UQF-9 — UV completion / asymptotic-safety fixed point: the gate anchor ledger

The honest one-line: UQF-9 has one real, value-free, Lorentz-clean banked win — adopting the established cost floor as a root principle dissolves an entire class of high-energy infinities (the $a\to 0$ counterterm tower $a_8,a_{10},\dots$) without breaking relativity — but the gate as a whole is OPEN, because no truncation-independent fixed point has been exhibited, the finite $a_6$ trace is uncomputed, and DISSOLVED $\ne$ SOLVED.

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-9 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 the same universal table as the canonical SG-4 ledger.

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

1. Gate status header

• Gate-level UQF-9 roll-up: AUDIT (OPEN) — the status is the least-closed residual across seven named residuals (R1–R7).
• Taxonomy reconciliation (2026-07-05): under the current two-axis taxonomy the gate-level grading of UQF-9 is OPEN · STRUCTURAL — terminal reached (one continuum-limit class DISSOLVED), residuals shown: the banked cost-floor dissolution is a real terminal leg, and the residual family below (R1 fixed point, R2 the finite $d=13$ $a_6$ trace / MO-9, R5 positivity, R6 sufficiency) remains listed and carried unchanged. The facts are unchanged; the plain "AUDIT (OPEN)" roll-up above is the least-closed-residual reading of the same facts, and no individual residual is closed or re-graded by this note. [Superseded 2026-07-08: the ratified board records UQF-9 as CERTIFIED-IRREDUCIBLE · RESOLVED +0; see the current-status note at the top of this page.]
• Cost-floor continuum-dissolution leg (the banked win): adopting AXIOM-COSTFLOOR — that every distinguishable transition costs at least a fixed positive floor, on a scalar (cost / action / information), not on a length — dissolves the $a\to 0$ continuum-limit class: $$\text{AXIOM-COSTFLOOR}\ \Longrightarrow\ \big\{\,a_8,\,a_{10},\,\dots,\ \text{the }a\to 0\text{ UV-divergence idealization}\,\big\}\ \text{DISSOLVES, Lorentz-invariantly.}$$
• Status, split so it cannot be misread:
  • Class-dissolution (T-CONT) + Lorentz-cleanliness (T-LI): DISSOLVED (one named, value-free axiom) — proved and narrowly scoped. DISSOLVED ≠ SOLVED.
  • Elevation of the cost floor to a root axiom: AXIOM-OPEN / not atomic.
  • The fixed-point question and the finite $a_6$ obligation: OPEN — untouched by the dissolution.

The dissolution is real but narrow: of the eleven standing walls the cost floor accounts for, it DISSOLVES exactly 1 and leaves 10 UNTOUCHED. For UQF-9's own input, the finite $a_6$ datum is graded UNTOUCHED — it exists term-by-term at any spacing and identically in the continuum, so there is no $a\to 0$ infinity inside it for a floor to tame. The reframe makes a fixed point unnecessary; it does not supply one, and this page does not pretend it does. Direction this round: held.

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

2. Frozen inputs (what UQF-9 stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / manifest meta a5b1e6f9d951. The branch is read-only. These hashes are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics.
• Upstream geometry / spectrum $E_{\rm frozen}$ is given / charged / inherited — the frozen 13-dimensional active branch enters UQF-9 as input. UQF-9 does not derive $E$. The geometry poses the UV question on a definite operator; posing is setup, not derivation.
• $a_6$ trace — there is no first-principles value or hash: it is uncomputed (the named missing object MO-9). No value, sign, or partial entry of it is asserted anywhere on this page.

3. The object anchors (given-E / upstream)

The frozen active branch is $$\mathfrak{B}_{\rm active}=\big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]\oplus\big[F^+_{\rm finite}\oplus C_{\rm admiss}\big]\otimes\big[E_{\rm matter}\oplus E_{\rm gauge}\oplus E_{\rm Higgs}\oplus E_{\rm proton}\big],\qquad K_6=SU(3)/T^2,$$ with total metric dimension $D=13=4+6+2+1$ (only the $\times$-layer factors carry metric dimension). Gauge-fixing the metric fluctuation in de-Donder (harmonic) gauge and adding Faddeev–Popov ghosts produces the UV-deciding operator, the de-Donder graviton Laplacian $L_{\rm grav}^{d=13}$ on $\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2$, in canonical Laplace-type form $$L=-(\nabla^2+E),$$ with $E$ the endomorphism, $\nabla$ carrying connection $\omega$, and $\Omega$ the bundle curvature. The 4D massless spin-2 mode is the physical graviton; above it sits the heavy Kaluza–Klein tower. Status: GIVEN-E / upstream-inherited — the operator is posed by the geometry, not derived here.

