circulated) A "derivation" giving η_B ≈ 6.0545×10⁻¹⁰ from a quarter-phase on the neutrino projectors True by construction — the load-bearing entries were reverse-engineered to land in the window; the value is provably the maximum of a one-parameter CP family, attained only at the chosen phase. RETRACTED; never cite.
Substitute the measured δ_CP^ℓ ≈ 260° Use the low-energy CP phase as the leptogenesis source Firewalled out by a theorem-level 6-vs-3 phase count: the high-scale source depends on three phases the low-energy data does not expose.
M_R = κ·M_U candidate A geometric recipe for the heavy scale Off by ~231× from the corpus's own stated seesaw scale; and even a derived M_R is degenerate with the Yukawa norm, so it fixes nothing alone.
Textbook factor-few estimate Run the standard chain to factor-few accuracy BLOCKED-AT-INPUTS — the high-scale CP phases, M_R spectrum, and absolute Y_ν are nowhere transcribed; ε₁ is identically inert. No value quoted.
The 2026-06-04 closure claim A first-pass "we closed it" Withdrawn the same day under a ten-agent audit; the mechanized exit-2 checker now blocks any repeat.

The honest summary: every shortcut either smuggles the answer in (κ³/π, the relabel candidates) or hits a genuine wall (the firewalled low-energy phase, the recipe-absent M_R). The program's contribution is to have named and proved exactly where each wall is, and to refuse the shortcuts.


3. The construction — rigorous math

This section shows the work: how the frozen branch relates to a leptogenesis calculation, the exact arithmetic proving the bare geometry sources zero CP, the textbook chain assembled and halted honestly at the first missing input, and the four-fix definition-lock.

3.1 The frozen anchor (carried unchanged, load-bearing)

𝔅_active = [M₄ × K₆ × S² × S¹_Y/ℤ₂]_×  ⊕  [F⁺_finite ⊕ C_admiss]_⊕
                                          ⊗  [E_matter ⊕ E_gauge ⊕ E_Higgs ⊕ E_proton]_⊗
K₆ = SU(3)/T²  (flag manifold);   D = 4+6+2+1 = 13;   ℤ₆ center;
spin-ℂ index χ(K₆,E) = −3  (three generations);
κ ≡ e^{−π√3} > 0;   M_U ~ 1.0×10¹⁶ GeV.
Frozen hashes:  dcc66f1b2685 (geometry anchor) / a5b1e6f9d951 (13D boundary package).

(Source: …/HANDOFF_SPECIALIST_2_BG10.md §2; …/BG10_FOUR_FIX_SPEC.md frozen anchor block.) The Majorana M_R declaration at the Cartan-torus modulus τ = ω is absorbed into the F⁺ chamber data. Any leptogenesis object here is the F⁺-projection of this frozen 13D branch, not "generic SU(3) seesaw."

3.2 The load-bearing / not-validation split

The geometry is load-bearing for the ingredients, not for the answer. Leptogenesis needs four magnitudes: a high-scale CP source, the heavy Majorana spectrum M_R, the absolute neutrino Yukawa norm, and the flavored washout/density-matrix kinetics. The frozen branch supplies the texture/structure of these (τ = ω Majorana texture; flavor eigenstructure with 3 distinct eigenvalues; low-energy CP phases) but not the magnitudes that decide η_B.

 frozen branch (τ=ω Majorana texture, flavor eigenstructure, low-energy CP phases)
        --[supplies texture, NOT magnitudes]-->  leptogenesis inputs
   high-scale CP invariant I_CP   [reduces to ONE mod-8 spin-c bit; default sign WRONG]
   neutrino Yukawa norm           [OPEN — only candidate axiom was a relabel of the answer]
   density-matrix / flavor proj.  [BLOCKED — flavor-democracy in tension with frozen data]
   seesaw scale M_R               [recipe-ABSENT; candidate M_R=κ·M_U off by ~231×; degenerate]
        --[four-fix thermal leptogenesis, NONE at certificate grade]-->  η_B  (0 of 4)
        --compare ONCE-->  η_B ∈ [6.0,6.2]×10⁻¹⁰  (pre-registered; falsification accepted)

(Source: …/01_DOSSIER.md §1.) The cardinal honest point: the magnitudes that decide η_B terminate on no derived geometric quantity. The one genuine structure-first datum is that the geometry's default spin-c sign bit gives the wrong sign — evidence the residue is a real structural fact, not a fitted parameter.

3.3 The bare-geometry CP source is identically zero (exact arithmetic)

This is the cleanest, fully-shown result of the gate. The geometry's only named direct holomorphic factor at the order-three fixed point is q₁(ω) = e^{2πiω}, with ω = e^{2πi/3} = −1/2 + i√3/2:

2πiω  = 2πi(−1/2 + i√3/2) = −πi − π√3
q₁(ω) = e^{−πi} · e^{−π√3} = (−1) · e^{−π√3} = −κ ,      κ = e^{−π√3} > 0.

So q₁(ω) = −κ is REAL (numerically Im ~ 6×10⁻¹⁸, i.e. zero). Its phase is π, not a quarter of π. If the CP-source entry is proportional to it, H₁₂ ∝ q₁(ω) ∈ ℝ, then H₁₂² ∈ ℝ and

Im(H₁₂²) = 0   ⟹   ε₁ ∝ Im(H₁₂²) = 0   ⟹   η_B = 0.

τ = ω fixes only the magnitude |H₁₂| ∝ κ; it contributes zero CP phase. (Source: …/HANDOFF_SPECIALIST_2_BG10.md §3.) This is the rigorous core of the headline: the bare geometry, as frozen, sources exactly zero asymmetry.

3.4 The low-energy CP firewall — 6-versus-3 phase counting

The measured low-energy δ_CP^ℓ ≈ 260° cannot be substituted for the high-scale invariant. A type-I seesaw with three generations carries 6 high-scale physical CP phases but the low-energy observables constrain only 3. Leptogenesis depends on the extra 3 invisible phases. Therefore measuring the low-energy phase does not fix the high-scale CP invariant. (Source: …/einstein_geometry_attack/EINSTEIN_GEOMETRY_WALLS_ATTACK.md:60, row B3, byte-verified in …/BG10_HIGHSCALE_DERIVATION_ATTEMPT.md §1(c) and …/BG10_HIGHSCALE_EXTRACTION_AND_ETAB.md §1.3.)

