UQF-10 — Compactification consistency: the gate anchor ledger

The honest one-line: UQF-10 asks a survival question, not a measurement one — does the already-frozen 13D compactification $K_6\times S^2\times S^1_Y/\mathbb{Z}_2$ stay a consistent quantum theory once loops are on? Two legs are real within-truncation wins (the single-modulus reduction S-1, scoped to admissibility; the perturbative no-tachyon result S-3); four load-bearing stability rows wait on one named, un-run FRG computation that is itself blocked by the shared UV-completion wall; the induced 4D $\Lambda$ is a computed honest FAIL; and the one cheap, UV-independent stability check already on the table — the shape-doublet Hessian — comes back a saddle, leaning REFUTED. That is the frozen mid-audit record; on the live board the gate stands RESOLVED +0 (DERIVED-GIVEN-anchor, ratified 2026-07-08), with these residual rows shown unchanged.

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the four bridges and applies them, object by object, to one gate. Every exact thing UQF-10 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape and universal table as the canonical SG-4 ledger.

1. Gate status header

• Gate-level UQF-10 roll-up: DERIVED-GIVEN-anchor — RESOLVED +0 on the live board (ratified 2026-07-08). Frozen mid-audit reading: OPEN (AUDIT) — certificate-conditional within the declared truncation, and BLOCKED by the shared UQF-9 / FRG UV-completion wall.
• Taxonomy reconciliation (2026-07-05): under the ratified closure taxonomy (board 2026-07-08) the gate-level grading is the campaign terminal carried on the scoreboard — RESOLVED +0 (DERIVED-GIVEN-anchor), read as TERMINAL + RESIDUALS-SHOWN. The residual family listed in this ledger (the un-run FRG computation, the shared UV wall, the live negatives including the shape-doublet saddle and the induced-$\Lambda$ FAIL, and the completeness walls) remains listed and carried unchanged. The facts are unchanged: the earlier OPEN roll-up above reflects the superseded least-closed-residual rule (one open piece forcing the whole gate open), not a different set of facts.
• The closed local/partial legs, stated so they cannot be misread:
  • Single-modulus reduction (S-1): SCOPED — AXIOM-FROZEN-BRANCH-ADMISSIBILITY, admissibility/scope only.
  • Perturbative no-tachyon (S-3): DERIVED-GIVEN-E within truncation — banked as CERT-S3-PERTURBATIVE-NO-TACHYON.
• The local survival predicate, written as the row-basis conjunction over the frozen branch: $$\mathrm{Survive}(\mathfrak{B}_{\rm active})\;\equiv\;\bigwedge_{i=1}^{7} S_i ,\qquad \text{gate status}=\text{least-closed }S_i .$$ Given $\mathfrak{B}_{\rm active}$, $S_3$ holds perturbatively and $S_1$ holds as a declared scope; $S_2,S_5,S_6,S_7$ are computation-debt behind an un-run FRG matching; $S_4$ ($\Lambda$) is a disclosed computed FAIL. By the superseded least-closed-residual rule, one open piece forced the mid-audit roll-up to OPEN; the ratified board carries the RESOLVED +0 terminal with these rows shown unchanged.

selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique.

UQF-10 audits a stability property of a frozen geometry, not a number. There is no measured invariant that "closes" it; the achievable ceiling within current reach is certificate-within-truncation on every row plus a proven composition theorem plus a built UV completion — and even that would be DERIVED-GIVEN-E within truncation, never DERIVED-CLOSED.

2. Frozen inputs (what UQF-10 stands on, not what it produces)

• Frozen branch hashes content dcc66f1b2685 / manifest meta a5b1e6f9d951, READ-ONLY. These are audit anchors — they certify which object was audited and that it cannot be quietly retuned. They do not validate the physics.
• The audited branch $\mathfrak{B}_{\rm active}$ enters frozen and unchanged as the input the gate audits, never as a result the gate validates: $$ \mathfrak{B}_{\rm active}=\big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]\ \oplus\ \big[F^+_{\rm finite}\oplus C_{\rm admiss}\big]\ \otimes\ \big[E_{\rm matter}\oplus E_{\rm gauge}\oplus E_{\rm Higgs}\oplus E_{\rm proton}\big], $$ with $K_6=SU(3)/T^2$ the flag manifold, $D=13=4+6+2+1$, radii pinned at the Weyl-rigid chamber center $\vec u=(1,1,1)$, $R_0\equiv(2\pi M_U)^{-1}=1.592\times10^{-17}\,\mathrm{GeV}^{-1}$, $M_U\sim1.0\times10^{16}\,\mathrm{GeV}$. A stable spectrum given the geometry is not proof the geometry is the one nature picks — uniqueness is an upstream SHAPE / SG-1 question.