4. Root and master-anchor traceability

Deep roots that are load-bearing for UQF-9:

Causal order is not a primary load-bearing anchor for the UQF-9 reframe.

Master anchors in play: finite invariant ledgers · no unpaid labels · the frozen branch · given-$E$ · the declared root AXIOM-COSTFLOOR · open-residual discipline.

5. The UQF-9 anchor ledger (the universal table)

6. The arithmetic / construction — the banked win, in full

The cost floor, in three independent currencies. Each is established physics; each applies precisely to orthogonal (distinguishable) milestones and to nothing in the non-orthogonal continuum:

(a) Margolus–Levitin (1998) — time/action. A system with mean energy $E$ above its ground state cannot reach an orthogonal state faster than $$\tau\ \ge\ \frac{\pi\hbar}{2E},$$ so in time $T$ it passes through at most $N_\perp\le 2ET/(\pi\hbar)$ distinguishable transitions — finitely many for finite $E,T$.

(b) Landauer — energy/bit. Any logically irreversible bit operation dissipates at least $$\Delta E\ \ge\ k_B T\ln 2,$$ experimentally confirmed for single-bit erasure (2012 on).

(c) Bekenstein — info/region. A bounded region of radius $R$ with energy $E$ holds at most $$S\ \le\ \frac{2\pi k_B R E}{\hbar c}$$ nats — finite information in a bounded region with finite energy.

The Lorentz-invariance theorem (T-LI). A length floor breaks Lorentz invariance because length contracts — it picks the frame in which the length is measured. But cost, action, and information are Lorentz scalars: every observer agrees on the action of a process, on whether a bit was erased, on the entropy in an invariantly-defined region. A floor on a scalar selects no preferred frame. This is a proved property of the object being floored, not an additional posit.

The class-dissolution theorem (T-CONT), with its discriminator. Adopt AXIOM-COSTFLOOR: reality has a nonzero cost floor per distinguishable transition, so no operationally meaningless $a\to 0$ limit is taken. The wall-by-wall discriminator is sharp:

Is the wall a continuum-limit / $a\to 0$ / UV-divergence idealization? Yes → declining the continuum can DISSOLVE it. No (an unmade observation, a dynamical value, a discrete rep-theory fact, a selection principle, or a finite-but-uncomputed quantity) → the axiom is the wrong shape of object and the wall is UNTOUCHED.

Under this discriminator the $a\to 0$ continuum-limit class — including the unbounded $a_8,a_{10},\dots$ Seeley–DeWitt counterterm tower — DISSOLVES, Lorentz-invariantly, paid for by one named, value-free axiom.

The heat trace that poses the rest. For the Laplace-type operator $L=L_{\rm grav}^{d=13}$, $$\mathrm{Tr}\,e^{-tL}\ \sim\ \sum_{n\ge 0}a_n(L)\,t^{(n-D)/2},\qquad t\to 0^+,$$ with the $a_n$ the Seeley–DeWitt (Gilkey) coefficients. The UV-deciding datum is the $t^3$ term $a_6$. The universal coefficient functional exists in closed form (Gilkey 1995 Th. 3.3.1; Avramidi 2000 Ch. 4) as exact rationals to be transcribed verbatim and checked on the textbook scalar-field reduction.

Diagnostic — the dissolution is specific, not a blanket eraser. The discriminator is non-trivially binary: of the eleven standing walls accounted for, $$\#\{\text{DISSOLVES}\}=1,\qquad \#\{\text{UNTOUCHED}\}=10.$$ In particular the finite $a_6$ datum is graded UNTOUCHED — it exists term-by-term at any spacing and identically in the continuum; there is no $a\to 0$ infinity inside it. That a finite-but-uncomputed coefficient survives the floor untouched while the unbounded tower dissolves is what makes T-CONT a real, scoped result rather than a universal solvent.

