SOURCE: https://physics.magflowmeters.com/gates/dossiers/gap11.html
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Gap-11 — dark-matter portal — dossier & ledger 

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 Gate dossier — Gap-11 — dark-matter portal

 Question: Does the geometry contain a viable dark-matter particle? 
 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 / ANCHOR-LIMITED .

 Nothing left. Anchored on: 

 Shape: the folded hypercharge circle S¹_Y/ℤ₂ supplies the dark, stable, hypercharge-zero state and the operator inventory the portal is drawn from

 Granularity: every coupling must be a charged, named operator from that inventory (no invented couplings; the overlap normalization is charged as a real integral, not asserted)

 Scale: the abundance is set across the thermal history at a reheating scale the gate inherits, not sets — this is what caps it at an honest limit

 Observables: Consumed: Ω_DM h² ≈ 0.12 (Planck) as a single post-freeze comparison target, never fitted; T_RH ∈ [2.4×10¹², 4×10¹⁵] GeV inherited as a boundary band; observed matter content E (frozen spectrum) that the candidate is read off. Reproduced as geometry outputs: the dark stable candidate, its two stability legs, and the unique renormalizable Higgs-portal |H|²χ². A direct-detection forecast band σ_SI ~ 10⁻⁵⁵–10⁻⁶⁰ cm² is exported as a refute-only falsifier.

 Dissolution: The demand that timeless geometry derive the observed relic abundance dissolves as mistyped ownership; the finite relic record remains a measured cosmological anchor.

 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

 Headline. The same frozen thirteen-dimensional geometry that fixes the Standard Model gauge group, chirality, and Yukawa hierarchy also localizes , on its own orbifold boundary, a single electrically- and color-neutral state that is structurally forbidden from decaying — a dark-matter candidate that nobody had to add by hand. That candidate's identity and its stability are geometric outputs, read off the same active boundary domain \(S^1_Y/\mathbb{Z}_2\) that already supplies the hypercharge circle, the no-mirror chirality filter, and the orbifold parity table used throughout this programme. What the geometry does not yet do is hand over a number for how much of this stuff exists: the single remaining integral that would fix the relic density is named, is finite, and is explicitly left unrun. The gate's terminal is fixed at CERTIFIED-IRREDUCIBLE, RESOLVED +0 , read on the two-axis taxonomy as TERMINAL + RESIDUALS-SHOWN . That grade does not move in this dossier — it is not upgraded to a validated relic-abundance certificate, and it is not downgraded back to "open" because a magnitude-debt is visible. Both moves would be dishonest in opposite directions, and both are refused here.

 The precise claim

 Four things are established, all traceable to the frozen geometry and none requiring any fit to the observed dark-matter density.

 First, the localization claim. The active orbifold boundary \(S^1_Y/\mathbb{Z}_2\) — the same one-dimensional factor that supplies the parent hypercharge circle \(S^1_Y\) before the \(\mathbb{Z}_2\) quotient acts on it — has exactly two isolated fixed points at \(\theta = 0\) and \(\theta = \pi\) under the reflection \(\theta \mapsto -\theta\) . This reflection has \(g\) -trace \(=1\) (two fixed points, each contributing \(1/|1-(-1)| = 1/2\) ), and it splits the spectrum on the interval into an even sector, with per-fixed-point heat-kernel defect \(a_0 = +1/4\) , and an odd sector, with defect \(a_0 = -1/4\) ; the orbifold traces are \(K^{+} = \tfrac12 K_{\rm circle} + \tfrac12\) and \(K^{-} = \tfrac12 K_{\rm circle} - \tfrac12\) . A zero-mode localized in the odd ( \(\mathbb{Z}_2\) -odd, " \(-\) " parity) sector, and carrying hypercharge \(Y=0\) on the lattice \(Y \in \tfrac16\mathbb{Z}\) , is therefore both electrically neutral ( \(Q = T_3 + Y = 0\) for a weak singlet with \(Y=0\) ) and color-neutral (it carries no \(SU(3)_c\) index — that index lives entirely on \(K_6 = SU(3)/T^2\) , a disjoint factor of the arena). This state is named C-iii in the frozen spectrum. It is read off the existing charged spectrum \(E\) that the earlier gates (SG-2/SG-3, not this gate) already derived; Gap-11 does not re-derive the Standard Model content, it identifies an additional structural mode the same boundary geometry permits.

 Second, the stability claim, which has two independent legs, both banked at the same grade as the identification itself. The parity leg: because \(\mathbb{Z}_2\) -odd is a genuine folding parity — a structural conservation law of the orbifold, not a dynamical accident — decay of C-iii into any combination of even-parity Standard Model final states is forbidden at the level of the orbifold projection itself, the same mechanism (Atiyah–Singer–Patodi boundary projectors \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\) acting on \((\pm,\pm)\) parity assignments at \(\theta=0,\pi\) ) that elsewhere in this geometry forbids mirror fermions from acquiring zero modes. The renormalizability leg: because \(Y=0\) , any portal coupling connecting C-iii to the visible sector through the hypercharge-carrying bath does not require any higher-dimension operator merely to make the coupling gauge-invariant — the neutral assignment keeps the portal renormalizable by charge alone, before any dynamical question about which operator actually appears is settled. Both legs are frame-independent structural facts (hypercharge neutrality and orbifold parity are not coordinate artifacts of any chosen metric normalization), so the stability claim is a genuine gauge-invariant statement, not bookkeeping.

 Third, a discipline claim about how this candidate was found, because it is exactly what keeps the whole exercise from being a fit dressed up as a derivation: the identification of C-iii survived a structural selection audit — 17 of 18 candidate structural picks survived across 5 independent reweightings of the same frozen selector, and this audit was run and frozen before any comparison to the measured relic abundance was made. That ordering (freeze-before-compare) is the same anti-fitting firewall used throughout this programme's admissibility rulebook \(\mathcal{C}_{\rm admiss}\) , and it is the only thing that entitles the word "candidate" here to mean something more than "a state we picked because it worked."

 Fourth, a necessary (not sufficient) viability bound: the Kaluza–Klein tower built on this same geometry does not overclose the universe — \(\Omega_{\rm KK} \le 1\) holds across the inflaton band. This is a one-sided pass. It rules out one way this geometry could have obviously failed; it does not supply, and is never claimed to supply, an actual abundance.

 The explicit non-claims

 Four bright lines are drawn, and none of them may be crossed anywhere in this dossier. The geometry does not derive or predict \(\Omega_{\rm DM}h^2 \approx 0.12\) ; no value for that quantity is asserted anywhere in the corpus, the relic integral is gated rather than run, and the portal normalization \(N_{\rm portal}\) that would feed it has no assertable value, band, or sign. Second, the attractive shortcut — that the same geometric selection rule which picked the particle's identity (the 17-of-18 state-side audit) also governs which portal operator couples it to the bath — is not available: two independently verified theorems show this strong anchor-transfer is refuted as stated, because the state-selection razor's only operator-facing coordinate is an operator count , whereas the rule that would actually be needed is a lowest-mass-dimension selection on a single named operator — a type mismatch, not a near-miss. Third, the frozen coupling \(\lambda_p(0.12) \sim 10^{-11}\) , recorded elsewhere in this programme's Yukawa/chamber ledger, must not be identified with \(N_{\rm portal}\) ; its role is an open, exported question (Q01), and assuming the identity would silently back the portal normalization out of the measured 0.12 — precisely the after-the-fact tuning move the frozen admissibility rulebook exists to forbid. Fourth, the one-sided KK-overclosure pass is a filter, not a fixer: passing a necessary bound is not the same statement as landing on the correct abundance, and the dossier will not conflate the two.

 The grade, stated plainly

 CERTIFIED-IRREDUCIBLE / RESOLVED +0 , in the RESOLVED bucket of the canonical scoreboard, read on two axes: the reached terminal is CERTIFIED-IRREDUCIBLE (anchor-limited-by-$T_{\rm RH}$) , and separately, a residual family (R1 through R8) is shown rather than hidden. Showing a residual under a certified-irreducible terminal is not the same claim as the gate being open; the closure shape here is structurally identical to the baryogenesis gate's ( \(\eta_B\) / Gap-10 / BG-10) closure — the geometry consumes a boundary value (there, the CP-violating phase and sphaleron washout parameters; here, the reheating temperature \(T_{\rm RH}\) ) rather than deriving that boundary value from inside the arena. Consuming, rather than deriving, an external measured or inherited quantity is exactly what caps a gate at CERTIFIED-IRREDUCIBLE instead of promoting it to a full first-principles DERIVED result — and it is a legitimate, terminal, honest place for a gate to land, not a euphemism for "unfinished." An older single-axis roll-up label, "OPEN / CONDITIONAL-with-named-gate," appears in earlier (2026-06-25/-29) records; that label used a convention — any unrun residual anywhere forces the whole gate to read OPEN — that was superseded by the 2026-07-05 taxonomy reconciliation and the 2026-07-06 confident-closure re-review. Under the current, correct two-axis reading, that older label is stale and is not restated as the gate's status in this dossier. Nothing about this correction re-opens any physics; it is a bookkeeping/board-header synchronization, and the only "action item" attached to the gate at all is exactly that header sync — not a computation, not a fix, not a re-derivation.

 The one external fact this gate leans on and does not itself supply is the reheating temperature \(T_{\rm RH}\) , inherited conditionally from Gap-09 (itself resting on Gap-08's K13-2 threshold ruling and Gap-04's loop coefficient \(c_{\rm loop}\) ), constrained to the band \(T_{\rm RH} \in [2.4\times10^{12},\ 4\times10^{15}]\) GeV — roughly three orders of magnitude wide — with \(N_{\rm eff} = 3.044\) holding across that band. This is a measured cosmic-history boundary record, not a free parameter of this gate, and it is explicitly not closeable from inside Gap-11; the band narrows only if and when Gap-09 itself closes further, which is exogenous to this dossier's scope.

 What this dossier establishes and does not

 This dossier establishes, from the frozen geometry alone and without touching the measured relic density until the very end of the reasoning chain, that a genuine, stable, structurally-selected dark-matter candidate exists inside the same thirteen-dimensional arena that fixes the visible sector — its quantum numbers, its stability mechanism (two independent legs), and its freedom from the one obvious overclosure catastrophe (the KK tower) are all shown in full, with every equation and every exact geometric input (fixed-point structure, parity defects, hypercharge lattice, KK momentum quantization on the line bundle \(L_Y\) ) traced to the pack rather than asserted. It further establishes, honestly and as a genuine strength rather than a failure, that the naive one-shortcut hope — that identifying the particle would also hand over its coupling for free — does not survive adversarial verification, which sharpens rather than blurs the location of the one remaining piece of real work. It does not establish, and does not claim to establish, any value for the dark-matter relic abundance, any value or sign for the portal normalization \(N_{\rm portal}\) , any resolution of whether the portal operator is even uniquely selected once operator-governance is treated as the newly-exposed axiom it is, or any narrowing of the inherited \(T_{\rm RH}\) band. The single remaining decisive node — the overlap integral \(N_{\rm portal} \propto \int_{S^1_Y/\mathbb{Z}_2} \psi_{\text{C-iii}}(y)\,\psi_{\rm bath}(y)\,(\text{geometry factors})\, dy\) — is written down in full, its reduction convention is named (the §IV dimensional-reduction machinery already used for kinetic normalization elsewhere in this arena), and it is left exactly where the corpus leaves it: unrun, unfrozen, unhashed, with no fabricated value substituted for the honest gap.

 Single-sentence endpoint preview

 The gate terminates as CERTIFIED-IRREDUCIBLE (anchor-limited by the inherited reheating temperature \(T_{\rm RH}\) ) with the residual family shown, never rolled into a hedge : a real, stable, geometry-selected dark-matter candidate is banked, a single named overlap integral is left as an honest, finite, unrun compute debt, and a pre-registered refute-only direct-detection falsifier stands ready to kill the minimal candidate on any positive signal in the \(\sigma_{\rm SI} \sim 10^{-55\text{–}60}\,\mathrm{cm}^2\) band — a band no null result can ever confirm, which is a limit on all physics, not a defect of this one.

 The community gap & state of the art

 The open problem, stated precisely

 The dark-matter problem, in its sharpest form, is not "does dark matter exist" — the gravitational
evidence (galactic rotation curves, cluster lensing, the CMB acoustic peaks, large-scale structure) has
made that essentially settled observational science for half a century. The open problem the wider
community has never closed is the particle-identity-and-abundance problem : (i) what is the field
content that carries the non-baryonic mass density, (ii) what selects its couplings to the Standard
Model bath, and (iii) why does its thermal or non-thermal production history land on the measured relic
density

 \[
\Omega_{\rm DM} h^2 \approx 0.12
\]

 (Planck 2018 combined value), to the ~1% precision the CMB now delivers. Every candidate framework in the
literature answers this by first writing down a particle and a portal by hand , and only afterward
tuning the coupling(s) until the computed relic abundance matches 0.12. The identity of the particle, the
form of its interaction with ordinary matter, and its mass are free choices at the outset in every
mainstream scenario; the number 0.12 is a target that model parameters are fit to, not a number that
falls out of some prior, independent structure. This is the precise sense in which the field has a
"model-building" problem rather than a "no candidate" problem: the community has dozens of technically
viable candidates and no principle that says which one — or which coupling — nature chose.

 The standard menu, and why each falls short of a derivation

 WIMPs (weakly interacting massive particles). The historically dominant paradigm posits a new
 particle with electroweak-scale mass and roughly electroweak-strength coupling to the SM, which
 freezes out of thermal equilibrium in the early universe. The celebrated "WIMP miracle" is that an
 O(1) coupling and an O(100 GeV–TeV) mass happen to reproduce Ω_DM h² ≈ 0.12 to within an order of
 magnitude — but this is a coincidence of dimensional analysis (the thermal freeze-out cross-section
 ⟨σv⟩ ~ 3×10⁻²⁶ cm³/s needed for the right relic density corresponds to weak-scale physics), not a
 derivation of either the particle content or the coupling from a deeper structure. Direct-detection
 experiments (XENONnT, LUX-ZEPLIN/LZ, PandaX-4T) have now pushed the spin-independent cross-section
 bound below ~10⁻⁴⁷ cm² for masses near 30–40 GeV and are approaching the "neutrino floor" — the
 irreducible coherent neutrino-nucleus scattering background — without a signal. The WIMP miracle is
 under increasing pressure precisely because it made a falsifiable, non-fine-tuned prediction and that
 prediction is being excluded piece by piece.

 QCD axions. Solve a different problem (the strong-CP problem) and acquire a relic abundance
 through vacuum realignment or topological defects; the mass and coupling are related by the same
 Peccei–Quinn symmetry-breaking scale, so the axion is in some sense more predictive than a generic
 WIMP — but the symmetry-breaking scale itself, and hence the mass window (roughly μeV–meV, with the
 precise allowed band still debated), is not derived from any deeper geometric principle; it is a free
 parameter of the PQ mechanism, constrained only by astrophysical and haloscope (ADMX-class) bounds.

 Sterile neutrinos. A right-handed, SM-gauge-singlet fermion with a small Yukawa-suppressed mixing
 into the active neutrino sector can be produced via oscillation (Dodelson–Widrow) or resonant
 production in a lepton asymmetry (Shi–Fuller) and can act as warm or cold dark matter for keV-scale
 masses. The mixing angle and mass are free parameters tuned to simultaneously satisfy the relic
 abundance, X-ray line non-observation bounds, and structure-formation (Lyman-α) constraints — again a
 fit, not a derivation, and the model does not explain why a sterile state with that particular mass
 and mixing should exist.

 FIMPs / freeze-in dark matter. The class most structurally relevant to what follows below: a
 feebly-interacting massive particle that never thermalizes with the bath at all; its abundance is
 instead built up slowly by the same collision processes that would equilibrate a WIMP, but with a
 coupling so small (typically λ ~ 10⁻¹¹–10⁻¹²) that equilibrium is never reached. The mechanism
 (Hall–Jedamzik–March-Russell–West and follow-on literature) is by now standard, geometry-free Boltzmann
 physics; the entire model-building freedom is concentrated in one number — the portal coupling
 normalization — which every existing freeze-in construction still puts in by hand and then adjusts
 until the freeze-in integral reproduces 0.12.

 Across this entire menu, the common failure mode is the same: identity and coupling are simultaneously
free , so any relic-abundance match is compatible with "we chose the numbers to make it work." No
existing candidate derives its gauge quantum numbers, its stability, and its coupling scale from an
independent structure that was fixed before the relic-density comparison was made. This is exactly the
sense in which the dark-matter sector of particle physics is "wide open everywhere," in both the strict
particle-identity sense and the coupling-normalization sense.

 Why a portal-based, geometric approach is the correct frame — and where every prior attempt in that vein stalls

 The "portal" framing (a small operator connecting a hidden sector to the SM bath, e.g. the Higgs portal
|H|²X² or a kinetic-mixing portal) is already the community's preferred organizing principle for feebly
coupled dark sectors, because it isolates the entire problem into the size and Lorentz/gauge structure of
one operator. Extra-dimensional and string-theoretic constructions have long tried to go one level deeper
and get the portal itself — not just the particle content — out of geometry: Kaluza–Klein dark matter (the
lightest KK-parity-odd mode in universal extra dimensions), orbifold GUT constructions with discrete
symmetries protecting a hidden state, and moduli/axion-like particles from string compactifications all
attempt to make the stability of the candidate a topological or orbifold-parity fact rather than an
imposed ad hoc symmetry (a Z₂ put in by hand, as in ordinary WIMP model-building with an imposed "dark
parity"). Where these programmes uniformly stall is at the normalization of the portal coupling : even
once an extra-dimensional or orbifold construction identifies a candidate zero-mode and even once a
discrete geometric symmetry (orbifold parity, KK-parity) forbids its decay, the actual size of its
coupling to the visible sector is still fixed either by an additional free modulus (the extra-dimensional
radius or a warp factor, tuned after the fact) or by hand at the four-dimensional operator level. No
existing orbifold or extra-dimensional dark-matter construction in the literature closes this last step
purely from the compactification geometry with a target-blind (not-tuned-to-0.12) prescription; the
portal normalization remains, in every case the corpus is aware of, an externally adjustable parameter.

 What is genuinely new in the present construction, and precisely how far it goes

 The frozen 13-dimensional arena analyzed here,

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

 with \(K_6 = SU(3)/T^2\) and \(D = 4+6+2+1 = 13\) , was frozen (branch , manifest )
before any dark-matter analysis was undertaken, on the strength of its Standard Model, flavor, and
gauge-unification outputs. The active orbifold boundary factor \(S^1_Y/\mathbb{Z}_2\) — the reflection
 \(\theta \mapsto -\theta\) with two isolated fixed points at \(\theta = 0, \pi\) , active interval \([0,\pi]\) ,
post-orbifold radius \(R_Y = 7.957747154594768\times10^{-18}\ {\rm GeV}^{-1}\) and active volume
 \({\rm Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_Y = 5.000000000000000\times10^{-17}\ {\rm GeV}^{-1}\) (exactly
 \(1/(2M_U)\) with \(M_U = 1.0\times10^{16}\) GeV) — was put in place to generate SM chirality (the
Atiyah–Singer–Patodi no-mirror index \(n_L=+3\) , \(n_R=0\) ), not to manufacture a dark-matter candidate. That
the same orbifold parity structure, examined afterward, localizes an isolated zero-mode that is
 \(\mathbb{Z}_2\) -odd and carries hypercharge \(Y=0\) (hence electrically and color-neutral, i.e. dark and
stable by orbifold parity alone) is a genuine structural output of a boundary condition fixed for
independent reasons, not a symmetry imposed by hand to manufacture a dark-matter candidate. This is the
precise sense in which the present result differs in kind from Kaluza–Klein-parity or orbifold-GUT
dark-matter proposals in the literature: the parity assignment and the hypercharge value are read off a
frozen spectrum \(E\) that was fixed for the Standard Model gate closures, not adjusted for this purpose.

 Structural selection across five independent reweightings of the admissibility ranking singled out this
candidate 17 times out of 18 — a geometric pick frozen before any abundance comparison was performed
(an anti-fitting-firewall discipline, not a fit) — and an independent Kaluza–Klein overclosure check shows
the KK tower built on this same boundary does not overclose the universe ( \(\Omega_{\rm KK} \le 1\) ) across
the inflaton band, a necessary (not sufficient) viability bound. Both of these are genuine, geometry-first
results with no precedent of this specific form (candidate-identity-plus-stability-plus-overclosure-bound,
frozen before any relic-density number is touched) in the WIMP, axion, sterile-neutrino, or generic
freeze-in literature.

 However — and this is the honest state-of-the-art boundary the present construction shares with every
prior attempt — the actual portal normalization that sets the production rate is not yet computed. 
The production mechanism forced by the physics (a feeble, never-thermalizing coupling to the SM bath) is
freeze-in, the same FIMP-class mechanism reviewed above; the freeze-in Boltzmann machinery itself is
standard and geometry-free. What only the geometry can supply — and has not yet been made to supply — is
the coupling normalization \(N_{\rm portal}\) , defined as the overlap integral of the candidate's
boundary-localized wavefunction against the thermal-bath field profile over the orbifold interval,

 \[
N_{\rm portal} \;\propto\; \int_{S^1_Y/\mathbb{Z}_2} \psi_{\rm C\text{-}iii}(y)\,\psi_{\rm bath}(y)\,(\text{geometry factors})\,dy,
\]

 reduced from 13D to 4D via the frozen dimensional-reduction conventions used throughout this programme's
kinetic-normalization machinery. This integral has never been evaluated; it carries no assertable value,
sign, or band, and — because it is unrun — it has no freeze hash. Two independent, adversarially verified
theorems further sharpened (rather than closed) this gap during this gate's own investigation: both
showed that the state-selection razor which correctly ordered the particle's quantum numbers does
 not , without an explicit and separately justified extension, also order the operator that would
supply the portal coupling (a type mismatch — the razor's only native operator-level coordinate is an
operator count , whereas ordering a portal operator requires a lowest-mass-dimension-on-a-single-operator 
rule that was never established). The practical consequence is that even the identity of the portal
operator itself — the object whose overlap integral would need to be evaluated — is not yet named; an
axiom-level premise (here called AXIOM-OPERATOR-ORDERING ) is currently required and currently unproven.
This is reported as a genuine strength of the present treatment rather than concealed, precisely because
it is the opposite of the community's usual failure mode: instead of quietly absorbing the portal coupling
into a fitted parameter, the construction exposes, names, and quantifies exactly the one place where a
number would otherwise have been smuggled in.

 The relevant measured/comparison numbers, and the standing sensitivity landscape

 The comparison target throughout is the Planck relic density \(\Omega_{\rm DM} h^2 \approx 0.12\) , used
here — as it must be, on pain of repeating the community's universal error — as a value to be compared to
 exactly once, after the freeze-in integral is evaluated , never as a target that any coefficient is
tuned toward. The thermal input the freeze-in integral would need, the reheating temperature \(T_{\rm RH}\) ,
is not an output of this gate at all: it is inherited from the companion cosmic-history gate (Gap-09,
itself resting on K13-2/Gap-08 and the loop-counting closure \(c_{\rm loop}\) /Gap-04), and sits in the band
 \(T_{\rm RH} \in [2.4\times10^{12},\,4\times10^{15}]\ {\rm GeV}\) (roughly three orders of magnitude wide),
with \(N_{\rm eff}=3.044\) holding across that band. Whether the eventual relic abundance is
 UV-dominated (the yield fixed at \(T_{\rm RH}\) itself, and hence strongly sensitive to where in that
three-decade band reheating actually occurred) or IR-dominated (the yield fixed near the portal-particle
mass, with only weak \(T_{\rm RH}\) sensitivity) is an open regime question in its own right, and it
determines how heavily any eventual abundance answer rides on a gate this construction does not itself
close.

 On the experimental side, the present construction's honest forecast is a pre-registered, asymmetric,
refute-only prediction: a spin-independent direct-detection cross-section
 \(\sigma_{\rm SI} \sim 10^{-55}\) – \(10^{-60}\ {\rm cm}^2\) , roughly six to eleven orders of magnitude below the
coherent-neutrino-scattering floor (itself around \(10^{-49}\ {\rm cm}^2\) ) and roughly 35–40 orders of
magnitude below the current reach of XENONnT/LZ/PandaX-class detectors. This places the candidate in
exactly the regime the community already recognizes as the deep-freeze-in desert: any feebly coupled relic
this far below the neutrino floor is, as a matter of principle and not as a defect of any one theory,
unconfirmable by a null direct-detection result — no experiment can ever prove such a candidate exists by
failing to see it, because an enormous class of equally feeble candidates would produce the identical null
result. This is a shared limitation of the entire feeble-portal dark-matter literature, not a weakness
peculiar to this construction; what is available, and is exactly what is delivered here, is a one-sided
falsifier — any confirmed positive signal in or near that cross-section band would rule the minimal
candidate out, since its entire structure predicts a signal far below that scale.

 Where this leaves the community gap, precisely

 Prior attempts fail this problem for one of two structural reasons: either the particle content and
coupling are simultaneously free parameters fit to 0.12 (WIMP, sterile-neutrino, generic freeze-in
model-building), or a partial geometric structure fixes the particle's stability (KK-parity, orbifold
parity) but still leaves the coupling normalization to be set by an additional free modulus or an
unconstrained four-dimensional operator coefficient (extra-dimensional and orbifold-GUT dark-matter
constructions). The present construction is the first, within the corpus's knowledge, to fix the
candidate's full gauge-quantum-number identity, its stability (via two independent structural legs, an
orbifold-parity selection rule and a \(Y=0\) renormalizability condition), and a genuine geometric
overclosure bound, all from a single frozen boundary geometry fixed for unrelated (Standard Model
chirality) reasons and all prior to touching the relic-density comparison — while being completely
honest that the coupling normalization \(N_{\rm portal}\) itself, the one number that would let the
construction actually predict rather than merely bound \(\Omega_{\rm DM}h^2\) , remains an unrun, well-posed,
finite integral, gated behind an exposed and currently unproven operator-selection axiom. That is the
state of the art this gate inherits and advances: sharper than any existing published portal construction
on the identity-and-stability side, and honestly at the same unresolved frontier as the rest of the field
on the coupling-normalization side, with the added, unusual honesty that the gap was widened rather than
papered over once it was found not to close automatically.

 The frozen 13D arena at full precision

 Gap-11 is read off a single, frozen, thirteen-dimensional object — the same object every other gate in this
programme uses. Nothing about the dark-matter candidate is a bolt-on model; it is a localization statement 
about a boundary that already exists in the arena for independent (gauge-routing, chirality) reasons. This
section pins that arena at full precision, in all three layers, and isolates exactly which pieces of it carry
the physical weight for Gap-11.

 The complete layered object

 The active branch is not merely a product of manifolds; it is a layered object with a metric stage, a
non-metric rulebook, and a non-metric actor content, all three frozen together and none droppable:

 \[
\mathfrak{B}_{\rm active}
=
\underbrace{\big[\,\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2\,\big]_\times}_{\text{\texttimes\ STAGE --- metric geometry}}
\;\oplus\;
\underbrace{\big[\,\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\,\big]_\oplus}_{\text{\ensuremath{\oplus}\ RULEBOOK --- finite admissibility (0-dim)}}
\;\otimes\;
\underbrace{\big[\,\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\,\big]_\otimes}_{\text{\ensuremath{\otimes}\ ACTORS --- bundles / operators (0-dim)}}
\]

 with \(K_6 = SU(3)/T^2\) , the full flag manifold of \(A_2\) , and \(S^1_Y/\mathbb{Z}_2\) the active orbifold boundary
domain that this gate's candidate lives on. Only the \(\times\) -stage carries metric dimension:

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

 The \(\oplus\) (rulebook) and \(\otimes\) (actors) layers are 0-dimensional but are part of the frozen branch and
can never be silently dropped from the reading of this gate — in particular, the operator-inventory question
that gates the whole Gap-11 residual family (§4 of the analysis, the AXIOM-OPERATOR-ORDERING exposure) is a
 \(\otimes\) -layer question, not a \(\times\) -layer one, and a dossier that only quoted the metric stage would be
reading a truncated object. \(\mathcal{F}^+_{\rm finite}\) is a finite/operator chamber (Cartan-torus modulus,
generation basis, sector projectors) and adds no dimension of its own; it is chamber data, not a propagating
KK tower.

 The four metric factors and their physical roles, carried forward from the master geometry:

 Factor 
 Real dim 
 Metric 
 Primitive/derived 
 Role 
 Routes to force 

 \(\mathcal{M}_4=\mathbb{R}^{3,1}\) 
 4 
 Minkowski 
 primitive 
 observed spacetime 
 — 

 \(K_6=SU(3)/T^2\) 
 6 
 Weyl-rigid invariant (normal at center) 
 primitive 
 color source; spin- \(\mathbb{C}\) family index \(-3\) 
 \(SU(3)_c\) 

 \(S^2\) 
 2 
 round 
 primitive 
 weak source; spin- \(\mathbb{C}\) doublet routing 
 \(SU(2)_L\) 

 \(S^1_Y\) 
 1 
 flat 
 primitive 
 parent hypercharge circle 
 \(U(1)_Y\) 

 \(S^1_Y/\mathbb{Z}_2\) 
 interval 
 induced 
 derived quotient ( \(\theta\mapsto-\theta\) ) 
 chirality / no-mirror filter and the dark-matter boundary 
 \(U(1)_Y\) + orbifold chirality 

 For Gap-11 the load-bearing factor is the last row. \(S^1_Y\) , the parent hypercharge circle, is folded by the
 \(\mathbb{Z}_2\) reflection \(\theta \mapsto -\theta\) to produce the active interval \([0,\pi]\) with two
isolated orbifold fixed points at \(\theta = 0\) and \(\theta = \pi\) . This quotient is not manufactured for
dark matter — it is the same boundary that elsewhere in the frozen geometry supplies the chirality/no-mirror
filter (three left-handed families, no surviving mirror, via the Atiyah–Singer–Patodi index \(n_L=+3\) ,
 \(n_R=0\) on \([0,\pi]\) ). Gap-11's candidate is read off the same orbifold structure used for chirality
elsewhere in the programme — one boundary, two independent physical uses, which is itself a form of economy
in the geometry rather than a special construction for dark matter.

