# AI CLOSURE COMPENDIUM — ARTICLE 2: QUANTUM THEORY OF GRAVITY

> **Discipline header.** No status was ever upgraded. Frozen branch `dcc66f1b2685` READ-ONLY (no geometry,
> coset, radius, code-distance, or boundary datum is altered by this document). This compendium
> RE-EXPRESSES the existing, frozen UQF/Gap status ledger in a closure vocabulary; it derives no
> physics, computes no number, and advances no tier. Every gate's underlying status text is the
> one already banked in `Quantum.md` §3.3 / §10 / §11 and the TOE gap docs. Where this document
> says "AXIOM-CLOSED:dissolved," it is a STATEMENT ABOUT THE TOE'S COST-FLOOR AXIOM, not a new
> result: the cost-floor declines (does not solve) a continuum/`a→0`/UV idealization. **dissolved
> ≠ solved.** The finite residual and any genuinely-different object stay OPEN even at the TOE level.
>
> **Source of truth.** `Quantum.md` (Paper 3, "13D Unified Quantum Force Completion"), headline
> **Σ = AUDIT** (min over 17 gates), plus the TOE gap docs under `rendered/TOE/` (gap_01, gap_13,
> gap_02, gap_03). EXACT paths cited per gate in the evidence lines.

---

## 0. Closure vocabulary (used EXACTLY)

- **DERIVED-CLOSED** — closed from the geometry+axioms with no remaining named gate.
- **DERIVED-GIVEN-E** — closed given the frozen geometry data E (the embedding/index data).
- **AXIOM-CLOSED:dissolved** — a continuum / `a→0` / UV idealization that the cost-floor (Finite
  Operational Cell Law) DECLINES. The infinite-refinement question stops being a live demand
  because the theory has a finite operational cell. **dissolved ≠ solved.** Only the
  continuum/idealization HALF is dissolved.
- **AXIOM-CLOSED:declared** — reduced to a single named posit, accepted as an axiom.
- **CERTIFICATE-CONDITIONAL** — a linearized/perturbative-grade certificate, conditional on a
  named OPEN gate.
- **PARTIAL** — partly closed, partly open, both halves named.
- **OPEN** — genuinely unfinished, with a named path; no fabricated number.

## 0.1 The three irreducible TOE posits (the level-(b) axiom set)

1. **GRANULARITY** — the cost-floor / Finite Operational Cell Law: physics is evaluated on a
   finite operational cell; there is no free `a→0`. This is the axiom that DISSOLVES continuum/UV
   idealizations.
2. **SCALE** — `M_Pl` set by Buckingham-π dimensional closure + `v_EW`.
3. **SHAPE** — the frozen 13D geometry `dcc66f1b2685` + the embedding/index data E.

Plus the ordinary axiom floor (the BRST/positivity/measure axioms of the construction).

**HARD RULE.** The TOE is NOT used to manufacture closure. A gate OPEN at level (a) becomes
AXIOM-CLOSED:dissolved at level (b) for its continuum half ONLY; its finite residual and any
genuinely-different object stay OPEN at BOTH levels. The following are NEVER fabricated and stay
OPEN at both levels: the **finite `a₆` VALUE**, the **finite uniform-gap-bridge** (UQF-11), the
**Born-rule weights** (Gap-14), the **strong-CP θ̄** (UQF-12), and the **interacting/nonperturbative
half** of positivity (UQF-3) and unitarity (UQF-14).

---

# THE 12 QG GATES

---

## GATE UQF-5 — Graviton certificate

**(i) REQUIREMENT.** Extract a physical, positive-norm, massless spin-2 graviton from the
linearized metric block `G_{μν}`: two physical polarizations, correct dispersion, correct
Newtonian limit, BRST/de-Donder gauge fixing; higher-curvature corrections enumerated; the
interacting/UV route named.

**(ii) STATUS ALONE (a): CERTIFICATE-CONDITIONAL** (banked tier CANDIDATE). The linearized
analysis is straightforward and reproduces textbook results; the certificate is conditional on
UQF-9 (UV completion).

**(iii) STATUS GIVEN TOE (b): CERTIFICATE-CONDITIONAL** — same. The two-polarization count and
Newtonian limit hold at the linearized level; the interacting graviton routes to UQF-9, whose
continuum-demand half is AXIOM-CLOSED:dissolved by the cost-floor but whose finite `a₆` residual
is OPEN. So the headline is unchanged: a linearized certificate gated on a residual that the TOE
does not close.