3. The object anchors (given the frozen branch)

The gauge routing audited (by isometry, GIVEN the branch):

The Cartan-torus modulus $\tau=\omega$ is absorbed into the $F^+$ chamber data (Option B) and is not a separately stabilized propagating modulus. Status: GIVEN / frozen-branch — not UQF-10-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for UQF-10:

Master anchors in play: finite invariant ledgers (the four-term $V(\sigma)$, the row basis) · no unpaid labels (no axiomatized FRG; no self-anchored $\mu_{\rm cell}$) · the frozen branch · given-the-geometry · the declared admissibility freeze · open-residual discipline (gate = least-closed residual).

5. The UQF-10 anchor ledger (the universal table)

6. The arithmetic — the computed legs, in full

6.1 The breathing-mode potential and the one open coefficient

The single-$\sigma$ stabilization object is the four-term potential: $$ V(\sigma)=c_{\rm KK}\,e^{-4\sigma}+c_{\rm bdry}\,e^{-2\sigma}+c_{\rm Wilson}\cos\theta_W\,e^{-4\sigma}+c_{\rm loop}\,e^{-6\sigma}. $$ One term from bulk KK-tower energetics, one from the boundary the fold creates, one from the Wilson/Hosotani sector, and one from loops. Three of the four coefficients are computed and banked (carried only to define the well; none is combined into $c_{\rm loop}$ — that combination is a forbidden shortcut):

$c_{\rm loop}$ is the coefficient of the $e^{-6\sigma}$ term: a one-loop / threshold-resolved radiative residue of the compact residual vacuum energy — not a count-only Casimir, and a distinct object from $c_{\rm KK}$.

6.2 The FRG matching that would fill it

The flow of the effective potential is the Wetterich equation projected onto constant fields: $$ k\,\frac{d}{dk}U_k=\tfrac12\,\mathrm{STr}\!\left[\frac{\partial_k R_k}{\Gamma_k^{(2)}+R_k}\right], $$ and the $\sigma^{-6}$ coefficient takes the threshold-resolved supertrace form (FRG-2, LPA, Litim): $$ c_{\rm loop}=\frac{1}{4(4\pi)^2}\sum_{\rm levels}\Big[\deg_B(\ell)\,l^0_B\!\big(m_B^2(\ell)/k^2\big)-\deg_F(\ell)\,l^0_F\!\big(m_F^2(\ell)/k^2\big)\Big]. $$ The normalization $1/(4(4\pi)^2)$ is fixed (the same one-loop measure as $c_{\rm KK}$). The scheme — truncation (strict LPA, $Z_k=1$, $\eta=0$), regulator (Litim, on truncation-quality grounds only), and normalization — is frozen before any number, with reverse-fitting, regulator-tuning toward $\Lambda$, FRG-4 promotion, and combination with $c_{\rm KK}$ all explicitly forbidden.

The standing blocker. The five source-hashed spectrum CSVs (the bosonic $K_6/S^2/S^1_Y$ towers, the twisted-Dirac $K_6$ tower, and the retained per-level enumeration) are MISSING; certificate criterion CC-1 (spectra present + source-hashed) is presently FALSE, so the supertrace is not yet evaluable.

The count diagnostic — and the forbidden shortcut it must not become. The retained content has $n_B=35$ bosons and $n_F=90$ fermions, so the count-only combination $$ \mathrm{Str}[1]=n_B-n_F=35-90=-55<0 . $$ The threshold-resolved supertrace can have either sign. Reading a sign off the bare count $-55$ is a forbidden count-only shortcut; the banked state is sign = UNDETERMINED.