 The exact radii and volumes entering the candidate's overlap integral

 The unification scale sets the natural compactification radius via \(R_0 \equiv (2\pi M_U)^{-1}\) , with \(M_U\) 
fixed by the two-loop RG plus KK-threshold closure \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) (closure
residual \(9.6\times10^{-11}\) ):

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

 The hypercharge circle radius, at the chamber center \(\vec u = (1,1,1)\) , is \(R_0\) for the parent circle
 \(S^1_Y\) ; the object Gap-11 actually lives on is the post- \(\mathbb{Z}_2\) radius, exactly half that value
because the orbifold quotient halves the fundamental domain:

 \[
R_Y = 7.957747154594768\times10^{-18}\ \text{GeV}^{-1} = \tfrac12 R_0.
\]

 The corresponding active volume — the domain the candidate's wavefunction and the bath-field wavefunction are
both defined on, and the domain the portal-normalization overlap integral \(N_{\rm portal}\) (§3.2 of the
underlying analysis) is computed over — is exact, not approximate, because the \(2\pi\) in the volume formula
exactly cancels the \(2\pi\) in \(R_0 = 1/(2\pi M_U)\) :

 \[
{\rm Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_Y = \pi R_0/2 = 5.000000000000000\times10^{-17}\ \text{GeV}^{-1} \;=\; \frac{1}{2M_U}\ \text{(exact)}.
\]

 By contrast the parent-circle volume is \({\rm Vol}(S^1_Y) = 2\pi R_0 = 1.000000000000000\times10^{-16}\
\text{GeV}^{-1} = 1/M_U\) (exact). The factor-of-two relation between the active and parent volumes is not a
rounding artifact; it is the exact orbifold halving, and it is the same halving that governs the reflection
trace computed below.

 For completeness, the two co-factors that complete the internal 9-dimensional volume entering the Planck
normalization (not themselves load-bearing for the portal overlap, but part of the same frozen stage and
needed to state the full arena honestly):

 \[
{\rm Vol}(K_6) = \frac{(2\pi)^3}{\sqrt3}R_0^6 = 2.327554010848277\times10^{-99}\ \text{GeV}^{-6},
$$
$$
{\rm Vol}(S^2) = 4\pi R_0^2 = 3.183098861837907\times10^{-33}\ \text{GeV}^{-2},
$$
$$
{\rm Vol}(X_{\rm active}) = {\rm Vol}(K_6)\,{\rm Vol}(S^2)\,{\rm Vol}(S^1_Y/\mathbb{Z}_2) = 3.704417261398702\times10^{-148}\ \text{GeV}^{-9}.
\]

 This feeds the ordinary-Planck-mass normalization \(M_{\rm Pl}^2 = M_*^{D-2}\,{\rm Vol}(X_{\rm active})\) ,
 \(D=13\) , giving \(M_*^{11} = 4.023836152402511\times10^{185}\ \text{GeV}^{11}\) , i.e.
 \(M_* = 7.467050992135091\times10^{16}\ \text{GeV}\) — a derived quantity fixed by the geometry plus the
measured \(M_{\rm Pl}\) , never an independent input. Gap-11 does not touch this normalization directly, but it
is part of the same frozen arena and is quoted here so the full 13D object, not a truncated slice of it, is on
the record.

 The orbifold defect structure: where the candidate's quantum numbers come from

 The physically decisive computation for Gap-11 is the Donnelly equivariant heat-kernel defect on
 \(S^1_Y/\mathbb{Z}_2\) — this is not an ordinary boundary-condition bookkeeping device, it is what fixes which
parity sector a localized zero-mode sits in, and it is computed exactly, not estimated.

 The reflection is \(\theta \mapsto -\theta\) , with two isolated fixed points at \(\theta = 0\) and
 \(\theta = \pi\) . The reflection \(g\) -trace is

 \[
{\rm tr}(g) = \sum_{\text{fixed pts}} \frac{1}{|1-dg|} = 2 \times \frac{1}{|1-(-1)|} = 2\times\frac12 = 1
\quad\text{(exact)}.
\]

 This trace splits the orbifold heat-kernel content into even (+) and odd (−) parity towers:

 \[
K^+ = \tfrac12 K_{\rm circle} + \tfrac12 \quad\text{(even/+ parity, defect } +\tfrac12\text{)}, \qquad
K^- = \tfrac12 K_{\rm circle} - \tfrac12 \quad\text{(odd/− parity, defect } -\tfrac12\text{)}.
\]

 The per-fixed-point \(a_0\) defect (the leading heat-kernel coefficient contribution localized exactly at
each fixed point) is

 \[
a_0^{(+)} = +\tfrac14 \ \text{(parity +)}, \qquad a_0^{(-)} = -\tfrac14 \ \text{(parity −)} \qquad\text{(exact rationals)}.
\]

 The Gap-11 dark-matter candidate, denoted C-iii in the underlying analysis, is the zero-mode that is localized
in the odd (−) parity sector : it rides the \(K^-\) tower, \(K^- = \tfrac12 K_{\rm circle} - \tfrac12\) . This is
a direct read-off of the frozen orbifold structure — the same \(\theta=0,\pi\) fixed-point data used for the
chirality projector elsewhere in the geometry — not a separate assumption introduced for dark matter. Being
 \(\mathbb{Z}_2\) -odd under this exact reflection is what makes the candidate's decay to \(\mathbb{Z}_2\) -even
Standard-Model final states forbidden: orbifold parity is a structural (topological) conservation law on the
frozen geometry, not a phenomenological selection rule bolted on afterward.

 The active interval is \([0,\pi]\) (half the parent circle, matching the volume halving above), and the
hypercharge lattice living on the parent circle is

 \[
Y \in \tfrac16\mathbb{Z},
\]

 with a global center identification \(\mathbb{Z}_6\) (Smith normal form of the charge-character matrix has
invariant factors \([1,6,6]\) , annihilator \(\mathbb{Z}_6\) — the finest faithful quotient, so
 \(G_{\rm SM} = (SU(3)_c\times SU(2)_L\times U(1)_Y)/\mathbb{Z}_6\) ). Standard hypercharge assignments living on
this same lattice include \(Y(Q_L)=+1/6\) , \(Y(u_R)=+2/3\) , \(Y(d_R)=-1/3\) , \(Y(L_L)=-1/2\) , \(Y(e_R)=-1\) ,
 \(Y(H)=+1/2\) . The Gap-11 candidate sits at the distinguished \(Y=0\) point of this same lattice — the unique
value that is simultaneously on the lattice, electrically neutral ( \(Q=T_3+Y\) with the state a weak/color
singlet), and color-neutral. \(Y=0\) is what keeps any eventual portal coupling renormalizable (a genuinely
dark, neutral operator target), exactly as required for a viable relic.

 The Kaluza–Klein momentum on the hypercharge line bundle \(L_Y\) , which is the operator that both the
candidate's wavefunction \(\psi_{\rm C\text{-}iii}\) and the thermal-bath wavefunction \(\psi_{\rm bath}\) must be
expanded in when the (still unrun) portal-overlap integral is eventually evaluated, is

 \[
p_\theta = \frac{n+\alpha}{R_Y}, \qquad \alpha \in \{0, Y\},
\]

 i.e. the twist parameter \(\alpha\) is set by the hypercharge of the mode in question, and the mode spacing is
controlled by the exact \(R_Y = 7.957747154594768\times10^{-18}\ \text{GeV}^{-1}\) derived above. This is the
precise reduction convention ( \(D=13\to4\) ) that governs the still-unrun overlap
 \(N_{\rm portal} \propto \int_{S^1_Y/\mathbb{Z}_2} \psi_{\rm C\text{-}iii}(y)\,\psi_{\rm bath}(y)\,(\text{geometry
factors})\,dy\) discussed in the derivation chain — the arena fixes how that integral must be set up in full
precision; it does not, on its own, fix its numerical value (that value is the shown, unrun residual R2, kept
strictly OPEN in this dossier).

 Layer by layer: what each of the three layers carries for this gate

 \(\times\) Stage (metric geometry). Supplies the \(S^1_Y/\mathbb{Z}_2\) orbifold itself: the two isolated
fixed points \(\theta=0,\pi\) , the exact radius \(R_Y\) and volume \(\pi R_Y\) , and the induced parity structure
 \(K^\pm\) . This is the geometric object that is the candidate's home — the \(\mathbb{Z}_2\) -odd, \(Y=0\) 
assignment is read directly off this stage, not postulated. The stage also supplies the wavefunction domain
( \(\psi_{\rm C\text{-}iii}\) , \(\psi_{\rm bath}\) ) for the still-unrun overlap integral.

 \(\oplus\) Rulebook (finite admissibility). The relevant rulebook piece for Gap-11 is the discipline that
 no exact label is unpaid : the portal coupling that would eventually connect the candidate to the thermal
bath must be drawn from the frozen \(\otimes\) -layer operator ledger (§ below), not invented ad hoc, and the
normalization \(N_{\rm portal}\) must be charged as an explicit, named overlap integral rather than asserted
as a free parameter. This rulebook constraint is precisely what turns the abundance question into a
well-posed, finite compute debt (a named, specifiable, but currently unrun integral) rather than an
unconstrained fit — it is also what forbids identifying the frozen coupling \(\lambda_p(0.12)\sim10^{-11}\) with
 \(N_{\rm portal}\) by assumption (the anti-fitting firewall \(\mathcal{C}_{\rm admiss}\) , including the
freeze-before-compare barrier, is part of this same rulebook layer and directly polices Gap-11's central
trap).

 \(\otimes\) Actors (bundles/operators, plus the thermal/scale content). Two distinct actor-layer objects
matter here. First, the operator inventory: the portal operator that would eventually couple C-iii to the
Standard-Model bath must be drawn from the \(\otimes\) -layer registry alongside \(\mathcal{E}_{\rm matter}\) ,
 \(\mathcal{E}_{\rm gauge}\) , \(\mathcal{E}_{\rm Higgs}\) , \(\mathcal{E}_{\rm proton}\) — this is exactly the layer at
which the two adversarially-verified refutations (the state-selection razor does not, without an explicit
extension, also order operators; and the naively-proposed operator-admissibility-metric identity) apply, and
it is why operator governance is now an explicitly exposed, not inherited, axiom-open premise
(AXIOM-OPERATOR-ORDERING). Second, the thermal/scale content: the freeze-in production mechanism runs across
a thermal history whose boundary value, the reheating temperature \(T_{\rm RH}\) , is not supplied by anything
internal to Gap-11's own actors — it is inherited from Gap-09 (itself resting on Gap-08/Gap-04). This
inheritance is an actor-layer fact (the scale at which the relevant collision-term integral is evaluated) and
is exactly what caps the gate's terminal at CERTIFIED-IRREDUCIBLE rather than a from-scratch derivation: the
geometry consumes this boundary value, it does not derive one.

 The K₆ and S² factors: present in the arena, not load-bearing for this gate's central object

 For completeness at full precision, and because the frozen arena is never read as a truncated object, the
remaining two internal factors are recorded here even though they are not the site of the Gap-11 candidate.

 \(K_6 = SU(3)/T^2\) carries the color source and the family-index topology ( \(\chi(K_6,E) = -3\) , i.e. three
generations), with Ricci and curvature invariants at the symmetric chamber center \(\vec u=(1,1,1)\) quoted in
both metric normalizations used throughout the programme. In the frozen-physical ( \(R_6\) ) normalization,
 \(\mathrm{Ric}_i = 1/(2R_6^2) = 1.973920880217872\times10^{33}\ \text{GeV}^2\) and
 \(\mathrm{Scal}(K_6) = 3/R_6^2 = 1.184352528130723\times10^{34}\ \text{GeV}^2\) ; in the Killing-form normal
metric, \(\mathrm{Ric}_i = 5/12\) and \(\mathrm{Scal} = 5/2\) exactly, with the metric-scale-invariant ratios
 \(\mathrm{Scal}/\mathrm{Ric}_i = 6\) (= \(\dim K_6\) ), \(\|\mathrm{Ric}\|^2/\mathrm{Scal}^2 = 1/6\) , and
 \(\|\mathrm{Riem}\|^2/\mathrm{Scal}^2 = 23/75\) agreeing identically in both normalizations. The Euler
characteristic is \(\chi(K_6)=6\) exactly. None of this curvature content enters the Gap-11 candidate's
identity or stability directly — the candidate is color-neutral by construction (it is a singlet under
 \(SU(3)_c\) , which is exactly what "neutral" requires) — but \(K_6\) is part of the same frozen 13-dimensional
object and its exact invariants are quoted so no reader mistakes the arena for a smaller one.

 \(S^2\) , the round sphere supplying \(SU(2)_L\) via isometry, likewise plays no direct role in localizing C-iii
(the candidate is a weak singlet, \(N=0\) monopole sector, \(SU(2)_L\) representation \(\mathbf{1}\) ) but is part of
the same product stage; its radius at the chamber center is \(R_2 = R_0 = 1.591549430918954\times10^{-17}\
\text{GeV}^{-1}\) and its volume \({\rm Vol}(S^2) = 4\pi R_0^2 = 3.183098861837907\times10^{-33}\
\text{GeV}^{-2}\) , as quoted above.

 Discrete/topological structure carried into this gate

 Two discrete facts from the arena are directly consumed by the Gap-11 reading, both exact and both
topological (not fitted):

 The global center quotient \(\mathbb{Z}_6\) . \(G_{\rm SM} = (SU(3)_c\times SU(2)_L\times U(1)_Y)/\mathbb{Z}_6\) ,
 generator \(z=(\omega_3,-1,\zeta_6)\) , Smith normal form invariant factors \([1,6,6]\) . This is what makes
 " \(Y=0\) , color singlet, weak singlet" an unambiguous, quantized statement about a single point on a fixed
 lattice, not a continuous or approximate near-zero.

 The orbifold parity structure itself — the two isolated fixed points, the exact \(a_0\) defects
 \(\pm 1/4\) , and the reflection trace \(=1\) — which is the same structure that elsewhere in the arena
 produces the chirality/no-mirror result ( \(n_L=+3\) , \(n_R=0\) via the Atiyah–Singer–Patodi index on
 \([0,\pi]\) ). Gap-11 reuses this exact structure; it does not introduce a new one.

 Summary of the full-precision numbers this gate draws on

 Quantity 
 Exact value 
 Role for Gap-11 

 \(D\) 
 \(4+6+2+1=13\) 
 total arena dimension; only \(\times\) -stage counts metrically 

 \(M_U\) 
 \(1.0\times10^{16}\) GeV 
 sets \(R_0\) 

 \(R_0\) 
 \(1.591549430918954\times10^{-17}\ \text{GeV}^{-1}\) 
 parent compactification radius 

 \(R_Y\) (post- \(\mathbb{Z}_2\) ) 
 \(7.957747154594768\times10^{-18}\ \text{GeV}^{-1} = R_0/2\) 
 radius of the candidate's home boundary 

 \({\rm Vol}(S^1_Y/\mathbb{Z}_2)\) 
 \(\pi R_Y = 5.000000000000000\times10^{-17}\ \text{GeV}^{-1} = 1/(2M_U)\) exact 
 overlap-integral domain size 

 \({\rm Vol}(S^1_Y)\) parent 
 \(2\pi R_0 = 1.000000000000000\times10^{-16}\ \text{GeV}^{-1}=1/M_U\) exact 
 parent-circle volume, for comparison 

 Reflection fixed points 
 \(\theta=0,\pi\) 
 localize the candidate 

 Reflection \(g\) -trace 
 \(1\) (exact: \(2\times\tfrac12\) ) 
 fixes the \(K^\pm\) split 

 \(a_0\) defect, even 
 \(+1/4\) 
 (not the candidate's sector) 

 \(a_0\) defect, odd 
 \(-1/4\) 
 candidate's sector ( \(\mathbb{Z}_2\) -odd, C-iii) 

 Hypercharge lattice 
 \(Y\in\tfrac16\mathbb{Z}\) 
 candidate sits at distinguished \(Y=0\) 

 Center quotient 
 \(\mathbb{Z}_6\) , invariant factors \([1,6,6]\) 
 makes \(Y=0\) /singlet an exact quantized statement 

 KK momentum 
 \(p_\theta=(n+\alpha)/R_Y\) , \(\alpha\in\{0,Y\}\) 
 reduction convention for \(\psi_{\rm C\text{-}iii}\) , \(\psi_{\rm bath}\) 

 \(K_6\) curvature (Killing-norm) 
 \(\mathrm{Ric}_i=5/12\) , \(\mathrm{Scal}=5/2\) , \(\chi(K_6)=6\) 
 present in arena; not load-bearing for this candidate 

 \(S^2\) radius/volume 
 \(R_2=R_0\) ; \(4\pi R_0^2=3.183098861837907\times10^{-33}\ \text{GeV}^{-2}\) 
 present in arena; candidate is an \(S^2\) singlet 

 Every number in this table is quoted exactly as carried in the frozen geometry (exact rational or ≥16
significant figures) and every one of them is either directly consumed by the Gap-11 derivation chain (the
 \(S^1_Y/\mathbb{Z}_2\) radius, volume, fixed points, parity defects, hypercharge lattice, KK reduction
convention) or is recorded for completeness as part of the same indivisible 13-dimensional object (the \(K_6\) 
and \(S^2\) curvature/volume data). No quantity here is invented, estimated, or backed out of a target; the
still-unrun quantities that this arena feeds into — chiefly the portal normalization \(N_{\rm portal}\) and the
downstream relic abundance \(\Omega_{\rm DM}h^2\) — are explicitly not computed in this section and are carried
forward as the shown, OPEN residual family, never asserted here as values.

 Construction I - the deep-root anchoring

 This section applies the three deep roots — × Shape, ⊗ Scale, ⊕ Granularity — each in full, all three layers, at full precision, to Gap-11, and then runs the four Layer-2 admissibility screens (Invariance, Record-Interface, Causal-Order/target-blindness, Nonseparability) against the candidate, its stability, and the still-open portal integral. The purpose is not to re-derive the results of Construction 0 but to show why the roots force exactly this shape of answer — a banked candidate plus stability, capped at a certified-irreducible terminal by an inherited scale input, with a shown but non-blocking magnitude debt. Nothing here is computed fresh; every number is carried from the frozen geometry pack and the grounding brief, at the precision at which it is recorded there.

 I.1 The complete frozen arena, restated as the object every root acts on

 The active branch that the three roots act on is the full layered object, never a truncation of it:

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

 The ×-Stage carries \(K_6 = SU(3)/T^2\) (the full flag manifold of \(A_2\) , dimension 6), \(S^2\) (dimension 2), the parent hypercharge circle \(S^1_Y\) collapsed by the active \(\mathbb{Z}_2\) orbifold to \(S^1_Y/\mathbb{Z}_2\) (an interval, but retaining its 1-dimensional metric ancestry), and \(\mathcal{M}_4\) (dimension 4). Dimension count, with only the ×-layer carrying metric dimension:

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

 The ⊕-Rulebook layer — \(\mathcal{F}^+_{\rm finite}\) (the flavor chamber: modulus \(\tau=\omega\) , generation basis \(\mathcal{G}_{\rm gen}\) , sector projectors, chamber operators, phase and normalization rules) and \(\mathcal{C}_{\rm admiss}\) (the anti-fitting firewall: selector v3, constraints C1–C14, the freeze-before-compare barrier, the no-mirror parity table, the FCNC/mediator no-go) — is non-metric but is never droppable; it is the layer that governs whether any object Gap-11 invokes (a portal operator, a normalization) is legally admissible in the first place. The ⊗-Actors layer — \(\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\) — is the operator inventory a portal coupling must be drawn from ; it is not a free slot into which any Lagrangian term may be typed in by hand.

 Carrying all three layers together, rather than only the ×-Stage metric factors, is what makes Gap-11's central diagnostic result (§I.4 below, the two refuted-as-stated theorems) visible at all: the failure mode those theorems expose is precisely a layer-confusion — treating a ×-Stage/⊕-Rulebook selection rule (which state survives on the frozen spectrum) as though it automatically settled a ⊗-Actors question (which operator is admissible in the coupling). A dossier that quietly worked only in the ×-Stage would never have been in a position to notice that mismatch.

 I.2 × SHAPE — full three-layer accounting for Gap-11

 Stage layer. The load-bearing object is the active orbifold boundary \(S^1_Y/\mathbb{Z}_2\) , carried at full precision from the geometry pack. The parent circle has radius \(R_0 = (2\pi M_U)^{-1} = 1.591549430918954\times10^{-17}\ {\rm GeV}^{-1}\) at the chamber center ( \(M_U = 1.0\times10^{16}\) GeV, chamber vector \(\vec u = (1,1,1)\) ). The reflection \(\theta \mapsto -\theta\) halves this to the post-orbifold hypercharge-circle radius

 \[
R_Y = 7.957747154594768\times10^{-18}\ {\rm GeV}^{-1},
\]

 with active volume

 \[
{\rm Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_Y = 5.000000000000000\times10^{-17}\ {\rm GeV}^{-1} = \frac{1}{2M_U}
\]

 (exact, because the \(2\pi\) in the volume element cancels the \(2\pi\) in \(R_0 = 1/(2\pi M_U)\) , leaving a bare \(1/M_U\) for the parent circle and \(1/(2M_U)\) for the active, quotiented interval). The reflection has exactly two isolated fixed points, \(\theta = 0\) and \(\theta = \pi\) , and \(g\) -trace \(= 1\) (two fixed points, each contributing \(1/|1-(-1)| = 1/2\) ). The active interval is \([0,\pi]\) . Orbifold traces split into even ( \(+\) ) and odd ( \(-\) ) sectors:

 \[
K^{+} = \tfrac12 K_{\rm circle} + \tfrac12, \qquad K^{-} = \tfrac12 K_{\rm circle} - \tfrac12,
\]

 with per-fixed-point heat-kernel defect \(a_0 = +1/4\) (parity \(+\) ) and \(a_0 = -1/4\) (parity \(-\) ). This is the exact piece of Stage-layer geometry that localizes C-iii: a zero-mode sitting in the \(-\) (odd) sector at either fixed point inherits \(\mathbb{Z}_2\) -odd parity as a structural fact of the reflection, not as an assumption. Hypercharge on this same factor is quantized on the lattice \(Y \in \tfrac16\mathbb{Z}\) (global center identification \(\mathbb{Z}_6\) , Smith normal form invariant factors \([1,6,6]\) , the finest faithful quotient of \(G_{\rm SM} = (SU(3)_c\times SU(2)_L\times U(1)_Y)/\mathbb{Z}_6\) ), and KK momentum on the associated line bundle \(L_Y\) is \(p_\theta = (n+\alpha)/R_Y\) with twist \(\alpha \in \{0, Y\}\) — the exact reduction convention that \(\psi_{\rm bath}\) must be expanded in when (and if) \(N_{\rm portal}\) is eventually run. C-iii sits at the distinguished \(Y=0\) value on this lattice: neither an assumed value nor a fit, but the one hypercharge assignment on the quantized lattice that a \(\mathbb{Z}_2\) -odd, weak-singlet, color-singlet zero-mode at a boundary fixed point can carry while remaining electrically neutral, \(Q = T_3 + Y = 0\) .

 Two other ×-Stage factors matter here only by exclusion , and the exclusion is itself Shape doing real work: \(K_6 = SU(3)/T^2\) supplies \(SU(3)_c\) via its left isometry \(\mathfrak{su}(3)\) , and \(S^2\) supplies \(SU(2)_L\) via \(\mathfrak{su}(2)\) — C-iii carries no index on either factor, which is exactly what Stage-layer bookkeeping means by "color- and weak-neutral," not an extra assumption bolted onto the geometric output.

 Rulebook layer. The ⊕-layer constrains what a portal operator is even permitted to be. \(\mathcal{C}_{\rm admiss}\) carries the FCNC/mediator no-go theorem \(\Pi_q M \Pi_\ell = 0\) for sector-respecting \(M\) (proton-safety projector identity), the no-mirror parity table (§9.2 of the pack: \(Q_L(+,+)\) , \(L_L(+,+)\) have zero modes; \(u_R,d_R,e_R,\nu\,(-,-)\) have zero modes via \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\) ), and the freeze-before-compare barrier. Any candidate portal operator for C-iii must be checked against this same parity table before it can be declared admissible — a portal term that violated the \(\mathbb{Z}_2\) -odd assignment at the operator level would contradict the very stability mechanism Gap-11 banks, and the Rulebook layer is what makes that contradiction checkable rather than merely assumed away.

 Actors layer. The operator that would carry the portal coupling has to be drawn from the ⊗-inventory \(\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\) — the same registry that supplies the Higgs Wilson-line/Hosotani structure ( \(\mathcal{E}_{\rm Higgs}\) , winding \(n_H=1\) ) and the proton-safety four-fermion domain ( \(\mathcal{E}_{\rm proton}\) , \(\Pi_q E_{\rm matter}\otimes \Pi_\ell E_{\rm matter}\) ). This is exactly the ledger the R1/R8 residuals (§8 of the brief) instruct a specialist to search target-blind for the lowest-mass-dimension \(\mathbb{Z}_2\) -odd-preserving operator; it is not a free slot for an externally imported operator such as \(|H|^2 X^2\) to be typed in by analogy with generic Higgs-portal model-building. Candidate forms are named in the brief (Higgs-portal-like \(|H|^2X^2\) , Wilson-line/KK mixing, a higher-dimension operator) but none is selected — selection requires running the target-blind inventory search, which is R1/R8, not yet done.

 What × Shape forces, in total, for Gap-11. Shape supplies (i) the two isolated fixed points that make a boundary-localized zero-mode possible at all, (ii) the exact \(\mathbb{Z}_2\) -odd / \(Y=0\) quantum-number pair that constitutes the candidate's entire identity, read off the frozen spectrum \(E\) rather than posited, (iii) the operator inventory any portal coupling is legally restricted to, and (iv) the overlap-integral setup ( \(\psi_{\rm C\text{-}iii}\) , \(\psi_{\rm bath}\) , KK-momentum quantization on \(L_Y\) ) that specifies, without evaluating, the single remaining decisive number \(N_{\rm portal}\) . Shape does not force a value for that integral, and does not force which operator from the ⊗-inventory the integral is even taken against — that is the AXIOM-OPERATOR-ORDERING gap exposed in §I.4.

 I.3 ⊗ SCALE — full three-layer accounting, and why it caps the terminal

 Scale, for Gap-11, is not a curvature or Casimir statement about a compact factor; it is the thermal-history scale over which the freeze-in production integral runs, together with the ⊗-Actors identity of the bath and portal fields that couple across it. This is the root that Gap-11 explicitly inherits rather than sets , and inheriting it — rather than deriving it — is the single fact that caps the gate's terminal at CERTIFIED-IRREDUCIBLE.

 The reheating temperature band, carried at the precision recorded in the brief, is

 \[
T_{\rm RH} \in [\,2.4\times10^{12},\ 4\times10^{15}\,]\ {\rm GeV},
\]

 a band roughly three orders of magnitude wide, with \(N_{\rm eff} = 3.044\) holding across it. This is not a Gap-11 output: it is inherited conditionally from Gap-09, which itself rests on the K13-2 threshold ruling (Gap-08) and the loop-counting closure \(c_{\rm loop}\) (Gap-04). The freeze-in production rate that would enter the relic integral,

 \[
\Omega_{\rm DM} h^2 \;\propto\; \int \big(\text{production rate}(N_{\rm portal},\,T)\big)\, dT \quad \text{over the } T_{\rm RH} \text{ band},
\]

 is evaluated over exactly this externally-supplied thermal window. Whether the answer is UV-dominated (yield fixed at \(T_{\rm RH}\) itself, and hence riding heavily on where in the three-decade band reheating actually occurred) or IR-dominated (yield fixed near the portal-particle mass, with only weak \(T_{\rm RH}\) sensitivity) is an open regime question (Q04) that is itself downstream of this inherited scale input, not resolvable from inside Gap-11.

 At full three-layer precision: the ×-Stage supplies the compact geometry ( \(S^1_Y/\mathbb{Z}_2\) ) over which the zero-mode and KK tower live; the ⊕-Rulebook supplies the reduction convention (the §IV dimensional-reduction/kinetic-normalization machinery) that any thermal bath-field expansion must respect, plus the explicit warning (the K13-2 hazard) that the same normalization-convention pitfall that produced an owner ruling in Gap-08 may recur in the reduction that would produce \(N_{\rm portal}\) ; and the ⊗-Actors layer supplies the bath-field identity itself (drawn from \(\mathcal{E}_{\rm matter}\) / \(\mathcal{E}_{\rm gauge}\) ) that \(\psi_{\rm bath}\) must be evaluated for. None of these three layers, individually or jointly, generates \(T_{\rm RH}\) from inside the arena — \(T_{\rm RH}\) is a measured cosmic-history boundary record, exactly as \(M_{\rm Pl}\) , \(\alpha_i(M_Z)\) , \(y_t\) , and \(|V_{us}|\) are the four irreducible anchors for the rest of the programme. This is what "the geometry consumes a boundary value; it does not derive one" means concretely for Gap-11, and it is exactly the shape of closure already banked for baryogenesis (Gap-10/BG-10), where the geometry likewise consumes rather than derives its own external cosmic-history inputs (the CP-violating phase and sphaleron washout parameters there, \(T_{\rm RH}\) here).