**(iv) ARGUMENT (the certificate).** In 4D de-Donder gauge the construction reproduces the standard
helicity-mode expansion; the polarization count is the textbook reduction `10 − 4 (gauge) − 4
(residual) = 2` transverse-traceless modes (a fast check verifiable by hand). The dispersion is
massless and the Newtonian limit is recovered. This is a genuine, free-field certificate — the
"surveyed-street" match where the construction touches known physics. It is FLAGGED
CERTIFICATE-tier at the linearized level (§8.6) and states plainly what it does NOT cover (the
interacting, nonperturbative graviton, routed to UQF-9). The TOE does not upgrade this: the
cost-floor dissolves the demand for a continuum UV fixed point, but the finite consistency
coefficient (`a₆`) the interacting graviton needs has only its **bulk** computed (cross-checked, as a labeled consistency coefficient, 2026-06-23) — its TOTAL (with the Z₂ orbifold-defect) is uncomputed and, by MO-12, necessary-not-sufficient regardless.

**(v) EXACT CAVEAT.** Linearized only. The certificate is CONDITIONAL on UQF-9; if UV completion
fails, UQF-5 loses CERTIFICATE tier (§3.4.2 coupling). The interacting graviton is NOT certified.
No claim about graviton scattering above the cutoff.

**(vi) EVIDENCE.** `Quantum.md` §8 / §8.4.3–§8.5 / §8.6; §3.3 UQF-5 (CANDIDATE, "CERTIFICATE
conditional on UQF-9"); §3.4.2 coupling row "UQF-9 → UQF-5 loses CERTIFICATE"; reader on-ramp
"the linearized graviton (§8.4.3–§8.5)".

---

## GATE UQF-9 — UV-completion certificate

**(i) REQUIREMENT.** A named UV-completion route with a frozen truncation (corpus default:
asymptotic safety — a non-Gaussian UV fixed point with finitely many relevant directions),
exhibited at declared truncation, with flow back to observed low-energy physics.

**(ii) STATUS ALONE (a): OPEN.** Banked tier AUDIT. The asymptotic-safety non-Gaussian fixed
point existence is a GLOBAL open problem; the paper does not claim closure.

**(iii) STATUS GIVEN TOE (b): AXIOM-CLOSED:dissolved (continuum UV-fixed-point demand) + OPEN
(finite `a₆` consistency residual).** This is the HEADLINE for this gate and the canonical
illustration of the dissolution mechanism.

**(iv) ARGUMENT (dissolution + residual).** The standard UV-completion DEMAND is: show the theory
is well-defined at *arbitrarily high* energy via an `a→0` continuum fixed point. The cost-floor
axiom (GRANULARITY) DECLINES exactly this demand: with a finite operational cell there is no free
`a→0` limit to take, so "is there a continuum non-Gaussian fixed point?" stops being a live
consistency requirement — it is dissolved (NOT solved; the fixed point is not exhibited, the
question is removed). **But the FINITE residual survives.** Even with a cost-floor, the theory must
pass its finite heat-kernel consistency coefficient — `tr[a₆(L_grav^{d=13,de-Donder})]` projected
onto the cubic-curvature basis with K₆×S²×S¹_Y holonomy — and that coefficient now has a computed, cross-checked **bulk** (a labeled consistency coefficient, `−2.817995812×10⁹⁴ GeV⁶`, 2026-06-23) but an **UNCOMPUTED TOTAL** (the Z₂-defect boundary piece is a named blocker). The
gap doc reifies UV completion into this named finite object and grades it "[S] — perturbative
but heavy" (Gilkey 1995 Th. 3.3.1; Avramidi 2000 Ch. 4). The TOE removes the infinite
half and leaves the finite half OPEN: even the cross-checked bulk is one term in the unbounded a₈/a₁₀ tower (MO-12 necessary-not-sufficient). **No closure-grade `a₆`/TOTAL is invented here, and the bulk is never read as gap-closing.**

**(v) EXACT CAVEAT.** dissolved ≠ solved: no UV fixed point is exhibited; the cost-floor only
removes the continuum-limit DEMAND. The finite `a₆` is a definite-but-heavy symbolic computation
(machine lane after Gap-04 infrastructure), NOT done. Falsifier rides on it: if computed `a₆`
violates positive-definiteness for physical-Hilbert closure, Gap 01 is refuted at decision grade.
The corpus Λ-line guard applies: nothing in the UV narrative may be read as a live Λ /
vacuum-energy cancellation (TOE_FINAL v12 supertrace FAIL, v14 chamber route THEOREM_REFUTED;
Λ stays Weinberg-open).

**(vi) EVIDENCE.** `Quantum.md` §14, §3.3 UQF-9 (AUDIT), §11.3 "UQF-9 correctly AUDIT", §11.2
Λ-row; `rendered/TOE/gap_01_uv_quantum_gravity/00_decomposition.md` (the `a₆` object, "perturbative
but heavy," walled at field grade); `.../subsystems/S2_a6_universal_machinery.md` (universal
template; "must be CITED and verified, never invented").

---

## GATE UQF-3 — Physical Hilbert space (positivity)

**(i) REQUIREMENT.** A positivity certificate: `⟨ψ|ψ⟩ ≥ 0` for all `|ψ⟩ ∈ H_phys`, equality only
for the zero vector; physical states = BRST cohomology `ker(s)/im(s)` on the BRST-extended
Fock-like space.

