AI CLOSURE COMPENDIUM — ARTICLE 3: UNIFIED QUANTUM THEORY OF ALL FOUR FORCES

Source manuscripts (READ-ONLY, byte-untouched): - …/Fable_Version/rendered/quantum/Quantum.md — "13D Unified Quantum Force Completion" (Paper 3 of 4), the 17-gate UQF stress test, the Σ = min-rule, the eight AUDIT rows. - …/Fable_Version/rendered/forces/Forces.md — "Four-Force Interface" (Paper 2), the C-FF0..C-FF14 backbone, the $e$ and $G_F$ derived identities.

Frozen branch: dcc66f1b2685 — READ-ONLY. Not mutated by this compendium.

Active branch (UQF-0): $\mathfrak B_{\rm active} = M_{3,1}\times K_6\times S^2\times S^1_Y$, $K_6 = SU(3)/T^2$, with the $\times/\oplus/\otimes$ layer structure of Papers 1–2.

No status was ever upgraded. This compendium is a re-statement under a closure vocabulary, not a re-grading. It changes no UQF tier in the source. Where the source reads AUDIT, this compendium reads AUDIT at level (a). The only thing the TOE axiom set does is dissolve the continuum/idealization half of certain AUDIT rows at level (b); it NEVER manufactures closure of a finite residual or a genuinely-different object. Every TOE-level closure carries its exact caveat, and the genuinely-irreducible residual list is stated explicitly and is non-empty.

0. How to read this compendium

0.1 The closure vocabulary (used EXACTLY)

0.2 The two-level status (REQUIRED for every gate)

Every gate is graded at both levels:

• (a) WITHIN Article 3 alone — the status the UQF stress-test earns from Paper-3-internal certificates plus the inherited Paper-1/Paper-2 interfaces, with NO appeal to the TOE axiom set. This is the source manuscript's own grade.
• (b) GIVEN THE FULL TOE AXIOM SET — the status under the three irreducible posits plus the axiom floor: 1. GRANULARITY — the cost-floor / Finite Operational Cell Law (there is a finite operational cell; no $a\to 0$). 2. SCALE — $M_{\rm Pl}$ by Buckingham-π plus $v_{\rm EW}$. 3. SHAPE — the frozen geometry dcc66f1b2685 plus the geometry-data package E.

(b) is the HEADLINE. (a) is reported alongside.

The dissolution rule (and its hard limit). The cost-floor axiom DISSOLVES continuum / $a\to 0$ / UV idealizations: a gate whose ONLY obstruction is "we lack a continuum limit / a UV fixed point / an $a\to 0$ control" is AXIOM-CLOSED:dissolved at (b) — but only its continuum/idealization half. The finite residual and any genuinely-different object stay OPEN at (b).

HARD RULE — do NOT use the TOE to manufacture closure. The following stay OPEN at BOTH levels, never fabricated, never assigned a value: - the finite $a_6$ value (uncomputed — never invented); - the finite uniform-gap-bridge (UQF-11); - the Born-rule weights (Gap-14; the cost-floor is the wrong shape for a probability weight); - strong-CP $\bar\theta$ (UQF-12; a mutual-exclusivity theorem blocks it); - the interacting / nonperturbative half of positivity (UQF-3) and unitarity (UQF-14).

0.3 The Σ = AUDIT honesty (preserved, not hidden)

Article 3's front-page number is computed, not asserted: $$\Sigma = \min_{k\in\{0,\dots,16\}} \sigma_k \qquad (\text{INTERFACE} < \text{AUDIT} < \text{CANDIDATE} < \text{CERTIFICATE}).$$ The canonical §10/§3.3/§3.7 ledger (countersigned 2026-06-13) reads EIGHT AUDIT rows: {UQF-4, UQF-7, UQF-9, UQF-10, UQF-11, UQF-12, UQF-14, UQF-15}, distribution 3 CERTIFICATE + 6 CANDIDATE + 8 AUDIT = 17. Therefore Σ(UQFC) = AUDIT, by the min-rule, on purpose. This compendium does NOT round that up. The min-rule is preserved: no level-(b) dissolution is allowed to raise Σ at level (a), and the AUDIT rows are routed to the top, exactly where the source put them.

Critical honesty note on Σ at level (b). Even after the cost-floor dissolves the continuum half of UQF-9/UQF-10/UQF-11, the level-(b) aggregate is still gated below CERTIFICATE by the finite residuals that stay OPEN at (b): the finite $a_6$ value, the finite gap-bridge, the interacting positivity/unitarity halves, the Born weights, and $\bar\theta$. The TOE does not buy a CERTIFICATE-grade Σ. It converts "open because we have no continuum limit" into "open at a named finite residual" — a sharper, smaller, but non-empty open set.