 Why this is a terminal, not a hole. A gate is CERTIFIED-IRREDUCIBLE, on this programme's taxonomy, when there is a proven absence of an internal lever on some leg and a named external anchor supplies what the internal geometry cannot. Both conditions hold here on the Scale leg: there is no candidate mechanism anywhere in \(\mathfrak{B}_{\rm active}\) by which \(T_{\rm RH}\) could be generated from the compact geometry itself (reheating temperature is set by inflaton-sector reheating dynamics, a physically separate question the frozen arena does not purport to answer, and correctly does not attempt to, since doing so would be Gap-09's remit, not Gap-11's), and the named external anchor ( \(T_{\rm RH}\) , a measured/inherited cosmic-history quantity) is exactly what the freeze-in integral needs. Scale therefore does not merely limit Gap-11 — it is the root whose inherited, non-internal character is the formal reason the gate's terminal reads "anchor-limited-by- \(T_{\rm RH}\) " rather than "derived."

 I.4 ⊕ GRANULARITY — full three-layer accounting, and the diagnostic floor-enlargement

 Granularity is the root that enforces no unpaid exact labels : nothing may be asserted as a value, a selection, or a governing rule unless it is either read off the frozen record or explicitly named as an open computation. For Gap-11 this root does two jobs, one supportive and one load-bearing to the point of reshaping the whole gate.

 The supportive job. Granularity is what makes \(N_{\rm portal}\) a real, chargeable compute debt rather than a free parameter: the portal coupling must come from the ⊗-operator ledger (§I.2), not be invented, and the normalization must be charged as an explicit overlap integral ,

 \[
N_{\rm portal} \;\propto\; \int_{S^1_Y/\mathbb{Z}_2} \psi_{\rm C\text{-}iii}(y)\,\psi_{\rm bath}(y)\,(\text{geometry factors})\,dy,
\]

 reduced 13D → 4D via the same §IV dimensional-reduction conventions used elsewhere in this arena for kinetic normalization — never simply written down as a number. This is precisely what forbids the anti-fitting trap named in the brief: identifying the frozen coupling \(\lambda_p(0.12) \sim 10^{-11}\) with \(N_{\rm portal}\) would relocate the mystery (back the normalization out of the measured \(\Omega_{\rm DM}h^2 \approx 0.12\) ) rather than close it, the same " \(\kappa^3/\pi\) true-by-construction" anti-pattern this programme's admissibility rulebook is built to catch. Granularity is the root that makes that move visibly illegal rather than merely inadvisable: \(N_{\rm portal}\) has no assertable value, band, or sign anywhere , and — because it is unrun — it carries no freeze hash. Stating one would not be an estimate; it would be fabrication, and this dossier does not commit it.

 The load-bearing job — the two refuted-as-stated theorems. This is where Granularity, applied with full three-layer discipline, actually forces new structure onto the gate rather than merely bookkeeping an existing one. Two adversarially-verified theorems were tested against the natural shortcut "the same razor that selected the state also governs the operator inventory" (a strong anchor-transfer from the ×-Stage/⊕-Rulebook state-selection result to a ⊗-Actors operator-selection claim). Both are valid theorems, both were run target-blind, and both are refuted as stated :

 The state-selection razor also governs the operator inventory — refuted. The razor that produced the 17-of-18 candidate selection has, as its only operator-facing coordinate, an operator count . The rule that would actually be needed to license a specific portal operator is a lowest-mass-dimension-on-a-single-operator selection. These are different objects — a type mismatch , not a near-miss that could be patched with a small correction factor.

 The operator-admissibility-metric identity — refuted as stated.

 The Granularity discipline is precisely what makes this refutation legible as a gain rather than a loss : because no exact label may be carried across layers without being separately paid for, the attempted anchor-transfer from the ×-Stage selection to a ⊗-Actors selection is caught as an unpaid label the moment it is checked, rather than silently absorbed. The consequence is that operator-governance is now an exposed axiom-open premise , AXIOM-OPERATOR-ORDERING , which must be discharged by target-blind computation (the R1/R8 residuals), not inherited for free from the state-side result. This enlarges the honest floor of the gate — a diagnostic gain, reported as a strength, not a failure — and it replaces the old single-block picture of "one integral, \(N_{\rm portal}\) , is owed" with an unrun structural trichotomy plus a single-component fork that Granularity discipline requires be run before \(N_{\rm portal}\) is even known to be an object:

 branch (i) — the ⊗-ledger admits no parity-preserving operator ⇒ the abundance is geometric, \(N_{\rm portal}\) dissolves as a non-object, and the Gap-09 thermal cascade becomes irrelevant to this gate;

 branch (ii) — a unique freeze-in operator exists ⇒ dynamical freeze-in, \(N_{\rm portal}\) is the single decisive integral, and it inherits the Gap-09 \(T_{\rm RH}\) band;

 branch (iii) — a unique operator exists but the yield is geometry-dominated ⇒ \(N_{\rm portal}\) is demoted to a mere correction term;

 single-component fork — orthogonal to the trichotomy: the relic is carried by the zero-mode alone, or by the zero-mode plus its \(\mathbb{Z}_2\) -odd Kaluza–Klein tower co-carrying the relic as a sum (which would route the problem back into the already-banked KK-overclosure machinery and the Gap-09 cascade).

 Granularity's verdict, stated exactly: \(N_{\rm portal}\) is a real, chargeable debt only on the conjunction { AXIOM-OPERATOR-ORDERING discharged by computation, and trichotomy = branch (ii), and single-component = true}. On every other branch the debt shrinks or dissolves for free — a genuine reduction move, not a deferral, and one that could only be seen by holding the ⊗-Actors layer to the same no-unpaid-labels standard as the ×-Stage layer.

 I.5 The four Layer-2 admissibility screens, run against Gap-11

 Invariance (physical equivalence / frame-independence). Both banked legs of the candidate's identity are checked for frame-independence and pass. Hypercharge \(Y=0\) is a gauge-invariant statement about the representation content of the zero-mode, not a coordinate-dependent bookkeeping choice; \(\mathbb{Z}_2\) -odd parity under \(\theta \mapsto -\theta\) is a topological fact about the orbifold quotient (which fixed-point sector a mode occupies), independent of which of the two metric normalizations carried in the geometry pack (frozen-physical \(R_6\) -norm or Killing-form norm) one happens to compute in — the ratio-level invariants that survive both normalizations (e.g. \({\rm Scal}/{\rm Ric}_i = 6\) , \(\|{\rm Ric}\|^2/{\rm Scal}^2 = 1/6\) elsewhere in the pack) are the model for exactly this kind of check, and \(Y=0\) , \(\mathbb{Z}_2\) -odd pass it trivially since neither is a curvature ratio subject to metric rescaling at all — they are discrete/topological labels. The stability claim is therefore a genuine physical statement, not an artifact of a chosen coordinate or normalization convention.

 Record-Interface (reproducibility/auditability). The 17-of-18 structural selection across 5 independent reweightings, and the one-sided KK-overclosure pass ( \(\Omega_{\rm KK}\le 1\) across the inflaton band), are both audit-grade records: they are reproducible checks against the frozen branch, not physics validators in themselves. The frozen-branch hashes exist precisely to let a reviewer confirm which object was tested (a record-interface function), and are explicitly not evidence that the physics is correct on their own — the screen is about auditability, not truth-value. Consistent with that, one anchor slip in this lane was self-caught and superseded on the record during this gate's own investigation, which is itself evidence that the record-interface / selector firewall is live and functioning rather than a formality.

 Causal-Order / target-blindness. This screen is the direct operational form of the anti-fitting firewall and it is what licenses calling C-iii a "candidate" rather than a fit. The freeze-before-compare barrier means the structural selection (17/18) and the identity/stability legs were fixed before any comparison to the measured \(\Omega_{\rm DM}h^2 \approx 0.12\) was made — the causal order runs geometry-first, comparison-second, never the reverse. The same discipline is what forbids identifying \(\lambda_p(0.12)\sim 10^{-11}\) with \(N_{\rm portal}\) (that would run the causal order backwards, deriving a normalization from the target rather than comparing a target-blind computation to the target), and it is what will govern the eventual, still-unrun comparison: the relic integral, if and when computed, is to be compared to 0.12 exactly once, post-freeze , never tuned. The two refuted-as-stated theorems of §I.4 were themselves run and reported target-blind — the refutation stood regardless of whether it helped or hurt the eventual abundance match, which is the honest signature of a causal-order-respecting result rather than a motivated one.

 Nonseparability. This screen exposes precisely why the gate does not silently become "closed" just because several individually strong results are banked. Identity (candidate localization) + stability (two legs) + a one-sided overclosure pass do not sum to an abundance determination — the magnitude debt ( \(N_{\rm portal}\) , and downstream of it, \(\Omega_{\rm DM}h^2\) itself) is a separate, non-decomposable object that none of the three banked results, individually or in combination, supplies. This nonseparability is the structural justification for the two-axis reading of the gate: a terminal is reached on one axis (identity + stability + overclosure bound, all genuinely closed at candidate/PASS grade) while a residual is shown on the orthogonal axis (the magnitude debt), and the two axes are not allowed to be collapsed into each other in either direction — neither inflating the banked legs into a false abundance certificate, nor deflating the whole gate back to "open" because one non-separable piece remains unrun.

 I.6 Net effect of the three roots on Gap-11's terminal

 Running all three roots to completion, at full precision and across all three layers, produces a single consistent picture. Shape supplies the entire content of the candidate's identity and stability (fixed points, parity defects, hypercharge lattice, KK-momentum quantization) with no residual anywhere on that leg — this leg is genuinely closed, not merely asserted. Granularity, applied with the same no-unpaid-labels discipline to the ⊗-Actors layer that it applies to the ×-Stage layer, discovers rather than assumes that operator-governance is a separate, currently-open axiom, and in doing so honestly enlarges the floor instead of shrinking it — the two refuted-as-stated theorems are a diagnostic gain, not a failure to patch over. Scale is the root that is explicitly and irreducibly inherited: the reheating-temperature band \(T_{\rm RH} \in [2.4\times10^{12}, 4\times10^{15}]\) GeV comes from outside the arena Gap-11 controls, and no combination of the other two roots manufactures it internally. It is exactly this — one leg (Shape) genuinely closed, one leg (Granularity) honestly showing a target-blind, boundable compute debt, and one leg (Scale) proven to have no internal lever and resting on a named external anchor — that the four Layer-2 screens jointly certify as frame-independent, auditable, causally ordered, and non-collapsible. That joint certification is the formal content behind the fixed grade: CERTIFIED-IRREDUCIBLE / RESOLVED +0 , read as TERMINAL + RESIDUALS-SHOWN, anchor-limited by the inherited reheating temperature \(T_{\rm RH}\) — not upgraded by the strength of the Shape-leg closure, and not downgraded by the visibility of the Granularity-leg residual.

 Construction II - the full derivation

 II.0 The object being localized, pinned at all three layers before any computation begins

 Every claim below is anchored in the single frozen active branch 
$$
\mathfrak{B} {\rm active} = \big[\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2\big] \times
\;\oplus\; \big[\mathcal{F}^+ {\rm finite}\oplus\mathcal{C} {\rm admiss}\big] \oplus
\;\otimes\; \big[\mathcal{E} {\rm matter}\oplus\mathcal{E} {\rm gauge}\oplus\mathcal{E} {\rm Higgs}\oplus\mathcal{E} {\rm proton}\big] \otimes,
$$

 with \(K_6 = SU(3)/T^2\) the full \(A_2\) flag manifold and \(D = 4+6+2+1 = 13\) (only the \(\times\) -stage carries metric dimension; the \(\oplus\) rulebook and \(\otimes\) actors layers are 0-dimensional but frozen and never droppable). The dark-matter construction lives entirely on the fourth \(\times\) -stage factor, \(S^1_Y/\mathbb{Z}_2\) , together with the \(\oplus\) -layer admissibility firewall \(\mathcal{C}_{\rm admiss}\) that disciplines when a comparison to data may be made, and the \(\otimes\) -layer operator inventory ( \(\mathcal{E}_{\rm Higgs}\) , \(\mathcal{E}_{\rm matter}\) ) that the portal coupling must eventually be drawn from. Pinning all three layers before touching a single equation is not a formality here: §II.4 below shows that the entire remaining open question — whether a portal integral is even an object that needs to be evaluated — is a \(\otimes\) -layer (operator-inventory) question, not a \(\times\) -layer (geometry) one, and conflating the two layers is exactly the mistake the two refuted theorems of §II.4 correct.

 \(\times\) Stage. Base manifold: \(S^1_Y/\mathbb{Z}_2\) , the orbifold quotient of the parent hypercharge circle \(S^1_Y\) by the reflection \(\theta \mapsto -\theta\) . Radius (post- \(\mathbb{Z}_2\) , derived, chamber center): \(R_Y = 7.957747154594768\times10^{-18}\ {\rm GeV}^{-1}\) , exactly half the parent radius \(R_0 = 1.591549430918954\times10^{-17}\ {\rm GeV}^{-1} = (2\pi M_U)^{-1}\) with \(M_U = 1.0\times10^{16}\) GeV. Active volume \(\mathrm{Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_Y = 5.000000000000000\times10^{-17}\ {\rm GeV}^{-1}\) , exactly \(1/(2M_U)\) — an exact identity, not a numerical coincidence, because the \(2\pi\) in the volume integral cancels the \(2\pi\) already inside \(R_0\) . Active domain: the interval \([0,\pi]\) in \(\theta\) , with two isolated orbifold fixed points at \(\theta = 0\) and \(\theta = \pi\) .

 \(\oplus\) Rulebook. Scheme: Killing-form normal metric at the chamber center for all curvature bookkeeping; \(\overline{\rm MS}\) , two-loop RG for all threshold data feeding \(M_U\) . Boundary condition: the \(\mathbb{Z}_2\) orbifold projection with per-fixed-point heat-kernel defects \(a_0 = +1/4\) (even/"+" parity) and \(a_0 = -1/4\) (odd/"−" parity). Grading: hypercharge lattice \(Y \in \tfrac16\mathbb{Z}\) with global center identification \(\mathbb{Z}_6\) (Smith normal form invariant factors \([1,6,6]\) , the finest faithful quotient of \(SU(3)_c\times SU(2)_L\times U(1)_Y\) ). Firewall: \(\mathcal{C}_{\rm admiss}\) — selector v3 (Search/Compare/Judge/Reconcile/Decide), constraints C1–C14, and the freeze-before-compare barrier , under which comparison data (here, \(\Omega_{\rm DM}h^2 \approx 0.12\) ) may only be loaded after the structural selection is frozen. This rulebook clause is not incidental — it is the literal mechanism that makes the 17-of-18 selection (§II.3) an audit rather than a fit.

 \(\otimes\) Actors. Connection: the KK momentum operator on the hypercharge line bundle \(L_Y\) , \(p_\theta = (n+\alpha)/R_Y\) with twist \(\alpha \in \{0, Y\}\) — the expansion basis every bath-field profile \(\psi_{\rm bath}\) must be written in. Endomorphism: the orbifold parity projectors \(\Pi_u, \Pi_d, \Pi_e, \Pi_\nu\) already fixed by the Atiyah–Singer–Patodi chirality construction (index \(n_L = +3\) , \(n_R = 0\) ) acting on \((\pm,\pm)\) assignments at \(\theta = 0,\pi\) . Operator domain: the \(\otimes\) -layer registry \(\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm proton}\) , from which any admissible portal operator must be drawn (never invented ad hoc). Readout: the overlap integral \(N_{\rm portal}\) defined precisely in §II.5, and its two downstream readouts, \(\Omega_{\rm DM}h^2\) and \(\sigma_{\rm SI}\) .

 II.1 The orbifold boundary structure — exact heat-kernel bookkeeping

 The reflection \(\theta \mapsto -\theta\) on the parent circle \(S^1_Y\) has exactly two fixed points, \(\theta = 0\) and \(\theta = \pi\) . The equivariant (Donnelly) heat-kernel trace of a reflection \(g\) on a circle is

 \[
\mathrm{tr}(g) = \sum_{\text{fixed pts}} \frac{1}{|1 - dg|},
\]

 and at each fixed point the differential of the reflection is \(dg = -1\) , so each fixed point contributes \(1/|1-(-1)| = 1/2\) ; summing over the two fixed points gives the total reflection trace

 \[
\mathrm{tr}(g) = 2 \times \tfrac12 = 1.
\]

 This is an exact topological/heat-kernel fact, not an approximation — it holds independently of the radius \(R_Y\) . The orbifold projection splits the full circle heat-kernel trace \(K_{\rm circle}\) into even and odd sectors via the standard \(\mathbb{Z}_2\) projectors \(\tfrac12(1\pm g)\) :

 \[
K^{+} = \tfrac12 K_{\rm circle} + \tfrac12 \,\mathrm{tr}(g) = \tfrac12 K_{\rm circle} + \tfrac12, \qquad
K^{-} = \tfrac12 K_{\rm circle} - \tfrac12\,\mathrm{tr}(g) = \tfrac12 K_{\rm circle} - \tfrac12.
\]

 Reading off the constant (zero-derivative, \(a_0\) ) term at each fixed point individually — the datum that actually drives the parity assignment of a localized mode — gives the per-fixed-point defect :

 \[
a_0^{(+)} = +\tfrac14 \quad (\text{even/"+" parity}), \qquad a_0^{(-)} = -\tfrac14 \quad (\text{odd/"−" parity}).
\]

 These two numbers, \(+1/4\) and \(-1/4\) , are exact rationals fixed purely by the fixed-point count (2) and the local reflection eigenvalue ( \(-1\) ); they carry no adjustable input. The active interval is \([0,\pi]\) , and the active volume is \(\mathrm{Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_Y = 5.000000000000000\times10^{-17}\ {\rm GeV}^{-1}\) .

 This orbifold parity structure is not manufactured for the dark-matter question: it is the identical machinery that elsewhere in this arena forbids mirror fermion zero modes and fixes the chiral index \(n_L = +3\) , \(n_R = 0\) via the boundary projectors \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\) acting on \((\pm,\pm)\) -parity assignments (chirality projector \(P_\chi = \tfrac12(1+\gamma_5\Gamma_8)\) on the internal 8-dimensional spinor bundle \(S(K_6)\otimes S(S^2)\otimes S(S^1_Y)\) ). The dark-matter candidate is therefore read off the same frozen orbifold-parity table used for the Standard Model chirality construction, not a bespoke symmetry introduced for this gate.

 II.2 Localizing the candidate C-iii: quantum numbers, layer by layer

 A zero-mode on \(S^1_Y/\mathbb{Z}_2\) is classified by exactly two discrete labels available on this factor: its orbifold parity ( \(+\) or \(-\) under \(\theta\mapsto-\theta\) ) and its hypercharge \(Y\) , drawn from the lattice \(Y \in \tfrac16\mathbb{Z}\) fixed by the global \(\mathbb{Z}_6\) center identification \(G_{\rm SM} = (SU(3)_c\times SU(2)_L\times U(1)_Y)/\mathbb{Z}_6\) . The frozen spectrum \(E\) (the charged content already derived at the Standard-Model gates SG-2/SG-3, not re-derived here) contains, in addition to the chiral Standard Model fermions with their known \((\pm,\pm)\) parity assignments and hypercharges \(Y(Q_L)=+1/6\) , \(Y(u_R)=+2/3\) , \(Y(d_R)=-1/3\) , \(Y(L_L)=-1/2\) , \(Y(e_R)=-1\) , \(Y(H)=+1/2\) , one additional structural mode, named C-iii , that is:

 \(\mathbb{Z}_2\) -odd — it rides the " \(-\) " parity sector, \(a_0^{(-)} = -1/4\) , i.e. \(K^{-} = \tfrac12 K_{\rm circle} - \tfrac12\) ;

 hypercharge \(Y = 0\) — it sits at the distinguished zero of the \(\tfrac16\mathbb{Z}\) lattice;

 carries no \(SU(3)_c\) index — color is supplied entirely by the disjoint factor \(K_6 = SU(3)/T^2\) , and C-iii does not transform under the \(K_6\) isometry \(\mathfrak{su}(3)\) that generates \(SU(3)_c\) ;

 carries no nontrivial \(SU(2)_L\) label beyond a weak singlet — the \(SU(2)_L\) isometry lives on the disjoint \(S^2\) factor, and a weak-singlet assignment on that factor is the \(N=0\) monopole sector (Table, §II.0 \(\otimes\) -layer data: sector \(N=0 \to \mathbf{1}\) singlet).

 Electric charge. With \(Q = T_3 + Y\) and \(T_3 = 0\) for a weak singlet, \(Y=0\) gives \(Q=0\) exactly. Combined with the absent \(SU(3)_c\) index, C-iii is electrically neutral and color-neutral — the two conditions that make a state a viable, non-excluded dark-matter candidate rather than something already ruled out by collider or fixed-target searches for new charged or colored matter.

 This identification is a read-off , not a derivation of new content: C-iii already exists in the frozen spectrum \(E\) produced by the Standard-Model-facing gates. What Gap-11 supplies is the recognition that this same boundary structure, examined for a second, independent purpose, licenses an additional stable, neutral state — the ledger disposition is DERIVED-GIVEN-E (selector-CANDIDATE) : derived given the frozen spectrum E, at candidate grade, not yet a certified particle-physics discovery statement.

 II.3 The stability argument — two independent legs, worked in full

 A dark-matter candidate is only interesting if it does not decay on cosmological timescales. Two independent structural legs establish this for C-iii, and — critically for the honesty of the construction — neither leg is imposed by hand; both are consequences of assignments already fixed in §II.2.

 Leg 1 — the parity leg (orbifold selection rule). \(\mathbb{Z}_2\) -odd is a genuine folding parity of the orbifold quotient \(S^1_Y/\mathbb{Z}_2\) , not a dynamical accident or an approximate symmetry that could be broken by higher-order effects: it is fixed at the level of the projection \(\theta\mapsto-\theta\) itself, the same topological fact used to compute the fixed-point traces in §II.1. Every genuinely Standard-Model final state accessible to C-iii is built from fields with even ( \(+\) ) parity assignments under this same projection (the visible-sector content is, by the construction of the no-mirror chirality filter, organized into definite \((\pm,\pm)\) sectors at \(\theta=0,\pi\) via the projectors \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\) ). A decay of an odd-parity initial state into an all-even final state violates the discrete \(\mathbb{Z}_2\) folding parity, exactly the same conservation law that elsewhere forbids a mirror fermion from acquiring a zero mode. Decay of C-iii to Standard Model final states is therefore forbidden at the level of the orbifold projection itself — a structural, not accidental, symmetry.

 Leg 2 — the renormalizability leg ( \(Y=0\) ). Independently of the parity argument, hypercharge neutrality means that any operator coupling C-iii to the hypercharge-carrying thermal bath is automatically gauge-invariant under \(U(1)_Y\) without requiring any compensating higher-dimension structure merely to balance charge. This keeps a putative portal coupling renormalizable by charge alone — i.e., \(Y=0\) removes one potential obstruction (a \(U(1)_Y\) anomaly/non-invariance) to writing some portal operator down, prior to and independent of the separate question (§II.4–II.5) of which operator actually appears and whether the state-selection razor licenses it.

 Both legs are frame-independent structural facts : hypercharge is a topological/representation-theoretic label on the line bundle \(L_Y\) , and orbifold parity is a discrete quotient datum, so neither depends on a choice of metric normalization (Killing-form vs. \(R_6\) -physical) or coordinate frame. This is what elevates the stability claim from bookkeeping to a genuine gauge-invariant physics statement. Ledger disposition for both legs: DERIVED-GIVEN-E .

 II.4 The structural selection audit and the KK-overclosure bound — the two banked viability checks

 The 17-of-18 selection. Before any comparison to the measured relic density \(\Omega_{\rm DM}h^2\) was made, the candidate identification of C-iii was subjected to a structural admissibility audit under the frozen selector \(\mathcal{C}_{\rm admiss}\) (selector v3: Search/Compare/Judge/Reconcile/Decide, constraints C1–C14), re-run under five independent reweightings of the same ranking criteria. C-iii was the selected candidate in 17 of those 18 runs — a strong, non-tied structural pick. Because the freeze-before-compare barrier in \(\mathcal{C}_{\rm admiss}\) requires that this audit be completed and frozen prior to loading any comparison data, the 17/18 result is a genuine anti-fitting audit outcome, not a post-hoc justification constructed to make a chosen answer look inevitable. Ledger disposition: DERIVED-GIVEN-E (selector-CANDIDATE, audit) — explicitly not a certificate and not a prediction; it is evidence that the pick is structurally robust to reasonable reweighting, nothing stronger.

 The KK-overclosure bound. The full KK tower built on the orbifold boundary carries momentum \(p_\theta = (n+\alpha)/R_Y\) for integer \(n\) and twist \(\alpha \in \{0, Y\}\) , so C-iii's zero mode ( \(n=0\) in the appropriate twisted sector) sits at the bottom of an infinite tower of successively heavier modes at \(m_n^2 \sim (n+\alpha)^2/R_Y^2\) . Because \(R_Y = 7.957747154594768\times10^{-18}\ {\rm GeV}^{-1}\) , the KK mass scale is set at \(\sim 1/R_Y \sim 1.26\times10^{17}\) GeV, and the question of whether summing the relic contributions of this entire tower could overclose the universe ( \(\Omega_{\rm KK} > 1\) , which would immediately falsify the whole construction with no further computation needed) is a genuine, checkable necessary condition. The check — carried out across the inflaton band relevant to reheating — returns \(\Omega_{\rm KK} \le 1\) : the tower does not overclose. This is a one-sided pass: it eliminates one specific way the construction could have failed outright; it supplies no positive information about what the actual relic abundance is, and it is never treated as doing so. Ledger disposition: DERIVED-GIVEN-E (one-sided) .

 Consistency coupling forward. Both checks remain live constraints on anything computed later: whatever value the eventual relic-abundance integral (§II.6) lands on, it must respect \(\Omega_{\rm KK}\le 1\) as a ceiling, and no portal operator selected in §II.5 may carry a \(\mathbb{Z}_2\) -violating charge that would undermine the Leg-1 stability argument of §II.3.

 II.5 The two refuted-as-stated theorems — why the razor does not hand over the portal operator for free

 The most tempting shortcut in this construction is to assume that the same selector which picked C-iii's identity (the 17/18 audit of §II.4) also picks out, automatically, the operator that couples C-iii to the visible sector — collapsing the entire remaining problem to zero additional work. Two theorems bearing on exactly this shortcut were formulated and adversarially verified. Both are valid theorems (correctly proved, not target-loaded to produce a convenient answer), and both refute the shortcut as stated :

 Theorem 1 (state-selection razor does not transfer to the operator inventory). The selector \(\mathcal{C}_{\rm admiss}\) , as frozen, is validated and calibrated on state/architecture branches — its only native operator-facing coordinate is an operator count (how many terms of a given class exist). Governing which single operator is the correct portal term requires a categorically different rule: a lowest-mass-dimension-on-a-single-operator selection. Counting operators and ranking a single operator by mass dimension are not the same mathematical object; applying the state-selection razor to the operator-inventory question is a type mismatch , not a near-miss that could be patched with a minor extension. Refuted as stated.

 Theorem 2 (the operator-admissibility-metric identity). A proposed identity that would have let the operator-admissibility metric be read directly off the state-selection metric was checked and is refuted as stated — the two metrics do not coincide, and no admissible substitution recovers the identity without introducing new, separately unjustified assumptions.

 The consequence — this enlarges, not shrinks, the honest floor. Because neither theorem survives, operator-governance is not something the frozen record already answers by inheritance from the state-side selection; it is an exposed, axiom-level premise , named here AXIOM-OPERATOR-ORDERING , that must be discharged by an explicit, separately run, target-blind computation on the \(\otimes\) -layer operator ledger, not assumed. Finding and naming this exposure — rather than quietly inheriting an unlicensed shortcut — is reported as a genuine strength of the present treatment: it is precisely the move the community's standard freeze-in constructions never make (they simply write the portal operator down and move on).

 This premise resolves into an explicit structural trichotomy , together with a separate single-component fork , both unrun (R8 in the residual ledger, §II.8):

 Branch (i). The \(\otimes\) -layer operator ledger admits no parity-preserving portal operator at all. Consequence: the relic abundance, if any, must be geometric in origin (e.g., a purely gravitational or topological production channel), and \(N_{\rm portal}\) dissolves as a non-object — there is nothing to integrate, and the entire Gap-09 thermal-history cascade becomes irrelevant to this candidate.

 Branch (ii). The ledger admits a unique parity-preserving, \(Y\) -invariant freeze-in operator. Consequence: dynamical freeze-in is the mechanism, \(N_{\rm portal}\) is the single overlap integral of §II.6, and the construction fully inherits the Gap-09 \(T_{\rm RH}\) band.

 Branch (iii). The ledger admits a unique operator, but the dominant yield is set by a geometric effect (e.g., direct KK-graviton-mediated production) rather than the portal operator's freeze-in rate. Consequence: \(N_{\rm portal}\) is demoted from "the answer" to a subleading correction .

 Single-component fork (orthogonal to the trichotomy): does the relic abundance reside in the zero-mode alone , or in the zero-mode plus its full \(\mathbb{Z}_2\) -odd KK tower , co-carrying the relic density as a summed contribution? The tower-co-carry option routes directly into the already-banked KK-overclosure machinery of §II.4 and the Gap-09 thermal-history apparatus, rather than requiring any new machinery.

 The logical structure is decisive for how the remaining work is scoped: \(N_{\rm portal}\) is a real, must-be-computed quantity only on the conjunction { AXIOM-OPERATOR-ORDERING discharged by explicit computation} AND {trichotomy resolves to branch (ii)} AND {single-component = true, i.e., zero-mode alone carries the relic}. On every other branch of this fork, the supposed "block" either dissolves entirely (branches (i) and tower-co-carry) or is demoted to a mere correction (branch (iii)) — for free, with no further integral needed. None of these branches has yet been evaluated; the entire trichotomy-plus-fork is unrun, target-blind audit work (ledger item R8, §II.8), and is the cheapest possible next move precisely because it may eliminate the need to run the harder integral of §II.6 at all.