This is theorem-level and is one of the two PROVED legs of the firewall. It is an honesty disclosure — it explains why low-energy data cannot close the high-scale source — not a closeable piece of physics.

3.5 The seesaw degeneracy firewall (the other PROVED leg)

The corpus mass map is M_ν^eff = −M_D M_R⁻¹ M_Dᵀ with M_D = N_ν⟨g_a|O_ν|g_b⟩ (GUT.md:5020, 5513). The light-sector outputs (Δm², PMNS; GUT.md:2202 list, NuFIT values in TOE_VALIDATION_ROUTE_DOSSIER.md:356–362) fix only the seesaw combination M_ν^eff, never M_D and M_R separately. The absolute normalization N_ν is listed as "N_ν via seesaw" (GUT.md:8307) — and a corpus-wide search for any N_ν numeral returns ZERO matches (verified). Hence:

(Source: …/BG10_HIGHSCALE_DERIVATION_ATTEMPT.md §1(a); …/BG10_HIGHSCALE_EXTRACTION_AND_ETAB.md §1.1.) This collapses two formerly-separate obstructions (the Yukawa-norm leg and the M_R leg) into one. It costs the floor nothing.

3.6 What IS recipe-derivable (the texture)

To be precise about what the geometry does supply: the relative O_ν texture is recipe-real and pinned to 16 significant figures:

(O_ν)^aa = diag(4.333420509983131e-3, 6.582872101129666e-2, 1.0),   hash 495ddbdcedb9   (GUT.md:4474)

The first entry is reproduced by κ = e^{−π√3} to <1e-15. The Yukawa overlap map is (Y_i)^ab = N_i⟨g_a|O_i|g_b⟩ (GUT.md:4475, 5000). This is the same machinery as O_e, which reproduces charged-lepton masses byte-equal. The second-cycle Berry phase φ_lept = +2π/3 (GUT.md:4478) is also computable — but the corpus pipes it straight to the low-energy output δ_CP^ℓ ≈ 260.2° (GUT.md:4479), which the firewall (§3.4) keeps OUT of the leptogenesis source.

(Source: …/BG10_HIGHSCALE_DERIVATION_ATTEMPT.md §1; …/BG10_HIGHSCALE_EXTRACTION_AND_ETAB.md §1.1, §1.3.) So: texture EXTRACTED; absolute Y_ν, M_R, and high-scale I_CP recipe-absent.

3.7 The pinned light-sector numerals (do not unblock anything)

From the validation route dossier (NuFIT-style, verified against the corpus):

Quantity Value Source
Δm²₂₁ (solar) 7.39×10⁻⁵ eV² TOE_VALIDATION_ROUTE_DOSSIER.md:356–363
|Δm²₃₁| (atm) 2.515×10⁻³ eV² dossier:356–363
sin²θ₁₂ 0.3032 dossier:356–363
sin²θ₁₃ 0.02216 dossier:356–363
sin²θ₂₃ 0.4493 (lower octant) dossier:356–363
δ_CP^ℓ (LOW-energy) 260.2° dossier:356–363; toe.md L396
δ_CKM (LOW-energy) 60° dossier:356–363
v (EW VEV) 246.02 GeV dossier:307,333

Derivable from the light sector (the only leptogenesis-relevant numerals that ARE computable), with the lightest mass m₁ = 0 taken only to state the bound's mass factor:

m₂ = √(Δm²₂₁)   = 8.5965×10⁻³ eV      (re-verified: 8.596511e-3 eV)
m₃ = √(|Δm²₃₁|) = 5.0150×10⁻² eV      (re-verified: 5.014978e-2 eV)
(m₃ − m₁)       = 5.0150×10⁻² eV      — the Davidson–Ibarra mass factor.

These reproduce exactly on independent recomputation. They are the light side only; every downstream stage also needs a high-scale numeral that is missing. (Source: …/BG10_ETAB_TEXTBOOK_ESTIMATE.md §1.1.)

3.8 The textbook chain — assembled correctly, halted honestly

The intended computation is standard flavored thermal leptogenesis. Each stage below is the correct textbook formula; each is guarded — it halts at the first genuinely-missing numeral rather than fabricating through.

S1 — CP asymmetry (the source).

ε₁ = (1/(8π(Y†Y)₁₁)) Σ_{j≠1} Im[((Y†Y)₁ⱼ)²] f(Mⱼ²/M₁²)
     f(x) = √x [ 1 − (1+x) ln((1+x)/x) + 1/(1−x) ]   (vertex + self-energy)

BLOCKED. Needs Im[((Y†Y)₁ⱼ)²] (high-scale CP phases) and Mⱼ/M₁ (M_R spectrum). Both missing. Without the phases the imaginary part is identically 0 → ε₁ = 0; the source is inert. This is the binding wall.

S1′ — Davidson–Ibarra upper bound (even the bound is blocked).

|ε₁| ≲ (3/16π) M₁ (m₃ − m₁)/v²

BLOCKED. The mass factor (m₃ − m₁) = 5.0150×10⁻² eV is pinned, but M₁ is MISSING, so even the upper bound cannot be stated.

S2/S3 — washout strength.

K = m̃₁/m*,   m̃₁ = (Y†Y)₁₁ v²/M₁,   m* ≈ 1.08×10⁻³ eV

BLOCKED. Needs (Y†Y)₁₁ (Y_ν texture absolute) and M₁ (M_R). Both missing.

Flavor regime (Fix #4 entry).

M₁ > 1e12 GeV → 1-flavor ;  1e9 < M₁ < 1e12 → 2-flavor ;  M₁ < 1e9 → 3-flavor

BLOCKED. The corpus's only M_R figure, the symbolic band M_R ~ 1e9–1e14 GeV (einstein_geometry_attack/EINSTEIN_GEOMETRY_WALLS_ATTACK.md:159), straddles all three regimes — so not even the regime is fixed, let alone the value.

S5 — assembly to η_B.

Y_B = (28/79) · κ · ε₁ / g*ˢ ,   g*ˢ = 106.75
η_B = 7.04 · Y_B

BLOCKED. The sphaleron factor 28/79, g*ˢ = 106.75, and η_B = 7.04·Y_B are correct method constants, but they are inert multipliers on an undetermined ε₁·κ.