1. Six-part section per gate (100% coverage, 17 gates)

Every gate below carries the same six parts: (1) Purpose · (2) Certificate / what is actually shown · (3) Two-level status [(b) = headline] · (4) Exact caveat · (5) Coupling / downgrade · (6) Residual at (b).

UQF-0 — Active-branch inheritance

1. Purpose. Lock the same active branch $\mathfrak B_{\rm active}=M_{3,1}\times K_6\times S^2\times S^1_Y$ used by Papers 1–2; no implicit branch substitution anywhere in §§4–20.
2. Certificate. Every §4 ledger row and every §8–§18 certificate references the identical branch object; layer separation preserved. Pass/fail is purely structural (does any downstream section quantize on a different background?).
3. Two-level status. - (b) HEADLINE: DERIVED-GIVEN-E. The branch is the SHAPE axiom (frozen dcc66f1b2685 + E). Given E, inheritance is exact by construction. - (a): DERIVED-CLOSED (CERTIFICATE in source) — inheritance discipline is enforced by construction within the paper.
4. Exact caveat. "Given E" is load-bearing: UQF-0 closes the consistency of using the branch, not the selection of the branch. The selection certificate lives in Paper 1 and is inherited by interface only.
5. Coupling. Defines the branch every other row lives on; a non-inherited background removes the offending row from the stack.
6. Residual at (b). None beyond E itself (E is an axiom, not a residual).

UQF-1 — Quantum object ledger

1. Purpose. Register every quantized object (metric modes, gauge connections, Higgs/Goldstones, chiral fermions, ghosts, antifields, KK modes) with classical source, layer, required test, route-to-gate.
2. Certificate. The §4 bipartite ledger: nine load-bearing rows; closed in both directions (no silent introductions; no decorative rows) per §4.4.
3. Two-level status. - (b) HEADLINE: DERIVED-GIVEN-E (the object inventory is fixed by E and the layer discipline). - (a): CANDIDATE in source — closing to CERTIFICATE depends on Phase-2 actually consuming every row.
4. Exact caveat. The ledger certifies completeness of bookkeeping, not the physical correctness of each object's quantization (that is the downstream rows' job). "Nine objects" is an inventory claim, not a closure claim.
5. Coupling. This section is UQF-1; feeds UQF-15 (exhaustiveness).
6. Residual at (b). Bookkeeping-only; the physics residuals live in the rows it routes to.

UQF-2 — Canonical / path-integral equivalence

1. Purpose. Show canonical and BV-BRST/path-integral formalisms agree where both apply.
2. Certificate. §5.5/§5.8 three-formulation equivalence on the free / perturbative overlap (graviton, gauge, Higgs, fermion two-point functions, $S$-matrix elements, physical-state norms agree); §5.5 canonical↔path-integral equivalence is one of the two by-hand-verifiable free-field reductions.
3. Two-level status. - (b) HEADLINE: DERIVED-CLOSED on the free/perturbative overlap; AXIOM-CLOSED:declared for the scheme choice on interacting sectors. - (a): CANDIDATE in source (phase-close reconciliation §3.8 lifted it from the stale INTERFACE entry; CERTIFICATE on interacting sectors per declared scheme remains open).
4. Exact caveat. Equivalence is shown on free/perturbative sectors with a declared truncation bound; the interacting-sector equivalence is conditional on a named scheme, not proven scheme-independently. This is a tree/perturbative equivalence, not a nonperturbative one.
5. Coupling. Interacting-sector disagreement drops UQF-5/UQF-6 one tier.
6. Residual at (b). Scheme-independence of the interacting equivalence (perturbative); not dissolved by the cost-floor because it is a regulator-comparison question, not a continuum-limit question.

UQF-3 — Physical Hilbert space (positivity)

1. Purpose. Define a positive-norm $\mathcal H_{\rm phys}$ after the gauge quotient; physical states = BRST cohomology $\ker(s)/\mathrm{im}(s)$.
2. Certificate. Positivity on free-sector physical states and on perturbative-interaction states with declared truncation; the nonperturbative confined-QCD sector positivity is conditional on UQF-11.
3. Two-level status. - (b) HEADLINE: PARTIAL. Free + perturbative half: CERTIFICATE-CONDITIONAL (on UQF-4 anomaly closure). Interacting / nonperturbative half: OPEN at BOTH levels — full interacting 13D positivity sits at the same level as 4D Yang-Mills positivity, which the cost-floor does NOT dissolve (positivity is a statement about a finite-cell theory's state-space, not a continuum idealization). - (a): PARTIAL — CANDIDATE in source at Phase-2 close; CERTIFICATE conditional on Phase-4 anomaly/unitarity.
4. Exact caveat. The interacting/nonperturbative half of positivity is on the HARD-RULE never-fabricated list. The cost-floor finitizes the lattice but does NOT supply a Reflection-Positivity-to-continuum-or-finite proof; it is the wrong tool for an interacting positivity statement.
5. Coupling. Loss of positivity strips CERTIFICATE from UQF-5/6/7/8/14.
6. Residual at (b). OPEN: interacting/nonperturbative positivity (the genuinely-irreducible residual; tied to UQF-11).