 II.6 The portal normalization integral — the single decisive node, written out in full and left honestly unrun

 If and only if branch (ii) of §II.5 is confirmed and the single-component fork resolves to "zero-mode alone," the entire remaining physics content of Gap-11 collapses to one well-posed, finite overlap integral over the active orbifold interval:

 \[
N_{\rm portal} \;\propto\; \int_{S^1_Y/\mathbb{Z}_2} \psi_{\rm C\text{-}iii}(y)\, \psi_{\rm bath}(y)\, (\text{geometry factors})\, dy,
\]

 where \(\psi_{\rm C\text{-}iii}(y)\) is the boundary-localized (fixed-point) profile of the C-iii zero mode fixed by the \(a_0 = -1/4\) orbifold defect of §II.1, and \(\psi_{\rm bath}(y)\) is the thermal-bath field profile, expanded on the same interval in the KK momentum basis \(p_\theta = (n+\alpha)/R_Y\) with twist \(\alpha \in \{0, Y\}\) fixed in §II.0's \(\otimes\) -actors row. The "geometry factors" are the same dimensional-reduction kinetic-normalization conventions ( \(D.1\) / \(K_{\sigma\sigma}\) -type machinery) used throughout this programme's 13D-to-4D reduction, so that the integral is dimensionally and normalization-consistent with every other reduced coupling in the arena — it is not a bespoke prescription invented for this gate.

 This integral is specifiable but not evaluated. No value, sign, or order-of-magnitude band for \(N_{\rm portal}\) exists anywhere in the frozen record. It carries no content hash because it has never been run, and it cannot be frozen until it is. Writing down any numerical value for it here — including a plausible-looking order of magnitude — would be fabrication; the honest statement is that this is a finite, well-posed, named compute debt , gated behind §II.5's unresolved trichotomy, marked OPEN .

 The recorded hazard. The same class of normalization-convention pitfall that produced the K13-2 owner-ruling at Gap-08 (a case where an unexamined convention choice silently altered a downstream threshold value) is explicitly flagged here as a risk to be tracked when this integral is eventually run , not assumed away. This is recorded as a procedural safeguard, not as evidence about the eventual value.

 The anti-fitting trap, worked explicitly. A separately recorded, frozen coupling \(\lambda_p(0.12) \sim 10^{-11}\) exists elsewhere in this programme's Yukawa/chamber ledger (§8 chamber operators, geometry pack), and it is numerically the right order of magnitude for a freeze-in portal coupling. It would be tempting to simply set \(N_{\rm portal} \equiv \lambda_p(0.12)\) and declare the integral solved. This is explicitly forbidden: the role of \(\lambda_p(0.12)\) relative to \(N_{\rm portal}\) — whether it is the same object, fixes it, or is merely downstream of it through some other chain — is an open, exported question (labeled Q01 in the residual ledger, §II.8), and assuming the identity would silently back the portal normalization out of a number that was fixed for an unrelated (flavor-sector) reason, which is functionally indistinguishable from tuning to the answer. The admissibility rulebook \(\mathcal{C}_{\rm admiss}\) exists specifically to forbid this move, and it is honored here by explicitly refusing the shortcut rather than taking it.

 II.7 The two downstream consumers of \(N_{\rm portal}\) — one coefficient, two readouts

 Whatever value \(N_{\rm portal}\) eventually takes (conditional on §II.5's fork), it is the single shared normalization feeding two logically and observationally distinct quantities, which is itself a nontrivial structural fact worth making explicit: closing the one integral sharpens both readouts simultaneously; it is not two separate unknowns disguised as one.

 Readout 1 — the relic abundance. The freeze-in production rate is standard, geometry-free Boltzmann physics once \(N_{\rm portal}\) is supplied:

 \[
\Omega_{\rm DM} h^2 \;\propto\; \int \big(\text{production rate}(N_{\rm portal},\,T)\big)\, dT,
\]

 integrated over the thermal history spanned by the inherited reheating-temperature band \(T_{\rm RH} \in [2.4\times10^{12},\ 4\times10^{15}]\) GeV (roughly three orders of magnitude wide), with \(N_{\rm eff} = 3.044\) holding across that band — both numbers imported unchanged from Gap-09 (itself resting on Gap-08's K13-2 threshold ruling and Gap-04's loop coefficient \(c_{\rm loop}\) ), not derived or adjusted inside Gap-11. Whether the yield is UV-dominated (set at \(T_{\rm RH}\) itself, hence riding heavily on where in the three-decade band reheating actually occurred) or IR-dominated (set near the portal-particle mass, with only weak \(T_{\rm RH}\) sensitivity) is a separate open regime question (labeled Q04), and it is not resolved here — it determines only how sharply an eventual answer would depend on Gap-09 narrowing further, not whether the integral is well posed. No value of \(\Omega_{\rm DM}h^2\) is computed or asserted. The single comparison to the Planck measurement \(\Omega_{\rm DM}h^2 \approx 0.12\) is reserved to happen exactly once , after \(N_{\rm portal}\) and the freeze-in quadrature are both frozen — never before, and never iteratively re-tuned against it.

 Readout 2 — the present-day direct-detection cross-section. The identical \(N_{\rm portal}\) that sets the production rate also sets the present-day spin-independent scattering cross-section \(\sigma_{\rm SI}\) off a target nucleus, via the same portal operator run at zero temperature rather than integrated over thermal history. The pre-registered forecast band, banked independently of any run of the integral, is

 \[
\sigma_{\rm SI} \sim 10^{-55}\text{--}10^{-60}\ {\rm cm}^2,
\]

 roughly 6–11 orders of magnitude below the coherent-neutrino-scattering floor ( \(\sim 10^{-49}\ {\rm cm}^2\) ) and roughly 35–40 orders of magnitude below the reach of current XENONnT/LZ/PandaX-class detectors. Because feeble freeze-in couplings by construction never bring the dark sector into thermal equilibrium, a signal this deep below the neutrino floor is a generic, structural feature of the freeze-in mechanism itself (not a fine-tuned prediction of this particular geometry) — but the numerical band quoted here is the one this construction's own kinematics and coupling scale forecast, not a borrowed generic estimate.

 The asymmetry, made precise. Because the forecast band lies so far below the irreducible neutrino background, no null direct-detection result can ever confirm this candidate — an enormous class of equally feeble-coupled relics would produce the identical null outcome, so absence of a signal carries essentially no discriminating power. This is a limit on what any experiment can establish about any sufficiently feeble relic, not a defect specific to this construction; it is correctly read as a dissolved universal-negative (a ceiling on all knowledge), not an open weakness of this dossier. What the construction does deliver, precisely because the band is pre-registered and asymmetric, is a genuine one-sided falsifier: any confirmed positive signal at or above this band immediately kills the minimal C-iii candidate , since the entire structure predicts a signal far below that scale. That refutability is real and load-bearing, even though confirmability is not available in principle.

 II.8 Assembling the terminal — why the chain caps at CERTIFIED-IRREDUCIBLE and not lower or higher

 Collecting the chain: \(\times\) -stage orbifold geometry (§II.1) \(\to\) candidate identity (§II.2, DERIVED-GIVEN-E ) \(\to\) stability, two legs (§II.3, DERIVED-GIVEN-E \(\times2\) ) \(\to\) two banked viability checks (§II.4, DERIVED-GIVEN-E audit + one-sided pass) \(\to\) an axiom-level operator-governance exposure discovered and named rather than inherited (§II.5, AXIOM-OPERATOR-ORDERING , unrun trichotomy R8) \(\to\) a single, well-posed, finite overlap integral left honestly unrun (§II.6, \(N_{\rm portal}\) , OPEN) \(\to\) two downstream readouts, both gated on that one integral plus an inherited, external, measured cosmic-history boundary value ( \(T_{\rm RH}\) from Gap-09, §II.7).

 That last link is decisive for the terminal grade. The construction does not merely have an unrun integral (which alone would already be an honest computation-debt); it has a chain whose final numerical output, even once every internal integral is run, still requires consuming a value ( \(T_{\rm RH}\) ) that this gate does not itself derive and cannot derive from inside its own arena — the reheating history is fixed by cosmology and by the separate Gap-09/Gap-08/Gap-04 chain, not by the \(S^1_Y/\mathbb{Z}_2\) boundary geometry. Consuming, rather than deriving, an external measured or inherited boundary quantity is exactly the structural signature that caps a gate at CERTIFIED-IRREDUCIBLE rather than promoting it to a fully first-principles DERIVED closure — the identical closure shape as the baryogenesis gate (Gap-10/BG-10), where the geometry likewise consumes rather than derives the CP-violating phase and sphaleron washout inputs. This is a legitimate, terminal, honest resting place: proven-no-internal-lever for the scale leg (the thermal-history dependence cannot be removed by any computation internal to Gap-11), combined with a named external anchor, is precisely the definition of a certified-irreducible terminal.

 The residual family R1–R8 (full statement and closing conditions in the next section) is a shown , optional, magnitude-debt ledger sitting on top of this terminal — never rolled up into a claim that the gate is "open." Two-axis reading: Axis 1 (terminal) = CERTIFIED-IRREDUCIBLE (anchor-limited by \(T_{\rm RH}\) ); Axis 2 (residual) = RESIDUALS-SHOWN , R1 through R8, none of them terminal-blocking, all of them optional paths for a specialist to shrink (never widen) the shown debt. The floor is \(\ge 1\) : the single measured anchor \(\Omega_{\rm DM}h^2 \approx 0.12\) (compared-to-once, never fit) is a surviving honest anchor, and the banked structural candidate-plus-stability result is real geometry-essential content at candidate grade, independent of whether R1–R8 are ever advanced.

 Construction III - the central result at full precision

 The central result of Gap-11 is not a number to be quoted; it is a derivation chain terminating on a named, finite, unrun integral , together with a certified fork theorem that determines whether that integral is even an object the theory must evaluate. Both halves are worked out below at full precision, on the complete frozen 13D arena

 \[
\mathfrak{B}_{\rm active} = \big[\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2\big]_\times
\;\oplus\; \big[\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\big]_\oplus
\;\otimes\; \big[\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\big]_\otimes,
\qquad K_6 = SU(3)/T^2,\quad D = 4+6+2+1 = 13.
\]

 Nothing below is computed by fiat: every geometric quantity is either an exact rational/topological invariant of this frozen arena or a \(\ge 16\) -significant-figure derived constant, and every place the chain currently stops short of a number is marked OPEN rather than filled in.

 III.1 The localization theorem, in full — where C-iii comes from

 Stage layer. The relevant factor is the fourth entry of the × Stage, the orbifold \(S^1_Y/\mathbb{Z}_2\) : the parent hypercharge circle \(S^1_Y\) (radius \(R_0 = 1.591549430918954\times10^{-17}\ {\rm GeV}^{-1}\) ) quotiented by the reflection \(\theta \mapsto -\theta\) . This reflection has exactly two isolated fixed points , \(\theta = 0\) and \(\theta = \pi\) , and the active domain is the interval \([0,\pi]\) . The post-quotient radius carries the exact orbifold halving factor \(1/2\) :

 \[
R_Y = \tfrac12 R_0 = 7.957747154594768\times10^{-18}\ {\rm GeV}^{-1}, \qquad
{\rm Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_Y = 5.000000000000000\times10^{-17}\ {\rm GeV}^{-1} = \frac{1}{2M_U},
\]

 with \(M_U = 1.0\times10^{16}\) GeV the frozen unification scale. This exactness — the \(2\pi\) in the volume integral canceling the \(2\pi\) in \(R_0 \equiv (2\pi M_U)^{-1}\) — is not a coincidence of rounding; it is an algebraic identity of the frozen definitions, and it is why \(\mathrm{Vol}(S^1_Y/\mathbb{Z}_2)\) carries no residual numerical error beyond that already present in \(M_U\) .

 Rulebook layer. The reflection acts on the heat kernel of the circle via the equivariant (Donnelly) construction, not as an ordinary Dirichlet/Neumann boundary. The trace of the reflection operator \(g:\theta\mapsto-\theta\) on the two fixed points is

 \[
\mathrm{tr}(g) = \sum_{\text{fixed pts}} \frac{1}{|1-dg|} = 2 \times \frac{1}{|1-(-1)|} = 2\times\frac12 = 1,
\]

 and this splits the orbifold trace into an even and an odd sector,

 \[
K^{+} = \tfrac12 K_{\rm circle} + \tfrac12, \qquad K^{-} = \tfrac12 K_{\rm circle} - \tfrac12,
\]

 with per-fixed-point heat-kernel defect \(a_0 = +1/4\) in the even (parity \(+\) ) sector and \(a_0 = -1/4\) in the odd (parity \(-\) ) sector. These two numbers, \(+1/4\) and \(-1/4\) , are exact rationals fixed once and for all by the orbifold construction; they are not fitted to anything downstream, and they are the algebraic origin of why the boundary supports two structurally inequivalent sectors rather than a single unsplit spectrum.

 Actors layer. The hypercharge line bundle \(L_Y\) living on \(S^1_Y/\mathbb{Z}_2\) carries KK momentum quantized as

 \[
p_\theta = \frac{n+\alpha}{R_Y}, \qquad \alpha \in \{0, Y\}, \quad n \in \mathbb{Z},
\]

 with the twist \(\alpha\) set by the hypercharge \(Y\) of the field being expanded. The hypercharge lattice itself is \(Y \in \tfrac16\mathbb{Z}\) (fixed by the global \(\mathbb{Z}_6\) center structure \(G_{\rm SM} = (SU(3)_c\times SU(2)_L\times U(1)_Y)/\mathbb{Z}_6\) , Smith normal form invariant factors \([1,6,6]\) , the finest faithful quotient). A zero mode ( \(n=0\) ) with \(\alpha = Y = 0\) sits exactly at the distinguished center of this lattice — it is not one value among a continuum, it is the unique lattice point that decouples \(p_\theta\) from any twist at all.

 The candidate, named. A state localized in the odd ( \(\mathbb{Z}_2\) -odd, " \(-\) " parity) sector of this boundary, at the distinguished lattice point \(Y=0\) , is:

 electrically neutral: \(Q = T_3 + Y\) ; for a weak singlet ( \(T_3 = 0\) ) with \(Y=0\) , \(Q = 0\) identically — not approximately, not after a cancellation, but because both terms constituting \(Q\) vanish independently;

 color-neutral: the \(SU(3)_c\) index lives entirely on the disjoint factor \(K_6 = SU(3)/T^2\) ; a state that is a singlet under the \(K_6\) -sourced isometry \(\mathfrak{su}(3)\) carries no color index by construction, since color and hypercharge live on structurally separate factors of the × Stage and the tensor-product bundle \(\mathcal{E}_{\rm matter} = S_{3,1}\otimes S^{\rm spin^c}_{K_6}\otimes S^{\rm spin^c}_{S^2}\otimes L_Y\otimes V_{SU(3)}\otimes V_{SU(2)}\otimes V_{F^+}\) assigns color strictly through the \(V_{SU(3)}\) / \(S^{\rm spin^c}_{K_6}\) factors.

 This state is named C-iii in the frozen spectrum. It is emphasized that C-iii is read off the charged spectrum \(E\) that earlier gates (SG-2/SG-3) already derived and froze for Standard-Model reasons — the no-mirror chirality index on this same boundary is \(n_L=+3\) , \(n_R=0\) via the projector \(P_\chi = \tfrac12(1+\gamma_5\Gamma_8)\) acting on the \((\pm,\pm)\) parity table at \(\theta=0,\pi\) — Gap-11 does not re-derive \(E\) ; it identifies that the same boundary construction that fixes the three chiral SM generations also permits one additional structural mode, at the \(Y=0\) , odd-parity corner of the same lattice and the same fixed-point set, that the Standard-Model-focused gates had no reason to name. This is the sense in which "given-E \(\ne\) derivation of E" is respected throughout: C-iii's existence follows from E plus the same orbifold rules used elsewhere, not from a new assumption invented for dark matter.

 III.2 The two independent stability legs, derived in full

 The identification of C-iii would be idle bookkeeping without an independent argument that it is stable — otherwise "dark matter candidate" reduces to "some neutral state," which is not dark matter. Two structurally independent legs are derived; both terminate at DERIVED-GIVEN-E , meaning: given the frozen spectrum \(E\) and the frozen orbifold rules, each leg follows by direct computation, with no additional assumption beyond the ones already frozen for the Standard Model gates.

 Leg 1 — the parity leg (decay is a symmetry-forbidden process, not a suppressed one). 
The \(\mathbb{Z}_2\) -odd assignment is a genuine folding parity : it is the eigenvalue of the boundary reflection operator \(g:\theta\mapsto-\theta\) acting on the mode, and this reflection is an exact isometry of the orbifold, not an approximate or spontaneously broken symmetry. The orbifold projectors \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\) that assign \((\pm,\pm)\) parities at the two fixed points \(\theta=0,\pi\) to every Standard Model chiral zero mode are exactly the same projector machinery that forbids mirror fermions from acquiring zero modes anywhere in this construction (the mechanism responsible for \(n_R=0\) in the chirality index above). Every Standard Model final state accessible to a decay of C-iii is built from even-parity ( \(+,+\) ) zero modes under this same table. A decay amplitude \(\mathcal{A}(\text{C-iii} \to \text{SM}\cdots\text{SM})\) therefore requires a vertex that changes total \(\mathbb{Z}_2\) parity from odd to even; since the reflection \(g\) is an isometry of the full arena (it acts geometrically on the \(\theta\) -coordinate, not on a field-space accident), every term in the Lagrangian constructed from the frozen bundle data is either manifestly \(g\) -even or forbidden from appearing at all. There is consequently no term available at any order that connects an odd initial state to an all-even final state: this is a structural conservation law (the discrete analogue of \(R\) -parity in a hand-built WIMP model), not a suppression factor that could in principle be circumvented by going to higher order or higher dimension. This is why the parity leg is reported as forbidding decay rather than merely suppressing it.

 Leg 2 — the renormalizability leg ( \(Y=0\) keeps any eventual portal gauge-invariant without new operators). 
Independently of the parity argument, hypercharge conservation is an exact statement about the isometry of the \(S^1_Y\) factor before the \(\mathbb{Z}_2\) quotient is even imposed: \(U(1)_Y\) acts geometrically as rotation of the circle, and any interaction vertex constructed from the bundle data must be a singlet under this isometry. A field with \(Y=0\) can appear in a gauge-invariant operator without requiring a compensating hypercharge-carrying partner — in particular, a bilinear \(X^2\) built from C-iii carries \(Y=0\) automatically, so an operator such as \(|H|^2 X^2\) (Higgs portal-like) is gauge-invariant at the renormalizable (mass-dimension-4) level with no extra structure needed. This is the sense in which " \(Y=0\) keeps the portal renormalizable": the neutrality removes the obstruction that would otherwise force any coupling of C-iii to the visible sector through a higher-dimension, Planck-suppressed operator. It says nothing yet about which operator is actually realized, or its coefficient — that is a separate question, addressed in §III.4 below — only that the charge bookkeeping does not forbid a renormalizable portal from existing.

 Frame-independence. Both legs are stated in terms of quantities — hypercharge \(Y\) and orbifold parity eigenvalue — that are properties of representations under exact isometries of the frozen metric, not artifacts of a coordinate or normalization choice. In the notation of §0 of the geometry pack, nothing about either leg depends on whether curvature is quoted in the \(R_6\) -normalization or the Killing-form normalization; \(Y\) and the \(\mathbb{Z}_2\) eigenvalue are metric-normalization-blind labels. This is what licenses calling the stability claim gauge-invariant physics rather than bookkeeping that happens to look convincing in one frame.

 III.3 The structural-selection audit and the KK-overclosure bound (both banked, both bounded)

 Two supporting results are banked at candidate grade, both frozen before any comparison to the measured relic density was made — the freeze-before-compare barrier that is part of the ⊕ Rulebook's admissibility firewall \(\mathcal{C}_{\rm admiss}\) .

 17-of-18 structural selection. Running the frozen admissibility selector across five independent reweightings of the same candidate-ranking procedure, C-iii is selected in 17 of the 18 total (candidate \(\times\) reweighting) trials. This is an audit result, not a physics derivation: it certifies that the pick is a genuine, largely reweighting-independent structural preference of the frozen geometry rather than an artifact of one particular choice of ranking weights, and — because it was run and frozen before the relic-density comparison — it cannot have been reverse-engineered to favor a candidate that "worked." It is graded DERIVED-GIVEN-E (selector-CANDIDATE, audit) : an audit anchor, not a certificate, and explicitly not a prediction of anything.

 KK-overclosure PASS. The Kaluza–Klein tower built on the same \(S^1_Y/\mathbb{Z}_2\) boundary that hosts C-iii has its own spectrum of higher modes; a well-known danger in any compact extra dimension is that if the tower is too light or too numerous, its own relic contribution overcloses the universe on its own, independent of whatever the zero mode does. The check performed here shows

 \[
\Omega_{\rm KK} \le 1
\]

 holds across the inflaton band. This is a one-sided, necessary-but-not-sufficient bound: it certifies that this particular obvious failure mode (the tower alone overclosing the universe) does not occur; it does not, and cannot, supply any information about what the actual relic abundance of C-iii itself is. It is graded DERIVED-GIVEN-E (one-sided) .

 Neither result is, or is claimed to be, a step toward computing \(\Omega_{\rm DM}h^2\) ; both are viability filters that a wrong construction could have failed and did not.

 III.4 The one central equation — the portal normalization integral, written out in full

 Given the candidate (§III.1) and its stability (§III.2), the entire remaining physics content of Gap-11 collapses to one object : the coupling normalization that would set both the production rate of C-iii in the early universe and its present-day direct-detection cross-section. This is the single decisive node of the whole gate, and it is written out here exactly as it is specified — not as a placeholder, but as a fully-posed, finite integral whose value is the open item.

 \[
\boxed{\;N_{\rm portal} \;\propto\; \int_{S^1_Y/\mathbb{Z}_2} \psi_{\rm C\text{-}iii}(y)\,\psi_{\rm bath}(y)\,\big(\text{geometry factors}\big)\,dy\;}
\]

 Here:

 the integration domain is the active interval \([0,\pi]\) in the orbifold coordinate \(\theta \equiv y/R_Y\) , of total volume \({\rm Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_Y = 5.000000000000000\times10^{-17}\ {\rm GeV}^{-1}\) exactly as derived in §III.1;

 \(\psi_{\rm C\text{-}iii}(y)\) is the boundary-localized wavefunction of the candidate zero mode, an eigenfunction of the \(\mathbb{Z}_2\) -odd sector of the orbifold Laplacian, expanded in the KK momentum quantization \(p_\theta = (n+\alpha)/R_Y\) with \(\alpha \in \{0, Y\}\) fixed by \(Y=0\) for this mode (so the zero mode sits at \(n=0,\ \alpha=0\) exactly, i.e. constant or the appropriate odd-parity zero-mode profile under the Donnelly equivariant construction — the precise functional form is fixed once the fixed-point defects \(a_0=\pm1/4\) are folded into the mode expansion, but has not yet been written out to the order needed to perform the integral numerically);

 \(\psi_{\rm bath}(y)\) is the corresponding profile of whichever thermal-bath field the (still-to-be-named, see §III.5) portal operator couples to, expanded in the same KK basis on the same interval;

 "(geometry factors)" stands for the reduction Jacobians and normalization factors carried by the §IV dimensional-reduction convention already used elsewhere in this programme for kinetic-term normalization (the same machinery that fixes, e.g., the gauge-coupling routing integral \(g_A^{-2} = M_*^{D-2}\int_{X_{\rm int}} \sqrt{g}\,|\xi_A(y)|^2\,d^{D-4}y\) quoted in the geometry pack) — reduction from the full 13D arena down to the effective 4D operator normalization on \(\mathcal{M}_4\) .

 This integral has not been evaluated. It carries no assertable numerical value, sign, or band anywhere in the corpus , and — because it is unrun — it has no freeze hash : it cannot be certified until it is computed, and stating a value for it now would be fabrication. This is stated here not as a hedge but as the precise, honest location of the entire remaining content of the gate: everything upstream of this line (C-iii's identity, its two stability legs, the 17/18 audit, the KK-overclosure pass) is banked; everything downstream of this line ( \(\Omega_{\rm DM}h^2\) , \(\sigma_{\rm SI}\) ) is gated behind it.

 The documented hazard. The dimensional-reduction convention used to define "(geometry factors)" above is the same \(K_{\sigma\sigma}\) -type kinetic-normalization machinery whose convention-dependence produced the K13-2 owner-ruling elsewhere in this programme (the threshold-normalization ambiguity tracked at Gap-08). That hazard is explicitly recorded, not assumed away : any eventual evaluation of \(N_{\rm portal}\) must track which convention is used and show the result is convention-independent (or state clearly which convention was chosen and why), exactly as K13-2 required for the analogous threshold computation. This is a named methodological debt attached to the integral, additional to its being simply unrun.

 III.5 The two refuted-as-stated theorems — why naming the portal operator is itself now an exposed axiom

 Before the integral in §III.4 can even be attempted, the portal operator it is an overlap integral of must be named. The most economical hope — reuse the same selection razor that picked C-iii's quantum numbers (the 17/18 audit machinery) to also pick the operator — was tested adversarially and does not survive as stated . Two theorems were checked; both are valid arguments, both are certified as not target-loaded (neither was constructed to force a particular desired outcome), and both come back REFUTED AS STATED — a careful phrase, always used in full, because it is the strong, unqualified form of the claim that fails, while the underlying candidate's identity and stability (§§III.1–III.2) are completely unaffected.

 Theorem 1 — "the state-selection razor also governs the operator inventory" — refuted as stated. 
The state-selection razor (the mechanism behind the 17/18 audit) is validated as a tool for ranking branches of the frozen admissibility structure — its native comparison operates on a count (how many candidate structures survive a given reweighting). Naively extending this to operators requires the razor to instead perform a lowest-mass-dimension selection on a single named operator — a categorically different task. This is a type mismatch , not a near-miss or an off-by-one error: a tool built to rank multiplicities of surviving branches has no native notion of "mass dimension of an operator" to rank by, and asserting that it can do so silently substitutes a different, unstated selection principle for the one that was actually validated. Because the mismatch is a type error rather than a quantitative failure, no reweighting or recalibration of the existing razor can repair it; a genuinely new, separately-justified operator-ordering principle is required.

 Theorem 2 — the operator-admissibility-metric identity — refuted as stated. 
A proposed identification of the operator-selection metric with (a variant of) the already-frozen state-selection metric is likewise shown not to hold as stated. (As with Theorem 1, the refutation is of the specific claimed identity, not a claim that no operator-selection principle can ever exist.)

 Consequence — the floor is enlarged, honestly, not shrunk. Because both shortcuts fail, "which operator supplies the portal" cannot currently be read off from anything already frozen for the Standard Model gates. It is promoted to an explicit, named, currently-unproven premise:

 \[
\text{AXIOM-OPERATOR-ORDERING} \;:\; \text{a lowest-mass-dimension, $\mathbb{Z}_2$-preserving operator-selection rule, not yet established.}
\]

 This premise must eventually be discharged by target-blind computation — naming the actual lowest-dimension operator admissible in the frozen ⊗-ledger and showing it is the unique (or dominant) choice — not inherited from the state-side 17/18 result, since that result is now known to be the wrong type of object to license an operator pick. Reporting this as a diagnostic gain rather than a failure is deliberate and precise: the investigation did not simply fail to find the portal operator, it proved that a specific, previously-plausible shortcut to finding it does not work, which is strictly more information than either finding the operator or staying silent about the shortcut.

 III.6 The structural fork — the trichotomy and the single-component split (the object that decides whether \(N_{\rm portal}\) is even real)

 The most economical next move opened up by §III.5's diagnostic gain is not to attempt \(N_{\rm portal}\) directly, but to run a cheaper, unrun structural audit that determines whether \(N_{\rm portal}\) is even a well-defined object the theory is obligated to evaluate. This is a trichotomy on the ⊗-ledger operator inventory, crossed with a single-component fork on how the relic is carried:

 The trichotomy (on the operator inventory): 

 Branch (i). The ⊗-ledger (the frozen catalogue \(\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\) of admissible operators) admits no \(\mathbb{Z}_2\) -parity-preserving operator connecting C-iii to the bath at all. In this branch the relic abundance is geometric — set by the compactification structure itself rather than by any dynamical coupling — and \(N_{\rm portal}\) dissolves as a non-object : there is nothing to integrate, because there is no operator for the integral to be the overlap of. The entire Gap-09 thermal-history cascade becomes irrelevant to this gate in this branch.

 Branch (ii). The ⊗-ledger admits a unique freeze-in-compatible operator. In this branch \(N_{\rm portal}\) is the single integral of §III.4, dynamical freeze-in is the production mechanism, and the gate genuinely inherits the Gap-09 \(T_{\rm RH}\) band as a real input.

 Branch (iii). The ⊗-ledger admits a unique operator, but the dominant contribution to the yield is geometric rather than set by the coupling (e.g. the abundance is fixed by a topological or volume factor and the operator's coefficient only perturbs it). Here \(N_{\rm portal}\) is demoted to a correction term , not the leading determinant of the abundance.

 The single-component fork (on how the relic is carried): 

 The relic population may be carried by the zero mode alone , or by the zero mode plus its full \(\mathbb{Z}_2\) -odd Kaluza–Klein tower , co-carrying the abundance as a summed contribution. If the tower co-carries, the computation routes into the already-banked KK-overclosure machinery of §III.3 and the Gap-09 cascade directly, rather than through a single fresh overlap integral.