Every formula stage is standard textbook leptogenesis and correctly applied; every one halts at a missing HIGH-scale numeral. No stage was fabricated through. (Source: …/BG10_ETAB_TEXTBOOK_ESTIMATE.md §2; re-verified …/BG10_HIGHSCALE_EXTRACTION_AND_ETAB.md §3; guarded script bg10_eta_B_textbook_BLOCKED.py → BLOCKED-AT-INPUTS output.)

3.9 The four-fix definition-lock (what a certificate-grade run must deliver)

A complete BG-10 attempt must execute four named, coupled fixes plus an assembly node:

S1  I_CP / ε        (canonical CP invariant → CP source)
   │
S2  ΔL = 2 washout  (RIS-subtracted collision terms)
   │
S3  N₂/N₃ damping   (three-state kinetic cascade)
   │
S4  density-matrix  (flavored kinetic equations, off-diagonal coherences)
   │
S5  assembly        (B−L → η_B via sphaleron + entropy/g_*; +3σ band; ONE comparison)

The fixes are coupled, not independent: S2 (ΔL = 2 washout) is the damping channel of S3 (N₂/N₃); both are flavor-resolved inside S4 (density matrix); S4 is sourced by S1 (I_CP). The falsifier is defined only on the joint solution assembled by S5.

Each fix carries a precise certificate criterion (CC-1…CC-4), stated symbolically and NOT asserted to hold:

(Source: …/BG10_FOUR_FIX_SPEC.md §§1–5.) Status: 0/4. None asserted to hold.

3.10 The illustrative S5 schematic (scaffolding, NOT a banked formula)

η_B  =  C_sphaleron · D_entropy(g_*) · Y_{B−L}^flavored
        ↑ ILLUSTRATIVE SCAFFOLDING — the standard leptogenesis schematic, NOT a
          corpus-pinned assembly map. The corpus names only "sphaleron conversion +
          entropy/g_* dilution" as the missing S5 object; the actual map (its precise
          functional form AND its factors) is the owner's S5 deliverable.

The comparison logic (performed ONCE, post-freeze):

if  [η_B − 3σ, η_B + 3σ] ⊆ nbhd of [6.0,6.2]×10⁻¹⁰   → PASS (consistency, NOT a promotion)
if  η_B outside [6.0,6.2]×10⁻¹⁰ at 3σ                  → BG-10 FALSIFIED (decision-grade)

(Source: …/BG10_FOUR_FIX_SPEC.md §5.2.) S5 may NOT run until all four fixes are certificate-grade; its premature execution is exactly what was withdrawn 2026-06-04.

3.11 The window-convention tension (carried verbatim, unresolved)

Owner-ruled 2026-06-14: the falsifier window η_B ∈ [6.0, 6.2]×10⁻¹⁰ and the harness band Y_B = (8.0 ± 0.3)×10⁻¹¹ are declared the same physical quantity in two conventions, related by η_B = 7.04·Y_B. But this identity does not arithmetically map the bands onto each other:

6.1×10⁻¹⁰ / 7.04 = 8.66×10⁻¹¹   — lies ABOVE the upper edge (8.3×10⁻¹¹) of the Y_B band
7.04 · 8.0×10⁻¹¹ = 5.63×10⁻¹⁰   — is NOT 6.1×10⁻¹⁰
7.04 · [7.7,8.3]×10⁻¹¹ = [5.42,5.84]×10⁻¹⁰  — below the window

This is a pre-existing owner/corpus datum carried verbatim from ST1:886. It is read into NO fix (post-comparison only), so it does not contaminate S1–S5 — but it must be reconciled by the owner before the single CC-5 comparison. (Source: …/BG10_FOUR_FIX_SPEC.md §0.3, §6.5.) It is surfaced here, not glossed.


4. The insights we used

These are the moves that produced the progress, shared at working-physicist depth so a reader can both check and build on them.

4.1 The κ³/π dismantling — finding the maximal-CP relabel

The circulated κ³/π manifest appeared to derive the leptogenesis CP invariant. The insight that dismantled it: carry the construction with a general phase θ in the CP-source entry, H₁₂ = a·e^{−iθ} (keystone magnitude a = 4κ/√3 held fixed):

H₁₂²     = a² e^{−2iθ}
Im(H₁₂²) = −a² sin(2θ)
⟹  I_BG(θ) = (κ³/π) · sin(2θ).

Then:

I_BG = κ³/π  ⟺  sin(2θ) = 1  ⟺  θ = π/4.

So κ³/π is the MAXIMUM of I_BG(θ) over θ, attained ONLY at θ = π/4. "The chamber selects θ = π/4" therefore means "choose MAXIMAL CP." The pre-registered κ³/π value sits exactly at the peak of a one-parameter family — a selection, not a derivation. (Source: …/HANDOFF_SPECIALIST_2_BG10.md §4.) This is the κ³/π falsification test in action: any object whose keystone equals the in-window value is a relabel of the answer.

4.2 The blind finite-structure test — Milgram / Gauss-sum no-go

To check whether the quarter-phase e^{±iπ/4} has a genuine structural origin, the program ran a blind test: compute the Gauss-sum / metaplectic phase of every finite quadratic module the frozen record names, then compare to π/4 — never assume π/4 and back-fill. The Milgram formula:

γ(A, q) = |A|^{−1/2} Σ_{x∈A} e^{2πi q(x)} = e^{2πi σ/8}    (σ = signature mod 8).
Finite structure (frozen origin) Gauss phase γ Verdict
A₂ — SU(3)/T² = K₆ root data; disc ℤ₃, σ=2 e^{iπ/2} = i 90° — NOT e^{iπ/4}
τ = ω order-3 fixed point; q₁(ω)=−κ, real phase π magnitude + sign only — no CP phase
A₁ / ℤ₂, q(1)=¼, σ=1 (1+i)/√2 = e^{iπ/4} THE ONE source of the needed phase
ℤ₆ ≅ ℤ₂ ⊕ ℤ₃ e^{3iπ/4} or e^{−iπ/4} NON-UNIQUE — needs a chosen sign/splitting
S₃ — Weyl group of SU(3) NOT a finite quadratic module; no definite Gauss phase
spin-ℂ index −3 e^{2πi(−3)/8} = e^{−3iπ/4} = −e^{iπ/4} eighth-root, but the WRONG sign

The no-go: the only structure yielding e^{iπ/4} is A₁/ℤ₂ — and it is not connected to the active neutrino projectors P_±^ν by any printed map in the frozen record. The projector algebra P_±(θ) is phase-degenerate (idempotent, orthogonal, complete for every real θ), so the algebra cannot pick out θ = π/4. (Source: …/HANDOFF_SPECIALIST_2_BG10.md §5A, independently reproduced 2026-06-22.)

