UQF-9 — UV completion / asymptotic-safety fixed point: full dossier

Gate: UQF-9 (UV completion / is there a genuine high-energy completion — a non-Gaussian fixed point?). Frozen branch: dcc66f1b2685 / manifest meta a5b1e6f9d951, READ-ONLY. Honest status (matches the live popup, unchanged): REFRAME / AXIOM-OPEN — the a→0 UV-divergence CLASS is dissolved Lorentz-cleanly by one established-physics floor; the finite obligation (a₆) and the fixed-point wall stay open. Dissolved ≠ solved. Direction: held. STATUS-UPGRADES:0. Nothing in this dossier upgrades a grade. Every number is traced to a corpus file or a standard reference. Where a quantity is uncomputed, it is marked OPEN — no value, sign, or partial of a₆ is asserted.

1. Executive summary + honest status

Headline. One established-physics fact — that every distinguishable step of any realizable process costs a minimum, and that this floor sits on a scalar (cost / action / information) rather than on a length — erases a whole class of high-energy infinities without breaking relativity. It does not finish quantum gravity, and we say so flat-out.

What this dossier establishes. The frozen 13-dimensional geometry of this framework poses the UV question on a single, definite operator — the de-Donder gauge-fixed graviton Laplacian at \(d=13\) plus its Faddeev–Popov ghosts — whose short-distance behaviour is carried by one named heat-kernel datum, the Seeley–DeWitt coefficient \(a_6\). On top of that setup it banks exactly one genuine, non-promoting result: elevating the cost floor (Margolus–Levitin / Landauer / Bekenstein) to a root principle dissolves an entire class of continuum artifacts — the unbounded \(a_8, a_{10}, \dots\) Seeley–DeWitt counterterm tower and the associated \(a\to 0\) UV-divergence idealization — at the cost of one named, value-free axiom, and it does so Lorentz-invariantly because the floored quantity is a scalar. That class-dissolution (we call it T-CONT) is proved and narrowly scoped. The Lorentz-cleanliness (T-LI) is a theorem of the construction, not a posit.

What this dossier does NOT establish. It does not exhibit a non-Gaussian / asymptotically-safe fixed point: no truncation-independent fixed point has been mounted, and current functional-renormalization-group (FRG) evidence trends against the naive-truncation route. It does not compute the finite \(a_6\) trace (that obligation is only partially discharged: the dimensionful bulk magnitude is route-inconsistent and, per GAP01.md, ill-posed at odd \(D=13\) — admissible at most as a labeled consistency coefficient; the orbifold-defect piece — correctly the Donnelly graded \(c_3^{\gamma}\), not a missing boundary coefficient — is blocked; and the positivity test is unevaluated). It does not settle whether "finite answers everywhere" constitutes UV completion — that is a field-level definitional judgement we deliberately decline. The cost floor leaves the cosmological-constant radiative-stability wall and the finite \(\Lambda\) value entirely untouched (the wall-impact ledger grades exactly 1 of 11 walls DISSOLVES and 10 UNTOUCHED).

The honest grade, stated positively. This is a serious, over-determined, falsifiable reframe — anchored on at least one measured invariant (the cost floor) — that removes the need for a continuum limit in one well-defined class of obstructions. It is not a validated UV completion. The reframe makes a fixed point unnecessary; it does not supply one, and we will not pretend it does. The confident testable bet is sharp: compute the finite \(d=13\) graviton+ghost \(a_6\) coefficient and check its positivity functional. A positivity violation would refute the companion Gap-01 at decision grade. The deepest residual — whether finite-answers-everywhere counts as UV completion — is a definitional line that no theory in any program can settle for everyone; that limit binds the entire field, not just us.

This grade is the least-closed residual across seven named residuals (R1–R7, §3, §6). With several non-terminal residuals open, the gate is AUDIT (OPEN), exactly as the popup states.

2. The community gap

2.1 The precise open problem

A predictive quantum theory of gravity has to remain sensible at arbitrarily high energy. There are only two defensible exits:

1. Exhibit a genuine high-energy completion — a truncation-independent, non-Gaussian (interacting) ultraviolet fixed point of the renormalization-group flow at which the dimensionless couplings approach finite values, so that the theory is asymptotically safe and predictive up to infinite energy with finitely many relevant directions; or
2. Replace "all the way to infinite energy" with something physically defensible — e.g., a principled reason the continuum limit need not be taken at all.