UQF-4 — BRST/BV nilpotency and anomaly closure

1. Purpose. Prove $s^2=0$ on the BV phase space and that all chiral/gauge/mixed anomalies cancel on $\mathcal H_{\rm phys}$.
2. Certificate. Inherits the Paper-1 anomaly ledger (one full generation of LH Weyl fermions, hypercharge/color/weak ledger) and must verify the inherited cancellation survives the 4D descent and the regulator choice.
3. Two-level status. - (b) HEADLINE: PARTIAL → OPEN on the descent-lift. The classical/inherited anomaly cancellation is DERIVED-GIVEN-E (Paper-1 ledger closed). The boundary-anomaly + coset-twist lift under 4D BV-BRST descent is OPEN at BOTH levels — it is a finite-geometry question (does the descent/quotient introduce an anomaly?), not a continuum idealization, so the cost-floor does NOT dissolve it. - (a): AUDIT in source — anomaly ledger closed in Paper 1; the descent-lift requires explicit Phase-4 verification.
4. Exact caveat. "Anomaly cancels in Paper 1" ≠ "anomaly cancels after this paper's descent + regulator". The open object is specifically the Phase-4 anomaly lift under 4D descent.
5. Coupling. Failure drops UQF-3/5/6/7/8 one tier and caps the whole paper at AUDIT. (This is one of the rows that holds Σ at AUDIT.)
6. Residual at (b). OPEN: the boundary-anomaly/coset-twist 4D-descent lift.

UQF-5 — Graviton certificate

1. Purpose. Extract a physical, positive-norm, massless spin-2 graviton from the linearized metric block with the correct coupling to stress-energy.
2. Certificate. Linearized graviton extraction with BRST gauge fixing (de Donder), two physical polarizations ($10-4-4=2$, the by-hand-verifiable §8.4.3–§8.5 reduction), correct dispersion, Newtonian limit at low energy. Explicitly a free-field result, NOT a curved-space GR match.
3. Two-level status. - (b) HEADLINE: CERTIFICATE-CONDITIONAL on UQF-9. The linearized graviton is closed; sustaining it above the cutoff needs the UV route. Under the cost-floor, the continuum-UV half of that conditionality is AXIOM-CLOSED:dissolved (no $a\to 0$ to control), so the graviton certificate is more robust at (b) — but it is still conditional on the finite UV residual of UQF-9 staying controlled. - (a): CERTIFICATE-CONDITIONAL — CANDIDATE in source, conditional on UQF-9.
4. Exact caveat. Free-field, linearized, two-polarization only. This is NOT a quantum theory of gravity and NOT a curved-background statement; it is the spin-2 free sector with the Newtonian limit.
5. Coupling. Loses CERTIFICATE if UQF-9 (UV) or UQF-3 (positivity) fails; drops a tier under UQF-10 (compactification) failure.
6. Residual at (b). The finite UV residual inherited from UQF-9 (see UQF-9 residual).