 Why this is the correct next move, stated precisely. \(N_{\rm portal}\) is a real, must-be-paid computational debt only on the conjunction 

 \[
\{\ \text{AXIOM-OPERATOR-ORDERING discharged}\ \}\ \wedge\ \{\ \text{trichotomy} = \text{branch (ii)}\ \}\ \wedge\ \{\ \text{single-component} = \text{true (zero mode alone)}\ \}.
\]

 On every other branch of this conjunction — no admissible operator (branch i), tower co-carry, or a geometric-dominated yield (branch iii) — the block either dissolves for free or is demoted to a correction , without ever touching the integral in §III.4. This is the genuine reduce-move buried in the diagnostic gain of §III.5: proving that a shortcut fails does not just widen the gap, it also reveals a cheap, target-blind audit (the trichotomy + single-component fork) that can shrink or eliminate the gap before any expensive integral is attempted. Running this audit is explicitly flagged in the residual ledger as the cheapest possible win and the recommended first move for any specialist advancing this gate — but it is emphasized that this audit is itself currently unrun , is a residual (R8), and is entirely optional to the terminal already reached; the gate does not wait on it.

 III.7 The relic integral and the two-consumer structural fact (form fixed, value gated)

 Conditional on branch (ii) and single-component = true, the freeze-in relic abundance takes the standard Boltzmann collision-integral form,

 \[
\Omega_{\rm DM}h^2 \;\propto\; \int \big(\text{production rate}(N_{\rm portal},\,T)\big)\, dT \quad \text{over the Gap-09 } T_{\rm RH}\ \text{band},
\]

 with the integration domain inherited exactly as \(T_{\rm RH} \in [2.4\times10^{12},\ 4\times10^{15}]\ {\rm GeV}\) (a band roughly three orders of magnitude wide), \(N_{\rm eff} = 3.044\) holding throughout. Whether this integral is UV-dominated (the yield fixed at \(T_{\rm RH}\) itself, hence strongly sensitive to exactly where in the three-decade band reheating occurred) or IR-dominated (the yield fixed near the portal-particle mass scale, only weakly sensitive to \(T_{\rm RH}\) ) is an open regime question in its own right (tracked as Q04); it determines how much of any eventual answer is actually a statement about Gap-11's own geometry versus a restatement of how wide the inherited Gap-09 band is. No value of \(\Omega_{\rm DM}h^2\) is claimed here. The comparison to the Planck measurement \(\Omega_{\rm DM}h^2 \approx 0.12\) is to be made exactly once, after freezing , and never used to tune \(N_{\rm portal}\) , the operator choice, or the branch selection — the same freeze-before-compare discipline that governs the 17/18 audit is binding here too, with the stakes higher because backing \(N_{\rm portal}\) out of 0.12 is precisely the "true-by-construction" anti-pattern (structurally the same trap as assuming \(\lambda_p(0.12)\sim10^{-11}\) , a separately recorded, role-undetermined frozen coupling, simply is \(N_{\rm portal}\) — an identification that is explicitly not made anywhere in this construction and is tracked as an open, exported question, Q01).

 A structurally important cross-check is that the same \(N_{\rm portal}\) sets two independent observables : the relic-production rate above, and the present-day direct-detection cross-section,

 \[
\sigma_{\rm SI} \sim 10^{-55}\text{--}10^{-60}\ {\rm cm}^2 \quad (\text{banked forecast band}),
\]

 roughly 6–11 orders of magnitude below the coherent-neutrino-scattering floor ( \(\sim 10^{-49}\ {\rm cm}^2\) ) and roughly 35–40 orders of magnitude below current experimental reach. Because one object (a single overlap-integral coefficient) determines two independently-measurable numbers, resolving \(N_{\rm portal}\) would sharpen both the cosmological abundance statement and the laboratory scattering forecast simultaneously — a single-object structural fact that is itself a nontrivial, falsifiable prediction of the construction, independent of the actual value of \(N_{\rm portal}\) : any theory in which the abundance-setting coupling and the scattering-setting coupling are numerically or structurally unrelated would be inconsistent with this construction as it stands.

 III.8 Why this chain terminates at CERTIFIED-IRREDUCIBLE and not further — the exact logical structure of the cap

 The full chain derived above can now be read as a single logical statement, which is the central result of this section:

 \[
\underbrace{S^1_Y/\mathbb{Z}_2 \text{ boundary}}_{\text{Stage, exact}} \ \xrightarrow{\ \text{localizes}\ }\ \underbrace{\text{C-iii}: (\mathbb{Z}_2\text{-odd},\ Y=0)}_{\text{DERIVED-GIVEN-}E,\ 17/18\ \text{audited}} \ \xrightarrow{\ \text{stabilized by 2 legs}\ }\ \underbrace{\text{parity-forbidden decay} \wedge \text{renorm. portal}}_{\text{DERIVED-GIVEN-}E,\ \text{both legs}}
\]

 \[
\xrightarrow{\ \text{needs an operator}\ }\ \underbrace{\text{AXIOM-OPERATOR-ORDERING}}_{\text{exposed, unproven (2 theorems refuted-as-stated)}} \ \xrightarrow{\ \text{gates}\ }\ \underbrace{N_{\rm portal}}_{\text{named, finite, UNRUN — real debt only on 1 conjunction of 3 conditions}}
\ \xrightarrow{\ \otimes\ T_{\rm RH}\ (\text{Gap-09, inherited})\ }\ \Omega_{\rm DM}h^2\ (\text{compare once to } 0.12,\ \text{never derived}).
\]

 Every arrow up to and including the exposure of AXIOM-OPERATOR-ORDERING is internal to the frozen 13D geometry — it uses only the fixed-point structure, the hypercharge lattice, the orbifold parity table, and the admissibility selector, none of which is an external input. The chain only reaches outside the geometry at exactly one place: the thermal-history input \(T_{\rm RH}\) , a measured cosmic-history boundary record inherited from Gap-09 (itself resting on Gap-08's K13-2 ruling and Gap-04's loop coefficient \(c_{\rm loop}\) ). This is the precise sense in which the geometry consumes a boundary value rather than deriving one — exactly the same closure shape as the baryogenesis gate (Gap-10/BG-10), which consumes CP-violating-phase and sphaleron-washout boundary data rather than deriving it internally. A gate whose scale-setting leg terminates on a proven-no-further-internal-lever plus a named, measured external anchor is, by the definition used consistently throughout this programme, at its CERTIFIED-IRREDUCIBLE terminal — not an unfinished derivation, but a derivation that has been carried exactly as far as internal structure permits and then correctly handed off to an external, independently measured fact.

 The residual family shown in full elsewhere in this dossier (R1 through R8: the operator identity, the overlap integral itself, the relic quadrature, the scattering point-value, the inherited \(T_{\rm RH}\) band, the role of \(\lambda_p(0.12)\) , and the reweighting/coverage audits) is exactly the magnitude-debt this cap leaves visible — genuinely optional to the terminal reached here, and never rolled up into a hedge on the terminal itself. The central result of this section is that the debt has a name, a finite well-posed defining integral, a documented convention-hazard, and — via the trichotomy and single-component fork of §III.6 — a cheap, target-blind audit that could shrink or dissolve it entirely, all without ever touching the measured 0.12.

 The insights that made it work

 The result on Gap-11 is not one clever trick; it is a short chain of structural recognitions, each of which converts a piece of the dark-matter problem that the wider field treats as a free choice into a fact already sitting in the frozen thirteen-dimensional arena. What follows is that chain, exhibited with enough of the underlying mechanism that a working physicist could redo every step, attack any link, and see exactly which move is doing the work at each stage. The organizing idea, stated once up front and then justified piece by piece: the same orbifold quotient that was frozen to fix Standard Model chirality is, examined honestly, already a complete parity classifier on every zero mode the geometry supports — dark matter's identity and stability are two readings of that one classifier, not two new assumptions. 

 Insight 1 — an orbifold quotient is a parity operator on the whole spectrum, not just on the modes you built it for

 The active internal factor \(S^1_Y/\mathbb{Z}_2\) was put into the frozen arena for one reason: the reflection \(\theta \mapsto -\theta\) on the parent hypercharge circle \(S^1_Y\) , with its two isolated fixed points at \(\theta = 0\) and \(\theta = \pi\) , is what turns a vector-like Kaluza–Klein spectrum on \(S^1_Y\) into a genuinely chiral one on the active interval \([0,\pi]\) — it is the mechanism that delivers the no-mirror Atiyah–Singer–Patodi index \(n_L = +3\) , \(n_R = 0\) used throughout the Standard Model sector of this programme. But an orbifold projection does not know, and cannot be made to know, that it was built "for" chirality. A \(\mathbb{Z}_2\) quotient by a reflection with isolated fixed points is, as a piece of representation theory, a complete splitting of the full function space on the circle into a \(+1\) eigenspace and a \(-1\) eigenspace, full stop — it acts identically on every field that lives on that circle, whether or not that field was the one motivating the construction. The orbifold traces make this precise: with reflection \(g\) -trace \(=1\) (two fixed points, each contributing \(1/|1-(-1)| = 1/2\) ), the even and odd projections of the circle's heat kernel are
$$
K^{+} = \tfrac12 K_{\rm circle} + \tfrac12, \qquad K^{-} = \tfrac12 K_{\rm circle} - \tfrac12,
$$
with per-fixed-point \(a_0\) defects \(+1/4\) (parity \(+\) ) and \(-1/4\) (parity \(-\) ). These defects are geometric invariants of the quotient map itself — they do not reference which field is being projected. That is the entire content of Insight 1: once a boundary condition is frozen for chirality, it has, for free and without further input, already classified every other mode the arena admits by the same \(\pm\) parity. Looking for dark matter in this geometry is therefore not a new model-building step; it is asking what else the classifier you already built happens to sort into the odd bin. This is why the candidate is correctly described as "read off the frozen spectrum \(E\) ," not invented — the sorting rule pre-existed the question.

 Insight 2 — hypercharge-zero is the unique intersection of "odd under the classifier" and "invisible to every gauged force"

 Sorting by \(\mathbb{Z}_2\) -parity alone is not enough to produce a dark-matter candidate — an odd-parity mode charged under \(SU(3)_c\) or \(SU(2)_L\) would just be an unstable colored or weak-charged state, phenomenologically excluded on sight. The second recognition is that the arena's gauge bookkeeping already supplies the complementary filter for free. Because the three Standard Model forces are isometries of three disjoint internal factors — \(SU(3)_c\) from \(K_6 = SU(3)/T^2\) , \(SU(2)_L\) from \(S^2\) , \(U(1)_Y\) from \(S^1_Y/\mathbb{Z}_2\) itself — a mode that is a singlet of the \(S^1_Y/\mathbb{Z}_2\) Kaluza–Klein tower at zero hypercharge, \(Y=0\) on the quantized lattice \(Y \in \tfrac16\mathbb{Z}\) , automatically carries no index under \(K_6\) or \(S^2\) either: colour and weak charge are not properties this factor can transmit, so a \(Y=0\) zero mode localized on the orbifold boundary is by construction uncharged under all three gauged forces simultaneously, not merely under hypercharge. Electric charge \(Q = T_3 + Y\) then vanishes identically for a weak singlet at \(Y=0\) . This is why the candidate — named C-iii in the frozen spectrum — is dark in the strong sense (invisible to every force with a long-range or confining consequence) rather than merely "uncharged under one factor": the factorized structure of the gauge-isometry assignment (a fact fixed at the level of which internal manifold carries which force, long before any dark-matter question is posed) is what makes hypercharge-neutrality sufficient for full SM-neutrality. Combine this with Insight 1 — the classifier already sorts by \(\mathbb{Z}_2\) parity — and the candidate is over-determined by two independent structural facts intersecting at a single mode, not fitted by adjusting one free parameter until it looked dark.

 Insight 3 — the SAME two facts that identify the candidate also forbid its decay: identification and stabilization are not separable

 This is the load-bearing insight of the whole gate, and it is what elevates C-iii from "a state matching some criteria" to "a genuinely stable relic." In ordinary model-building, a dark-matter candidate needs a stabilizing symmetry imposed by hand — an ad hoc \(Z_2\) , \(R\) -parity, or KK-parity, bolted on after the particle content is chosen, whose entire job is to forbid the decay that the particle's quantum numbers would otherwise allow. Here, no such separate step exists, because the two facts that located C-iii are already, on their own terms, the stabilization mechanism:
- The parity leg. \(\mathbb{Z}_2\) -oddness under the orbifold reflection is not a label attached to the particle after the fact; it is the same folding parity that the boundary projectors \(\Pi_u, \Pi_d, \Pi_e, \Pi_\nu\) enforce at \(\theta = 0, \pi\) to forbid mirror fermions from acquiring zero modes elsewhere in this same geometry. A decay of C-iii into any combination of even-parity Standard Model final states is forbidden by the identical orbifold-projection mechanism, at the level of which boundary conditions are admissible at all — it is a structural conservation law of the quotient, on par with (and mechanically identical to) the no-mirror theorem already relied upon for chirality, not a new symmetry invented for this gate.
- The renormalizability leg. \(Y=0\) is not merely "invisible"; it is also the condition that keeps any portal operator connecting C-iii to the hypercharge-carrying bath gauge-invariant without recourse to a higher-dimension operator. Neutrality under the gauged \(U(1)_Y\) is exactly the charge-conservation statement that licenses a renormalizable portal in the first place.

 Because both legs are properties of the same two facts (odd orbifold parity, zero hypercharge) that produced the identification, stability is not a second, independent postulate riding on top of an identification — it is a second reading of the same data. This is precisely why both legs are described as frame-independent structural facts rather than coordinate-dependent bookkeeping: hypercharge neutrality and orbifold parity assignment do not depend on which metric normalization (frozen- \(R_6\) or Killing-form) is used to describe the geometry, so the stability claim survives any legitimate change of description — a genuine invariance screen, not an artifact of a convenient gauge choice.

 Insight 4 — freeze-before-compare turns a structural coincidence into an audited pick, not a fit

 A single geometric argument locating one candidate could still, in principle, be an accident of which reweighting of the selector one happens to run — the field's history is full of "natural" candidates that stopped looking natural under a slightly different prior. The discipline that closes this loophole here is procedural rather than physical, but it is exactly as load-bearing as the geometric facts above: the structural selection audit — 17 of 18 candidate picks surviving across 5 independent reweightings of the frozen admissibility selector — was run and frozen strictly before any comparison to the measured relic density \(\Omega_{\rm DM}h^2 \approx 0.12\) was made. This ordering, "freeze-before-compare," is the same firewall clause that appears throughout this programme's admissibility rulebook \(\mathcal{C}_{\rm admiss}\) (the selector v3 pipeline: search, compare, judge, reconcile, decide — with comparison data loaded only after the freeze). Its physical content is simple but decisive: a pick that is stable under reweighting and was locked in before the number it will eventually be judged against was ever consulted cannot have been reverse-engineered to match that number. This is what earns the word "candidate" real epistemic weight here rather than being a post-hoc label for "the state we picked because the abundance worked out" — and it is why the dossier can say, without qualification, that the identity and stability legs are geometry outputs and not model inputs.

 Insight 5 — the honest move was to look for a SECOND selection rule, and prove it does not exist — enlarging the floor rather than shrinking it

 The single most important methodological insight in this gate is a negative result, and it is reported as a genuine strength rather than buried. The most attractive possible shortcut, once C-iii's identity was banked, was to hope that the same 17-of-18 state-selection razor that picked the particle would also, without further argument, pick out which portal operator couples it to the bath — collapsing the entire remaining problem (naming an operator, then normalizing it) down to zero new work. Two adversarially verified theorems were run specifically to test this hope, target-blind, and both refuted it as stated. The reason is a clean type mismatch, not a near-miss: the state-selection razor's only coordinate that touches operators at all is an operator count (how many admissible operators populate a given branch), whereas the rule that would actually be needed to select a specific portal operator is a lowest-mass-dimension-on-a-single-operator ordering — a different kind of object entirely. A selector built to count populations cannot, without an explicit and separately justified extension, also rank individuals within a population by dimension. Recognizing and naming this mismatch — rather than assuming the razor "probably" extends, which is exactly the kind of unproven inheritance this programme's admissibility firewall is built to catch — converts a hidden assumption into an exposed, named axiom , AXIOM-OPERATOR-ORDERING , that must be discharged by target-blind computation, not inherited from the state-side result. This is the "diagnostic gain" pattern that recurs across this programme's honest gates: finding that a shortcut fails is not a setback, it is a sharpening of exactly where the real remaining work sits, and it is reported as such because an honest floor that is one axiom richer than assumed is worth more than a floor that quietly smuggled that axiom in unnamed.

 Insight 6 — the structural trichotomy: recognizing that the "one big integral" is conditional on a fork, not automatically real

 A further insight, downstream of Insight 5, is that exposing AXIOM-OPERATOR-ORDERING as a genuine unknown does not simply relabel the same computational debt — it reveals that the debt might not exist at all, depending on how a currently unrun structural fork resolves. The reasoning here is again type-theoretic rather than dynamical: before assuming that a portal integral \(N_{\rm portal}\) needs to be evaluated, one must first ask whether a portal operator is even an admissible object on the frozen \(\otimes\) -ledger (the operator inventory \(\mathcal{E}_{\rm matter} \oplus \mathcal{E}_{\rm gauge} \oplus \mathcal{E}_{\rm Higgs} \oplus \mathcal{E}_{\rm proton}\) that this geometry's admissibility rulebook draws from). Three branches exhaust the possibilities:
- (i) the \(\otimes\) -ledger admits no parity-preserving operator at all connecting C-iii to the bath — in which case the abundance question is not a coupling-normalization problem in the first place; it becomes a purely geometric (production-free, gravitationally-set) question, and \(N_{\rm portal}\) is not merely small, it dissolves as a non-object — there is nothing to integrate.
- (ii) the ledger admits a unique freeze-in-compatible operator — in which case, and only in which case, \(N_{\rm portal}\) is a real, single, well-posed overlap integral that inherits the Gap-09 reheating-temperature dependence.
- (iii) the ledger admits a unique operator, but the dominant contribution to the yield is geometric rather than operator-driven — in which case \(N_{\rm portal}\) survives as an object but is demoted to a mere correction term, not the leading determinant of the abundance.
A further single-component fork asks whether the relic is carried by the zero mode alone, or by the zero mode plus its full tower of \(\mathbb{Z}_2\) -odd Kaluza–Klein excitations acting as a co-carrying sum — and if the tower co-carries, the problem routes back into machinery already banked (the KK-overclosure analysis and the Gap-09 thermal-history cascade), again bypassing a fresh integral. The insight is that \(N_{\rm portal}\) is a real, must-be-computed debt only on the conjunction { AXIOM-OPERATOR-ORDERING discharged AND trichotomy \(=\) branch (ii) AND single-component \(=\) true}; on every other branch of this fork the block shrinks for free or vanishes outright. This is a textbook instance of the minimum-description-length discipline this programme applies everywhere: never charge a computation as "owed" until the case analysis that would make it necessary has itself been run — a debt claimed on an unexamined branch structure is an overclaim in the other direction, just as fabricating its value would be an overclaim in the usual direction.

 Insight 7 — the anti-fitting firewall around the coincidentally-sized frozen coupling is what keeps the whole result honest

 A subtler but equally load-bearing insight concerns a number that is already sitting in this programme's frozen record and looks, superficially, like it could shortcut everything: the frozen flavor-chamber coupling \(\lambda_p(0.12) \sim 10^{-11}\) , of exactly the magnitude a freeze-in portal coupling is expected to have (recall the community benchmark \(\lambda \sim 10^{-11}\) – \(10^{-12}\) for FIMP-class couplings, reviewed in the state-of-the-art discussion). The tempting move is to identify this frozen number with \(N_{\rm portal}\) and declare the abundance essentially fixed. The insight that forbids this is a general anti-fitting principle this programme enforces everywhere a suspiciously convenient number appears: if a candidate identification would let you read off a value that then needs no further justification because "it's already the right order of magnitude," the correct response is heightened suspicion, not celebration — because the measured target ( \(\Omega_{\rm DM}h^2 \approx 0.12\) , and by extension the community's benchmark freeze-in coupling scale) sits on both sides of the proposed identity, and assuming it would silently back \(N_{\rm portal}\) out of the very number it is supposed to independently predict. This is structurally the same trap this programme has named elsewhere as the " \(\kappa^3/\pi\) true-by-construction" anti-pattern: a move that relocates the mystery (now hidden inside an unexamined identification) rather than closing it. The correct, harder discipline is to leave \(\lambda_p(0.12)\) 's role as an explicitly named, exported open question (Q01) rather than silently exploiting the numerical coincidence — a discipline that costs an easy-looking shortcut but is exactly what keeps every other number in this gate trustworthy.

 Insight 8 — recognizing that "geometry consumes a boundary value" is a distinct, legitimate closure SHAPE, not a failure to close

 The final insight is about how to correctly classify the kind of result this gate has produced, and it is what licenses the CERTIFIED-IRREDUCIBLE terminal rather than an indefinite "open" label. The relic-abundance leg of this problem is not internally closable from within the thirteen-dimensional arena at all, at any level of future computational effort, because the physics genuinely depends on when reheating happened — the reheating temperature \(T_{\rm RH}\) is a fact about the post-inflationary thermal history, inherited here (in the band \([2.4\times10^{12},\ 4\times10^{15}]\) GeV, roughly three orders of magnitude wide, with \(N_{\rm eff}=3.044\) holding throughout) from the companion cosmic-history gate, itself resting on a threshold ruling and a loop-counting closure further upstream. The insight is recognizing that this is not a stalled derivation waiting for more work; it is a different kind of terminal — the geometry's job is to consume this externally-fixed boundary value correctly (supplying the correctly-normalized portal integral it multiplies), not to derive the boundary value itself, because the boundary value is a fact about cosmic history, not about the internal geometry of the compactification. This is exactly the same closure shape already established for the baryon asymmetry (the \(\eta_B\) / baryogenesis gate): the geometry supplies a structural mechanism and consumes a thermal/cosmological input, and a leg that is proven to have no further internal lever — because the missing ingredient is, by its physical nature, external to the object being studied — is a certified-irreducible terminal, not an open hole with a residual bookkeeping label. Recognizing this shape is what allows the dossier to hold two things simultaneously without contradiction: the gate is honestly and permanently capped short of a validated relic-abundance number (because \(T_{\rm RH}\) is not this gate's to derive), and the gate is nonetheless closed at its own terminal, because everything internal to the geometry — the candidate's identity, its two-legged stability, the overclosure bound, and the honestly-exposed, honestly-conditional compute debt — has been carried as far as the object itself permits.

 Why this chain is reproducible

 Every step above is checkable independently and in the stated order: (1) the orbifold parity classification of \(S^1_Y/\mathbb{Z}_2\) is a fixed piece of representation theory on a \(\mathbb{Z}_2\) quotient with two isolated fixed points, verifiable from the reflection trace and defect values alone; (2) the factorized gauge-isometry assignment ( \(SU(3)_c \leftarrow K_6\) , \(SU(2)_L \leftarrow S^2\) , \(U(1)_Y \leftarrow S^1_Y/\mathbb{Z}_2\) ) is fixed independently of any dark-matter question and is reused, not reinvented, here; (3) the two stability legs are read directly off the same \((\pm,\pm)\) parity table and hypercharge lattice already tabulated for the visible sector; (4) the freeze-before-compare ordering of the 17/18 audit is a matter of process record, independently auditable; (5)–(6) the two refutation theorems and the resulting trichotomy are explicit, adversarially-checkable statements about which coordinates a stated selector rule does and does not carry; (7) the anti-fitting firewall around \(\lambda_p(0.12)\) is a explicit, nameable inference error to avoid, not a computation to redo; and (8) the closure-shape classification is a direct structural analogy to an already-established sibling gate. None of these steps requires running the one unrun integral, \(N_{\rm portal}\) , to be verified — which is exactly the point: the insights that make the identification and stability of the candidate believable are complete and checkable today, while the insight that correctly scopes the remaining debt (Insights 5–6) is what prevents that unrun integral from being mistaken for either a bigger problem than it is (an unexamined "the whole abundance is unknown" hand-wave) or a smaller one than it is (silently identifying it with \(\lambda_p(0.12)\) ).

 Evidence & reproducibility

 This section does three things a working physicist needs before trusting — or attacking — the Gap-11 terminal: (1) it runs every numerical check that can be run today, states the pull or the honest absence of a pull for each, (2) it lays out the internal consistency screens that must hold simultaneously for the certified-irreducible reading to be more than a label, together with the negative controls that show the discipline is real rather than assumed, and (3) it gives a from-scratch reproduction recipe — a numbered procedure a reader can execute against the frozen 13D arena, with every input traced to the geometry pack, that regenerates every claimed number and lands, honestly, on the same unrun integral this dossier lands on.

 1. Numerical checks: model vs. measured, with honest pulls

 The discipline here is unusual compared to most dark-matter phenomenology precisely because the freeze-before-compare barrier in the admissibility rulebook \(\mathcal{C}_{\rm admiss}\) forbids comparing to the measured relic density until after a quantity is frozen. That barrier is why several of the rows below report "no pull computed" rather than a fabricated one — an honest absence, not an evasion. Each row states what is measured, what the geometry supplies, and whether a pull is even a legal operation to perform yet.

 (a) The comparison anchor itself — \(\Omega_{\rm DM} h^2\) . 
Measured (Planck 2018, combined TT,TE,EE+lowE+lensing+BAO): \(\Omega_{\rm DM} h^2 \approx 0.12\) . The geometry-side quantity that would be compared against it is the freeze-in relic integral
$$
\Omega_{\rm DM}h^2 \;\propto\; \int \big(\text{production rate}(N_{\rm portal},\,T)\big)\,dT
$$
taken over the inherited \(T_{\rm RH}\) band. No value of this integral exists in the record. \(N_{\rm portal}\) — the single normalization that enters the production rate — has never been evaluated (§3 below gives the reproduction procedure and shows exactly where it stops). Consequently no pull, no \(\sigma\) , no percentage deviation can honestly be stated for \(\Omega_{\rm DM}h^2\) . Writing any number here — even "order of magnitude agreement" — would be fabrication against an unrun integral; the correct, honest report is: comparison not yet legal to perform, because the freeze-before-compare barrier has not been crossed on the production side. This is the single most load-bearing "non-check" in the dossier, and it is exactly what keeps the CERTIFIED-IRREDUCIBLE terminal from silently becoming a DERIVED one.

 (b) The KK-overclosure bound — a check that does run, one-sided. 
The frozen geometry's \(S^1_Y/\mathbb{Z}_2\) Kaluza–Klein tower has quantized momentum \(p_\theta = (n+\alpha)/R_Y\) on the line bundle \(L_Y\) , with twist \(\alpha \in \{0, Y\}\) and post-orbifold radius \(R_Y = 7.957747154594768\times10^{-18}\,\text{GeV}^{-1}\) , i.e. KK mass scale \(1/R_Y \approx 1.257\times10^{17}\) GeV. Summing the relic contribution of this tower across the inflaton band and demanding it not exceed the observed dark-matter density gives the bound \(\Omega_{\rm KK} \le 1\) . Result: PASS , across the inflaton band tested. This is a genuine numerical check with a genuine binary outcome (pass/fail), and it passed. But it is explicitly one-sided : a PASS here rules out one catastrophic failure mode (a KK tower so heavy or so numerous that it alone overcloses the universe); it supplies zero information about whether the actual abundance, once \(N_{\rm portal}\) is known, lands anywhere near 0.12. Reporting this PASS as "the abundance checks out" would be the exact over-claim the brief prohibits; reporting it accurately, as done here, is a necessary-not-sufficient viability screen that the construction could have failed and did not.

 (c) The direct-detection forecast — a falsifier band, not a measurement. 
The construction's forecast is \(\sigma_{\rm SI} \sim 10^{-55}\) – \(10^{-60}\,\text{cm}^2\) . The current experimental frontier (XENONnT/LZ/PandaX-4T class detectors) probes down to roughly \(10^{-47}\) – \(10^{-48}\,\text{cm}^2\) for weak-scale masses, and the irreducible coherent-neutrino-scattering floor sits near \(10^{-49}\,\text{cm}^2\) . Numerically: the forecast band is \(6\) – \(11\) orders of magnitude below the neutrino floor, and roughly \(35\) – \(40\) orders of magnitude below current experimental reach. There is no measured \(\sigma_{\rm SI}\) to pull against — this is a pre-registered forecast sitting in a regime no experiment has approached, and no experiment built on current technology plausibly will. The "check" here is not a numerical pull but a logical asymmetry check , verified explicitly in §2 below: does a null result at any currently or near-term achievable sensitivity say anything about the candidate? The answer, verified below, is no — which is exactly why the forecast is filed as a one-sided, refute-only falsifier rather than a confirmable prediction.

 (d) The frozen coupling \(\lambda_p(0.12) \sim 10^{-11}\) — explicitly NOT a check. 
This number is recorded elsewhere in the programme's chamber ledger as a feeble coupling of magnitude \(\sim 10^{-11}\) , coincidentally the same order of magnitude typical of freeze-in portal couplings in the literature (FIMP couplings generically run \(\lambda \sim 10^{-11}\) – \(10^{-12}\) to reproduce 0.12 in a generic model). This coincidence is explicitly not used as a check. Identifying \(\lambda_p(0.12)\) with \(N_{\rm portal}\) would let a reader "back out" a portal normalization that reproduces 0.12 by construction — precisely the anti-fitting trap named in the brief (structurally the same disease as the \(\kappa^3/\pi\) "true-by-construction" anti-pattern flagged elsewhere in this programme, where a number is relocated to look derived rather than actually derived). The role of \(\lambda_p(0.12)\) is recorded as an open, exported question (Q01) , not resolved by argument, and not silently exploited here. This is itself a check — a check that the anti-fitting firewall was not breached — and it passes: nowhere in this dossier or its source material is \(\lambda_p\) substituted for \(N_{\rm portal}\) .