This is the insight behind the honest endpoint: the quarter-phase is underivable from the current F⁺ authority, so it can only be declared (path ii), not derived.

4.3 The structure-first wrong-sign datum — why the residue is real, not fitted

The most important credibility move: the spin-ℂ index −3 itself produces e^{−3iπ/4} = −e^{iπ/4}, an eighth-root phase in the same family but with the wrong sign. So eighth-root phases are the natural currency of the frozen record (making e^{iπ/4} plausible/in-family), yet the default sign points the wrong way for generating the observed asymmetry.

This is the gate's strongest honest asset: the required quarter-phase departs from the target rather than being read off it. A parameter reverse-engineered to hit η_B would never come out wrong-signed. The wrong default sign is therefore evidence the residue is a real structural fact, not a fitted knob. (Source: …/HANDOFF_SPECIALIST_2_BG10.md §5B; …/01_DOSSIER.md §2.1 and cross-gate note.)

4.4 The cross-gate σ_ν unification (Dai–Freed)

A propagation finding (2026-06-29, not a promotion): BG-10's posited high-scale CP source reduces to one mod-8 spin-c/Pin reflection bit — σ_ν — and that bit is the same orbifold Dai–Freed object (the neutrino-plane Pin/APS phase on S¹_Y/ℤ₂ × K₆ × S²) that UQF-4 carries, that SG-4 inherits, and that Gate-5 quantum-consistency rides. Its default value, fixed by the global index χ = −3, is σ = 5 mod 8 = e^{−3iπ/4} — the wrong sign. Pinning σ_ν once moves Gate-5, UQF-4, SG-4 and BG-10 together. (Source: …/01_DOSSIER.md cross-gate propagation note, CONCERN 2(d).)

4.5 The seesaw degeneracy as a unifier, not just an obstruction

Recognizing that M_R and the absolute Y_ν enter physics only through M_ν^eff = −M_D M_R⁻¹ M_Dᵀ turned two separate "missing magnitude" problems into one. This is why the consolidated ledger can close R2 and R4 by collapse into the firewall (T1'): the light-sector data provably cannot fix either magnitude alone — a theorem, not a gap. The insight: a proved degeneracy is honest progress even though it closes no magnitude, because it tells the specialist exactly what new input is needed (something that breaks the N_ν ↔ M_R degeneracy). (Source: …/CONSOLIDATED_OPEN_GATES_HANDOFF_2026-06-25/BG10.md per-residual ledger; …/BG10_HIGHSCALE_EXTRACTION_AND_ETAB.md §4.)


5. Evidence & reproducibility

5.1 The certificates and frozen records

Object Where What it certifies
Frozen geometry anchor hashes dcc66f1b2685 / a5b1e6f9d951 the read-only 13D branch all inputs project from
O_ν texture GUT.md:4474, hash 495ddbdcedb9 the recipe-real relative texture (16 sig figs)
Seesaw mass map GUT.md:5020, 5513 M_ν^eff = −M_D M_R⁻¹ M_Dᵀ (the degeneracy source)
6-vs-3 phase count einstein_geometry_attack/EINSTEIN_GEOMETRY_WALLS_ATTACK.md:60, row B3 theorem-level firewall of δ_CP^ℓ from I_CP
Exit-2 checker https://physics.magflowmeters.com/scripts/ (v15, EXTERNAL) mechanized refusal of exit-2 until S1–S4 all certificate-grade
Guarded textbook script bg10_eta_B_textbook_BLOCKED.py + …_output.txt BLOCKED-AT-INPUTS halt with no fabrication
Recipe-trace script bg10_highscale_derivation_ATTEMPT.py + …_output.txt BLOCKED-AT-RECIPE with byte-verified line refs

5.2 Numerical checks (model vs measured, with honest pulls)

There is no model η_B to compare — the source ε₁ is identically inert without the high-scale inputs (§3.3, §3.8). So the honest "pull" is not a fit residual; it is the statement that the comparison is not yet runnable. The only measured anchor is η_B ≈ 6.1×10⁻¹⁰ (CMB + BBN), held as the floor.

The light-sector cross-checks that do reproduce exactly: - m₂ = √(Δm²₂₁) = 8.596511×10⁻³ eV — reproduces on independent recomputation. - m₃ = √(|Δm²₃₁|) = 5.014978×10⁻² eV — reproduces. - κ = e^{−π√3} reproduces the first O_ν entry (4.333420509983131e-3) to <1e-15.

The one honest tension surfaced (not laundered): the η_B = 7.04·Y_B convention does not reconcile the two bands (§3.11): 6.1×10⁻¹⁰/7.04 = 8.66×10⁻¹¹ lies above the Y_B band [7.7,8.3]×10⁻¹¹.

5.3 How a reader re-runs / re-derives

  1. Re-derive the zero-CP result (§3.3): compute q₁(ω) = e^{2πiω} with ω = e^{2πi/3}; confirm it equals −κ (real, phase π); conclude Im(H₁₂²) = 0 ⇒ ε₁ = 0. One line of complex arithmetic.
  2. Re-run the blind Gauss-sum test (§4.2): for each finite quadratic module (A₂, τ=ω, A₁/ℤ₂, ℤ₆, spin-ℂ −3), evaluate γ(A,q) = |A|^{−1/2} Σ e^{2πiq(x)} via the Milgram formula; confirm only A₁/ℤ₂ gives e^{iπ/4} and that spin-ℂ −3 gives −e^{iπ/4}. Reproduced from scratch 2026-06-22.
  3. Re-run the guarded textbook chain: execute bg10_eta_B_textbook_BLOCKED.py; confirm it halts at the first missing high-scale numeral with BLOCKED-AT-INPUTS.
  4. Confirm the recipe-absence: grep the corpus for any N_ν or M_R GeV numeral; confirm zero matches (the binding wall).
  5. Confirm the exit-2 checker blocks: it must refuse exit-2 while any of S1–S4 is below certificate grade — its correct steady state while the gate is open.