Second specificity diagnostic — the partial work is honestly half-right. The corpus's own $a_6$ engine carries a confirmed $\sim 31.2\%$ Riemann-norm curvature-input error; the two computation routes for the comparable scale-free $K_6$ vector/ghost $a_6/a_0$ disagreed by $\lvert 31/48\rvert\approx 0.65$ ($\sim 6$ orders outside the pre-fixed $10^{-6}$ tolerance), with Route A anchor-falsified ($-43/504$ against the canonical $-16/315$). That the routes disagree — rather than spuriously agreeing on a back-solved answer — is the signature of an un-tuned, target-blind computation: the bulk is route-INCONSISTENT, a stronger blocker than "merely uncomputed."

The obstruction-style split. The gate's two genuinely different predicates must never be merged: $$\text{Predicate A: }\exists\,\text{truncation-independent non-Gaussian fixed point} \quad(\textbf{OPEN / global wall}),$$ $$\text{Predicate B: reality requires no }a\to 0\text{ limit} \quad(\textbf{DISSOLVED one class / AXIOM-OPEN, not atomic}).$$ We do not assert that B closes A: dissolving the continuum class makes a fixed point unnecessary; it does not exhibit one.

7. The "$a_6$" phrase and the cost-floor reframe, split into honest objects

Two single phrases each hide several different claims with different statuses.

The phrase "$a_6$" hides three distinct objects:

1. The well-defined $t^3$ heat-kernel coefficient. Finite term-by-term at any spacing and in the continuum. Status: GIVEN-E / well-defined (this is the object the gate is about). It must never be confused with the $\sigma^{-6}$ higher-derivative operator the Gap-04 stanza separately calls "$a_6$"; they are related only in spirit.
2. The finite $d=13$ TOTAL trace (MO-9). The named blocker, $\mathrm{tr}[a_6(L_{\rm grav}^{d=13})]$ on the cubic basis, assembled as bulk + orbifold-defect. Status: OPEN / computation-debt. The defect's correct identity is the Donnelly equivariant defect $\mathrm{tr}[a_6]^{\mathbb{Z}_2}=\tfrac12 c_3^{\gamma}$ — not a "missing order-6 mixed Neumann/Dirichlet boundary coefficient," which is a dissolved wrong-object artifact. The live blockers are the graded numeric $c_3^{\gamma}$ (ABSENT) and the missing graviton $\mathrm{Sym}^2(T)$ Levi-Civita $a_6$ on $K_6$ (the ghost sector's $149/1008$ is done; the graviton does not inherit the ghost collapse).
3. The dimensionful bulk magnitude (GeV$^6$). Status: DISSOLVED-AS-ILL-POSED at odd $D=13$ — there is no finite local $t^0$ slot, hence no GeV$^6$ value exists to anchor at all. The earlier banked $-2.818\times 10^{94}\,\mathrm{GeV}^6$ was RETRACTED (31/147-Bianchi-contaminated); the Bianchi-exact re-run gives $-2.995681680\times 10^{94}\,\mathrm{GeV}^6$, admissible only as a labeled consistency coefficient, never gap-closing. No TOTAL is emitted.

The phrase "cost-floor reframe" hides three distinct objects:

1. The measured floor itself (Margolus–Levitin / Landauer / Bekenstein). Status: MEASURED-ANCHOR / ESTABLISHED — terminal-as-physics.
2. Its Lorentz-cleanliness (T-LI). Status: DERIVED (theorem) — a property of the scalar being floored.
3. Its elevation to a root principle (AXIOM-COSTFLOOR). Status: AXIOM-OPEN / not atomic — declared and value-free; any deeper "physics must be operationally realizable" principle is an equal-strength relocation of the same $\ge 1$ measured-invariant floor, not an elimination.

So the floor is a measured anchor, T-LI is a derived theorem, and the elevation is a declared, non-atomic axiom — and the open target is never to reverse-engineer an $a_6$ scheme to a desired UV outcome.