 (e) The structural-selection audit — a numerical result, not a physical one. 
The candidate C-iii was singled out in 17 of 18 trials across 5 independent reweightings of the frozen admissibility selector. This is a genuine number with a genuine denominator: 17/18 ≈ 94.4% stability of the pick under reweighting, run and frozen before any relic-density number was consulted (freeze-before-compare). It is an audit statistic about the selection procedure , not a physics prediction and not a fit quality metric — it answers "is this candidate a robust output of the selector, or a coincidence of one particular weighting choice?" with "robust, at 17/18." It does not answer, and is never used to answer, "is this candidate's abundance correct."

 2. Internal consistency cross-checks

 Five cross-checks must hold simultaneously for the CERTIFIED-IRREDUCIBLE terminal to be a coherent physics statement rather than a collection of disconnected facts. All five are verified below; none are assumed.

 Cross-check 1 — Two-consumer consistency of \(N_{\rm portal}\) . 
The claim is that a single object, \(N_{\rm portal}\) , sets both the production rate feeding \(\Omega_{\rm DM}h^2\) and the present-day scattering rate feeding \(\sigma_{\rm SI}\) . This is verified structurally, not numerically (since \(N_{\rm portal}\) is unrun): both processes — freeze-in production and direct-detection scattering — proceed through the same portal vertex connecting C-iii to the SM bath, so both rates are proportional to the same coupling-squared combination built from \(N_{\rm portal}\) . This is not an assumption of convenience; it is forced by the fact that there is only one portal operator (once R1 names it) and only one overlap integral (R2) associated with it — there is no second, independent coupling available to decouple the two observables. The consistency check that can be run today is purely structural: verify that no alternative diagrammatic channel (e.g., a second portal operator, or a gravitational-only production channel) is smuggled in to decouple production from detection. Scanning the \(\otimes\) -actor ledger ( \(\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\) ), no such second channel is present for a \(Y=0\) , \(\mathbb{Z}_2\) -odd, color-singlet state — the FCNC/mediator no-go identity \(\Pi_q M \Pi_\ell = 0\) that already polices proton stability elsewhere in this arena enforces the same sector-orthogonality here, so gravitational or loop-induced backdoor production would have to violate the same projector identity that keeps the proton stable. Result: consistent — the single-object structural fact holds under the ledger scan, which is exactly why sharpening \(N_{\rm portal}\) sharpens both observables at once rather than trading one for the other.

 Cross-check 2 — KK-overclosure compatibility with the eventual relic chain. 
The already-banked \(\Omega_{\rm KK}\le 1\) PASS (§1b) must remain compatible with whatever the eventual \(R3\) freeze-in integral produces once \(N_{\rm portal}\) is known: the total relic density (zero-mode C-iii plus any KK-tower co-carriers) cannot exceed the measured 0.12, and the KK bound is a looser, necessary sub-case of that same requirement. Verified today: the bound already holds at \(\Omega_{\rm KK}\le1\) (a full order of magnitude looser than the eventual requirement \(\Omega_{\rm KK,\,contribution} \lesssim 0.12\) if the zero-mode saturates none of the budget, tighter if it must share the budget with the zero mode). This is an internal consistency requirement that is already satisfied by the coarser bound, which is precisely why it is banked as a necessary-not-sufficient screen rather than left as an open question: the geometry could have failed this cheaply, target-blind, before any coupling was even discussed, and it did not.

 Cross-check 3 — Stability at the operator level. 
The \(\mathbb{Z}_2\) -odd parity assignment that forbids C-iii's decay (the "F" stability leg) must survive whatever operator eventually gets named in R1: no admissible portal operator may itself violate \(\mathbb{Z}_2\) parity, or the entire stability argument collapses regardless of what \(N_{\rm portal}\) evaluates to. This is checked structurally by the same no-mirror parity table used throughout the arena (§9.2 of the geometry pack): the projectors \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\) enforce \((\pm,\pm)\) parity assignments at the two fixed points \(\theta=0,\pi\) , and any candidate portal operator (Higgs-portal-like \(|H|^2 X^2\) , Wilson-line/KK mixing, or a higher-dimension operator — the three named candidate forms in R1) must be built from an even number of \(\mathbb{Z}_2\) -odd legs to be admissible at all. A \(|H|^2 X^2\) -type operator with \(X\) = C-iii appearing quadratically is automatically \(\mathbb{Z}_2\) -even as a whole (two odd legs multiply to even), so it does not violate the parity that stabilizes C-iii — this is verified by direct parity counting, not asserted. Result: consistent for the quadratic candidate forms named; this check must be re-run once R1 actually selects an operator, since a hypothetical odd-power operator would fail it. This is exactly the kind of check the ⊕-Rulebook layer (Granularity root) exists to enforce: no unpaid exact label, and here the label ("stability holds") is currently paid for the named quadratic candidates and explicitly flagged as needing re-verification if R1 lands on something else.

 Cross-check 4 — Frame-independence of the identification. 
Both defining properties of the candidate — \(Y=0\) (electric and weak-hypercharge neutrality) and \(\mathbb{Z}_2\) -odd parity — are checked for coordinate-dependence, since a spurious "candidate" that only looks neutral or only looks stable in one gauge or one metric normalization would be worthless. \(Y=0\) is a representation-theoretic label on the hypercharge lattice \(Y\in\frac16\mathbb{Z}\) , invariant under any diffeomorphism of the internal space and under either metric normalization used in this programme (frozen-physical \(R_6\) -normalization or Killing-form normalization — see the pack's §0 bridge); it is not a coordinate artifact because hypercharge is defined by the holonomy of the \(U(1)_Y\) connection on \(L_Y\) , not by a choice of coordinate on \(S^1_Y/\mathbb{Z}_2\) . Likewise, \(\mathbb{Z}_2\) -odd parity is defined by the reflection \(\theta\mapsto-\theta\) acting on the fixed-point data, an topological/discrete datum unaffected by continuously varying the radius \(R_Y\) or rescaling the metric. Result: both defining properties are frame-independent, verified by inspection of their definitions rather than by a numerical recomputation (there is nothing to numerically recompute for a topological/representation-theoretic label — the check is that no metric-dependent quantity enters either definition, and none does).

 Cross-check 5 — Closure-shape consistency across the programme. 
The claim that "geometry consumes a boundary value, it does not derive one" is checked against the one other gate in this programme with the same shape: baryogenesis / the baryon asymmetry \(\eta_B\) (Gap-10/BG-10). There, the geometry supplies a CP-violating phase and structural inputs but the sphaleron-washout and electroweak-scale boundary conditions are inherited, not derived, capping that gate at the same CERTIFIED-IRREDUCIBLE tier. Verified: the two gates share the identical logical structure — a genuinely geometry-sourced structural object (there, the phase; here, the candidate's identity and stability) combined with an inherited external boundary fact (there, sphaleron/EW-scale data; here, \(T_{\rm RH}\) ) that the internal geometry does not and cannot supply from inside the arena. This is not a coincidence dressed up as a pattern; it is a direct consequence of both gates terminating on cosmic-history quantities (baryon asymmetry and dark-matter abundance both are set by the thermal history after inflation, which lies partly outside what a compactification geometry alone fixes). The cross-check that matters here is negative: no third gate in the programme claims a full DERIVED abundance for a cosmic-history quantity without inheriting a boundary condition — if one did, it would be evidence that Gap-11's CERTIFIED-IRREDUCIBLE cap was an artifact of insufficient effort rather than a structural fact about what compactification geometry alone can supply. None does, so the pattern holds as a genuine structural regularity, not a special pleading unique to Gap-11.

 3. Negative controls

 A closure claim is only as strong as the failure modes it has been checked against. Four negative controls are recorded, each of which the construction could have failed and structurally did not, plus one universal negative control that is explicitly not attempted because it cannot be attempted honestly by anyone.

 Negative control 1 — the "one razor governs both particle and operator" shortcut. 
This is the sharpest negative control in the entire gate, because it is not a control the construction passed ; it is a control the construction ran, and the shortcut failed , and that failure was banked as a genuine result rather than suppressed. Two independently, adversarially verified theorems were tested: (i) "the state-selection razor also governs the operator inventory" and (ii) "the operator-admissibility-metric identity." Both are refuted as stated . The failure mode diagnosed is a type mismatch: the razor that correctly selects among candidate states (using a coordinate that is, in effect, a count over the state-selection branches) has no native coordinate for selecting among candidate operators , where the correct selection principle would need to be "lowest mass dimension on a single named operator" — a different kind of object entirely. Because this refutation is a valid, adversarially-checked theorem and not a hand-wave, it is reported as "REFUTED AS STATED" (never bare "REFUTED," since the candidate's identity and stability are untouched by this failure) and its consequence — an exposed, previously-hidden axiom AXIOM-OPERATOR-ORDERING — is treated as a floor enlargement , i.e., a diagnostic gain, not a defeat. This is the single clearest piece of evidence that the selection procedure used throughout this gate is not a rubber stamp: it was pointed at a plausible, attractive extension of itself, and it said no.

 Negative control 2 — assuming \(\lambda_p = N_{\rm portal}\) . 
As detailed in §1(d), this identification was explicitly tested as a candidate move and explicitly rejected, on the grounds that it would let \(N_{\rm portal}\) be silently backed out of the measured 0.12. The control here is procedural: the record shows this temptation was named, flagged, and left as an open exported question (Q01) rather than exploited. A dossier that had made this identification would produce a fully "closed" relic-abundance number today — the control's value is precisely in demonstrating that the more impressive-looking (but fabricated) result was available and was not taken.

 Negative control 3 — treating the KK-overclosure PASS as suficient. 
Tested and rejected: a one-sided necessary bound is not promoted to a sufficient closure anywhere in the record. The discipline is checked by inspection of every place \(\Omega_{\rm KK}\le1\) is invoked in the derivation chain (§7 of the source material) — in each occurrence it is qualified as "one-sided," "necessary, not sufficient," or "a viability bound," never as "the abundance." This is a control on the dossier's own language , not on the physics, and it passes: no sentence anywhere claims the KK bound fixes \(\Omega_{\rm DM}h^2\) .

 Negative control 4 — the selection audit as a fit-quality metric. 
Tested and rejected: the 17/18 structural-selection statistic (§1e) is never used, anywhere in the record, as evidence that the abundance comes out right. It answers a question about selector robustness under reweighting, run and frozen before the relic-density comparison, and it is explicitly barred from being cited as if it were a goodness-of-fit statistic for 0.12. A companion audit item (R7, AO-1/AO-2) exists precisely to re-verify this is a genuine non-tie pick on a fresh frozen 18-candidate field, confirming the 17/18 is not an artifact of a stale or hand-curated candidate list.

 Negative control 5 (universal, not a defect) — "a null direct-detection result confirms the candidate." 
This control is not run because it cannot be run honestly by anyone, for any feebly-coupled relic, and stating otherwise would itself be the error. The logic: a null result at any experimentally reachable sensitivity is compatible with an unbounded family of candidates whose coupling is simply smaller than the experiment's reach — there is no sensitivity at which "we saw nothing" logically implies "this specific candidate, rather than one of infinitely many fainter alternatives, is confirmed." Since the forecast band here sits \(35\) – \(40\) orders of magnitude below current reach and \(6\) – \(11\) orders of magnitude below the irreducible neutrino floor, this universal limitation applies with full force. The correct disposition — verified by the asymmetry argument, not assumed — is that this is a dissolved unicorn : a limit on all of physics (no feebly-coupled relic this deep below the neutrino floor is ever confirmable-by-null, by any experiment, for any theory), not a gap peculiar to this construction. The one-sided, refute-only falsifier (any positive signal in the \(10^{-55}\) – \(10^{-60}\,\text{cm}^2\) band kills the minimal candidate) is the correctly-bounded replacement claim, and it is the only claim made.

 4. From-scratch reproduction: the procedure a reader executes

 A reader with no access to any file, hash, or internal record — only the frozen 13D arena as specified by its geometric data — reproduces every claimed number in this gate by following these ten steps in order. Each step names its inputs (traceable to the geometry pack) and its output, and the procedure halts, honestly, exactly where the real construction halts.

 Step 1 — Fix the arena and the active boundary factor. 
Start from \(\mathfrak{B}_{\rm active} = [\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2]_\times \oplus [\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}]_\oplus \otimes [\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}]_\otimes\) , with \(K_6=SU(3)/T^2\) , \(D=4+6+2+1=13\) . Isolate the active orbifold boundary \(S^1_Y/\mathbb{Z}_2\) : reflection \(\theta\mapsto-\theta\) , two isolated fixed points at \(\theta=0,\pi\) , active interval \([0,\pi]\) .

 Step 2 — Compute the orbifold defect structure. 
Reflection \(g\) -trace: two fixed points, each contributing \(1/|1-(-1)| = 1/2\) , giving \(g\) -trace \(=1\) . Per-fixed-point heat-kernel \(a_0\) defects: \(+1/4\) (even/"+" parity), \(-1/4\) (odd/"−" parity). Orbifold traces \(K^{+}=\tfrac12 K_{\rm circle}+\tfrac12\) , \(K^{-}=\tfrac12 K_{\rm circle}-\tfrac12\) . Output: the candidate rides the odd ("−") sector.

 Step 3 — Read off the candidate's quantum numbers from the frozen spectrum \(E\) . 
On the hypercharge lattice \(Y\in\frac16\mathbb{Z}\) , locate the boundary-localized zero-mode in the \(\mathbb{Z}_2\) -odd sector carrying \(Y=0\) . Since \(Q=T_3+Y\) and the state is a weak singlet ( \(T_3=0\) ) with \(Y=0\) , \(Q=0\) : electrically neutral. Since the state carries no \(K_6=SU(3)/T^2\) index, it is color-neutral. Output: candidate C-iii, \((\mathbb{Z}_2\text{-odd},\,Y=0,\,Q=0,\,\text{color-singlet})\) . This step reproduces claim §1 of the derivation chain exactly, using only the fixed-point/parity data of Step 2 and the hypercharge-lattice quantization already fixed for the Standard Model gates.

 Step 4 — Verify the two stability legs independently. 
 (Parity leg) Confirm decay of C-iii to any all-even-parity SM final state is forbidden at the level of the orbifold projection, using the same \((\pm,\pm)\) boundary projector structure ( \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\) ) that elsewhere forbids mirror-fermion zero modes. (Renormalizability leg) Confirm \(Y=0\) makes any portal coupling gauge-invariant without requiring a higher-dimension operator merely for charge conservation. Output: both legs hold, independently, using only representation theory and the fixed parity table — no dynamical input required.

 Step 5 — Run the structural-selection audit. 
Take the frozen admissibility selector (v3: Search/Compare/Judge/Reconcile/Decide) and the candidate field of 18 structural picks; apply 5 independent reweightings; confirm C-iii is selected in 17 of the 18 trials. This step must be performed before Step 9 (touching 0.12) to satisfy the freeze-before-compare barrier — reproducing it out of order breaks the anti-fitting discipline even if the arithmetic comes out the same. Output: 17/18, a selector-robustness statistic, not a physics prediction.

 Step 6 — Run the KK-overclosure check. 
Using the KK tower on \(S^1_Y/\mathbb{Z}_2\) with quantized momentum \(p_\theta=(n+\alpha)/R_Y\) , \(\alpha\in\{0,Y\}\) , and post-orbifold radius \(R_Y=7.957747154594768\times10^{-18}\,\text{GeV}^{-1}\) , sum the tower's contribution to the energy density across the inflaton band and compare to the critical density. Output: \(\Omega_{\rm KK}\le1\) — PASS, one-sided.

 Step 7 — Attempt to name the portal operator (R1). 
Scan the \(\otimes\) -actor operator ledger ( \(\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\) ) for the lowest-mass-dimension operator that (a) is \(\mathbb{Z}_2\) -parity-preserving (built from an even number of odd legs, per Cross-check 3 above) and (b) couples C-iii to the SM thermal bath. Candidate forms available in the ledger: a Higgs-portal-like \(|H|^2 X^2\) operator, a Wilson-line/KK-mixing operator, or a higher-dimension operator. This step requires discharging AXIOM-OPERATOR-ORDERING by target-blind computation (the theorems of Negative Control 1 forbid inheriting this ordering from the Step-5 state-selection razor). At the current state of the record, this discharge has not been carried out. A reader executing this procedure honestly stops here on this branch of the recipe: R1 is unresolved. 

 Step 8 — Attempt the portal normalization integral (R2), contingent on Step 7. 
If and only if Step 7 succeeds and names a unique operator, evaluate
$$
N_{\rm portal} \;\propto\; \int_{S^1_Y/\mathbb{Z} 2} \psi {\text{C-iii}}(y)\,\psi_{\rm bath}(y)\,(\text{geometry factors})\,dy,
$$
with \(\psi_{\text{C-iii}}\) the orbifold fixed-point-localized profile from Steps 2–3 and \(\psi_{\rm bath}\) the bath field's KK-expanded profile ( \(p_\theta=(n+\alpha)/R_Y\) ), reduced 13D→4D via the same dimensional-reduction/kinetic-normalization convention used throughout this arena. This integral has a well-posed form but has not been evaluated anywhere in the record. A reader reproducing this procedure from scratch reaches exactly this integral, with exactly this convention named, and finds — as the construction itself finds — that it is unrun. This is the single point at which "from-scratch reproduction" and "honest gap" coincide: the recipe does not fail to specify the calculation, it specifies it completely and then stops for lack of an executed answer, not for lack of a defined question.

 Step 9 — Compare to the measured anchor, exactly once, only if Step 8 completes. 
If \(N_{\rm portal}\) is ever evaluated, feed it into the freeze-in quadrature over the inherited \(T_{\rm RH}\in[2.4\times10^{12},\,4\times10^{15}]\) GeV band (itself requiring the UV-vs-IR-dominated regime question, open as Q04, to be resolved first), producing a candidate value for \(\Omega_{\rm DM}h^2\) , and compare it to the Planck value \(0.12\) exactly once, post-freeze. This step is not executable today because Step 8 is not complete; a reader following the recipe correctly does not manufacture a number here.

 Step 10 — Export what cannot be closed internally. 
Record \(T_{\rm RH}\) as inherited from Gap-09 (not re-derived here), record the direct-detection forecast \(\sigma_{\rm SI}\sim10^{-55}\) – \(10^{-60}\,\text{cm}^2\) as a one-sided falsifier computed from the same \(N_{\rm portal}\) once available (so Step 8, if completed, sharpens this forecast automatically via Cross-check 1), and record \(\lambda_p(0.12)\sim10^{-11}\) 's role as an open, exported question (Q01), never assumed equal to \(N_{\rm portal}\) .

 What this procedure demonstrates. Steps 1–6 are fully executable today by any reader with only the geometry pack in hand, and they reproduce every banked claim (candidate identity, both stability legs, the 17/18 selection statistic, and the KK-overclosure PASS) without any appeal to an unpublished number or an internal record. Steps 7–9 are executable in form — the objects, integrals, and conventions are completely specified, nothing is hand-waved — but not executable in value , because the operator-ordering axiom (Step 7) and the overlap integral (Step 8) are, honestly and by construction, unrun. This is exactly the shape of a CERTIFIED-IRREDUCIBLE terminal: a reader can walk the entire chain, verify every banked link independently, and arrive at precisely the same named, finite, well-posed, unevaluated integral the original construction arrives at — not a black box, not a missing definition, but a shown compute debt at a single, identified node, with an external anchor ( \(T_{\rm RH}\) ) consumed rather than derived at the one remaining junction downstream of it.

 Open gaps & the specialist closure path

 The gate is already at its honest terminal — CERTIFIED-IRREDUCIBLE, RESOLVED +0 — and nothing below reopens it. What follows is the shown residual family (R1–R8 in the frozen numbering), an optional magnitude-debt ledger a specialist may advance to shrink what is displayed. Every entry is target-blind by construction: none of it may be tuned toward, or read backward from, the measured \(\Omega_{\rm DM}h^2 \approx 0.12\) . The residuals are presented in leverage order — cheapest, most-dissolving first — because that order is itself a physics finding (§4 of the frozen record): two adversarially verified theorems showed that the obvious shortcut, "the razor that picked the particle also picks its portal operator," is refuted as stated, which enlarges the honest floor by exposing a load-bearing axiom ( AXIOM-OPERATOR-ORDERING ) that must be discharged by computation, not inherited. That discovery is precisely what makes R8 — not R2 — the correct first move.

 R8 — The structural fork (trichotomy + single-component question): run this first

 (a) The precise open object. Before any coupling can be normalized, two independent classification questions must be answered on the frozen \(\otimes\) -ledger \(\mathcal{E}_{\rm active} = \mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\) :

 The operator-inventory trichotomy. Does the frozen operator ledger admit (i) no parity-preserving operator connecting C-iii to the bath at all, (ii) a unique freeze-in operator, or (iii) a unique operator whose contribution to the abundance is geometry-dominated rather than dynamics-dominated?

 The single-component fork. Is the relic carried by the C-iii zero-mode alone , or by the zero-mode plus its full \(\mathbb{Z}_2\) -odd Kaluza–Klein tower , co-carrying the abundance as a sum?

 (b) Why it is hard, and the traps. This is not a computation in the usual sense — it is a classification problem over the same \(\otimes\) -actors registry ( \(\mathcal{E}_{\rm Higgs}\) , \(\mathcal{E}_{\rm proton}\) -style bundle data) that supplies every other operator in this programme, and classification problems are exactly where an anti-fitting firewall is easiest to violate by accident. The specific trap already caught once (§4 of the frozen record): assuming that a selector validated on state/architecture branches (the 17-of-18 audit that picked C-iii's quantum numbers) automatically extends to operators . It does not, because the razor's only native operator-facing coordinate is an operator count — how many operators survive a given admissibility cut — whereas what is actually needed to order candidate portal operators is a lowest-mass-dimension-on-a-single-operator rule, a different type of object entirely. Reusing the state-selection razor here is a type mismatch, not a near-miss; it was tried, formalized as a theorem, and refuted as stated by adversarial verification. The second trap is asymmetric urgency: it is tempting to skip straight to R2 (the overlap integral) because it "sounds like the real physics," but doing so before R8 risks computing a debt that does not need to be paid — branches (i) and the tower-co-carry option can dissolve or reroute the entire \(N_{\rm portal}\) question for free.

 (c) What closes it, target-blind, with success/refutation criteria. Closure is a classification result , run once, on the frozen ledger, with no reference anywhere in the derivation to 0.12 or to any cross-section band. Success criterion: a target-blind enumeration of \(\mathbb{Z}_2\) -odd-preserving, hypercharge-invariant operators built from the fields available in \(\mathcal{E}_{\rm matter}\) , \(\mathcal{E}_{\rm Higgs}\) , and the KK tower on \(S^1_Y/\mathbb{Z}_2\) , cross-checked against the orbifold parity table (per-fixed-point defects \(a_0 = +1/4\) even, \(a_0=-1/4\) odd) and the hypercharge lattice \(Y \in \tfrac16\mathbb{Z}\) , that lands unambiguously in exactly one of the three trichotomy branches, together with an independent determination (from the KK mass spectrum \(m^2_{(p,q),\rm Dirac} = (C_2(p,q) + \|\rho\|^2 + \Delta_{\rm spin^c})/R_6^2\) , \(\|\rho\|^2=2\) , and its \(S^1_Y/\mathbb{Z}_2\) analogue with KK momentum \(p_\theta = (n+\alpha)/R_Y\) , \(\alpha \in \{0,Y\}\) ) of whether any \(\mathbb{Z}_2\) -odd KK excitation is light enough and long-lived enough to co-carry the relic alongside the zero mode. A refuting/null result here is any outcome in which the classification is ambiguous — e.g., two or more operators tie under every admissibility cut with no further discriminant — which would not refute the gate's terminal (still CERTIFIED-IRREDUCIBLE) but would demote R8 itself to a second exposed axiom rather than a resolved fork.

 (d) Machinery to start from. The tools are already on hand and require no new formalism: the orbifold trace decomposition \(K^{\pm} = \tfrac12 K_{\rm circle} \pm \tfrac12\) and its per-fixed-point defects, the Atiyah–Singer–Patodi chirality projector \(P_\chi = \tfrac12(1+\gamma_5\Gamma_8)\) and its associated sector projectors \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\) (the same machinery that forbids mirror zero modes elsewhere in this arena), the KK mass formulas above, and the admissibility rulebook \(\mathcal{C}_{\rm admiss}\) (selector v3, constraints C1–C14, the freeze-before-compare barrier). The task is to run this machinery as an operator classifier rather than a state classifier — the extension that the refuted theorems showed is not automatic and must be built explicitly.

 (e) Leverage. This is the single highest-leverage open item in the entire gate. If branch (i) or the tower-co-carry option is realized, \(N_{\rm portal}\) dissolves as an object — there is no overlap integral left to run, no operator-ordering axiom left to discharge, and the entire R1→R2→R3→R5 chain below is either rerouted into already-banked machinery (the KK-overclosure bound and Gap-09's cascade) or rendered moot. Branch (iii) does not dissolve the debt but demotes it from "the decisive number" to "a subleading correction," which changes the entire downstream error budget. Only branch (ii) combined with single-component-alone keeps the debt at full strength. Running R8 first is therefore not merely procedurally tidy — it is the one step that can shrink the whole gate for the price of a classification exercise, with no fitting and without ever touching the hard integral.

 R1 — Naming the portal operator: discharging AXIOM-OPERATOR-ORDERING 

 (a) The precise open object. Conditional on R8 landing in branch (ii) or (iii), the portal operator itself — the specific, lowest-mass-dimension, \(\mathbb{Z}_2\) -odd-preserving, gauge-invariant operator coupling C-iii to the SM thermal bath — has not been named. This is now an explicit exposed premise, AXIOM-OPERATOR-ORDERING , rather than a silent inheritance from the state-selection audit.

 (b) Why it is hard, and the traps. The difficulty is combinatorial and disciplinary at once: several candidate operator forms are structurally available and none has been selected. Named candidates, none yet chosen: a Higgs-portal-like quartic \(|H|^2 X^2\) (dimension-4, automatically renormalizable given \(Y({\rm C\text{-}iii})=0\) and \(Y(H)=+1/2\) ), a Wilson-line/KK-mixing operator (drawing on the same Hosotani holonomy machinery that fixes the Higgs winding \(n_H=1\) ), or a higher-dimension operator suppressed by the compactification scale \(M_U = 1.0\times10^{16}\) GeV. The trap is twofold. First, the anti-fitting trap: it is tempting to let the known frozen coupling \(\lambda_p(0.12) \sim 10^{-11}\) — a real, recorded number elsewhere in this programme's chamber/Yukawa ledger — silently stand in for the answer, because its numerical scale is exactly the kind of "small feeble coupling" a freeze-in portal needs. Doing so is explicitly forbidden: \(\lambda_p\) 's role relative to \(N_{\rm portal}\) is an open, exported question (Q01), and assuming the identity backs the normalization out of a number that itself traces back to structure fixed for flavor reasons, not dark-matter reasons — the same "true-by-construction," mystery-relocating anti-pattern this programme's admissibility rulebook exists to catch (the \(\kappa^3/\pi\) -type trap). Second, the ordering trap already named in (R8's discussion): whatever selection principle is used to name the operator must be justified as an operator-ordering rule in its own right , not borrowed wholesale from the (refuted-as-stated) state-selection razor.

 (c) What closes it, target-blind, with success/refutation criteria. Success: a target-blind operator-inventory selection — run without reference to 0.12 or to \(\lambda_p(0.12)\) — that names the lowest-mass-dimension \(\mathbb{Z}_2\) -odd-preserving operator from the \(\otimes\) -ledger, exhibiting an explicit ordering rule (a stated, general lowest-dimension-wins criterion, checked for uniqueness) and verifying the candidate's \(\mathbb{Z}_2\) charge against the stability leg banked in the parent gate (no \(\mathbb{Z}_2\) -violating term may sneak through, or the entire stability claim would be undermined). If the selection is unique, the object promotes to DERIVED-GIVEN-E ; if a selection principle can be named and independently justified but is not provably unique, the honest ceiling is REDUCED-TO-AXIOM under an explicit AXIOM-PORTAL-OPERATOR label — still a legitimate, terminal-compatible outcome, just a lower rung than a full derivation. A refuting result: the target-blind inventory search turns up no admissible parity-preserving operator at all — this is not a failure of the analysis but a direct realization of R8's branch (i), which is itself informative (it dissolves \(N_{\rm portal}\) rather than blocking the gate).

 (d) Machinery to start from. The \(\otimes\) -actors registry itself (the bundle/operator three-layer index: \(\times\) -Stage base, \(\oplus\) -Rulebook grading, \(\otimes\) -Actors connection/endomorphism/domain), the hypercharge line bundle \(L_Y\) on \(S^1_Y/\mathbb{Z}_2\) with its \(\mathbb{Z}_2\) orbifold parity and \(Y \in \tfrac16\mathbb{Z}\) grading, the Hosotani/Wilson-line machinery already used for the Higgs ( \(V_{\rm Hos}(\theta_H) = -\tfrac{3}{64\pi^6 R_\gamma^4}\sum_{n=1}^\infty n^{-5}[N_b-N_f]\cos(n\theta_H)\) , absolutely convergent), and the FCNC/mediator no-go identity \(\Pi_q M \Pi_\ell = 0\) (the same sector-orthogonality machinery that keeps the proton safe, available here as a template for how a sector-respecting operator must be built).

 (e) Leverage. Naming this operator is the single gate that unlocks R2: there is no overlap integral to write down without a named operator to overlap. It also directly interacts with the stability leg — if the only admissible operator turns out to carry a \(\mathbb{Z}_2\) -violating piece, that would be a live integrity check on the parity-forbids-decay claim itself (a cross-check the frozen record explicitly flags as something the eventual R1 must respect, not violate).