**(ii) STATUS ALONE (a): PARTIAL.** Banked tier CANDIDATE. Positivity holds on free-sector
physical states and on perturbative-interaction states with declared truncation; pass on the
nonperturbative confined-QCD sector is conditional on UQF-11.

**(iii) STATUS GIVEN TOE (b): linearized/perturbative CERTIFICATE-CONDITIONAL; interacting half
OPEN even given the TOE.** The free + perturbative positivity is a CERTIFICATE-CONDITIONAL on
UQF-4 (BRST nilpotency) and UQF-9. The nonperturbative half is NOT dissolved by the cost-floor:
full interacting 13D positivity sits at the same level as 4D Yang-Mills reflection positivity — a
genuinely-different object the cost-floor does not supply — so it stays OPEN at level (b).

**(iv) ARGUMENT.** On the free sectors, the BRST quotient yields positive-norm states and the
ghost/antifield rows are removed from `H_phys` — a real perturbative certificate. The cost-floor
does NOT manufacture interacting positivity: reflection positivity of a nonperturbative measure
is a constructive-QFT object (OS-2), and granularity (a finite cell) does not by itself prove the
finite-cell measure is reflection-positive. That is the same hard core as the mass-gap problem's
MO-2. So the honest split: perturbative positivity = CERTIFICATE-CONDITIONAL; interacting
positivity = OPEN at both levels.

**(v) EXACT CAVEAT.** Perturbative/truncation grade only. Conditional on UQF-4 and (for the QCD
sector) UQF-11. If a negative-norm physical state survives the quotient, UQF-3 drops to INTERFACE
and UQF-5/6/7/8/14 each lose CERTIFICATE. The interacting positivity certificate is NOT supplied
by the TOE.

**(vi) EVIDENCE.** `Quantum.md` §6, §3.3 UQF-3 (CANDIDATE; "pass on the nonperturbative confined-QCD
sector is conditional on UQF-11"), §11.3 "UQF-3 (physical Hilbert space, full interacting) —
correctly AUDIT … same level as 4D Yang-Mills positivity"; §3.4.2 coupling rows.

---

## GATE Gap-01 — tr[a₆] (the Seeley-DeWitt coefficient)

**(i) REQUIREMENT.** Compute `tr[a₆(L_grav^{d=13, de-Donder})]` — the Seeley-DeWitt a₆ heat-kernel
coefficient at d=13 on the de-Donder graviton+ghost operator, projected onto the cubic-curvature
basis with K₆×S²×S¹_Y holonomy — and verify its positive-definiteness functional for physical-
Hilbert-space closure.

**(ii) STATUS ALONE (a): OPEN.** Walled at field grade. *(2026-06-23 preservation-grade update: a **bulk** value HAS since been computed and cross-checked — `bulk tr[a₆] = −2.817995812×10⁹⁴ GeV⁶`, COMPLETE_CROSSCHECKED — but strictly as a labeled **consistency coefficient**; the Z₂-orbifold-defect TOTAL is uncomputed and, by MO-12, even a finished a₆ is necessary-not-sufficient. The status is therefore unchanged: OPEN.)*

**(iii) STATUS GIVEN TOE (b): OPEN at BOTH levels for the closure-grade VALUE.** The "is the theory
UV-complete?" CONTINUUM question is AXIOM-CLOSED:dissolved by the cost-floor (same dissolution as
UQF-9). The FINITE `a₆` **bulk** is now a computed, cross-checked consistency coefficient — but the
**TOTAL** (bulk + Z₂ defect) is UNCOMPUTED, the cost-floor does not compute it, and a finite a₆ is
one term in the unbounded a₈/a₁₀ tower (MO-12 necessary-not-sufficient). So the headline is OPEN: **the
closure-grade `a₆` is never invented, no TOTAL is emitted, and the bulk is never read as gap-closing.**

**(iv) ARGUMENT (why OPEN at both, no fabrication).** The a₆ functional EXISTS and is universal
(Gilkey Th. 3.3.1 / Avramidi Ch. 4): a finite linear combination of a fixed ~46-term cubic-
curvature basis with KNOWN rational coefficients. What is missing is its SPECIALIZATION to the d=13
graviton+ghost operator on the frozen geometry and the heavy fiber trace + basis projection (gap
doc nodes S1→S3.3, the wall). This is graded [S] — "set-up-able but must not be claimed to run";
fabricating any entry "would be the worst outcome." The cost-floor removes the question of whether
infinitely many higher coefficients must also be finite (the continuum half), but a₆ itself is one
finite coefficient that must be evaluated and checked for positivity. No amount of TOE axiom
supplies that number.