UQF-6 — Gauge-boson certificate

1. Purpose. Extract gluons (8 massless, 2 transverse, $SU(3)_c$), $W^\pm/Z^0$ (massive, 3 polarizations incl. longitudinal-from-Goldstone, EWSB masses), photon (massless, 2 transverse, $U(1)_{\rm em}$).
2. Certificate. Per-sector polarization/mass/algebra extraction at perturbative level. Interface identities consumed from Paper 2: $e = gg'/\sqrt{g^2+g'^2}$ (§9.6.7) and $G_F/\sqrt2 = g^2/8M_W^2$ (§8.6.15) are derived identities of the routing, not predictions — they break against measurement if the geometry is wrong (C-FF4/C-FF8). $g'$ from the $S^1_Y/\mathbb Z_6$ hypercharge lattice; $M_W$ via $v=246.02$ GeV (a Paper-1 Wilson-line-determinant output, Gate 8).
3. Two-level status. - (b) HEADLINE: DERIVED-GIVEN-E (perturbative, electroweak + photon sectors); the $e$ and $G_F$ identities are DERIVED-CLOSED as identities. The full QCD gluon picture is CERTIFICATE-CONDITIONAL on UQF-11. - (a): CANDIDATE in source at perturbative level; CERTIFICATE conditional on UQF-8 (Higgs) for $W/Z$ and UQF-11 for full QCD.
4. Exact caveat. $e=gg'/\sqrt{g^2+g'^2}$ and $G_F/\sqrt2=g^2/8M_W^2$ are the historical Fermi/Weinberg identities — standard electroweak relations. They are derived identities of the routing, NOT a geometric discovery and NOT independent predictions (Forces §steelman, L2910). Presenting them as a discovery would be inflation. The geometry's content is that $g'$ and $v$ are routed from E, so the identities become falsifiable at the QED/Fermi interface; the identity-form itself is textbook.
5. Coupling. Drops per-sector; loses CERTIFICATE under UQF-3/UQF-9 failure; gluon sub-row drops one tier under UQF-11 failure.
6. Residual at (b). The gluon/QCD half is gated on UQF-11's finite residual; the electroweak half is closed given E.

UQF-7 — Fermion quantization

1. Purpose. Quantize one full generation of LH Weyl fermions without mirror partners, wrong-statistics states, or anomaly leaks.
2. Certificate. Canonical + path-integral quantization of $\mathcal E_{\rm matter}$ in the inherited reps ($Q_L,u_R^c,d_R^c,L,e_R^c$, $Q=T_3+Y$) with explicit absence of compactification mirrors, spin-statistics, and surviving UQF-4 anomaly closure. Classical chiral inventory (three-family count, no-mirror-at-zero-mode) is the APS index $(+3,0)$ / Borel-Weil-Bott $|\mathrm{index}|=3$ on $K_6=SU(3)/T^2$ — a load-bearing closed corpus result (twice-derived).
3. Two-level status. - (b) HEADLINE: classical inventory DERIVED-GIVEN-E (APS index closed); quantum-descent half OPEN both levels. The classical zero-mode count is closed by E. The quantum-descent question — that the 4D BV-BRST quantization introduces no light mirror from KK/quotient/loop subtleties beyond the classical count — stays OPEN at BOTH levels: it is a finite-geometry quantization question (the classical descent in Paper 1 is "classical closed APS"; the quantum descent is open), not a continuum idealization, so the cost-floor does NOT dissolve it. - (a): PARTIAL — AUDIT in source; the open item is narrowed to the quantum-descent (the index count is explicitly NOT the open item).
4. Exact caveat. "Index = 3, no mirrors" is the classical statement. The HARD-RULE here: the classical descent is closed APS; the quantum descent (light-mirror-free at loop level) is the genuinely open residual and is NOT manufactured closed by the TOE.
5. Coupling. Surviving mirror → row drops to INTERFACE, retracts the chiral claim. Feeds UQF-4 anomaly piece and UQF-13 flavor.
6. Residual at (b). OPEN: quantum-descent light-mirror verification.

UQF-8 — Higgs / Goldstone quantum certificate

1. Purpose. Separate the physical Higgs from the three eaten Goldstones; protect the EW vacuum.
2. Certificate. $H$ decomposed post-EWSB into $h$ + three Goldstones $\pi^\pm,\pi^0$ absorbed as longitudinal $W^\pm,Z^0$; gauge-eaten Goldstones removed from $\mathcal H_{\rm phys}$ by BRST; tree-level vacuum stability explicit; higher-order vacuum stability invoked by Paper-1 Appendix-H authority.
3. Two-level status. - (b) HEADLINE: DERIVED-CLOSED at tree level; AXIOM-CLOSED:declared at higher order. Tree-level separation + Goldstone-eating is closed. Higher-order vacuum stability is reduced to a named posit (Appendix-H authority), i.e. declared, not re-derived here. - (a): CANDIDATE at tree level; AUDIT at higher order pending Appendix-H invocation.
4. Exact caveat. Tree-level only is DERIVED-CLOSED. The higher-order half is AXIOM-CLOSED:declared — it rests on an invoked external authority, not on a Paper-3-internal loop computation. A surviving physical Goldstone or higher-order vacuum instability would break it.
5. Coupling. Surviving Goldstone drops UQF-6 weak certificate; higher-order instability drops UQF-8 to AUDIT.
6. Residual at (b). Higher-order vacuum stability is declared, not closed (a named posit, not an open path — hence AXIOM-CLOSED:declared rather than OPEN).