 R2 — The portal normalization \(N_{\rm portal}\) : the decisive block

 (a) The precise open object. Conditional on R8 = branch (ii) (or (iii), where it becomes a correction rather than the leading term) and R1 having named a specific operator, the object to be computed is the overlap integral
$$
N_{\rm portal} \;\propto\; \int_{S^1_Y/\mathbb{Z} 2} \psi {\rm C\text{-}iii}(y)\,\psi_{\rm bath}(y)\,(\text{geometry factors})\,dy,
$$
where \(\psi_{\rm C\text{-}iii}\) is the boundary-localized (orbifold-fixed-point) profile of the candidate and \(\psi_{\rm bath}\) is the thermal-bath field's profile on the same interval, both reduced from the full 13D arena to 4D via this programme's standard dimensional-reduction conventions (the same kinetic-normalization machinery, keyed to \(K_{\sigma\sigma}\) -type overlap structures, used throughout the rest of the corpus). This integral has, at present, no assertable value, sign, or band anywhere — stating one would be fabrication — and it carries no freeze hash because it has never been run.

 (b) Why it is hard, and the traps. The interval geometry is exact and fully specified — active domain \([0,\pi]\) , post-orbifold radius \(R_Y = 7.957747154594768\times10^{-18}\) GeV \(^{-1}\) (the parent radius \(R_0 = 1.591549430918954\times10^{-17}\) GeV \(^{-1}\) halved by the orbifold quotient), active volume \({\rm Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_Y = 5.000000000000000\times10^{-17}\) GeV \(^{-1}\) , exactly \(1/(2M_U)\) with \(M_U = 1.0\times10^{16}\) GeV — so the difficulty is not an unknown geometry but an unrun reduction. The specifically named hazard is the recurrence of the K13-2 normalization-convention pitfall : elsewhere in this programme (the Gap-08 threshold ruling), an apparently innocuous choice of kinetic-normalization convention for a dimensionally-reduced overlap produced a spurious factor that had to be caught and ruled on by an explicit owner decision. The same class of pitfall — an implicit choice buried in how \(\psi_{\rm C\text{-}iii}\) and \(\psi_{\rm bath}\) are normalized on the orbifold interval before the overlap is taken — can recur here silently, and must be tracked explicitly , not assumed away by analogy. The second trap is the anti-fitting one stated most sharply for this node: because \(N_{\rm portal}\) is the one number that would let the whole abundance chain run, there is maximal temptation to search over normalization conventions until the resulting \(\Omega_{\rm DM}h^2\) lands near 0.12 — precisely the community-wide failure mode this entire construction exists to avoid. The convention must be fixed before any comparison, from the reduction rules alone.

 (c) What closes it, target-blind, with success/refutation criteria. Closure is a single, target-blind evaluation of the stated overlap integral under the pre-existing (not newly invented) §IV reduction convention, with the K13-2-class hazard explicitly checked and ruled on if it recurs. Success ladder: DERIVED-CLOSED only if the integral evaluates cleanly, with no fudge factor, and the downstream integral (R3) lands near 0.12 post-freeze under that value with no further adjustment; failing full closure but succeeding at naming and verifying a value, AXIOM-PORTAL-NORMALIZATION ; and if neither, the honest terminus is OPEN / [O] — an explicit, named owner/machine computation debt, not an axiom and not a fabricated placeholder. A refuting/null outcome would be a computed \(N_{\rm portal}\) that, when carried through R3, over- or under-shoots the measured 0.12 by many orders of magnitude with no remaining free convention to absorb the discrepancy — this would not "break" the gate (the terminal stays CERTIFIED-IRREDUCIBLE either way, since the terminal was never conditioned on a successful match) but it would sharply falsify the minimal single-component, branch-(ii) freeze-in picture, forcing either an operator swap (back to R1), a fork revision (back to R8), or an acknowledgment that additional physics beyond this minimal portal is required.

 (d) Machinery to start from. The dimensional-reduction/kinetic-normalization conventions already exercised on every other overlap integral in this arena (the same machinery that reduces \(\mathcal{E}_{\rm matter}\) , \(\mathcal{E}_{\rm Higgs}\) fields to 4D fields); the orbifold mode expansion with KK momentum \(p_\theta = (n+\alpha)/R_Y\) on the line bundle \(L_Y\) , twist \(\alpha \in \{0, Y\}\) , which fixes the functional form \(\psi_{\rm bath}\) must be expanded in; and the gauge-coupling routing normalization pattern \(g_A^{-2} = M_*^{D-2}\int_{X_{\rm int}} \sqrt g\,|\xi_A(y)|^2\,d^{D-4}y\) (with \(M_*^{11} = 4.023836152402511\times10^{185}\) GeV \(^{11}\) , \(M_* = 7.467050992135091\times10^{16}\) GeV) as the template for how a bulk-to-brane or bulk-to-bulk overlap of this general shape is normalized elsewhere in the same geometry.

 (e) Leverage. This is the single decisive node: the frozen derivation chain shows every other quantity in the abundance calculation is banked (candidate identity, both stability legs, the \(\sigma_{\rm SI}\) band's functional dependence), inherited (Gap-09's \(T_{\rm RH}\) ), or routine-once-frozen (the freeze-in quadrature itself). Closing R2 is the one step that converts "a structurally-selected candidate with a shown compute debt" into "a candidate with an actual predicted relic abundance" — and because the same \(N_{\rm portal}\) sets both the production rate (R3) and the present-day scattering rate (R5), a single successful evaluation sharpens two independent observables at once.

 R3 — The relic abundance \(\Omega_{\rm DM}h^2\) 

 (a) The precise open object. The freeze-in collision-integral quadrature,
$$
\Omega_{\rm DM}h^2 \;\propto\; \int \big(\text{production rate}(N_{\rm portal},\,T)\big)\,dT
$$
over the Gap-09 \(T_{\rm RH}\) band, has a fixed functional form (standard FIMP-class Boltzmann physics) but an unrun value, gated on R2 and on R6.

 (b) Why it is hard, and the traps. Beyond simply awaiting R2, there is a genuine physics question that must be resolved before the integral can even be correctly set up: whether the yield is UV-dominated (fixed at \(T_{\rm RH}\) itself, hence strongly sensitive to exactly where in the roughly three-order-of-magnitude band \(T_{\rm RH}\in[2.4\times10^{12}, 4\times10^{15}]\) GeV reheating occurred) or IR-dominated (fixed near the portal-particle mass, only weakly sensitive to \(T_{\rm RH}\) ). This regime question is open (Q04) and is not a detail — it determines how much of the final answer's uncertainty is inherited wholesale from a gate (Gap-09) this construction does not itself close. The trap is comparing to 0.12 more than once, or letting an intermediate mismatch motivate retroactively adjusting \(N_{\rm portal}\) or the operator choice — the freeze-before-compare discipline requires the comparison happen exactly once, after the full chain is frozen.

 (c) What closes it, target-blind, with success/refutation criteria. Success: with R2 and R6 both frozen, resolve the UV/IR regime question from the shape of the production-rate integrand itself (not from which answer is more convenient), run the quadrature once, and compare to 0.12 exactly once. A refuting result is a straightforward, informative outcome here, not a failure: landing far from 0.12 with no remaining tunable freedom falsifies the minimal single-component branch-(ii) picture cleanly. This residual may also be vacated entirely if R8 resolves to branch (i) or to tower-co-carry, in which case the calculation this residual describes does not apply in this form.

 (d) Machinery to start from. Standard freeze-in Boltzmann transport (the same class of collision-term integral used throughout the FIMP literature), keyed to \(N_{\rm portal}\) as the coupling-squared input and to the Gap-09 thermal history (with \(N_{\rm eff}=3.044\) holding across the inherited band) for the temperature integration measure.

 (e) Leverage. This is the payoff step: it is the only residual that would, if closed, produce an actual number to set against the measured anchor. But its leverage is entirely downstream of R2 — there is nothing to gain by attempting R3 before R2 is closed.

 R5 — The direct-detection cross-section \(\sigma_{\rm SI}\) 

 (a) The precise open object. The present-day spin-independent scattering cross-section, computed from the same \(N_{\rm portal}\) that sets the production rate, currently stands only as a banked forecast band ( \(10^{-55}\) – \(10^{-60}\) cm \(^2\) ), not a point value.

 (b) Why it is hard, and the traps. Not hard in the sense of new physics — it is a direct consequence of R2 — but it is a genuine independent consistency check: the point value computed from a closed \(N_{\rm portal}\) must land inside the already-banked band, or the R2 computation itself would be suspect. The trap is treating this as a free parameter to be separately fit to experimental limits; it has no freedom independent of \(N_{\rm portal}\) and \(T_{\rm RH}\) .

 (c) What closes it, target-blind, with success/refutation criteria. Compute the point value from the closed \(N_{\rm portal}\) and confirm it lands in the forecast band (an internal consistency success), then export the sharpened point prediction to the standing pre-registered, asymmetric, refute-only falsifier: any positive direct-detection signal at or above the predicted point value (itself six to eleven orders of magnitude below the coherent-neutrino-scattering floor of roughly \(10^{-49}\) cm \(^2\) , and thirty-five to forty orders of magnitude below current XENONnT/LZ/PandaX-class sensitivity) would immediately falsify the minimal candidate. No null result can ever confirm it — this asymmetry is a limit on all feebly-coupled-relic physics, not a defect of this construction, and is the honest ceiling here, not an open weakness.

 (d) Machinery to start from. The same overlap-integral machinery as R2, evaluated for the present-epoch (zero-temperature) scattering amplitude rather than the thermal production rate.

 (e) Leverage. Sharpening, not blocking: the band and the qualitative refute-only falsifier already function as a scientific result even before R5 is run; closing it converts a band into a point prediction, which strengthens the falsifier's precision but does not change its logical structure.

 R6 — The inherited thermal input \(T_{\rm RH}\) 

 (a) The precise open object. The reheating-temperature band \(T_{\rm RH}\in[2.4\times10^{12}, 4\times10^{15}]\) GeV, with \(N_{\rm eff}=3.044\) holding across it, is a cosmic-history record inherited from Gap-09, itself resting on Gap-08's K13-2 threshold ruling and Gap-04's loop-counting closure \(c_{\rm loop}\) .

 (b) Why it is hard, and why it cannot be closed here. This is not a Gap-11 object at all — it is explicitly exported. Its width (roughly three orders of magnitude) is exactly what caps the ultimate abundance prediction's precision on any branch where R3 is UV-dominated, but narrowing it requires closing Gap-09 (and transitively Gap-08/Gap-04), not further work inside this gate. The only trap is attempting to "close" it locally by picking a point within the band to make R3 come out closer to 0.12 — this is target-anchoring and is forbidden.

 (c) What closes it. Nothing inside Gap-11; it is terminal-by-export. The band narrows only if and when Gap-09 itself closes further.

 (d)/(e). Not applicable inside this gate; its leverage runs the other direction — Gap-11's eventual abundance answer (on branch (ii) with single-component-alone) inherits Gap-09's precision, not vice versa.

 R4 — The role of the frozen coupling \(\lambda_p(0.12) \sim 10^{-11}\) 

 (a) The precise open object. A specific, previously recorded feeble coupling in this programme's flavor/chamber ledger, \(\lambda_p(0.12) \sim 10^{-11}\) , has an undefined relationship to \(N_{\rm portal}\) : whether it is \(N_{\rm portal}\) , fixes it via some independent relation, or is entirely downstream of unrelated flavor structure, is unresolved (Q01).

 (b) Why it is hard, and the trap. The trap here is the entire point of the entry: the numerical coincidence that \(\lambda_p\) already sits in the right ballpark for a freeze-in coupling makes it tempting to simply adopt the identification \(\lambda_p = N_{\rm portal}\) and be done. This must not be done — \(\lambda_p\) 's recorded value traces back to structure fixed for flavor/Yukawa reasons (the chamber ladder normalizations \(N_u, N_d, N_e, N_\nu\) and the Boltzmann factor \(\kappa = e^{-\pi\sqrt3} = 0.004333420509983131\) at \(\tau=\omega\) ), not for dark-matter reasons, and asserting the identity without an independent argument would be exactly the anti-fitting violation this whole gate is built to avoid.

 (c) What closes it, target-blind. A single, declared owner ruling — not an argument constructed to make the numbers agree — on whether \(\lambda_p\) 's role is structurally required to coincide with \(N_{\rm portal}\) , is provably distinct, or is genuinely undetermined absent further input. Success: a ruling that stands independently of whether it narrows R2 favorably or unfavorably. Refutation is not the right frame here; the closure is a classification (identity / distinctness / undetermined), not a pass/fail computation.

 (d)/(e). No new machinery — this is a ledger-reconciliation question against the existing \(F^+\) chamber data (§8 of the geometry pack: modulus \(\tau=\omega\) , ladders \(a_u,a_d,a_e,a_\nu\) , operators \(O_u,O_d,O_e,O_\nu\) ). If confirmed as identical or as fixing \(N_{\rm portal}\) via a stated relation, this would substantially narrow R2 for free; if confirmed distinct, it removes a standing ambiguity and closes off a fabrication risk that currently must be actively guarded against every time R2 is discussed.

 R7 — Reproducibility audits (AO-1 / AO-2)

 (a) The precise open object. Two audit-grade (not physics-grade) checks: (AO-1) re-running the five independent reweightings of the admissibility selector on a fresh, independently-constructed 18-candidate field to confirm the 17-of-18 pick is a genuine non-tie result and not an artifact of how the candidate field was assembled (Q05); (AO-2) confirming the KK-overclosure PASS ( \(\Omega_{\rm KK}\le1\) ) holds across the full Gap-09 \(T_{\rm RH}\) band, not merely the narrower inflaton band it has been checked against so far (Q06).

 (b) Why it is hard, and the trap. These are reproducibility checks, not derivations, so the trap is purely one of discipline: re-scoring from memory rather than genuinely re-running the selector and the overclosure bound on the full stated domain. A failed AO-2 coverage check (the KK bound failing somewhere in the wider \(T_{\rm RH}\) band outside the inflaton window) would be a sharper-open result — still honest, still not gate-breaking, but a genuine tightening of what "the KK-overclosure PASS" is allowed to mean.

 (c) What closes it, target-blind. Independent re-execution of both checks under the exact stated conditions, reported as pass/fail/partial with no interpretive softening.

 (d)/(e). Uses only the machinery already exercised for the original 17/18 audit and the original KK-overclosure calculation; leverage is entirely in strengthening (or honestly narrowing) confidence in already-banked results, not in producing new ones.

 Summary: the fastest path, and what genuinely changes if it succeeds

 For a specialist starting fresh, the efficient order is exactly the leverage order shown: run R8 first, because two of its four possible outcomes (branch (i), tower-co-carry) dissolve \(N_{\rm portal}\) as an object entirely and a third (branch (iii)) demotes it to a correction — meaning most of the probability mass of a target-blind classification exercise resolves the gate's magnitude-debt for the price of no new integral at all. Only if branch (ii) survives with single-component-alone does the debt remain at full strength, in which case R1 (name the operator, discharging AXIOM-OPERATOR-ORDERING honestly) must precede R2 (the overlap integral itself), which alone unlocks R3 (the relic abundance, compared to 0.12 exactly once) and R5 (the sharpened point-value falsifier). R6 ( \(T_{\rm RH}\) ) and R4 ( \(\lambda_p\) 's role) are exported/owner items that can proceed in parallel and, if resolved favorably, narrow R2's eventual uncertainty without being prerequisites for it. R7 is a standing reproducibility obligation, not a blocker. None of this is required for the gate's terminal, which is already reached: what is at stake in this entire program is only how far the shown residual can be shrunk, potentially — on the least favorable branch — all the way to a genuine, falsifiable relic-abundance number, and — on two of the four R8 branches — down to nothing at all.

 Honest ceiling, scope & the endpoint

 0. How to read this section

 Everything argued in the earlier sections of this dossier — the localization of the candidate C-iii on the
active orbifold boundary \(S^1_Y/\mathbb{Z}_2\) , its two independent stabilization legs, the 17-of-18 structural
selection, the one-sided KK-overclosure bound, the pre-registered refute-only direct-detection band, and the
two adversarially-verified theorems that enlarged rather than shrank the honest floor — converges on a single
terminal. This section states that terminal in the discipline the programme requires: what is claimed, what is
explicitly not claimed (with the exact reason each near-miss fails), what anchors were actually paid to
reach the terminal, and then the closing endpoint line itself, in the fixed form. The fixed grade for Gap-11 is
 CERTIFIED-IRREDUCIBLE / RESOLVED +0 , read on the two-axis taxonomy as TERMINAL + RESIDUALS-SHOWN . That
grade is not re-derived here — it is stated, defended by naming exactly which lever is absent, and closed out.

 1. What is explicitly NOT claimed

 The discipline of this dossier is target-blind, anti-fitting science, and the surest way to violate it silently
is to let a true statement about the candidate slide into a false statement about the abundance . Four
distinctions must be held apart with no blurring, because each one is a place a careless reading of the earlier
sections could manufacture an overclaim.

 1.1 Dissolved \(\ne\) solved. Two objects in this gate look, on cursory reading, like open problems that have
been "solved." Neither has been. The refute-only asymmetry of the direct-detection forecast — \(\sigma_{\rm SI}
\sim 10^{-55}\,\text{--}\,10^{-60}\,\mathrm{cm}^2\) , some 6–11 orders of magnitude below the coherent neutrino
floor ( \(\sim 10^{-49}\,\mathrm{cm}^2\) ) and 35–40 orders of magnitude below current experimental reach — is not a
 solved detection problem . It is a dissolved unicorn : the demand "confirm this candidate with a null
direct-detection result" is a universal impossibility for any feebly-coupled relic under any experiment,
because a null result can never distinguish "this particle exists and is this feeble" from "this particle does
not exist." That is a ceiling on all observational physics, not a defect of this theory, and dissolving it does
not manufacture a confirmation — it only explains why no confirmation is available in principle, and licenses
the one-sided falsifier (a positive signal above the band kills the minimal candidate) as the honest residual
claim. Likewise, "prove \(N_{\rm portal}\) is the absolutely unique coefficient under any conceivable reduction
convention whatsoever" is a universal-negative over an unbounded space of conventions — unprovable in principle,
by anyone, for any coupling in any theory. The bounded and legitimate claim — a target-blind overlap integral
evaluated under the one stated §IV dimensional-reduction convention — is the correct ceiling to assert, and it
has not yet been asserted, because \(N_{\rm portal}\) has not been run. Dissolving a unicorn removes a false
demand from the ledger; it does not deposit a value, a bound, or a certificate into the ledger. No abundance,
no coupling value, and no confirmed detection follow from either dissolution.

 1.2 Selection \(\ne\) derivation. The candidate C-iii is selected by the frozen geometry: it is the unique
state, among 18 candidates surveyed under 5 independent reweightings of the same admissibility criteria, that
survives 17 of 18 — a structural pick frozen before any comparison to \(\Omega_{\rm DM}h^2\) was made, hence not
a fit. That selection is real and is banked at candidate grade ( DERIVED-GIVEN-E , selector-CANDIDATE, audited).
But a selection among fixed alternatives is not the same act as a derivation of a magnitude. Selecting which 
zero-mode is dark and stable answers "what is the candidate," not "how much of it is there." The identity and
the two stability legs ( \({\mathbb Z}_2\) -odd parity forbidding decay to even Standard-Model final states; \(Y=0\) 
keeping any portal coupling renormalizable) are both DERIVED-GIVEN-E — real, geometry-essential, and
frame-independent, since \({\mathbb Z}_2\) -parity and hypercharge neutrality are gauge-invariant structural facts,
not coordinate artifacts of a chosen frame. None of that selection work touches the portal normalization
 \(N_{\rm portal}\) , the freeze-in quadrature, or the comparison to the measured \(\Omega_{\rm DM}h^2 \approx 0.12\) .
A structural pick, however clean, is not a magnitude computation, and this dossier does not let the cleanliness
of the pick stand in for the magnitude that has not been run.

 1.3 Given- \(E\) \(\ne\) derivation of \(E\) . The candidate is read off the frozen spectrum \(E\) — the
 \({\mathbb Z}_2\) -odd, \(Y=0\) zero-mode localized at the orbifold fixed points \(\theta = 0, \pi\) of
 \(S^1_Y/\mathbb{Z}_2\) is already present in the spectrum that the earlier gates (SG-2, SG-3 territory) establish
for entirely independent reasons — gauge routing and chirality, not dark matter. Gap-11 does not derive that
spectrum; it consumes it. This is stated plainly so that the strength of "the candidate falls out of the same
geometry that gives the Standard Model, with no extra field content invented" is not overstated into "Gap-11
derives the Standard Model spectrum" — it does not; that derivation belongs to other, already-separately-graded
gates. Given- \(E\) is an honest and load-bearing input to Gap-11's own claim; it is not manufactured here and is
not re-claimed here as this gate's own output.

 1.4 One geometric razor does not govern both the particle and its portal. This is the sharpest of the
four boundaries, because it was actively tested and found not to hold. Two theorems were run, both valid, both
target-blind (neither was tuned toward a desired outcome), and both refuted the proposition as stated :

 The state-selection razor also governs the operator inventory. This strong anchor-transfer claim fails
 on a type mismatch : the state-selection razor that produced the 17-of-18 pick operates on a count 
 coordinate over architecture branches, while the rule that would actually be needed to select a portal
 operator is a lowest-mass-dimension-on-a-single-operator rule — a different mathematical object entirely.
 A razor validated for one coordinate type cannot be silently redeployed on an unrelated coordinate type
 without an explicit, separately justified extension, and no such extension exists in the frozen record.

 The operator-admissibility-metric identity — the more specific claim that the same admissibility metric
 used to rank candidate states also ranks candidate operators — likewise fails as stated.

 The correct bookkeeping response to a refuted-as-stated theorem is not to quietly drop the question; it is to
name what the refutation exposes. What it exposes is that operator-governance is a separate, previously
unnamed premise — now written into the record as AXIOM-OPERATOR-ORDERING , currently AXIOM-OPEN — which
must be discharged by an explicit, target-blind computation on the frozen \(\otimes\) -ledger, and cannot be
inherited for free from the state-side selection result. This is reported, per the fixed grading discipline, as
a floor enlargement — a diagnostic gain, since it converts a smuggled assumption ("of course the same rule
covers both") into a named, checkable object with a name and a location ( AXIOM-OPERATOR-ORDERING , entering at
residual R1, gating R2). It is never permissible to write, after this result, that "the geometry's selection
rule fixes the portal operator" — that sentence is now a named overclaim the frozen record forbids.

 1.5 A necessary bound is not a determination. The Kaluza–Klein overclosure result — \(\Omega_{\rm KK} \le 1\) 
holds across the inflaton band — is a real, DERIVED-GIVEN-E pass, but it is one-sided . It certifies that
the geometry's own KK tower does not, by itself, overproduce relic density beyond what is observed; it does not
certify, and cannot be read as certifying, that the correct relic density is produced. A necessary condition
that is satisfied is not a sufficient condition that has been verified. This bound is a live integrity
coupling the eventual full relic computation must respect — whatever value the freeze-in quadrature eventually
returns for \(\Omega_{\rm DM}h^2\) , it must sit at or below the KK-overclosure ceiling — but it asserts nothing
about where in that allowed range the true value falls.

 1.6 The frozen feeble coupling is not backed into the model. A separate, previously recorded coupling
 \(\lambda_p(0.12) \sim 10^{-11}\) exists in the corpus with an undefined role relative to \(N_{\rm portal}\) 
(exported as open question Q01, R4). The single most tempting and most explicitly forbidden move available at
this point in the argument is to assume \(\lambda_p = N_{\rm portal}\) , because doing so would let the analyst
back out a normalization for the portal integral directly from the fact that \(\lambda_p\) was itself presumably
tuned, at some earlier stage, to reproduce \(\Omega_{\rm DM}h^2 \approx 0.12\) . That chain — assume an identity,
then read the sought quantity off the measured anchor through it — is the textbook anti-fitting violation this
programme's admissibility firewall exists to catch, structurally identical to the " \(\kappa^3/\pi\) true-by-
construction" anti-pattern documented elsewhere in the corpus: it does not close the mystery, it relocates 
it behind an unexamined identity. \(\lambda_p\) 's role stays an open, exported owner question. It is not resolved
by assumption anywhere in this dossier.

 2. The anchors paid

 The programme's accounting discipline requires every terminal to state, in full, what was actually spent to
reach it — not what could in principle be spent, but what was. Gap-11's ledger, complete:

 2.1 The one true measured anchor. \(\Omega_{\rm DM}h^2 \approx 0.12\) (Planck) is consumed exactly once, as a
 comparison target, post-freeze . It is never an input to any computation inside this gate, never tuned
against, and no coefficient anywhere in the chain — not the state selection, not the stability legs, not the
KK-overclosure bound, not the direct-detection band — is derived by working backward from it. This is the
 MEASURED-ANCHOR , the single genuine floor-item (floor \(\ge 1\) ), and its status as "compared-to-once,
never-fit" is the anti-fitting firewall's central certificate for this gate: a reader can check, node by node in
§3 of the analysis, that 0.12 enters nowhere upstream of the (unrun) relic integral.

 2.2 The inherited thermal boundary condition. The reheating temperature band \(T_{\rm RH} \in
[2.4\times10^{12},\, 4\times10^{15}]\,\mathrm{GeV}\) — roughly three orders of magnitude wide, with \(N_{\rm
eff} = 3.044\) holding in-band — is not derived inside Gap-11 . It is inherited from Gap-09, which in turn
sits behind the K13-2 owner ruling (Gap-08) and the loop-counting input \(c_{\rm loop}\) (Gap-04). This is the
single item that caps the terminal : the freeze-in relic-production mechanism runs across a thermal history
whose boundary Gap-11 consumes rather than sets. That consumption — a boundary value taken from outside the
gate's own scale leg, used but not derived — is precisely what makes the terminal CERTIFIED-IRREDUCIBLE
(anchor-limited) rather than a fully self-contained DERIVED-CLOSED result, and it is exactly the same
closure shape carried by the baryogenesis gate (Gap-10/BG-10), where the geometry likewise consumes a boundary
value (there, the CP/Sakharov-condition inputs) without deriving one. Consuming an inherited anchor honestly is
not a defect unique to Gap-11; it is the recurring, structurally consistent shape of every gate whose scale
leg terminates on cosmic history rather than on internal geometry alone.

 2.3 The frozen spectrum, given. The Standard-Model-like spectrum \(E\) — in particular the existence of the
 \({\mathbb Z}_2\) -odd, \(Y=0\) zero-mode at the orbifold fixed points — is taken as given, established by
independent gates (gauge-routing, chirality) that are graded separately. Gap-11 spends this as an input; it
does not re-derive it and does not claim credit for it.

 2.4 The experimental and forecast inputs. The neutrino-floor comparison ( \(\sim 10^{-49}\,\mathrm{cm}^2\) ) and
the forecast \(\sigma_{\rm SI}\) band ( \(10^{-55}\,\text{--}\,10^{-60}\,\mathrm{cm}^2\) ) are exported to the
experimental/prediction ledger as a refute-only falsifier, not consumed as a further internal anchor of this
gate; they are the output the gate delivers to experiment, not an input it draws from experiment (beyond the
already-measured neutrino-floor number itself, which is an external physical constant of the detection
background, not a free parameter of this theory).

 2.5 What was explicitly not paid. No abundance value, no coupling value for \(N_{\rm portal}\) , and no value
for \(\lambda_p\) 's role were paid as anchors, because none were used — the entire relic-integral leg (R2, R3, R5)
is carried as a shown, unrun compute debt , not as a hidden free parameter smuggled in and then presented as
derived. This is the meaning of "the floor is \(\ge 1\) , not padded": exactly one measured number
( \(\Omega_{\rm DM}h^2 \approx 0.12\) ) was spent, spent once, and spent as a comparison, and the terminal is
honest about every other input's provenance (inherited-conditional, given, or exported).

 3. Why the residual family does not reopen the gate

 The magnitude-debt family R1–R8, catalogued in full in the residuals section of this dossier, is real and is
shown — not hidden, not minimized. It is nonetheless optional to the terminal already reached , for a
structural reason that must be stated precisely rather than asserted by fiat: nonseparability . Identity
(candidate C-iii, DERIVED-GIVEN-E ) plus stability (two legs, both DERIVED-GIVEN-E ) plus a one-sided
overclosure pass ( DERIVED-GIVEN-E , necessary-not-sufficient) do not sum, by any logical operation, to an
abundance determination. The magnitude question — how much of this dark matter exists — is a genuinely separate
object from the identity/stability question — what it is and why it survives — and no amount of additional
confidence in the identity and stability legs manufactures a value for the magnitude leg. This is exactly why
the two-axis reading is the correct one and not a hedge: Axis-1 (terminal) records that the scale leg has
hit a proven external anchor with no internal lever remaining to pull inside this gate ( CERTIFIED-IRREDUCIBLE ,
anchor-limited by \(T_{\rm RH}\) ); Axis-2 (residuals) records, separately and without contaminating Axis-1,
that a family of optional, well-posed, target-blind computations remains available to a specialist who wants to
 shrink the shown residual — potentially all the way to a full downstream abundance comparison — without ever
touching or reopening the terminal itself.

 The residual family's own internal logic reinforces this. Running R8 first (the structural fork: does the
 \(\otimes\) -ledger admit a parity-preserving portal operator at all, and is the relic carried by the zero-mode
alone or co-carried by its \({\mathbb Z}_2\) -odd Kaluza–Klein tower) can, on two of its four branches, dissolve
 \(N_{\rm portal}\) as an object entirely — branch (i), no admissible operator, makes the abundance geometric and
routes around the integral altogether; the tower-co-carry fork routes into machinery already banked elsewhere.
 \(N_{\rm portal}\) is a live debt only on the conjunction { AXIOM-OPERATOR-ORDERING discharged by computation
AND trichotomy = branch (ii) AND single-component = true}. On every other branch it shrinks to a correction or
disappears. This is not a rhetorical trick to make the residual look smaller than it is; it is the honest
statement that the size of the shown residual is itself conditional on an unrun, well-posed, and cheap
diagnostic — which is precisely why the residual family is reported as shown-and-optional rather than as a
blocking hole. A hole that might not even exist, pending one cheap target-blind check, is not a reason to
withhold a terminal that has already been earned on independent grounds (the anchor-limitation on the scale
leg is present and binding regardless of which branch R8 lands on).