**(v) EXACT CAVEAT.** The **bulk** a₆ has been computed as a cross-checked consistency coefficient (`−2.817995812×10⁹⁴ GeV⁶`, 2026-06-23); the **TOTAL** (bulk + Z₂ orbifold-defect) is NOT computed and is NOT invented here, and the bulk is reported as a consistency coefficient ONLY — never as gap-closing (MO-12 necessary-not-sufficient). Owner = the machine
lane (after Gap-04 infrastructure). Decision-grade FALSIFIER: the positivity functional `P(a₆) ≥ 0` has NOT been adjudicated (no sign asserted); if a computed a₆ violates it, Gap 01 is refuted. A separate scope question (S6, [O]/[X]):
even a finite, positive a₆ is ONE consistency coefficient — whether it *constitutes* UV completion
is an owner/field judgment the corpus has NOT made (a₆ is the missing object *for* UV completion,
not *the* UV completion).

**(vi) EVIDENCE.** `rendered/TOE/gap_01_uv_quantum_gravity/00_decomposition.md` (the gap as stated;
the 12-leaf tree; "No part of Gap 01 is AI-closeable"; S3.3 = the wall; S6 = scope question);
`.../subsystems/S2_a6_universal_machinery.md` ("coefficients are literature values — never
invented"); `Quantum.md` §14 / UQF-9 (downstream consumer).

---

## GATE Gap-13 — BH entropy / Page curve

**(i) REQUIREMENT.** (a) A geometry-native microstate-counting theorem giving `log N(A) = A/(4G) +
subleading` for the right reason; (b) a unitary-evaporation (Page-curve) mechanism for the
radiation entanglement entropy.

**(ii) STATUS ALONE (a): PARTIAL.** A value-match consistency datum is banked at DIAGNOSTIC tier
(the 4D zero-mode sector reproduces `A/(4G)` to 0.0028% — a DIAGNOSTIC value-match only (asserts
NOTHING about a microstate count), under freeze-before-compare); the
microstate COUNT and the Page MECHANISM are explicitly NOT claimed (PC-1/PC-2/PC-3).

**(iii) STATUS GIVEN TOE (b): UV leg AXIOM-CLOSED:dissolved; microstate-count + Page-mechanism
OPEN.** The "need a UV-defined evaporation/continuum" leg (the Gap-01 cascade) has its
CONTINUUM half dissolved by the cost-floor; but the geometry-native degeneracy theorem and the
Page-turnover mechanism are genuinely-different objects the cost-floor does not supply, so they
stay OPEN at level (b).

**(iv) ARGUMENT.** The banked datum is a number-matching-a-number consistency check, correctly
graded DIAGNOSTIC — it asserts NOTHING about *why* there are that many states (no count) or about
*unitary evaporation* (no Page curve). The state-counting theorem (S2.c) would have to be a
geometry-native count over the M_GUT degrees of freedom (Strominger–Vafa analog), gated on Gap 01
(Hilbert space) and a source+activation gate; graded [O]/[X] — "a genuine theorem, not a finite
computation … not even AI-setup-able until the Hilbert space (Gap 01) exists." The Page mechanism
(S3) is doubly downstream — Gap 01 AND Gap 14 (system-bath split on the 13D fields) — and the
corpus claims NONE (PC-3). The cost-floor dissolves only the continuum-UV leg that flows in through
Gap 01; the count and the turnover curve remain open objects with named paths.

**(v) EXACT CAVEAT.** The 0.0028% match is a DIAGNOSTIC consistency over-determination, NOT a
derivation of BH entropy; PC-1/PC-2 forbid reading it as a microstate count. No Page-curve
mechanism is claimed (PC-3); Gap 13 has "no falsifier of its own yet" precisely because the
disclaimers forbid a premature claim. No microstate number is invented.

**(vi) EVIDENCE.** `rendered/TOE/gap_13_bh_entropy_page/subsystems/S1a_consistency_claim.md`
(DIAGNOSTIC value-match, asserts no count/no Page); `.../S2c_counting_theorem.md` (geometry-native
degeneracy theorem OPEN, [O]/[X], gated on Gap 01); `.../S3_page_curve_mechanism.md` (Page mechanism
OPEN, double cascade Gap 01 + Gap 14, PC-3).

---

## GATE UQF-10 — Quantum compactification stability

**(i) REQUIREMENT.** Show the compact factors `K₆ × S² × S¹_Y` are stable under quantum
corrections: effective potential at declared loop order with a stable minimum at inherited volumes,
no tachyonic moduli, no light moduli violating fifth-force/EP bounds, Casimir accounted.

**(ii) STATUS ALONE (a): PARTIAL.** Banked tier AUDIT; the §3.3 row defers to UQF-9 ("without a
UV-completion certificate the moduli effective potential at higher loops is not controllable").

**(iii) STATUS GIVEN TOE (b): DERIVED-CLOSED (decision-grade) for stability EXISTENCE; certificate
still gated by the UQF-9 finite residual.** This is the strongest QG headline: the corpus's
native three-layer stabilization object discharges the stability-EXISTENCE question at DECISION
grade independent of asymptotic safety; what remains is the paper's own certificate-grade FRG
verification, which inherits the UQF-9 finite-`a₆` residual.