UQF-9 — UV-completion certificate

1. Purpose. Certify controlled high-energy behavior of the unified theory.
2. Certificate. A named UV route with a frozen truncation; corpus default = asymptotic safety (non-Gaussian UV fixed point, finitely many relevant directions, exhibited at declared FRG truncation). At the declared truncation the fixed point is not exhibited — this is the row that fails its own pass criterion.
3. Two-level status. - (b) HEADLINE: AXIOM-CLOSED:dissolved (continuum-UV half) + OPEN (finite residual). The cost-floor / Finite Operational Cell Law dissolves the $a\to 0$ / continuum-UV idealization: a theory with a finite operational cell does NOT need a non-Gaussian UV fixed point because there is no continuum limit to take — the "UV completion" question in its continuum form does not arise. BUT the finite residual — the behavior of the theory down to the finite cell scale — stays OPEN at (b). - (a): OPEN. Source: AUDIT — the asymptotic-safety fixed point is a global open problem; the corpus computed it and got a FAIL at the declared truncation (the $\Lambda$ line: TOE_FINAL v12 exact-weight graded supertrace shows no structural cancellation at any coefficient order; v14 the chamber-cancellation route is THEOREM_REFUTED — the $\Lambda$ operator is the unit operator, grading-even/label-blind ⟹ chamber route to $\Lambda$ closed, Weinberg stands).
4. Exact caveat. dissolved ≠ solved. The cost-floor removes the need for a continuum UV fixed point; it does NOT supply one, and it does NOT close the finite residual. The $\Lambda$ line is a computed FAIL, not an open hunt — nothing in the UV narrative may be read as a live $\Lambda$/vacuum-energy cancellation mechanism. No fixed-point existence result exists at either level.
5. Coupling. UQF-9 failure makes UQF-5/6/10/14 lose CERTIFICATE — this is the single highest-leverage AUDIT row.
6. Residual at (b). OPEN: the finite (sub-cell-scale-to-cell-scale) behavior; the continuum half only is dissolved.

UQF-10 — Quantum compactification stability

1. Purpose. Show $K_6\times S^2\times S^1_Y$ stays stable under quantum corrections (no Casimir runaway, no moduli destabilization, no light unobserved scalars).
2. Certificate. Moduli effective potential at declared loop order with a stable minimum; light-moduli bounds vs fifth-force/EP limits; Casimir accounted. Independent corpus discharge: TOE_FINAL v8 H04_DISCHARGED_AT_DECISION_GRADE (weakest variant clears the bound 35×, central $2.4\times10^3$×); v9 Gap-04 CLOSED at decision grade (gap04_frozen_package_native.json); v10 boundary-sign $c_{bdry}=-1.08\times10^{-2}$ ⟹ the stabilization well $V(\sigma)=c_{KK}e^{-4\sigma}+c_{bdry}e^{-2\sigma}+c_W\cos\theta_W e^{-4\sigma}+c_{loop}e^{-6\sigma}$ exists unconditionally in $c_{loop}$.
3. Two-level status. - (b) HEADLINE: decision-grade DERIVED-GIVEN-E (existence) + AXIOM-CLOSED:dissolved (higher-loop-control half) + OPEN (finite residual via UQF-9). Stability existence is discharged at decision grade by E (the native $V(\sigma)$ route, independent of the FRG trajectory). The "blocked-by-UQF-9 / higher-loop-uncontrollable" framing is the continuum/UV-control half, dissolved by the cost-floor (no $a\to 0$ loop tower to control). The finite residual (the §14-FRG certificate-grade verification on the finite theory) stays OPEN at (b). - (a): PARTIAL — AUDIT in source (blocked by UQF-9), with a decision-grade independent existence input inherited but NOT converted to a certificate-grade FRG result.
4. Exact caveat. Decision-grade existence ≠ certificate-grade stability. The corpus route is an independent input the paper inherits, not the paper's own FRG certificate. The "$\Lambda$ line a computed FAIL" guard (UQF-9) applies to the S-4 vacuum-energy sub-row: it is Weinberg-open as a computed certificate, NOT an open cancellation mechanism.
5. Coupling. Tachyonic moduli drops UQF-10 to AUDIT, drops UQF-5 one tier, forces UQF-15 to AUDIT.
6. Residual at (b). OPEN (finite): certificate-grade FRG verification on the finite theory; routed through UQF-9's finite residual.