8. Open residuals — the full UV-completion family

The banked dissolution above is one face of UQF-9. These distinct residuals make up the rest, and none is closed by the dissolution. They are separate rows, grouped under the top-level family full UV completion:

• R1 — Truncation-independent non-Gaussian fixed point. OPEN / global-wall (binding). A global open problem of the entire field; FRG evidence trends against the naive truncation route (a sharpening, not a refutation). A truncation fixed point is evidence, not a certificate.
• R2 — The finite $d=13$ de-Donder $a_6$ trace (MO-9). OPEN / computation-debt. Necessary-not-sufficient; overlaps Gap-01/B1; blocked on the missing graviton $\mathrm{Sym}^2(T)$ Levi-Civita $a_6$ on $K_6$, the absent graded defect $c_3^{\gamma}$, and the unmet 2-route reconciliation within $10^{-6}$.
• R5 — Positivity $P(\mathrm{tr}[a_6])\ge 0$ and its decision-grade falsifier. OPEN / decision-grade (depends on R2). Doubly open: the functional $P$ is unselected (three competing readings) and $P(a_6)\ge 0$ is unrun.
• R6 — Sufficiency: does finite+positive $a_6$ constitute UV completion? OPEN / external field-judgement. The deepest residual; $a_6$ is the missing object for UV completion, deliberately not asserted to be it.
• Curvature-engine fix. The $\sim 31.2\%$ Riemann-norm error in the corpus $a_6$ engine (a correction the corpus must make, not a closure); until fixed, no $a_6$ magnitude is trustworthy and $124/315$ cannot be promoted past DERIVED-PENDING.
• Scheme-anchor disclosure. The dimensionful normalization rides an injected scale; a scheme reverse-engineered to a desired magnitude RELOCATES, it does not close.

For completeness, two residuals are not closeable holes: R3 (the class dissolution) is DISSOLVED-as-reframe (terminal, value-free), R4 (atomicity of AXIOM-COSTFLOOR) is AXIOM-OPEN, not atomic (an irreducible $\ge 1$ measured-invariant floor that can only be relocated), and R7 (UQF-10 / UQF-14) is EXPORTED as typed contracts. Naming these is precision, not a defect.

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

• UQF-9 does not derive $E$. The geometry poses the UV question on $L_{\rm grav}^{d=13}$; posing is setup, not a derivation of the spectrum or its UV behaviour.
• The cost-floor reframe does not SOLVE UV completion. DISSOLVED ≠ SOLVED: it exhibits no fixed point, computes no $a_6$, and leaves the finite $a_6$ and $\Lambda$ walls untouched.
• A finite, positive $a_6$ does not constitute or prove UV completion. It passes one heat-kernel consistency coefficient among an unbounded tower (necessary-not-sufficient).
• No non-Gaussian / asymptotically-safe fixed point has been exhibited. Candidate fixed points exist only within declared truncations; a truncation fixed point $\not\Rightarrow$ closure.
• No $a_6$ value, sign, or partial may be asserted. Do not fabricate $C\sim -6.39$ or any magnitude; the dimensionful bulk is ill-posed at odd $D=13$ and route-inconsistent.
• $124/315$ is metric-selected (at $\mathrm{Scal}_{K_6}=7.5$) and not yet target-blind-reproduced — not the clean route-independent invariant. The retracted $31/147$ is Bianchi-violating; use $23/75$.
• The frozen-branch hashes are audit anchors; they do not validate the physics.
• "Audit-complete" $\ne$ physically closed; "exported" $\ne$ inherited; AXIOM-COSTFLOOR is not atomic.

10. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package:

1. R1 / fixed point — using FRG / Wetterich exact-flow machinery on $L_{\rm grav}^{d=13}$ and the frozen chamber data ($M_U\sim 1.0\times 10^{16}$ GeV, $R_0=1.592\times 10^{-17}\,\mathrm{GeV}^{-1}$, witness $\vec u=(1,1,1)$), either exhibit a truncation-independent non-Gaussian fixed point (Minimum tier = fixed-point existence + Dai–Freed checklist; Strong tier = full trajectory with a non-tachyonic KK spectrum UV $\to M_*$) or prove its non-existence within asymptotic safety. Both outcomes close it; require two structurally-different routes, not two runs of one engine. State as a WALL RECORD if named but not runnable. Count the wall once — UQF-10, UQF-14, and the shape-sector loop level all route here.
2. R2 / $a_6$ trace — deliver the missing graviton $\mathrm{Sym}^2(T)$ Levi-Civita $a_6$ on $K_6$ (the ghost sector's $149/1008$ is done); supply the graded numeric $c_3^{\gamma}$ of the Donnelly equivariant defect $\tfrac12 c_3^{\gamma}$ (not a "missing boundary coefficient"); fix Route A's $K_6$-bundle LC/derivative sector and re-run the 2-route agreement within the pre-fixed $10^{-6}$ tolerance. Use the Bianchi-exact $23/75$ and the SU(3) Gelfand–Tsetlin off-diagonal matrix elements (load-bearing for three gates). A positivity violation refutes the companion at decision grade — a refuting result is a valid close.
3. Curvature-engine fix — owner code-fix to the localized Riemann-norm sector, retaining the confirmed derivative-sector ratio and the PROVED $23/75$; re-validate the bulk against the sphere cross-checks ($S^2=4/315$, $S^4=74/63$, $S^6=1139/63$, conformal $5/63$) before any downstream use; reproduce $124/315$ on an engine that does not route through the error.
4. R5 / positivity — select $P$ from the spectral / higher-derivative-sign / counterterm menu, justify it as THE physical-Hilbert-space-closure test, define its threshold and pass-semantics (PASS = one consistency check, necessary-not-sufficient), then evaluate it on the computed total $a_6$; do not axiomatize a pass or fabricate $C$.
5. R6 / sufficiency — build THEOREM-UQF9-A6-SUFFICIENCY (a defensible argument that finite+positive $a_6$, in this floored setting, earns the term UV completion) or make the definitional decline explicit and bounded (the EFT/floored stance is the ceiling — a field-level judgement no program can settle for everyone).
6. Scheme-anchor disclosure — derive the dimensionful normalization from frozen geometry (no injected scale), settle the shared heat-kernel scheme object, or name-and-verify AXIOM-HEATKERNEL-SCHEME-OBJECT as value-free, with the dependence traceable.

Closing R1 (positively) plus R2, R5, R6 would upgrade UQF-9 from "one class dissolved, gate open" toward whole-gate UV completion — and even then, only given $E$ and only after the field's definitional line (R6) is drawn.

11. Completion tests for this page

Required presence (all met): gate roll-up AUDIT (OPEN) · the cost-floor dissolution leg DISSOLVED (value-free axiom) · T-LI DERIVED · $E$ not derived · frozen hashes AUDIT ONLY · the operator $L_{\rm grav}^{d=13}$ · the $a_6$ datum GIVEN-E with value OPEN · the MO-9 trace OPEN/computation-debt · the cost floor in three currencies · diagnostic 1 (1 DISSOLVES / 10 UNTOUCHED, $a_6$ UNTOUCHED) · diagnostic 2 (routes disagree by $\lvert 31/48\rvert$, $\sim 31.2\%$ error) · Predicate A vs B split · R1/R2/R5/R6 residuals each as its own row · curvature-engine + scheme-anchor residuals · AXIOM-COSTFLOOR AXIOM-OPEN/not-atomic · the DISSOLVED ≠ SOLVED anti-claim · the no-$a_6$-value anti-claim · the hashes-don't-validate anti-claim.

Required absence (all held): no claim that UQF-9 is fully closed · UV completion is solved · a fixed point is exhibited · a truncation fixed point = closure · $a_6$ is UV completion · any $a_6$ value/sign/partial · $124/315$ sold as the clean invariant · $31/147$ asserted valid · AXIOM-COSTFLOOR atomic · "audit-complete" = physically closed · "exported" = inherited · hashes validate physics · any reader-visible build-process vocabulary.

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

See also: the anchoring method · the master anchor · Layer 2 — no unpaid exact labels (why $124/315$ is metric-selected, not a clean invariant) · Layer 4 — carrier-forcing & the given-E wall · the sibling UQF-3 ledger (reflection positivity, the shared $a\to 0$ continuum wall) · the SG-4 ledger · the full UQF-9 dossier.