**(iv) ARGUMENT.** TOE_FINAL v8 `H04_DISCHARGED_AT_DECISION_GRADE` (weakest variant clears the
operative bound 35×, central 2.4×10³); v9 Gap 04 CLOSED at decision grade (D15-native runner PASS,
package `gap04_frozen_package_native.json`); v10 boundary-sign event `c_bdry = −1.08×10⁻²` ⟹ the
stabilization well `V(σ) = c_KK e^{−4σ} + c_bdry e^{−2σ} + c_W cosθ_W e^{−4σ} + c_loop e^{−6σ}`
exists UNCONDITIONALLY in `c_loop` (the `c_loop` window vacated; self-caught). This is a decision-
grade EXISTENCE discharge by a route the paper did NOT use — making UQF-10's "blocked-by-UQF-9"
framing too pessimistic. Hence DERIVED-CLOSED (decision-grade) for the existence question. It is
NOT a promotion: the paper inherits this as a decision-grade INPUT (the background is stable) but
does not convert it into its own certificate-grade FRG-6 result, which remains gated by the UQF-9
finite residual.

**(v) EXACT CAVEAT.** Decision-grade EXISTENCE only, via the corpus stabilization object — NOT the
§14 FRG-6 trajectory this paper's certificate is conditioned on. The paper's tier stays AUDIT
(zero promotion); certificate-grade FRG verification remains the paper's open obligation, carrying
the UQF-9 finite-`a₆` residual.

**(vi) EVIDENCE.** `Quantum.md` §15, §3.3 UQF-10 (AUDIT), §11.2 UQF-10 row (v8/v9/v10 discharge;
"AUDIT — unchanged tier … cited as an independent route … but does not convert into its own
certificate").

---

## GATE UQF-14 — Unitarity / causality / locality

**(i) REQUIREMENT.** Per-sector unitarity bound check (partial-wave on WW→WW, gg, graviton–graviton
at perturbative scale); causality via Lorentz covariance + spectral conditions; locality via
cluster decomposition on `H_phys`.

**(ii) STATUS ALONE (a): PARTIAL.** Banked tier AUDIT. Per-sector bounds are routine where the
sector is perturbative; the graviton-sector unitarity ABOVE the cutoff is conditional on UQF-9.

**(iii) STATUS GIVEN TOE (b): perturbative CERTIFICATE-CONDITIONAL; high-energy half OPEN.** The
below-cutoff per-sector unitarity is a perturbative CERTIFICATE-CONDITIONAL on UQF-3/UQF-4. The
above-cutoff (graviton-sector) unitarity inherits the UQF-9 dissolution: the continuum-completion
demand is dissolved by the cost-floor, but the finite high-energy unitarity statement is a
genuinely-different object the cost-floor does not certify, so it stays OPEN at level (b).

**(iv) ARGUMENT.** Where the sector is perturbative, the partial-wave/Lorentz/cluster checks are
routine and constitute a real conditional certificate. Above the cutoff the graviton-sector
unitarity needs the UV completion; the cost-floor removes the `a→0` demand but does not deliver a
finite proof that high-energy graviton amplitudes are unitary — that proof is the same finite
residual class as `a₆`/interacting positivity. So: perturbative half certified-conditional;
high-energy half open.

**(v) EXACT CAVEAT.** Perturbative/below-cutoff grade only. High-energy graviton-sector unitarity
is OPEN (gated on UQF-9). Λ-line guard applies (no high-energy reading implies Λ/vacuum-energy
cancellation; TOE_FINAL v12/v14). Unitarity violation in the graviton sector at `Λ_⋆ < M_Pl` drops
UQF-5 one tier (§3.4.2).

**(vi) EVIDENCE.** `Quantum.md` §17, §3.3 UQF-14 (AUDIT; "graviton sector's unitarity above the
cutoff is conditional on UQF-9"; corpus Λ guard); §3.4.2 coupling.

---

## GATE UQF-4 — BRST/BV nilpotency & anomaly descent

**(i) REQUIREMENT.** Prove `s² = 0` on the BV-extended phase space and that all chiral/gauge/mixed
anomalies cancel on `H_phys`, with the inherited Paper-1 anomaly ledger surviving the 4D descent
and the regulator choice.

**(ii) STATUS ALONE (a): PARTIAL.** Banked tier AUDIT. The Paper-1 anomaly ledger is closed; lifting
it under the 4D-descent + BRST quantization here "requires explicit verification, deferred to
Phase 4."

**(iii) STATUS GIVEN TOE (b): PARTIAL — boundary-anomaly + coset-twist rows OPEN.** The cost-floor
does NOT dissolve this gate: anomaly descent is a finite, structural quantization question, not a
continuum idealization. The bulk nilpotency + inherited anomaly cancellation is the closed part;
the boundary-anomaly and coset-twist descent rows remain OPEN with their named pathway, at both
levels.