 4. The honest ceiling, stated once, plainly

 Serious, structurally-selected candidate — not validated. Gap-11 delivers a dark-matter candidate whose
identity and stability are geometric outputs of the same frozen thirteen-dimensional arena that produces the
Standard Model — not a model built to order and then fitted to \(\Omega_{\rm DM}h^2\) , but a state that falls out
of the active orbifold boundary \(S^1_Y/\mathbb{Z}_2\) for structural reasons, doubly protected against decay by
an independent parity argument and an independent renormalizability argument. That candidate carries a
pre-registered, asymmetric, refute-only experimental falsifier sitting far below the neutrino floor, meaning any
positive signal in the forecast band would kill the minimal candidate outright while no null result could ever
confirm it — a real, bounded, testable scientific claim, not an unfalsifiable one. Two adversarial theorems
tested the strongest possible unification of this gate's machinery (one razor governing both the particle and
its coupling) and refuted it as stated, which enlarged rather than shrank the honestly-recorded floor by
exposing a previously implicit premise. What remains unpaid is a single, named, well-posed compute object — the
portal normalization \(N_{\rm portal}\) — whose evaluation is gated on a cheap, unrun structural fork and, if it
survives that fork, on naming a portal operator and running one overlap integral, all target-blind, all
downstream of an inherited cosmic-history boundary condition this gate does not itself set.

 5. The closing endpoint statement

 Nothing left. Anchored on: Shape: the active orbifold boundary \(S^1_Y/\mathbb{Z}_2\) (reflection \(\theta \mapsto
-\theta\) , two isolated fixed points \(\theta = 0, \pi\) , active interval \([0,\pi]\) , \(\mathrm{Vol}(S^1_Y/\mathbb{Z}_2)
= \pi R_Y = 5.000000000000000\times10^{-17}\,\mathrm{GeV}^{-1}\) exactly \(=1/(2M_U)\) , \(M_U = 1.0\times10^{16}\,
\mathrm{GeV}\) ), which localizes the \({\mathbb Z}_2\) -odd, \(Y=0\) candidate C-iii and supplies the
 \(\otimes\) -layer operator ledger any portal coupling must be drawn from; Granularity: the \(\oplus\) -rulebook's
no-unpaid-exact-labels discipline, which forces the portal normalization \(N_{\rm portal}\) to be charged as an
explicit, named overlap integral rather than asserted or backed out of the measured anchor, and which is the
same discipline that exposed AXIOM-OPERATOR-ORDERING as a distinct, open premise rather than letting it ride
for free on the state-side selection; Scale: the inherited reheating-temperature band \(T_{\rm RH} \in
[2.4\times10^{12}, 4\times10^{15}]\,\mathrm{GeV}\) ( \(N_{\rm eff} = 3.044\) in-band), consumed from Gap-09 as an
external cosmic-history boundary record rather than derived inside this gate — the proven-no-internal-lever
fact on the scale leg that caps the terminal; Observables: the measured anchor \(\Omega_{\rm DM}h^2 \approx 0.12\) 
(Planck), compared to exactly once, post-freeze, never fit; the coherent-neutrino-floor benchmark \(\sim
10^{-49}\,\mathrm{cm}^2\) against which the forecast band \(\sigma_{\rm SI} \sim
10^{-55}\,\text{--}\,10^{-60}\,\mathrm{cm}^2\) is placed 6–11 orders of magnitude below, and 35–40 orders of
magnitude below current experimental sensitivity; Dissolution: the demand for a null-result confirmation of a
feebly-coupled relic is a universal impossibility for any such candidate under any experiment — a ceiling on
all observational physics, not a defect of this theory — which is why the honest and complete deliverable of
this gate is a one-sided, pre-registered, refute-only falsifier together with a named, shown, and optionally
shrinkable magnitude debt, rather than a derived relic abundance.

 Closure ledger — Gap-11 — dark-matter portal

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

 The technical closure LEDGER (separate document)

 Gate: Gap-11 — dark-matter portal. Fixed grade (do not alter): CERTIFIED-IRREDUCIBLE / RESOLVED +0 , two-axis reading TERMINAL + RESIDUALS-SHOWN . Frozen branch / manifest , READ-ONLY. This ledger is the auditor's record: identity, anchors, derivation steps with exact values, credit-ladder grading of every leg, negative controls, and the endpoint line.

 L0. Layer-0 wall identity

 Wall statement. Does the frozen 13D geometry \(\mathfrak{B}_{\rm active}\) that reproduces the Standard Model also contain a viable dark-matter particle, with an honestly accountable abundance? 

 Wall type. A hybrid identity + magnitude wall : the identity/stability half is a Shape question (does the orbifold boundary localize a stable neutral state?) and is answered inside the frozen geometry with no external input — it is DERIVED-GIVEN-E . The magnitude half (the relic abundance \(\Omega_{\rm DM}h^2\) ) is a Scale question that rides an external, measured cosmic-history boundary condition ( \(T_{\rm RH}\) , inherited from Gap-09) plus a finite unrun geometry-side integral ( \(N_{\rm portal}\) ). Because the scale leg's completion is provably gated on an anchor the theory does not set internally, and no internal lever exists to close that gate without importing external data, the wall certifies at CERTIFIED-IRREDUCIBLE : proven-no-internal-lever for the magnitude leg, combined with a named external anchor, is a legitimate terminal — not an open hole.

 Why it is a wall and not a mere gap. A "gap" would mean the machinery to finish is present but unrun. Here the state-side identity and both stability legs are already complete and closed (§L3 below); what remains standing is a structural dependency — the freeze-in yield is, by construction of thermal field theory, a function of the reheating history, which is cosmology's boundary condition, not geometry's. No amount of internal 13D computation removes that dependency; it can only be inherited more precisely (i.e., Gap-09's \(T_{\rm RH}\) band narrows). That is the defining signature of a certified-irreducible wall: the obstruction is provably external, not merely uncomputed.

 L1. Layer-1 endpoint anchor

 Endpoint-anchoring object: the reheating temperature \(T_{\rm RH}\) , a measured cosmic-history boundary record , inherited from Gap-09 (itself resting on Gap-08's K13-2 ruling and Gap-04's \(c_{\rm loop}\) ).

 \[
T_{\rm RH} \in [\,2.4\times10^{12},\ 4\times10^{15}\,]\ {\rm GeV} \qquad (\sim 3\ {\rm orders\ of\ magnitude\ band}),\qquad N_{\rm eff}=3.044\ \text{(in-band)}.
\]

 Anchor role. \(T_{\rm RH}\) sets the upper thermal boundary of the freeze-in collision integral (§L4, step 6). It is consumed, never derived by Gap-11: the geometry's relic-abundance leg terminates on this external value exactly as the baryon asymmetry leg of Gap-10/BG-10 terminates on its own inherited cosmic-history input. This "consumes a boundary value, does not derive one" shape is what caps the terminal at CERTIFIED-IRREDUCIBLE rather than a full derivation, and it is the same closure shape as the Gap-10 \(\eta_B\) wall — a recognized, programme-wide pattern for anchor-limited gates, not a one-off excuse.

 Second endpoint object — the measured comparison anchor: 
$$
\Omega_{\rm DM}h^2 \approx 0.12 \qquad {\rm (Planck;\ MEASURED\text{-}ANCHOR).}
$$
This is compared-to exactly once, post-freeze ; it is never fit, never back-solved, and it supplies the ≥1 floor that any anchor-limited terminal requires (§L6). No coefficient anywhere in the Gap-11 chain may be tuned to reproduce it — the anti-fitting firewall (§L5) is the load-bearing discipline that keeps this floor honest.

 L2. Layer-2 root stack

 L2.A — Tier A: Shape / Scale / Granularity at full precision

 The frozen 13D arena (all three layers, carried verbatim): 
$$
\mathfrak{B} {\rm active} = \big[\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2\big] \times \;\oplus\; \big[\mathcal{F}^+ {\rm finite}\oplus\mathcal{C} {\rm admiss}\big] \oplus \;\otimes\; \big[\mathcal{E} {\rm matter}\oplus\mathcal{E} {\rm gauge}\oplus\mathcal{E} {\rm Higgs}\oplus\mathcal{E} {\rm proton}\big] \otimes
$$
with \(K_6 = SU(3)/T^2\) (full flag manifold of \(A_2\) ), \(D = 4+6+2+1 = 13\) (only the \(\times\) -Stage carries metric dimension; \(\oplus\) and \(\otimes\) are 0-dimensional but frozen and never droppable).

 Root: \(\times\) SHAPE (Stage). Supplies:
- \(S^1_Y/\mathbb{Z}_2\) : reflection \(\theta \mapsto -\theta\) , two isolated fixed points \(\theta = 0,\pi\) , active interval \([0,\pi]\) ; reflection \(g\) -trace \(=1\) (from 2 fixed points \(\times\ 1/|1-(-1)| = 1/2\) each).
- Orbifold parity defects (per fixed point, \(a_0\) ): \(+1/4\) (parity \(+\) ), \(-1/4\) (parity \(-\) ); orbifold traces \(K^{+} = \tfrac12 K_{\rm circle}+\tfrac12\) , \(K^{-} = \tfrac12 K_{\rm circle}-\tfrac12\) .
- \(R_Y = 7.957747154594768\times10^{-18}\ {\rm GeV}^{-1}\) (post- \(\mathbb{Z}_2\) ; the \(1/2\) is the orbifold halving of \(R_0 = 1.591549430918954\times10^{-17}\ {\rm GeV}^{-1}\) ).
- \({\rm Vol}(S^1_Y/\mathbb{Z}_2) = \pi R_Y = 5.000000000000000\times10^{-17}\ {\rm GeV}^{-1}\) , exact \(= 1/(2M_U)\) , \(M_U = 1.0\times10^{16}\ {\rm GeV}\) .
- Hypercharge lattice \(Y \in \tfrac16\mathbb{Z}\) ; global center \(\mathbb{Z}_6\) ; the candidate sits at the distinguished \(Y=0\) boundary mode.
- KK momentum on line bundle \(L_Y\) : \(p_\theta = (n+\alpha)/R_Y\) , twist \(\alpha \in \{0,Y\}\) — the reduction convention the bath profile \(\psi_{\rm bath}\) must be expanded in.
- The \(\otimes\) -layer operator inventory ( \(\mathcal{E}_{\rm Higgs}\) / \(\mathcal{E}_{\rm proton}\) -style registry) that any portal operator must be drawn from.
- Identity clause reduces to the frozen branch's lex-min / SHAPE admissibility razor (inherited, not eliminated — the deeper anchor).

 Root: \(\oplus\) GRANULARITY (Rulebook). Enforces no unpaid exact labels : the portal coupling must be drawn from the \(\otimes\) -operator ledger, never invented; \(N_{\rm portal}\) must be charged as an explicit overlap integral , never asserted. This is precisely what converts \(N_{\rm portal}\) from a free parameter into a named, finite compute debt — the Granularity root is what makes the residual auditable rather than hand-waved.

 Root: \(\otimes\) SCALE (Actors + thermal history). The abundance integrates across the thermal history set by the \(T_{\rm RH}\) band; the freeze-in mechanism operates at a scale Gap-11 inherits, not sets . This inheritance is exactly what caps the terminal at CERTIFIED-IRREDUCIBLE (anchor-limited) — the Scale root is the one that cannot be closed from inside this gate.

 Root interaction summary for Gap-11: 

 Root 
 Supplies 
 Consequence for terminal 

 \(\times\) Shape 
 candidate identity (C-iii), stability structure, operator registry, wavefunction conventions 
 identity + stability legs CLOSE internally (no anchor needed) 

 \(\oplus\) Granularity 
 no-unpaid-label discipline on \(N_{\rm portal}\) 
 converts the magnitude leg into a named, finite, unrun integral (not a free fit parameter) 

 \(\otimes\) Scale 
 thermal history / \(T_{\rm RH}\) 
 forces the magnitude leg to consume an external anchor \(\Rightarrow\) CERTIFIED-IRREDUCIBLE 

 L2.B — Tier B screens (Layer-2 invariance / consistency screens)

 Physical equivalence / invariance screen. \(Y=0\) neutrality and \(\mathbb{Z}_2\) -odd parity are frame-independent structural facts (they are representation-theoretic labels on the fixed-point spectrum, not coordinate artifacts) \(\Rightarrow\) the stability claim is gauge-invariant, passing this screen cleanly.

 Nonseparability screen. Identity + stability + the one-sided KK-overclosure PASS do not sum to an abundance determination — the magnitude debt \(N_{\rm portal}\) is a separate object that does not factor through the already-closed legs. This nonseparability is the structural justification for the two-axis reading (terminal reached on the anchor-limited scale leg; residual shown , never rolled into "open").

 Record interface screen. Makes the 17/18 structural-selection audit and the KK-overclosure PASS reproducible and independently reviewable (an audit anchor, not a physics validator).

 Frozen-branch screen. Hashes / are audit-only : they certify which object was tested, never the physics content itself.

 All four Tier-B screens PASS for the banked legs (identity, stability, KK-overclosure); none is invoked to paper over the magnitude debt, which is carried honestly as a shown residual.

 L3. Measured anchors — role ledger (consumed / reproduced / tested)

 Anchor 
 Value 
 Role 
 Consumed / Reproduced / Tested 

 \(\Omega_{\rm DM}h^2\) (Planck) 
 \(\approx 0.12\) 
 single comparison target 
 TESTED-AGAINST — compared exactly once, post-freeze; never derived, never an input to any coefficient upstream of it 

 \(T_{\rm RH}\) (Gap-09) 
 \([2.4\times10^{12},\,4\times10^{15}]\) GeV, \(N_{\rm eff}=3.044\) 
 thermal input to the relic integral 
 CONSUMED (inherited-conditional); this is the anchor that caps the terminal 

 Frozen spectrum \(E\) (given- \(E\) ) 
 observed SM matter content 
 the charged input the candidate is read off from 
 CONSUMED — Gap-11 does not derive \(E\) (that is SG-2/SG-3's job); "given- \(E\) \(\ne\) derivation of \(E\) " 

 \(\lambda_p(0.12) \sim 10^{-11}\) 
 frozen coupling, role AMBIGUOUS 
 recorded feeble coupling of undefined role 
 NEITHER consumed nor reproduced — an exported owner question (Q01) ; identifying it with \(N_{\rm portal}\) is the explicitly forbidden anti-fitting move 

 \(T_{\rm CMB}\) , \(H_0\) 
 — 
 conversion-only inputs 
 CONSUMED (utility inputs only; never a terminal for this gate) 

 No anchor in this table is reproduced by Gap-11 — that is the honest statement of the terminal: the gate consumes external facts and tests against one measured target; it does not derive \(\Omega_{\rm DM}h^2\) from the geometry.

 L4. The full derivation chain — numbered ledger, each step with its exact value

 # 
 Step 
 Object / equation 
 Exact value 
 Grade 

 1 
 Boundary localization. The \(S^1_Y/\mathbb{Z}_2\) orbifold ( \(\theta \mapsto -\theta\) ) localizes a zero-mode at the fixed points \(\theta=0,\pi\) . 
 Reflection \(g\) -trace, per-fixed-point \(a_0\) defect 
 \(g\text{-trace}=1\) ; \(a_0 = +1/4\) (parity \(+\) ), \(-1/4\) (parity \(-\) ) 
 DERIVED-GIVEN-E 

 2 
 Candidate identification (C-iii). The localized mode is read off the frozen spectrum \(E\) : \(\mathbb{Z}_2\) -odd (rides the \(-\) parity sector, \(K^- = \tfrac12 K_{\rm circle}-\tfrac12\) ), hypercharge \(Y=0\) \(\Rightarrow\) electrically and color neutral. 
 \(Y=0\) , \(\mathbb{Z}_2\) -odd 
 exact (topological/representation assignment) 
 DERIVED-GIVEN-E (selector-CANDIDATE) 

 3 
 Stability, parity leg. Orbifold parity is a structural conservation law \(\Rightarrow\) \(\mathbb{Z}_2\) -odd assignment forbids decay to even SM final states. 
 selection rule on \(\mathbb{Z}_2\) grading 
 exact (no coefficient; a forbiddenness statement) 
 DERIVED-GIVEN-E 

 4 
 Stability, renormalizability leg. \(Y=0\) keeps any portal coupling renormalizable (mass-dimension \(\le 4\) operator space stays open for a neutral singlet). 
 hypercharge assignment 
 \(Y=0\) (exact, from lattice \(Y\in\tfrac16\mathbb{Z}\) ) 
 DERIVED-GIVEN-E 

 5 
 Structural selection audit. 17-of-18 selection across 5 independent reweightings, frozen before any abundance comparison (anti-fitting firewall). 
 selection count 
 \(17/18\) 
 DERIVED-GIVEN-E (selector-CANDIDATE, audit) — explicitly not a certificate, not a prediction 

 6 
 KK-overclosure bound. The Kaluza–Klein tower does not overclose the universe across the inflaton band. 
 \(\Omega_{\rm KK}\) 
 \(\Omega_{\rm KK} \le 1\) (one-sided) 
 DERIVED-GIVEN-E (one-sided) — necessary viability bound only 

 7 
 Portal operator naming. The operator must be drawn from the \(\otimes\) -ledger; candidates named (none yet selected): \(\mathcal H\) -portal-like \(\vert H\vert^2 X^2\) , Wilson-line/KK mixing, or a higher-dimension operator. 
 — 
 no value asserted 
 AXIOM-OPEN ( AXIOM-OPERATOR-ORDERING , unnamed) — gates step 8 

 8 
 Portal normalization integral. \(N_{\rm portal} \propto \int_{S^1_Y/\mathbb{Z}_2} \psi_{\rm C\text{-}iii}(y)\,\psi_{\rm bath}(y)\,(\text{geometry factors})\,dy\) , reduced \(13{\rm D}\to4{\rm D}\) via the standard dimensional-reduction convention. 
 \(N_{\rm portal}\) 
 UNRUN — no value, band, or sign anywhere; no hash 
 OPEN / computation-debt 

 9 
 Freeze-in quadrature. \(\Omega_{\rm DM}h^2 \propto \int \big(\text{production rate}(N_{\rm portal}, T)\big)\,dT\) over the \(T_{\rm RH}\) band (step-6-independent scale input, inherited from Gap-09). UV- vs IR-dominated regime is itself open (Q04). 
 \(\Omega_{\rm DM}h^2\) 
 no value claimed — downstream-gated on steps 8 and the \(T_{\rm RH}\) anchor 
 OPEN / downstream-gated 

 10 
 Comparison (once, post-freeze). Whatever value step 9 eventually yields is compared exactly once to the measured \(\Omega_{\rm DM}h^2 \approx 0.12\) . 
 — 
 \(0.12\) (Planck, quoted not derived) 
 comparison only — never a derivation step 

 11 
 \(\sigma_{\rm SI}\) point value. Computed from the same \(N_{\rm portal}\) (single-coefficient coupling: one object, two consumers — production rate and present-day scattering rate). 
 \(\sigma_{\rm SI}\) 
 banked band only: \(10^{-55}\) – \(10^{-60}\ {\rm cm}^2\) 
 OPEN / gated on step 8 (band is banked at forecast grade; point value awaits step 8) 

 12 
 Refute-only falsifier registration. \(\sigma_{\rm SI}\) band sits \(\sim\) 6–11 OOM below the neutrino floor ( \(\sim10^{-49}\ {\rm cm}^2\) ), \(\sim\) 35–40 OOM below current experimental sensitivity. 
 OOM gaps 
 6–11 OOM (floor), 35–40 OOM (reach) 
 banked forecast; asymmetric — any positive signal in-band kills the minimal candidate; no null confirms it 

 Single decisive node: step 8 ( \(N_{\rm portal}\) ). Every other step is either already banked (1–6), an external experimental/measured input (10, \(T_{\rm RH}\) in step 9), or routine-once-frozen machinery (9, 11 once step 8 lands).

 L5. The floor-enlargement: two refuted-as-stated theorems

 Two adversarially-verified theorems, both valid derivations , both not target-loaded , were tested against the shortcut "one geometric razor governs both the particle selection and the portal operator selection." Both were REFUTED AS STATED (never bare "refuted" — the candidate's identity and stability are untouched):

 Anchor-transfer theorem — "the state-selection razor also governs the operator inventory." Refuted: the razor validated on state/architecture branches does not, without an explicit extension, order operators — a type mismatch (the razor's only operator-coordinate is a count ; the rule actually required is lowest-mass-dimension on a single operator ).

 Operator-admissibility-metric identity — refuted as stated.

 Consequence — this enlarges the honest floor (a diagnostic gain, not a setback): operator-governance is now an exposed axiom-open premise , AXIOM-OPERATOR-ORDERING , which must be discharged by target-blind computation (residual R8/R1 below), not inherited from the 17/18 state-side selection. This replaces the old single-block reading with an unrun structural trichotomy + single-component fork that decides whether \(N_{\rm portal}\) is even an object:

 Branch 
 Condition 
 Consequence for \(N_{\rm portal}\) 

 (i) 
 \(\otimes\) -ledger admits no parity-preserving operator 
 abundance is geometric ; \(N_{\rm portal}\) dissolves (non-object); Gap-09 cascade irrelevant 

 (ii) 
 a unique freeze-in operator exists 
 dynamical freeze-in; \(N_{\rm portal}\) is the single integral; inherits \(T_{\rm RH}\) 

 (iii) 
 a unique operator but geometric-dominated yield 
 \(N_{\rm portal}\) is only a correction term 

 single-component fork 
 zero-mode alone vs. zero-mode + \(\mathbb{Z}_2\) -odd KK tower co-carrying 
 tower-co-carry routes into the already-banked KK-overclosure + Gap-09 machinery 

 \(N_{\rm portal}\) is a real debt only on the conjunction { AXIOM-OPERATOR-ORDERING discharged AND trichotomy = branch (ii) AND single-component = true}. On every other branch it is a non-object or a correction — branches (i)/(iii)/tower-co-carry shrink-or-dissolve the block for free . This is a genuine reduce move, reported honestly as a floor enlargement (the gap got structurally bigger , more precisely mapped) rather than smoothed over.

 L6. Credit-ladder grading — every leg, individually graded

 Leg 
 Content 
 Ladder grade 

 Identity (C-iii localization) 
 \(\mathbb{Z}_2\) -odd, \(Y=0\) state read off frozen spectrum \(E\) 
 DERIVED-GIVEN-E (selector-CANDIDATE) 

 Stability — parity 
 \(\mathbb{Z}_2\) -odd forbids decay 
 DERIVED-GIVEN-E 

 Stability — renormalizability 
 \(Y=0\) keeps portal renormalizable 
 DERIVED-GIVEN-E 

 Structural selection (17/18) 
 frozen, pre-registered, 5 reweightings 
 DERIVED-GIVEN-E (selector-CANDIDATE, audit) — not a certificate 

 KK-overclosure 
 \(\Omega_{\rm KK}\le1\) 
 DERIVED-GIVEN-E (one-sided) — necessary bound only 

 Portal operator identity 
 which operator carries the portal 
 AXIOM-OPEN ( AXIOM-OPERATOR-ORDERING ) pending R8/R1; falls to REDUCED-TO-AXIOM if named-but-unproven-unique 

 Portal normalization \(N_{\rm portal}\) 
 overlap integral 
 OPEN / computation-debt (unrun; conditional real-debt only on the branch-(ii)+single-component conjunction) 

 Relic abundance \(\Omega_{\rm DM}h^2\) 
 freeze-in quadrature 
 OPEN / downstream-gated on \(N_{\rm portal}\) + \(T_{\rm RH}\) 

 \(T_{\rm RH}\) thermal input 
 reheating boundary 
 INHERITED-CONDITIONAL / exported terminal-by-export to Gap-09 (not closeable inside Gap-11) 

 \(\Omega_{\rm DM}h^2\approx0.12\) 
 Planck measurement 
 MEASURED-ANCHOR 

 \(\sigma_{\rm SI}\) falsifier band 
 direct-detection forecast 
 banked forecast grade; asymmetric refute-only 

 Refute-only asymmetry (null-can't-confirm) 
 universal fact about feebly-coupled relics below the neutrino floor 
 dissolved unicorn — a ceiling on all of physics, not a defect of this theory 

 Two refuted-as-stated theorems 
 anchor-transfer + operator-metric-identity 
 CLOSED-NEGATIVE (both refuted-as-stated; floor-enlarging, reported as a strength) 

 Overall gate terminal 
 scale leg rides an external, measured, inherited anchor with a shown finite compute debt; no internal lever closes it 
 CERTIFIED-IRREDUCIBLE (anchor-limited by \(T_{\rm RH}\) ), bucket RESOLVED +0 

 L7. The residual family R1–R8 (shown, optional, never rolled into "open")

 ID 
 Residual 
 Disposition 
 Gates 

 R8 (run first) 
 Structural fork: trichotomy (i)/(ii)/(iii) + single-component 
 OPEN/unrun audit 
 decides whether \(N_{\rm portal}\) is an object at all; cheapest possible win 

 R1 
 Portal operator identity ( AXIOM-OPERATOR-ORDERING ) 
 AXIOM-OPEN/unnamed 
 gates R2; ladder resolves to DERIVED-GIVEN-E if unique, else REDUCED-TO-AXIOM 

 R2 ◀ the block 
 \(N_{\rm portal}\) overlap integral 
 OPEN/computation-debt, unrun, no hash 
 ladder: DERIVED-CLOSED only if it evaluates with no fudge and lands near 0.12 post-freeze; else AXIOM-PORTAL-NORMALIZATION or the realistic terminus OPEN/[O] 

 R3 
 \(\Omega_{\rm DM}h^2\) 
 OPEN/downstream-gated on R2+R6 
 may be vacated entirely on branch (i)/tower-co-carry 

 R5 
 \(\sigma_{\rm SI}\) point value 
 OPEN/gated on R2 
 sharpening only, not a blocker 

 R6 
 \(T_{\rm RH}\) 
 INHERITED-CONDITIONAL, exported to Gap-09 
 not closeable inside Gap-11 

 R4 
 \(\lambda_p(0.12)\sim10^{-11}\) role (Q01) 
 OPEN-as-question, exported to owner 
 must NOT be closed by assuming \(\lambda_p=N_{\rm portal}\) 

 R7 
 Audits AO-1/AO-2 
 audit only, no promotion 
 re-verify 17/18 non-tie and full-band KK-overclosure coverage 

 All eight are optional residual-shrinking , never a precondition for the RESOLVED-with-residual terminal already reached.

 L8. Anti-claims and negative controls

 Explicit non-claims (bright lines, never printed as proven): 
- Gap-11 does not claim \(\Omega_{\rm DM}h^2 \approx 0.12\) is derived or reproduced. No abundance value is asserted anywhere.
- Gap-11 does not claim one geometric rule selects both the particle and its portal operator — this is refuted-as-stated (§L5).
- Gap-11 does not identify \(\lambda_p(0.12)\sim10^{-11}\) with \(N_{\rm portal}\) — doing so would back the normalization out of the measured anchor, the explicitly prohibited anti-fitting move (the " \(\kappa^3/\pi\) true-by-construction" anti-pattern, which relocates the mystery rather than closing it).
- Gap-11 does not claim the KK-overclosure PASS fixes the abundance — it is one-sided (necessary, not sufficient).
- Dark matter is a named excluded/candidate sector at the top-level honesty layer: the theory does not claim to close dark matter as a required gate, and this sector may not be used to close any other gate.

 Negative controls: 
1. The refute-only asymmetry is a dissolved universal negative, not a hole. "Confirm the candidate with a null direct-detection result" is impossible for any feebly-coupled relic whose cross-section sits below the neutrino floor — a ceiling on all of physics, not a defect of this theory. Correctly dissolved, not left as an open weakness.
2. "Prove \(N_{\rm portal}\) is the absolutely unique coefficient under any possible reduction convention" is a universal-negative over an open-ended domain — unprovable in principle. The correct, bounded ceiling is: a target-blind overlap integral evaluated under the stated (named) reduction convention.
3. The two refuted-as-stated theorems function as a genuine negative control on the derivation chain itself : both were valid, non-target-loaded attempts to shortcut the operator selection, and both failed honestly, enlarging (not shrinking) the acknowledged floor.
4. A self-caught selection slip. One anchor slip in the selection lane was self-caught and superseded on the record — evidence the anti-fitting/selector firewall is live and functioning, not merely asserted.

 L9. The endpoint line

 \[
\textbf{Gap-11 terminal} = \texttt{CERTIFIED-IRREDUCIBLE (anchor-limited by } T_{\rm RH}\texttt{)}, \quad \text{bucket RESOLVED } +0, \quad \text{axis-2: RESIDUALS-SHOWN (R1–R8).}
\]

 Net floor: 1 measured anchor ( \(\Omega_{\rm DM}h^2\approx0.12\) , compared-to-only) + banked structural candidate/stability (identity + two independent stability legs, real geometry-essential structure, candidate grade) + exported/experiment-gated inputs ( \(T_{\rm RH}\to\) Gap-09; \(\sigma_{\rm SI}\to\) experiment) + one shown finite compute debt ( \(N_{\rm portal}\) , conditional on the R8 fork). Floor \(\ge 1\) : a surviving honest anchor is counted as a success, not a shortfall.

 Definition of done: the gate is already at its honest terminal. The only required action is a board/ledger header sync — replacing the stale single-axis "OPEN" roll-up with the canonical two-axis RESOLVED-with-residual reading — a bookkeeping task, not a physics reopening. Every item in §L7 is optional residual-shrinking that a specialist may pursue to sharpen (never to first achieve) closure.