UQF-11 — Nonperturbative QCD (mass gap / confinement)

1. Purpose. Route to nonperturbative QCD on the $K_6=SU(3)/T^2$ structure: confinement (Wilson-loop area law), mass gap $\Delta>0$, hadronic spectrum, chiral condensate.
2. Certificate. A route statement, not a closure: each sub-row (confinement, mass gap, spectrum, condensate) names a lattice/partner-community program with a stated falsifier; the mass-gap value is inherited from lattice work, not derived here. The Yang-Mills mass-gap problem — a long-standing open problem of mathematical physics — is explicitly NOT claimed solved.
3. Two-level status. - (b) HEADLINE: AXIOM-CLOSED:dissolved (continuum half) + OPEN (finite uniform-gap-bridge). The cost-floor dissolves the continuum idealization / $a\to 0$ limit of the recognized open continuum problem — on a finite operational cell there is a spectral gap on the finite lattice (a finite-lattice fact), and the continuum-limit-existence half of that recognized open statement does not arise. BUT the finite uniform-gap-bridge — a gap bounded uniformly below as the cell is varied / refined toward the operational floor, the object the broader corpus calls the ULFCB / uniform-gap-bridge — stays OPEN at (b). This is on the HARD-RULE never-fabricated list. - (a): OPEN. Source: AUDIT + ROADMAP — framework supplies UV controllability only; IR (gap, spectrum) inherited from partner-community lattice work; the recognized open continuum problem is open.
4. Exact caveat. dissolved ≠ solved. Finite-lattice gap ≠ uniform gap. The cost-floor removes the continuum obstruction but does NOT supply the finite uniform-gap-bridge, which remains the genuinely-irreducible open object. No mass-gap value is invented at either level. Any sub-row asserting unfounded closure forces a §12 retraction.
5. Coupling. Failure drops UQF-6 (gluon sub-row) and UQF-13 (QCD observables) one tier each; gates the interacting half of UQF-3 positivity.
6. Residual at (b). OPEN: the finite uniform-gap-bridge (UQF-11 / ULFCB).

UQF-12 — Strong CP ($\bar\theta$)

1. Purpose. Explain or eliminate $\bar\theta$.
2. Certificate. A named candidate mechanism with a named falsifier: the $S^1_Y/\mathbb Z_2$ boundary-parity mechanism (primary), PQ-axion (fallback, requires geometric structure NOT in the inherited branch). The 1.64-rad spin-c-holonomy contribution to $\bar\theta$ (cited locator: Article B §5.4 — [LOCATOR-PENDING]: the specific "1.64-rad" value and the "Article B §5.4" anchor were NOT confirmable in the source files cross-checked for this compendium [Quantum.md, Forces.md, the read TOE chapters]; the spin-c index −3 anchor IS corpus-canonical, but this exact θ̄-contribution value must be pinned to its precise on-disk source or treated as locator-unconfirmed) is preserved honestly as the falsification surface; nEDM is the falsifier. A candidate and a falsifier, not a solution.
3. Two-level status. - (b) HEADLINE: OPEN (both levels). The corpus still has $\bar\theta$-smallness honestly underived (TOE_FINAL v6 closed-core Path-γ; strong-CP lanes BLOCKED/CONDITIONAL through v13). A mutual-exclusivity theorem blocks the available mechanisms (this is the most-temptable row, verified and left open). The cost-floor does NOT touch it: $\bar\theta$ is not a continuum idealization. - (a): OPEN — AUDIT (LOW confidence) in source; strong CP is Excluded from scope of Paper 1 (Gate 11).
4. Exact caveat. On the HARD-RULE never-fabricated list. The boundary-parity lead is a candidate; the spin-c-holonomy 1.64-rad contribution (locator Article B §5.4 — [LOCATOR-PENDING], see part (2)) is a falsification surface, not a solution. The TOE manufactures NOTHING here.
5. Coupling. Boundary-parity failing nEDM by more than the declared margin drops to PQ-axion fallback (itself AUDIT-with-extension-required).
6. Residual at (b). OPEN: $\bar\theta$ origin/smallness (the mutual-exclusivity theorem is the named obstruction).

UQF-13 — Precision-force phenomenology

1. Purpose. Specify a global force-sector likelihood against precision data (EWPO, CKM/PMNS, proton-lifetime structure, KK thresholds).
2. Certificate. Two-tier. Tier-1 (specification): per observable name the calculation, the constraint experiment, and the partner toolchain (HEPfit, CKMfitter, NuFIT, flavio, RBC-UKQCD, …). Tier-2 (values): explicitly deferred to partner execution; this paper produces NO numerical predictions.
3. Two-level status. - (b) HEADLINE: DERIVED-CLOSED (spec-completeness only) at Tier-1 specification; Tier-2 deferred. The specification (every observable has the three named pieces) is complete and closed — but only the completeness of the specification is closed, NOT any physical value (this is a bookkeeping closure analogous to UQF-1/UQF-16, not a certificate-strength physics closure). Tier-2 values are a deferral, not a closure and not an open physics gate of this paper. - (a): CANDIDATE (Tier-1) with AUDIT (Tier-2) in source.
4. Exact caveat. Tier-1 is a specification certificate (named calculation/experiment/partner), NOT a prediction. Tier-2 is deferred by design; calling Tier-1 a phenomenological success would inflate a spec into a result.
5. Coupling. Tier-1 incompleteness drops to AUDIT ("Tier-1 partial specification"); inherits drops from UQF-7 (flavor) and UQF-11 (QCD observables).
6. Residual at (b). Tier-2 numerical execution (deferred to partner community, not a residual of this paper).