6.3 The one UV-independent stability piece, computed — the specificity diagnostic

The most important computed content is a live, UV-completion-independent falsifier computed below the wall. The shape-doublet moduli-potential Hessian at $\vec u=(1,1,1)$ is $\Lambda$-free, $\mu_{\rm cell}$-free, and computable without the FRG run. Along the doublet ray $u=(1+\varepsilon,1-\varepsilon,1)$ on $SU(3)/T^2$: $$ \left.\frac{d^2R}{d\varepsilon^2}\right|_0=+4-5=-1<0 , $$ diagonal convexity $+4$ versus the structure-constant triangle $-5$. Unit-normalized, the $\varepsilon^2$-coefficient is $+2-5/2=-1/2$. Curvature is negative in both conventions — the sign is direction- and normalization-invariant. This is the gate's specificity diagnostic: it is a non-trivial, sign-definite result reproduced independently in sympy and cross-confirmed in the SG-6 certificate, not an artifact. Its honest disposition is LIVE FALSIFIER — leaning REFUTED, banked as a first-class negative certificate CERT-SHAPE-DOUBLET-SADDLE. Two watch-items keep it from being over-read: it is a single-slice result that refutes positive-definiteness (one negative direction suffices) but is not yet a full $2\times2$ signature certificate; and the saddle → mass² map needs canonical kinetic normalization and the R6 graded-Casimir net sign (a contribution $\mu_{\rm cell}\cdot q\gtrsim0.5$ would be required to overcome the $-1$) — both absent. Hence leaning refuted, not fully refuted. Critically: UV completion would not rescue a confirmed saddle.

7. Declared-structure splits (single phrases hiding several claims)

• "S-1 is closed" splits into three: (1) the single-modulus reduction holds within the declared truncation — SCOPED AXIOM-FROZEN-BRANCH-ADMISSIBILITY; (2) the freeze is not "by a positive-definite Hessian" — that reading is a claim-boundary violation; (3) the forcing upgrade THEOREM-TARGET-RESIDUAL-ISOMETRY-FORCES-HESSIAN-SIGN is a separate object, OPEN / leaning REFUTED.
• "S-2 positivity" splits into the shape-doublet half (computable below the wall — leaning REFUTED, §6.3) and the volume-singlet half (B3-blocked + the $\mu_{\rm cell}$ self-anchoring firewall T1 — OPEN). They are different objects with different blockers; collapsing them hides the falsifiable prediction.
• "the well exists" splits from "the gate is stable": the inherited existence result is decision-grade context, but it is cited input ONLY, closes no FRG row, and supplies no value or sign for $c_{\rm loop}$. Existence ≠ stability.
• "S-3 no tachyon" splits into perturbative (the banked win) and nonperturbative tunneling exclusion (OPEN, beyond truncation).
• "the $\Lambda$ anchor" splits into the measured $\Lambda\approx(2.3\,\mathrm{meV})^4$ (a separate measured-but-irreducible terminal anchor) and the gate's own induced $\Lambda$ line (a computed FAIL). The two are never laundered into each other.

8. Open residuals — the families

The residuals group into four families; none is collapsed into a vague item.

A. The un-run computation (computation-debt). - $c_{\rm loop}$ value + sign — BLOCKED_MISSING_FRG_MATCHING / sign UNDETERMINED; CC-1 (spectra) is FALSE. The five source-hashed CSVs must be produced blind before the supertrace is evaluable. - S-2 (volume), S-5, S-7 — all collapse onto the one missing object, the FRG-controlled moduli effective potential; behind the UV build.

B. The shared global wall (BLOCKED). - B3 = UQF-9 FRG-branch construction — a genuine non-Gaussian asymptotic-safety fixed point for the frozen branch, currently un-built (SHARED_BLOCKER_BUILDS/ holds only B1, B2; the heat-kernel coefficient $a_6$ is uncomputed). UQF-10 is coupled to and not independent of UQF-9 and cannot close ahead of it; the blocked-by edge is mutually recorded and no theorem is stretched across the wall.

C. The live negatives (first-class, banked). - Shape-doublet Hessian — CERT-SHAPE-DOUBLET-SADDLE, $d^2R/d\varepsilon^2|_0=-1$, leaning REFUTED; resolving the full $2\times2$ signature is the highest-value runnable move. - S-4 / $\Lambda$ — a disclosed computed FAIL, Weinberg-OPEN, no live cancellation mechanism; the $\mathrm{Str}[1]=-55$ consilience is structural consilience only, NOT an adopted axiom. - S-1 forcing theorem — leaning REFUTED (the saddle is evidence against the isometry forcing a positive Hessian).