UQF-9 asks whether the frozen the framework geometry survives the ultraviolet by one of these two routes — and, honestly, whether the question is even posed correctly for a theory with a fundamental cost floor.

2.2 History and state of the art

The asymptotic-safety scenario for gravity originates with Weinberg's 1979 proposal that gravity might possess a non-trivial UV fixed point of the Wilsonian flow, with a finite-dimensional critical surface. The modern program is built on the functional / exact renormalization group (the Wetterich equation), and the bulk of the positive evidence comes from truncated flows: one projects the effective average action onto a finite operator basis (Einstein–Hilbert, then \(R^2\), then higher-curvature truncations) and finds an interacting fixed point with an apparently small number of relevant directions. The signature difficulty is that these fixed points are exhibited only at a declared truncation; truncation-independence — that the fixed point and its critical exponents survive as the operator basis is enlarged without bound — is the open object, and it is precisely what no construction in the field has delivered.

State of the art, stated without spin:

• Best positive bound: interacting fixed points are robust across many truncations with a stable count of relevant directions, which is genuine, suggestive evidence — but it is within-truncation evidence.
• Best negative signal: as truncations grow, control of the scheme- and gauge-dependence becomes delicate; in the regime relevant here, current FRG evidence trends against the naive truncation route (recorded in this gate's residual ledger as a sharpening of R1, not a refutation).
• The universal status: exhibiting a truncation-independent non-Gaussian fixed point for quantum gravity is a global open problem of the entire field — unsolved everywhere, not just on this branch. UQF-9 inherits that wall; it does not create it.

2.3 Prior attempts and why each falls short

• Pure asymptotic safety (truncated FRG). Falls short of closure because truncation-independence is unproved; a fixed point at a chosen truncation is evidence, not a certificate.
• A literal smallest length (lattice / minimal-length) regulator. This is the classical way to stop infinite descent, but it picks out a preferred frame — the frame in which the length is measured — and therefore breaks Lorentz invariance. It pays the standard, expensive relativity-breaking bill, which is exactly why "spacetime is discrete at the smallest scale" has never been made to sit cleanly with relativity. (the framework's reframe sidesteps this; see §3.3.)
• Perturbative counterterm bookkeeping. Gravity is perturbatively non-renormalizable; the Seeley–DeWitt counterterm tower \(a_8, a_{10}, \dots\) is unbounded, each order introducing new independent couplings. Treating that tower as a literal physical requirement is exactly the \(a\to 0\) idealization this framework declines as out of scope (the class T-CONT dissolves) — but declining it is a scoping move, not a fixed-point theorem.

The honest position UQF-9 takes is therefore neither "we solved it" nor "we are stuck": it is "we dissolved one well-defined class of the difficulty Lorentz-cleanly, localized the finite remainder to one named refutable object, and we keep the global wall openly on the books."

3. The construction — rigorous math

3.0 The pipeline at a glance

 frozen 13D branch  --de-Donder gauge-fix + FP ghosts-->  L_grav^{d=13}   (the UV-deciding operator)
        --short-time heat-kernel expansion-->  Tr e^{-tL} = Σ_n a_n t^{(n-13)/2}
        --the t^3 term-->  a_6   (the finite short-distance datum;  necessary-not-sufficient)
        --positivity test P(a_6) ≥ 0-->  physical-Hilbert-space closure?   (criterion unselected)

 cost-floor lens (ℏ>0 on a SCALAR)  --declared root AXIOM-COSTFLOOR-->  no a→0 limit is taken
        ⟹  the unbounded counterterm tower a_8,a_10,…  and the a→0 UV-divergence class  DISSOLVE
        ⟹  but a_6 (a finite term-by-term object) is UNTOUCHED;  no fixed point is exhibited

(Source: …/PER_GATE_DOSSIERS/UQF9_COMPLETION_HANDOFF/01_DOSSIER.md §1.)