5.4 Discipline record (the same-day withdrawal)

A 2026-06-04 first-pass closure claim was withdrawn the same day under a ten-agent audit, under the explicit rule "silent promotion = structural FAIL." The mechanized exit-2 checker is ARMED and BLOCKING. This is the standing model for refusing premature closure and is itself part of the evidence that the 0/4 status is enforced, not merely asserted. (Source: …/BG10_FOUR_FIX_SPEC.md §0.2, §6.3.)


6. Open gaps + closure path (the specialist work plan)

This is the load-bearing section. For each open hole: the precise statement, why it is hard plus the traps to avoid, exactly what closes it (target-blind, with the refuting outcome named), the machinery and inputs (cited paths), and the cross-gate leverage.

Binding discipline for every work-package below. Derive BLIND, then compare, accept falsification. Never fit to the η_B window [6.0,6.2]×10⁻¹⁰. The κ³/π manifest is the true-by-construction cautionary example — the thing to dismantle, never to reproduce. Any object whose keystone equals the in-window value is a relabel and must be rejected. Frozen branch READ-ONLY; STATUS-UPGRADES:0.

6.1 Hole #1 — Fix #1 / CC-1: the canonical CP invariant I_CP

(a) Precise statement. Deliver I_CP[Y_ν, M_R] in closed form: a rephasing-/basis-invariant functional of the frozen (Y_ν, M_R); prove it is the unique basis-invariant source of ε at the relevant order; tie it to ε via a cited leptogenesis CP-asymmetry relation; evaluate it on the frozen inputs with no per-entry fitting.

(b) Why it's hard / traps. The invariant must be a structural functional of the FROZEN data, not reverse-fit. Trap 1: substituting the measured δ_CP^ℓ ≈ 260° — firewalled out by the 6-vs-3 count (§3.4); this is an explicit corpus firewall violation. Trap 2: the κ³/π relabel — any I_CP whose value equals the maximal-CP κ³/π is just sin(2θ)=1 in disguise (§4.1). Trap 3 (named so the specialist does not repeat it): I_CP cannot be evaluated at all without M_R and absolute Y_ν, both of which are recipe-absent (Hole #3) — so CC-1's evaluation is downstream of Hole #3; the form and invariance proof are independently writable now.

(c) Exactly what closes it. Write I_CP as a Jarlskog-type rephasing-invariant combination of (Y_ν, M_R); prove rephasing/basis invariance algebraically; cite the ε ← I_CP relation at the relevant order. Success: a closed-form, provably-invariant I_CP tied to ε by a cited relation. Refuting outcome (valid close): if one can prove no basis-invariant functional of the frozen-fixable data alone can be non-zero (the bare-geometry zero-CP result, §3.3, already shows the bare holonomy gives Im = 0), that strengthens path (ii) — the CP source is genuinely an added axiom.

(d) Machinery & inputs. Standard leptogenesis CP-invariant theory (Davidson–Nardi–Nir). Frozen inputs: the O_ν texture (GUT.md:4474/4475), the seesaw map (GUT.md:5020, 5513). The AI lane may draft the symbolic invariant form labeled "SETUP — NOT A VALUE" but must NOT evaluate it. Start from …/BG10_FOUR_FIX_SPEC.md §1; S1_fix1_canonical_CI.md.

(e) Leverage. I_CP is the matrix source feeding Fix #4 (density matrix); closing CC-1 unblocks CC-4's source term. It is also the high-scale invariant whose phase is the σ_ν bit shared with UQF-4/SG-4/Gate-5 (§4.4).

6.2 Hole #2 — the high-scale CP phase / the σ_ν sign bit (the binding wall on CP)

(a) Precise statement. The high-scale leptogenesis CP phase has no computable corpus recipe (RECIPE-ABSENT). The continuum CP phase compresses to one mod-8 spin-c/Pin sign bit (σ_ν, BG10-Axiom-SpinC-QP / BG10-Axiom-FQM); the geometry's default value (χ = −3 ≡ 5 mod 8 → e^{−3iπ/4} = −e^{iπ/4}) gives the WRONG sign; the needed +e^{iπ/4} is not derivable from the frozen record.

(b) Why it's hard / traps. Path (i) — derive the quarter-phase from a finite Weil/metaplectic structure — was systematically tested against every finite structure the frozen record names (ℤ₂/ℤ₃/ℤ₆/S₃/A₂) and FAILED (§4.2): the sole e^{iπ/4} source (A₁/ℤ₂) has no printed map to the active neutrino projectors P_±^ν. Trap 1: reinstating the retracted spin-c construct without record support — it is RETRACTED as frozen-record-unsupported analyst conjecture (grep=0; cited Gap-03 import is a banked NEGATIVE). Trap 2: choosing the finite structure / Gauss sum because it yields π/4 — that is target-fitting; if you find yourself doing it, that is evidence FOR path (ii), not against it. Trap 3: treating in-family as derived — eighth-root phases being natural currency (§4.3) makes e^{iπ/4} plausible but NOT forced; the spin-ℂ index produces −e^{iπ/4}, the wrong sign.

(c) Exactly what closes it. EITHER (path i) supply a frozen-record-fixed finite Weil/metaplectic structure that blindly computes the phase — show the active P_±^ν eigenspaces carry the Weil representation of a structure the frozen branch actually fixes, then evaluate the Gauss sum/Maslov phase and compare to π/4 (the sharp checkable conjecture: "the spin-ℂ lift of the S¹_Y/ℤ₂ orbifold reflection acts on the neutrino CP two-plane as the A₁ Weil representation, yielding e^{iπ/4}" — pinning away the index's extra minus sign and fixing the module); OR (path ii) formally DECLARE BG10-Axiom-QP/FQM as a named added axiom and downgrade BG-10 to "conditional on a CP-phase axiom." Success (path i): a blind Gauss-sum computation yielding exactly +e^{iπ/4} from a frozen-fixed structure ⇒ the axiom becomes a theorem. Refuting outcome (valid close): a blind computation yielding a phase ≠ π/4, or −e^{iπ/4}, or proving the chamber is fundamentally continuous ⇒ path (ii) is the honest endpoint, and the wrong-sign default may even disfavor the gate.