D. The completeness walls (theorem-debt, co-equal with B3). - P0 predicate-completeness — it is not proven that S-1…S-7 are a necessary + sufficient failure-mode basis (candidate missing mode: the Gap-15 state-space measure $\mu$). A BUILT B3 alone would still not close the gate. - The row-composition theorem — even if every row certifies, it is not proven they jointly imply the survival predicate (cross-terms boundary ↔ KK ↔ moduli may spoil the conjunction). - S-6 boundary — B2 exists as a shared build, but the branch-pullback row is un-run; "anomaly-free" must not be read as "physically absent." - S-3 nonperturbative — tunneling / false-vacuum-decay exclusion is open.

9. Anti-claims (what this page refuses to say)

• UQF-10 does not prove the compactification quantum-stable. Zero of the seven rows reaches full quantum closure. Formally, $\mathfrak{B}_{\rm active}$ passing $S_1,S_3$ within truncation is not $\bigwedge_i S_i$ proven.
• The audit is not a selector. A stable spectrum given the geometry is not proof the geometry is the one nature picks; selection is an upstream SHAPE / SG-1 question, OPEN.
• Existence is not stability. The inherited compactification-existence result is cited input only; it closes no FRG row and supplies no value or sign for $c_{\rm loop}$.
• $\Lambda$ is not controlled. S-4 is a computed honest FAIL with no live cancellation mechanism; the measured $\Lambda$ and the gate's induced-$\Lambda$ FAIL are never laundered into each other.
• The sign of $c_{\rm loop}$ is not known. Reading it off the bare count $\mathrm{Str}[1]=-55$ is a forbidden count-only shortcut; the threshold-resolved supertrace can have either sign.
• An un-run calculation is not an axiom. Axiomatizing the un-run FRG would be reverse-engineering from the measured value.
• $\mu_{\rm cell}$ may not self-anchor. Anchoring the volume-singlet eigenvalue on the EW hierarchy via $\kappa^3/\pi$ is circular (the DERIVED no-go T1).
• Within-truncation is not full; perturbative is not nonperturbative; single-slice is not the full signature. The S-3 win is perturbative-only; the shape-doublet $-1$ refutes positive-definiteness but is not yet the full $2\times2$ Hessian certificate.
• The frozen-branch hashes are audit anchors; they do not validate the physics.
• B3 un-built ≠ B3 a unicorn. The specific branch fixed-point construction is bounded and stays in the work plan; only the absolute "no framework could ever prove nonperturbative stability for any realistic space" version is the shared, dissolved ceiling.

10. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package; ordering is falsifier-first, shared-blocker-first, predicate-first, wall-first.

1. Resolve the shape-doublet Hessian signature (HIGHEST PRIORITY, runnable now, UV-independent). Build the full $2\times2$ shape-doublet Hessian in canonical kinetic-normalized coordinates plus the R6 graded-Casimir net sign, target-blind. A confirming negative signature is a valid close — it hardens the endpoint to a structural moduli-stability FINDING that UV completion would not rescue; a positive-definite signature lifts the shape half of S-2. Do not over-read the single slice; do not tune the Casimir $q$ to clear the $-1$.
2. Produce the five source-hashed spectrum CSVs (CC-1). Generate the bosonic towers + the twisted-Dirac $K_6$ tower + the retained ledger at $\vec u=(1,1,1)$ blind from the frozen bundle data ($c_1(L_{K_6})$ read off the geometry), then source-hash. Prerequisite artifact — no "refuting" outcome.
3. Compute $c_{\rm loop}$ to a convergent number (CC-2). Run the Wetterich/LPA/Litim flow UV → $M_*$ with normalization $1/(4(4\pi)^2)$. A non-convergent result is a valid finding (leaves the well at decision grade). Do not combine with $c_{\rm KK}$; do not promote FRG-4 operators.
4. Determine the sign of $c_{\rm loop}$, FRG-4-stable (CC-3). Compute the twisted-Dirac eigenvalue shifts, evaluate the Litim threshold supertrace level-by-level, verify stability under the FRG-4 scan without tuning. "Sign UNDETERMINED / tracks $-55$ / FRG-4-unstable" is a legitimate terminal, not a step toward closure.
5. S-2 volume-singlet positivity (Hole 5). Run the Strong-tier UQF-9 trajectory (needs B3) and verify positivity by two structurally-different routes; retire T1 with a stability-independent $\mu_{\rm cell}$ readout. A negative eigenvalue is a valid FINDING.
6. S-5 KK trajectory stability (Hole 6). Run the Min-tier UQF-9 pass and verify the KK tower stays non-tachyonic along the flow. A tachyonic crossing is a valid FINDING.
7. S-6 orbifold / boundary (Hole 7). Pull B2 back to the frozen branch; execute S-6A/B/C each by two independent routes. A boundary-anomaly obstruction is a valid FINDING.
8. S-7 runaway $V_{\rm eff}(\sigma)$ (Hole 8) — the most consequential falsifier. Compute $V_{\rm eff}$ under FRG control; show bounded-below with a stable minimum at the frozen radii. An unbounded-below result is a valid (and consequential) FINDING — it downgrades the gravity interface, not merely this gate.
9. Nonperturbative tunneling exclusion (Hole 9). A bounce / false-vacuum-decay computation showing sufficient lifetime; lifts S-3 beyond truncation.
10. Build UQF-9 / B3 (Hole 10). Exhibit the non-Gaussian fixed point and mount the four FRG object-certificates (theory-space, branch-pullback, projection/readout, scheme-robustness), target-blind, with negative controls that FAIL on ≥3 known-bad inputs. Highest cross-gate leverage — but do Hole 1 first.
11. Predicate-completeness P0 (Hole 11) and the composition theorem (Hole 12). Prove the seven rows exhaust quantum-survival within scope (resolve the Gap-15 measure candidate), and prove the row certificates plus cross-terms compose to the predicate. Co-equal with B3; a discovered eighth mode or a counterexample is a valid FINDING.
12. S-1 forcing theorem (Hole 13). Prove or refute THEOREM-TARGET-RESIDUAL-ISOMETRY-FORCES-HESSIAN-SIGN; a proof upgrades S-1 scope → DERIVED, the current lean keeps it at admissibility-scope. Tied to Hole 1.