**(iv) ARGUMENT.** `s² = 0` is verifiable order-by-order on the BV action, and the inherited
one-generation anomaly ledger cancels in the bulk. What is NOT discharged is that the cancellation
SURVIVES the descent to 4D through the `S¹_Y/Z₂` boundary and the K₆ coset twist — a finite but
unperformed verification. Because it is a finite structural object (not a continuum limit), the
cost-floor offers no dissolution; the gate stays PARTIAL at level (b), with the boundary+coset rows
named OPEN.

**(v) EXACT CAVEAT.** Bulk nilpotency + inherited cancellation only; boundary-anomaly + coset-twist
descent NOT verified. An uncancelled anomaly drops UQF-7 and UQF-6 one tier and strips UQF-3 of
CERTIFICATE; the paper as a whole cannot exceed AUDIT until resolved (§3.4.2).

**(vi) EVIDENCE.** `Quantum.md` §7, §3.3 UQF-4 (AUDIT), §11.3 "UQF-4 (BRST/anomaly closure, boundary
rows) — correctly AUDIT … boundary-anomaly + coset-twist rows remain open with their named
pathway"; §3.4.2 coupling table.

---

## GATE UQF-7 — Fermion chirality / quantization

**(i) REQUIREMENT.** Quantize one full generation of left-handed Weyl fermions on the active branch
in the inherited reps, with NO mirror partners, spin-statistics satisfied, anomaly closure
surviving the quantization.

**(ii) STATUS ALONE (a): PARTIAL.** Banked tier AUDIT. The classical no-mirror inventory is
inherited; demonstrating the 4D-descent preserves chirality at the QUANTUM level (no light mirror
from KK/quotient subtleties) requires explicit verification.

**(iii) STATUS GIVEN TOE (b): classical inventory DERIVED-GIVEN-E (APS index `(+3,0)`); quantum
descent OPEN.** The classical chiral inventory is DERIVED-GIVEN-E — closed and twice-derived from
the frozen geometry data E (the APS one-sided index `(+3,0)`, "the three families are the spin-
structure witness," cross-checked against the BWB scan with `|index| = 3`). The QUANTUM-descent
no-light-mirror question is a finite structural object the cost-floor does not dissolve, so it
stays OPEN at both levels.

**(iv) ARGUMENT.** The classical zero-mode count is load-bearing-closed in the corpus: index
`D_{K₆} = 3`, APS `(+3,0)`, no mirrors at the zero-mode level — twice-derived (APS + BWB). This is
DERIVED-GIVEN-E (it is read off the frozen branch + index data, not the cost-floor). The index
count is therefore NOT the open item. The RESIDUAL — that the 4D BV-BRST quantization of the
descent introduces no light mirror from KK/quotient/loop subtleties beyond the classical count —
is genuinely open. It is finite/structural, so the cost-floor gives no dissolution; OPEN at (b),
with a named verification path.

**(v) EXACT CAVEAT.** Only the CLASSICAL inventory is closed (DERIVED-GIVEN-E); the QUANTUM-descent
no-light-mirror verification is open. Tier stays AUDIT (open-item NARROWED to quantum descent, NOT
promoted). A surviving mirror drops the row to INTERFACE and downgrades the unified-completion claim
to "non-chiral candidate."

**(vi) EVIDENCE.** `Quantum.md` §11 (UQFC-11) / §3.3 UQF-7 (AUDIT, with the 2026-06-12 open-item
narrowing: classical inventory inherits APS `(+3,0)` EXTERNAL; residual = quantum descent); §11.2
UQF-7 row.

---

## GATE Gap-14 — Born rule

**(i) REQUIREMENT.** Derive the Born-rule probability weights (the `|ψ|²` measure on outcomes) from
the framework — the CH-2 KK-tower decoherence channel that fixes the probability assignment.

**(ii) STATUS ALONE (a): OPEN.** The corpus honest-halts (NotImplementedError) on the CH-2 channel;
"cascades via Gap 01's a₆."

**(iii) STATUS GIVEN TOE (b): OPEN.** The cost-floor is the WRONG SHAPE for a probability weight:
a finite operational cell / cost functional does not, by itself, produce the `|ψ|²` measure. So the
TOE does NOT dissolve and does NOT close this gate — OPEN at both levels. **Never fabricated.**

**(iv) ARGUMENT (why OPEN at both, no fabrication).** The Born rule is a statement about the WEIGHTS
on the physical Hilbert space, not about the existence/finiteness of an operator spectrum. The
cost-floor axiom constrains operational cost (finiteness of refinement), which is a different
mathematical object from a normalized probability measure on outcomes; one cannot manufacture the
quadratic measure from a cost floor without a separate posit. The decoherence-channel route
(CH-2) additionally cascades on Gap 01's a₆ (the UV-defined Hilbert space). So even given the full
TOE axiom set the Born weights are not derived; they remain an OPEN object with a named (decoherence/
system-bath) path. No probability rule is invented.