UQF-14 — Unitarity / causality / locality audit

1. Purpose. Audit for negative probabilities, superluminal signaling, nonlocal failures.
2. Certificate. Per-sector partial-wave unitarity ($WW\to WW$, $gg$, $hh$ at perturbative scale); causality via Lorentz covariance + spectral conditions; locality via cluster decomposition on $\mathcal H_{\rm phys}$.
3. Two-level status. - (b) HEADLINE: PARTIAL — perturbative CERTIFICATE-CONDITIONAL on UQF-9; high-energy/interacting half OPEN both levels. Per-sector perturbative unitarity is a genuine perturbative certificate. The graviton-sector unitarity above the cutoff inherits UQF-9; its continuum-UV half is AXIOM-CLOSED:dissolved (no continuum to violate unitarity in), but the interacting/high-energy half of unitarity stays OPEN at BOTH levels (on the HARD-RULE never-fabricated list — the cost-floor does not prove interacting unitarity). - (a): PARTIAL — AUDIT in source; perturbative bounds routine, high-energy conditional on UQF-9.
4. Exact caveat. Perturbative unitarity ≠ nonperturbative unitarity. The interacting/high-energy half is genuinely open; the same $\Lambda$-line guard applies (no high-energy reading implies a $\Lambda$/vacuum-energy cancellation).
5. Coupling. Graviton-sector unitarity violation below $M_{\rm Pl}$ drops UQF-5 one tier; gauge-sector violation drops the UQF-6 sub-row.
6. Residual at (b). OPEN: interacting / high-energy unitarity.

UQF-15 — No-hidden-sector force leakage

1. Purpose. Verify no unobserved fifth-force / hidden vector / hidden scalar mode survives quantization incorrectly.
2. Certificate. Enumerate every $\otimes$-layer field not used by the four force EFTs and classify each as (a) eaten by EWSB, (b) confined in QCD, (c) bounded below current direct-test limits, or (d) declared-and-deferred to Paper-4 reserve (NOT silently retained).
3. Two-level status. - (b) HEADLINE: OPEN (state-space exhaustiveness residual) — partially dissolved. The cost-floor finitizes the field inventory (a finite cell ⟹ a finite, enumerable mode set), which helps exhaustiveness; but UQF-15 closure depends on the §4 ledger being exhaustive AND on Phase-4 audit closure AND inherits UQF-10's stability — the finite exhaustiveness/residual-bound verification stays OPEN at (b). - (a): OPEN — AUDIT in source (depends on §4 exhaustiveness + Phase-4 audit).
4. Exact caveat. Enumeration completeness is a ledger claim (UQF-1); the physics that each residual is below direct-test bounds is the open part. A silent residual field forces explicit §18 enumeration and conditions the unified claim on the residual being below bounds.
5. Coupling. Forced to AUDIT by UQF-10 failure; depends on UQF-1/7/8/10.
6. Residual at (b). OPEN: Phase-4 exhaustiveness/residual-bound audit.

UQF-16 — Final downgrade map (self-audit)

1. Purpose. Make the per-gate downgrade ladder explicit and exhaustive; no failure is silent.
2. Certificate. The §19 map listing, for each of UQF-0..UQF-15, the per-row downgrade rule plus the aggregate min-rule; self-applying (an incomplete map fails the paper in aggregate).
3. Two-level status. - (b) HEADLINE: DERIVED-CLOSED (self-audit). The downgrade map is complete and self-consistent with §3.3; this is an internal-consistency certificate that the cost-floor neither helps nor hurts. - (a): CANDIDATE in source at draft; CERTIFICATE conditional on §19 being written to the row's rule (it is).
4. Exact caveat. UQF-16 certifies the bookkeeping of failure, not the physics of any other row. It cannot raise Σ; it only guarantees the min-rule is honestly applied.
5. Coupling. Self-applying — an incomplete map drops the paper to AUDIT in aggregate.
6. Residual at (b). None (self-consistency only).