Closing all twelve — and only with B3 built, P0 proved, and the composition theorem proved — would move UQF-10 toward whole-gate closure, and even then only DERIVED-GIVEN-E within truncation.

11. Completion tests for this page

Required presence (all met): gate roll-up RESOLVED +0 (frozen mid-audit reading: OPEN (AUDIT)) · the local survival-predicate conjunction · S-1 SCOPED + S-3 DERIVED-GIVEN-E-within-truncation labels · "geometry not derived/selected" · frozen hashes (AUDIT ONLY) · every exact object as its own row · the specificity diagnostic ($d^2R/d\varepsilon^2|_0=-1$, sign-invariant) · the count diagnostic $\mathrm{Str}[1]=-55$ · the four-term $V(\sigma)$ + banked coefficients · $c_{\rm loop}$ OPEN / sign UNDETERMINED · S-4/$\Lambda$ computed FAIL · B3 / UQF-9 BLOCKED · P0 + composition open · every open residual as its own row · the gate's anti-claims.

Required absence (all held): no claim that UQF-10 is physics-closed or proven stable · geometry derived or selected · the audit sold as a selector · existence sold as stability · $\Lambda$ sold as controlled · the $c_{\rm loop}$ sign read off $-55$ · the un-run FRG axiomatized · $\mu_{\rm cell}$ self-anchored · within-truncation = full · perturbative = nonperturbative · single-slice = full Hessian signature · hashes validate physics · B3 called built or called a unicorn · any reader-visible build-process vocabulary.

Tests passed: all required-presence items present; all required-absence items held. Tests failed: none. Open items: the twelve closure-plan work-packages (§10), led by the runnable, UV-independent shape-doublet Hessian signature. Assumptions made: none beyond the dossier — every status matches the dossier's frozen audit record (mid-audit roll-up OPEN/AUDIT — on the ratified board the gate stands RESOLVED +0; S-1 admissibility-scope; S-3 within-truncation; S-2 shape-half leaning REFUTED; S-4 disclosed FAIL; $c_{\rm loop}$ OPEN; B3 BLOCKED; P0/composition open); every number is traced to the dossier or corpus; nothing fabricated; nothing upgraded.

This gate follows the same eleven-part shape and universal table as the canonical SG-4 anchor ledger.

See also: the anchoring method · A0 — the master anchor · Layer 1 — the metric anchor (the frozen geometry the survival audit stands on) · Layer 4 — carrier-forcing & the given-E wall (why the geometry is load-bearing but not certifying) · SG-6 anchor ledger (the shape-doublet Hessian — the $-1$ slice retired there as a volume-contaminated artifact; corrected doublet mass² $\{+1,+1\}$) · UQF-3 anchor ledger · SG-4 — the canonical gate ledger · the full UQF-10 dossier.