(d) Machinery & inputs. Weil/metaplectic representation theory; Milgram's formula γ(A,q) = |A|^{−1/2} Σ e^{2πiq(x)} = e^{2πiσ/8}; the frozen finite structures (ℤ₆ center, K₆ Weyl/A₂ data, τ=ω, N=1 spin-ℂ flux, S¹_Y/ℤ₂ orbifold reflection). Start from …/HANDOFF_SPECIALIST_2_BG10.md §5/§5A/§5B and …/BG10_THREE_RULE_FPLUS_AUTHORITY_APPENDIX.md.

(e) Leverage. σ_ν is shared: pinning it once moves Gate-5, UQF-4, SG-4, and BG-10 together (§4.4). This is the single highest-leverage object in the gate.

6.3 Hole #3 — M_R recipe-absence (the single binding wall) + Yukawa degeneracy

(a) Precise statement. The heavy Majorana spectrum M_R = (M₁,M₂,M₃) and absolute scale are RECIPE-ABSENT — the seesaw mechanism is NAMED ("the heavy Majorana scale, if present, is fixed by the Cartan-torus modulus and the spin-ℂ flux N=1", GUT.md:9584) but no formula converts (τ=ω, N=1, R_{T²,Cartan}) into a number. The only corpus figure is a band M_R ~ 1e9–1e14 GeV that straddles all three flavor regimes. Separately, the absolute neutrino Yukawa is mathematically degenerate with M_R (only y_ν²/M_R is constrained).

(b) Why it's hard / traps. This is the binding wall — it cascades to block absolute Y_ν and I_CP's evaluation. Trap 1: the candidate M_R = κ·M_U is off by ~231× from the corpus's own stated seesaw scale — reverse-engineering it to land η_B is the κ³/π pattern; do not. Trap 2: the seesaw degeneracy means even a derived M_R does not by itself fix the Yukawa norm; you need two independent inputs, or one that breaks the N_ν ↔ M_R degeneracy. Trap 3: fabricating any GeV numeral is forbidden and would invalidate the entire η_B.

(c) Exactly what closes it. Owner derivation of an actual M_R formula/spectrum from (τ=ω, N=1, Cartan-torus modulus), target-blind; resolve the ~231× tension or document it as a genuine obstruction; then check whether the seesaw-fixed Yukawa norm is forced or free. Success: a non-relabel, non-tuned recipe converting frozen data into a spectrum with no reference to η_B. Refuting outcome (valid close): proving no such recipe exists in the frozen record ⇒ M_R is an honestly-named added axiom, and the gate is conditional on it.

(d) Machinery & inputs. Type-I seesaw; the Cartan-torus modulus R_{T²,Cartan} = 1.710231e-17 GeV⁻¹ (cited in …/BG10_HIGHSCALE_EXTRACTION_AND_ETAB.md §4) and spin-ℂ flux N=1; GUT.md:9584 (the named mechanism), GUT.md:8307 (N_ν "via seesaw"). Start from …/BG10_HIGHSCALE_DERIVATION_ATTEMPT.md §1(b).

(e) Leverage. M_R fixes the flavor regime (which decides whether Fix #4's density-matrix treatment is even needed), the washout scale (Fix #2), and the loop-function f(Mⱼ²/M₁²) in ε₁ (Fix #1). Closing it unblocks the evaluation of CC-1, CC-2, CC-3. It is entangled with the Yukawa norm: closing one without the other closes nothing.

6.4 Hole #4 — Fix #2 / CC-2: RIS-subtracted ΔL = 2 washout

(a) Precise statement. The RIS-subtracted (real-intermediate-state-subtracted), unitarity/CPT-consistent ΔL = 2 collision terms (ℓH ↔ ℓ̄H̄ and associated ΔL = 2 scatterings) are absent — no rate, washout factor, or named subtraction scheme is banked for this model.

(b) Why it's hard / traps. Trap 1: without RIS subtraction, on-shell N exchange is double-counted, spuriously erasing or inflating the asymmetry. Trap 2: the subtraction must be shown unitarity/CPT-consistent and the on-shell-N piece shown to avoid double-counting — a schematic estimate is not a certificate. Trap 3: the scheme must be named/cited; the corpus does not pin one, so "RIS subtraction" alone is insufficient.

(c) Exactly what closes it. Write the RIS-subtracted collision terms γ_{ΔL=2}^{RIS-sub}(T) = γ_full(ℓH ↔ ℓ̄H̄, ΔL=2 scatt.) − γ_{on-shell N}, with the on-shell-N subtraction shown to avoid double-counting, evaluated over the Gap-09 T_RH window, with a named/cited subtraction scheme. Success: unitarity/CPT-consistent washout rates over the thermal window. Refuting outcome: if the washout is so strong it suppresses any asymmetry below the window regardless of ε, that bounds the gate.

(d) Machinery & inputs. Standard leptogenesis subtraction schemes (Giudice et al.; Buchmüller–Di Bari–Plümacher). Frozen M_R sets the mediator scale (recipe-absent — entangled with Hole #3). Upstream: the Gap-09 T_RH window (CONDITIONAL, constraint only, never promoted through). Start from …/BG10_FOUR_FIX_SPEC.md §2.

(e) Leverage. S2 (ΔL = 2 washout) is the damping channel of S3 (Hole #5); closing CC-2 partially unblocks CC-3.

6.5 Hole #5 — Fix #3 / CC-3: the N₂/N₃ damping ledger

(a) Precise statement. The three-state (N₁,N₂,N₃) kinetic cascade / N₂–N₃ damping ledger is absent — only a DIAGNOSTIC floor (which merely passes the T_RH range) is banked, and it must not be promoted.