**(v) EXACT CAVEAT.** The cost-floor cannot be used to manufacture the Born weights (wrong shape).
The decoherence route is an honest halt cascading on Gap 01. No `|ψ|²` derivation is fabricated.

**(vi) EVIDENCE.** `Quantum.md` Gap-14 references (CH-2 KK-tower decoherence honest halt);
`rendered/TOE/gap_01_uv_quantum_gravity/00_decomposition.md` (S5.2 Gap-14 consumer contract; "Gap
14 honest-halts (NotImplementedError) and cascades via Gap 01's a₆"); `gap_13.../S3_page_curve_
mechanism.md` (system-bath split ≡ Gap 14 first action).

---

## GATE UQF-11 — Yang-Mills mass gap (nonperturbative QCD)

**(i) REQUIREMENT.** A nonperturbative QCD route compatible with `K₆ = SU(3)/T²`: confinement
(Wilson-loop area law), a mass gap `Δ > 0`, hadronic spectrum, lattice–continuum bridge, chiral
condensate. The continuum mass-gap existence problem: a non-trivial OS/Wightman continuum SU(3) Yang-Mills with unique
vacuum, `H ≥ 0`, and a gap `Δ > 0` SURVIVING the continuum limit.

**(ii) STATUS ALONE (a): OPEN.** Banked tier AUDIT + ROADMAP; this long-standing open problem of mathematical physics is
explicitly NOT claimed solved. The corpus banks only a CANDIDATE compatibility (structurally
compatible with a gapped nonperturbative phase, explicitly NOT a proof).

**(iii) STATUS GIVEN TOE (b): continuum half AXIOM-CLOSED:dissolved; finite uniform-gap-bridge
OPEN.** The cost-floor dissolves the CONTINUUM-LIMIT demand (the `a→0` uniform gap that survives the
continuum limit — MO-2/MO-5/6/7). But the FINITE uniform-gap-bridge — that a finite-lattice gap is
uniform across the operationally-relevant cells — is a genuinely-different, finite object the
cost-floor does not supply. So OPEN at level (b).

**(iv) ARGUMENT (dissolution + residual).** The hard core of the continuum mass-gap existence problem is the conjunction of a
nonperturbative continuum CONSTRUCTION + a gap that survives `a→0` and `volume→∞` (reflection
positivity of the continuum measure, OS→Wightman reconstruction, uniform `Δ>0`). The cost-floor
(GRANULARITY) declines the `a→0` half: with a finite operational cell there is no continuum limit
to control, so "does the gap survive `a→0`?" is dissolved (NOT solved — no proof is produced; the
demand is removed). What the cost-floor does NOT give is the FINITE uniform-gap-bridge: the
statement that the gap is uniform (bounded below by a single `Δ>0`) across the finite family of
operational cells. That is exactly the UQF-11 open object. The geometry contributes only the INPUT
specification (which group SU(3)_c, which `(+3,0)` field content, no architectural obstruction —
grade CANDIDATE); it does NOT supply the IR dynamics. The seed-case (ii) is the SAME recognized open continuum problem,
if anything HARDER (it inherits all 4D difficulty plus the compact-boundary/parity sector). No
measure, bound, or positivity estimate is fabricated.

**(v) EXACT CAVEAT.** dissolved ≠ solved: no continuum construction, no uniform gap, no reflection
positivity is proven; the cost-floor only removes the `a→0` DEMAND. The finite uniform-gap-bridge
is OPEN and is never fabricated. Closure owner EXTERNAL (constructive QFT). Geometry supplies an
input specification, not a method; (ii) is no easier than (i).

**(vi) EVIDENCE.** `Quantum.md` §12, §3.3 UQF-11 (AUDIT + ROADMAP), §11.3 "UQF-11 correctly
AUDIT + ROADMAP … long-standing open problem of mathematical physics; closes nothing here"; `rendered/TOE/gap_02_yang_mills_
mass_gap/04_step0_theorem_statement.md` (continuum mass-gap existence problem (i) + seed (ii); MO-1…MO-7; "(ii) is no
easier than (i)"; fabrication guard).

---

## GATE UQF-12 — Strong CP (θ̄)

**(i) REQUIREMENT.** Explain or eliminate `θ̄` — a named structural mechanism with a named
falsifier (corpus default: `S¹_Y/Z₂` boundary-parity; fallback: PQ axion, which needs geometry not
in the active branch).

**(ii) STATUS ALONE (a): OPEN.** Banked tier AUDIT (LOW confidence). `θ̄`-smallness is honestly
UNDERIVED; the paper provides "a candidate and a falsifier, not a solution." The geometric route
is PROVEN BLOCKED by a mutual-exclusivity theorem.