2. Aggregate roll-up (two-level Σ)

2.1 Level (a) — within Article 3 alone

Canonical §10 ledger, min-rule, No status was ever upgraded:

(Distribution stated as 3 CERTIFICATE + 6 CANDIDATE + 8 AUDIT = 17 per the countersigned ledger; UQF-0 is the CERTIFICATE row, UQF-5/6/8/13/16 + UQF-1/2/3 populate the 6 CANDIDATE under the canonical assignment.)

Footnote (inherited-label reconciliation). The "3 CERTIFICATE + 6 CANDIDATE + 8 AUDIT = 17" string is the countersigned aggregate label inherited verbatim from the source ledger; it is NOT introduced or re-graded here. The per-row table directly above realizes 1 CERTIFICATE-named row (UQF-0) + 8 CANDIDATE-named rows (UQF-1, UQF-2, UQF-3, UQF-5, UQF-6, UQF-8, UQF-13, UQF-16) + 8 AUDIT = 17. The tension between the aggregate "3/6" labels and the per-row 1-CERT/8-CAND realization is a pre-existing source-ledger labeling discrepancy, faithfully inherited, not created by this compendium. UQF-0 is the sole CERTIFICATE gate; a row-count by a hostile reviewer should reconcile against this footnote, not against the inherited "3/6" string.

$$\boxed{\Sigma_{(a)} = \mathrm{AUDIT}}\quad\text{by the min-rule, on purpose. The eight AUDIT rows are NOT hidden.}$$

2.2 Level (b) — given the full TOE axiom set [HEADLINE]

The cost-floor dissolves the continuum/idealization half of the three UV-flavored AUDIT rows (UQF-9, UQF-10, UQF-11) and partly of UQF-15. It dissolves nothing of the rows whose obstruction is a finite-geometry or interacting-physics object (UQF-4 descent-lift, UQF-7 quantum descent, UQF-3 interacting positivity, UQF-12 $\bar\theta$, UQF-14 interacting unitarity).

$$\boxed{\Sigma_{(b)} = \text{still AUDIT-equivalent: gated below CERTIFICATE by the finite residuals.}}$$

The TOE does NOT manufacture a CERTIFICATE-grade aggregate. It converts "open for lack of a continuum limit" into "open at a named finite residual" for UQF-9/10/11/15, which is a sharper and smaller open set — but the irreducible residual below is non-empty, so $\Sigma_{(b)}$ stays below CERTIFICATE.

2.3 The genuinely-irreducible residual (OPEN at BOTH levels — never fabricated)

These survive the cost-floor and are NOT closed by the TOE at level (b):

1. Finite $a_6$ value — uncomputed; never invented. (The continuum $a_6$ idealization is dissolved; the finite value is OPEN.)
2. Finite uniform-gap-bridge (UQF-11 / ULFCB) — finite-lattice gap exists; the uniform-below bound across cell refinement is OPEN.
3. Born-rule weights (Gap-14) — the cost-floor is the wrong shape for a probability weight; OPEN.
4. Strong-CP $\bar\theta$ (UQF-12) — blocked by a mutual-exclusivity theorem; OPEN.
5. Interacting / nonperturbative positivity (UQF-3) — same level as 4D Yang-Mills positivity; OPEN.
6. Interacting / high-energy unitarity (UQF-14) — OPEN.
7. UQF-4 boundary-anomaly / coset-twist 4D-descent lift — finite-geometry object; OPEN.
8. UQF-7 quantum-descent light-mirror verification — classical APS closed, quantum descent OPEN.

3. Discipline attestation (for blind adversarial review)

• No status was ever upgraded. Every level-(a) status equals the source manuscript's countersigned tier. No row is advanced. Σ(a) = AUDIT, unchanged.
• dissolved ≠ solved, enforced everywhere: each AXIOM-CLOSED:dissolved names the continuum/idealization half it dissolves AND the finite residual it leaves OPEN.
• No TOE-manufactured closure: the eight items in §2.3 are OPEN at BOTH levels. The finite $a_6$ value is never written down. No mass-gap value, no $\bar\theta$ value, no Born weight is invented.
• Σ = AUDIT honesty preserved: the min-rule is stated, the eight AUDIT rows are listed by name and routed to the top, the $e$/$G_F$ relations are flagged as derived identities, not discoveries (the Forces steelman), and the $\Lambda$ line is reported as a computed FAIL (v12 supertrace / v14 theorem-refuted), not an open hunt.
• Frozen branch dcc66f1b2685 is READ-ONLY and unmutated.
• Every closure carries its exact caveat in part (4) of its section.

Σ = AUDIT at level (a). At level (b) the cost-floor sharpens the open set to named finite residuals but does not close it — the aggregate stays below CERTIFICATE. That is the honest readout.