(b) Why it's hard / traps. Trap 1 (the cardinal one): promoting the diagnostic floor to "the result" reproduces the withdrawn 2026-06-04 move and is a structural FAIL. Trap 2: the M₂ = M_U ~ 10¹⁶ GeV state against a maximal reheat gives M₂/T_RH^max ~ 2.5, so N₂ is Boltzmann-suppressed but not to mathematical zero — it cannot be treated as exactly absent (the κ³/π manifest's D_N = diag(1,0,0) assumption is unjustified). Trap 3: whether η_B is N₁-set or carries N₂/N₃ memory is undetermined until the cascade is computed.

(c) Exactly what closes it. Quantify N₂/N₃ production–decay rates and inter-state flavor-resolved washout W_{i←j} on the frozen spectrum, superseding (not promoting) the diagnostic floor with the actual damped surviving Y_{B−L}. Success: a quantified three-state cascade with the surviving asymmetry. Refuting outcome: if N₂/N₃ memory dominates and pushes the result outside the window, that is a valid falsifying close.

(d) Machinery & inputs. Three-state Boltzmann/kinetic leptogenesis (Di Bari; Blanchet–Di Bari). Frozen M_R spectrum (recipe-absent — entangled with Hole #3); Gap-09 T_RH window (constraint only). Start from …/BG10_FOUR_FIX_SPEC.md §3.

(e) Leverage. Consumes CC-2's washout terms; feeds CC-4's matrix washout.

6.6 Hole #6 — Fix #4 / CC-4: flavored density-matrix kinetics

(a) Precise statement. The flavored density-matrix kinetic treatment (off-diagonal coherences, charged-lepton-Yukawa decoherence) is absent. The flavor-diagonal approximation can shift η_B by an order-one factor — decisive against a 3σ window.

(b) Why it's hard / traps. This is the most coupled fix (sourced by S1, coupled to S2/S3). Trap 1: off-diagonal-coherence handling is exactly where order-one errors hide. Trap 2: the proposed flavor-democracy symmetry (rank-1 P_s = (1/3)𝟙𝟙ᵀ) is in direct tension with the frozen flavor data — it implies a 2-dimensional commutant, but the frozen flavor data carries 3 distinct eigenvalues (incompatible). This is the R3 BLOCK. Trap 3: flavor-democracy is forced only if the leptogenesis density-matrix generator is S₃-equivariant ([L_BG, S₃] = 0), which the frozen "3 generations + no free per-family norm" is strictly weaker than.

(c) Exactly what closes it. Either (a) prove the flavor-democracy/eigenvalue tension is fatal (a clean no-go, terminal as a no-go), or (b) supply the actual density-matrix inputs consistent with 3 distinct eigenvalues and solve dρ_{αβ}/dz = S_{αβ}[I_CP] − {W,ρ}_{αβ} − λ_Y[decoherence(ρ)]_{αβ} over the T_RH equilibration schedule to Y_{B−L}^flavored, with cited density-matrix-leptogenesis references and the equilibration boundaries stated. Success: a solved flavored kinetic system. Refuting outcome: proving flavor-democracy is incompatible with the frozen data is itself a valid terminal no-go.

(d) Machinery & inputs. Density-matrix leptogenesis (Blanchet–Di Bari–Jones–Marzola; De Simone–Riotto). Matrix source from I_CP (Hole #1); frozen flavor eigenstructure; Gap-09 T_RH window. Start from …/BG10_FOUR_FIX_SPEC.md §4.

(e) Leverage. CC-4 is the last fix before assembly; its order-one factor is decisive against the 3σ window. Resolving the flavor-democracy tension also retires the R3 BLOCK in the residual register.

6.7 Hole #7 — S5 assembly / CC-5: the sphaleron + entropy map and 3σ band

(a) Precise statement. The sphaleron + entropy/g* conversion map (its functional form AND factors) is not corpus-pinned at certificate grade; the propagated 3σ-band construction is undocumented. The illustrative η_B = C_sphaleron · D_entropy · Y_{B−L} (§3.10) is scaffolding, not a banked formula.

(b) Why it's hard / traps. Trap 1: S5 may NOT run until all four fixes hold at certificate grade — premature execution is exactly the withdrawn 2026-06-04 move. Trap 2: a partial delivery of any subset does NOT move the ledger off 0/4 (the falsifier is defined only on the joint solution). Trap 3: the window-convention tension (§3.11) must be reconciled before the single comparison — it is read into no fix but is decisive at CC-5.

(c) Exactly what closes it. Deliver the corpus-pinned B−L → η_B assembly map (functional form + factors) and the 3σ-band method, runnable only after CC-1…CC-4 hold, then perform exactly ONE post-freeze comparison. Success: a single η_B ± 3σ compared once. Inside the window ⇒ consistency (NOT a promotion); outside at 3σ ⇒ BG-10 falsified (a decision-grade verdict, accepted).

(d) Machinery & inputs. Standard sphaleron conversion (28/79) and entropy/g dilution (gˢ = 106.75); the η_B = 7.04·Y_B convention (with its unresolved band tension). Start from …/BG10_FOUR_FIX_SPEC.md §5.

(e) Leverage. CC-5 is the terminal node — it converts all upstream work into the single falsifiable number. Nothing else in the gate produces a comparison.

6.8 Hole #8 — the wrong-sign default (the baryogenesis sign problem)

(a) Precise statement. The frozen spin-ℂ index −3 yields −e^{iπ/4}, not +e^{iπ/4} — the natural in-family phase points the wrong way for generating the observed asymmetry.

(b) Why it's hard / traps. This is a genuine wrong-sign obstruction, not a closure. Trap: treating "in-family" (eighth-root phases are natural currency) as "derived" — the sign and module are not pinned (§4.3). The ℤ₆ non-uniqueness finding mirrors this: the sign depends on a chosen refinement.

(c) Exactly what closes it. A mechanism that pins the module/sign so the spin-c lift delivers +e^{iπ/4} (the sharp falsifiable conjecture in HANDOFF_SPECIALIST_2_BG10, §5B). Success: a blind computation pinning away the extra minus sign. Refuting outcome: if the sign is genuinely fixed to −e^{iπ/4} by the geometry, the bare theory predicts the wrong-sign asymmetry — a falsifying datum, honestly reportable.

(d) Machinery & inputs. Spin-ℂ/Pin⁻ index theory; the Dai–Freed σ_ν object; the S¹_Y/ℤ₂ orbifold reflection. Start from …/HANDOFF_SPECIALIST_2_BG10.md §5B and the cross-gate note in …/01_DOSSIER.md.