**(iii) STATUS GIVEN TOE (b): OPEN.** The cost-floor does NOT dissolve and does NOT close this gate:
`θ̄`-smallness is a finite structural CP question with a banked NO-GO, not a continuum idealization.
OPEN at both levels. **Never fabricated.**

**(iv) ARGUMENT (why OPEN at both, no fabrication).** The geometric route is REFUTED by a
representation-theory mutual-exclusivity theorem (owner-ratified 2026-06-14): a Realization-B Z₂
sign-lever on `arg det(M_u M_d)` exists ONLY IF Q_L and u_R/d_R sit on K₆ bundles with OPPOSITE
determinant-character; but every dim-3 su(3) irrep is `3` or `3̄`, each carrying the SAME sign
det-character — so chiral asymmetry (opposite det-character) and the three-family witness
`|index| = 3` are MUTUALLY EXCLUSIVE. The committed `[−6,6]` weight scan finds zero escapes,
consistent with the proof. Any escape would require structure OUTSIDE the homogeneous-line-bundle
setting (DECLINED, fail-closed). The cost-floor (a finiteness/cost axiom) has no bearing on this
determinant-character obstruction; it cannot manufacture the small `θ̄`. OPEN at both levels, with
nEDM as the experimental adjudicator.

**(v) EXACT CAVEAT.** A CANDIDATE + a falsifier, NOT a solution; `θ̄`-smallness honestly underived.
The geometric route is PROVEN BLOCKED (mutual-exclusivity theorem) — this is a banked NEGATIVE, not
progress toward closure. nEDM is the falsifier on the `1.64-rad` spin-c-holonomy surface. No `θ̄`
value or mechanism is invented.

**(vi) EVIDENCE.** `Quantum.md` §13, §3.3 UQF-12 (AUDIT LOW), §11.3 "UQF-12 correctly AUDIT (LOW)
… most temptable row to complete was verified and left open"; `rendered/TOE/gap_03_strong_cp/
subsystems/G03.A.1.c_realization_B_mutual_exclusivity_refuted.md` (the mutual-exclusivity theorem,
REFUTED, owner-ratified).

---

# SUMMARY LEDGER

| Gate | alone (a) | given TOE (b) — HEADLINE |
|---|---|---|
| UQF-5 graviton | CERTIFICATE-CONDITIONAL on UQF-9 | CERTIFICATE-CONDITIONAL on UQF-9 (2 pols + Newtonian hold; interacting → UQF-9) |
| UQF-9 UV completion | OPEN | AXIOM-CLOSED:dissolved (continuum UV-fixed-point demand) + OPEN (finite a₆ residual) |
| UQF-3 positivity | PARTIAL | perturbative CERTIFICATE-CONDITIONAL; interacting half OPEN |
| Gap-01 tr[a₆] | OPEN | OPEN (continuum question dissolved; finite a₆ VALUE uncomputed → OPEN at both) |
| Gap-13 BH-entropy/Page | PARTIAL | UV leg AXIOM-CLOSED:dissolved; microstate-count + Page-mechanism OPEN |
| UQF-10 compactification | PARTIAL | DERIVED-CLOSED (decision-grade existence) (banked tier AUDIT; existence-only, not a promotion); certificate gated by UQF-9 finite residual |
| UQF-14 unitarity | PARTIAL | perturbative CERTIFICATE-CONDITIONAL; high-energy half OPEN |
| UQF-4 BRST/anomaly descent | PARTIAL | PARTIAL — boundary + coset-twist OPEN |
| UQF-7 fermion chirality | PARTIAL | classical DERIVED-GIVEN-E (APS +3); quantum descent OPEN |
| Gap-14 Born rule | OPEN | OPEN (cost-floor is wrong shape for a probability weight) |
| UQF-11 mass gap | OPEN | continuum half AXIOM-CLOSED:dissolved; finite uniform-gap-bridge OPEN |
| UQF-12 strong CP | OPEN | OPEN (mutual-exclusivity theorem; cost-floor cannot manufacture θ̄) |

## Genuinely-irreducible residual (gates OPEN given the full TOE)

These survive as OPEN at level (b) — the cost-floor neither dissolves nor manufactures them:

1. **Finite `a₆` VALUE** (Gap-01) — uncomputed; never invented.
2. **Finite uniform-gap-bridge** (UQF-11) — the finite residual after the continuum half is
   dissolved.
3. **Born-rule weights** (Gap-14) — cost-floor is the wrong shape for a probability measure.
4. **Strong-CP `θ̄`** (UQF-12) — mutual-exclusivity theorem blocks the geometric route.

Additionally, the INTERACTING/NONPERTURBATIVE HALVES of UQF-3 (positivity) and UQF-14 (unitarity)
stay OPEN given the TOE; their perturbative halves are CERTIFICATE-CONDITIONAL. UQF-4's
boundary+coset-twist descent and UQF-7's quantum descent are finite structural OPEN items the
cost-floor does not touch.

> **Σ remains AUDIT.** Zero promotions. Every closure above carries its exact caveat; dissolved ≠
> solved throughout; no `a₆`, gap value, Born weight, or `θ̄` is fabricated.