(e) Leverage. Same as Hole #2 — σ_ν is shared with UQF-4/SG-4/Gate-5. The sign is the same bit.

6.9 Hole #9 — the window-convention arithmetic tension

(a) Precise statement. The owner-pinned identity η_B = 7.04·Y_B does NOT map the harness band Y_B = (8.0±0.3)×10⁻¹¹ onto the falsifier window (6.1×10⁻¹⁰/7.04 = 8.66×10⁻¹¹ sits above the band; 7.04·8.0×10⁻¹¹ = 5.63×10⁻¹⁰ misses 6.1×10⁻¹⁰).

(b) Why it's hard / traps. Trap: letting the convention contaminate S1–S5 — it must be read into NO fix (anti-reverse-fit), carried verbatim, and resolved only at CC-5.

(c) Exactly what closes it. Owner reconciliation of the two conventions/bands into one consistent quantity before the single post-freeze comparison. Success: one consistent quantity. Refuting outcome: none — this is a convention bookkeeping issue, not a physics falsifier; but it must be settled so the CC-5 comparison is well-defined.

(d) Machinery & inputs. The two pinned bands (ST1:886). Start from …/BG10_FOUR_FIX_SPEC.md §0.3, §6.5.

(e) Leverage. Settling it makes the CC-5 comparison unambiguous; it contaminates nothing if held out of S1–S4.

6.10 Attack order (leverage ranking)

  1. Hole #2 / #8 (σ_ν CP phase + sign) — the genuine reduction already in hand; the highest-leverage object (shared with Gate-5/UQF-4/SG-4). Derive the mod-8 bit blind, or declare the axiom.
  2. Hole #3 (M_R recipe) — the single binding wall; cascades to unblock CC-1/CC-2/CC-3 evaluation. Entangled with the Yukawa norm.
  3. Holes #1, #4, #5, #6 (the four fixes) — the assembled-answer debt; runnable target-blind once Hole #3 supplies the magnitudes. Mind the coupling (S2 = S3's channel; S4 sourced by S1).
  4. Hole #7 (S5 assembly) — terminal; runnable only after 1–6 hold.
  5. Hole #9 (window convention) — bookkeeping; resolve before CC-5.

7. Honest ceiling & scope

Ceiling: serious candidate, NOT validated — an honestly-held open gate with a pre-registered falsifier, not a solved one.

What is explicitly NOT claimed: - NOT claimed: BG-10 / baryogenesis η_B is CLOSED or DERIVED. The 2026-06-04 closure claim was withdrawn the same day; every current doc holds the ledger at 0/4 with STATUS-UPGRADES:0. - NOT claimed: the quarter-phase e^{iπ/4} / I_BG = κ³/π is derived from the frozen geometry. RETRACTED as frozen-record-unsupported analyst conjecture; path (i) systematically tested and FAILED; κ³/π is true-by-construction and NEVER banked. - NOT claimed: the measured low-energy δ_CP^ℓ ≈ 260° supplies the leptogenesis CP source. It is firewalled OUT by the theorem-level 6-vs-3 phase count; substituting it is a corpus firewall violation. - NOT claimed: a textbook/factor-few η_B value already exists for this theory. BLOCKED-AT-INPUTS: ε₁ is identically inert without the missing high-scale phases; no η_B value is quoted.

The honest distinctions, held throughout: dissolved ≠ solved; selection ≠ derivation; given-E ≠ derivation of E; AXIOM-CLOSED ≠ proven; ANCHORED ≠ DERIVED; a captured log ≠ an independent reproduction.

The dissolved unicorns (shared ceilings, never open weaknesses, never claimed as proven): - "The geometry's high-scale CP phase is THE unique one no other math could produce" — proving a universal negative over all possible finite structures is unprovable in principle. The bounded result is the blind finite-structure test already run (ℤ₂/ℤ₃/ℤ₆/S₃/A₂) showing none of the named frozen structures forces e^{iπ/4}. - "No future theory could ever derive a parameter-free η_B" — an open-ended universal negative; not a hole. The bounded, honest claim is the pre-registered falsifier on THIS theory's four-fix computation. - "κ³/π is THE forced leptogenesis invariant" — would require proving forced-ness against any reparametrization; correctly dissolved by showing it is true-by-construction (maximal CP at θ = π/4), and it is NEVER banked.

The anchors paid. Exactly one atomic measured anchor is isolated and irreducible: η_B ≈ 6.1×10⁻¹⁰ (CMB + BBN), the floor ≥ 1, serving as the single pre-registered post-freeze comparison — an input to test against, never an output of this gate.

The genuine, non-promoting wins: (a) the reduction of the CP source to one mod-8 spin-c bit with a wrong-default-sign (structure-first, not fitted); (b) the dismantling of the κ³/π manifest as true-by-construction; (c) the precise location of the residue (M_R recipe-absence, the Yukawa/M_R degeneracy, the 6-vs-3 CP firewall); (d) the proof that the bare geometry sources exactly zero CP (q₁(ω) = −κ, real). None is a closure. The gate stays OPEN / Diagnostic, 0 of 4, STATUS-UPGRADES:0, frozen branch READ-ONLY.


Source root for all citations: C:/Users/cberg/Desktop/VS Code Projects Random/physics_Journal_and_patents/Fable_Version/rendered/. Primary sources synthesized: TOE/PER_GATE_DOSSIERS/BG10_COMPLETION_HANDOFF/01_DOSSIER.md; TOE/PER_GATE_DOSSIERS/CONSOLIDATED_OPEN_GATES_HANDOFF_2026-06-25/BG10.md; TOE/bg10_fourfix_scaffold/{BG10_FOUR_FIX_SPEC, BG10_ETAB_TEXTBOOK_ESTIMATE, BG10_HIGHSCALE_DERIVATION_ATTEMPT, BG10_HIGHSCALE_EXTRACTION_AND_ETAB}.md; TOE/HANDOFF_SPECIALIST_2_BG10.md; the dossier-build handoff TOE/30pagedoc/handoffs/GAP10_BG_10.md. This dossier asserts no η_B / ε / M_R / Y_ν / phase value; closes no gate; promotes nothing.