3.1 The UV-deciding operator (geometry → operator)

The frozen active branch is \[ \mathfrak{B}_{\rm active}=\big[\mathcal{M}_4 \times K_6 \times S^2 \times S_Y^1/\mathbb{Z}_2\big]_\times \;\oplus\; \big[F^+_{\rm finite}\oplus C_{\rm admiss}\big]_\oplus \;\otimes\; \big[E_{\rm matter}\oplus E_{\rm gauge}\oplus E_{\rm Higgs}\oplus E_{\rm proton}\big]_\otimes, \] with \(K_6 = SU(3)/T^2\) the flag manifold and total dimension \(D = 13 = 4+6+2+1\) (only the \(\times\)-layer factors carry metric dimension). (Source: 03_FROZEN_GEOMETRY_CONTEXT.md §1.)

Gauge-fixing the metric fluctuation in de-Donder (harmonic) gauge and adding Faddeev–Popov ghosts produces the de-Donder graviton Laplacian \(L_{\rm grav}^{d=13}\) on \(\mathcal{M}_4\times K_6\times S^2\times S_Y^1/\mathbb{Z}_2\), in canonical Laplace-type form \[ L = -(\nabla^2 + E), \] where \(E\) is the endomorphism (bundle "potential"), \(\nabla\) carries a connection \(\omega\), and \(\Omega\) is the bundle curvature. The 4D massless spin-2 mode of this operator is the physical graviton; above it sits the heavy Kaluza–Klein tower. (Sources: gap_01_uv_quantum_gravity/subsystems/S1.1_dedonder_graviton_operator.md, S1.3_laplace_type_data.md; GATE_REGRADE 5A linearized-certificate stanza.)

The named missing objects on the operator side are catalogued precisely in Gap-01: - MO-1 — the de-Donder graviton operator at \(d=13\) in Laplace form (the endomorphism \(E_{MN}{}^{PQ}\)); - MO-2 — the ghost-sector operators (FP vector ghost + any third ghost) with the subtraction sign and multiplicity; - MO-3 — the canonical \((E,\omega,\Omega)\) data plus the background Riemann tensor of the frozen geometry; - MO-4 — the holonomy / representation decomposition of the graviton+ghost fibers under \(K_6\times S^2\times S^1_Y\).

(Source: gap_01_uv_quantum_gravity/01_named_missing_objects.md.) These are all "certificate (symbolic)" grade — definite computations, not undecided existence problems. The ghost/BRST subtraction is well-posed (its sign and multiplicity are forced by BRST nilpotency), but — important honesty flag — it has not yet been verified against an actual computed \(a_6\) (residual R5 of the graviton sector). (Source: GATE_REGRADE 5A/5B stanza.)

3.2 The a₆ heat-kernel datum (operator → short-distance number)

For a Laplace-type operator \(L\), the heat trace has the short-time asymptotic expansion \[ \mathrm{Tr}\,e^{-tL} \;\sim\; \sum_{n\ge 0} a_n(L)\, t^{(n-D)/2}, \qquad t\to 0^+, \] where the \(a_n\) are the Seeley–DeWitt (Gilkey) coefficients — integrals of local curvature invariants built from \((E,\Omega,\text{Riem})\). The UV-deciding short-distance datum here is the \(t^3\) coefficient, \(a_6\) (the \(n=6\) term). It is the finite \(t^3\) term of a short-time expansion — it exists term-by-term at any spacing and identically in the continuum; there is no \(a\to 0\) infinity inside \(a_6\) itself. (Sources: 00_decomposition.md §0 disambiguation; COST_FLOOR_WALL_IMPACT_LEDGER.md §1 Gap-01 row.)

⚠ Disambiguation (carried verbatim from the corpus). Gap-01's "\(a_6\)" is the Seeley–DeWitt / heat-kernel coefficient (the \(t^3\) term). It is not the same object as the \(\sigma^{-6}\) higher-derivative operator the Gap-04 stanza also calls "\(a_6\)" ("\(a_6\) beats \(a_4\)" in the internal-volume effective potential). They are related only in spirit and must never be cross-cited as the same object. (Source: 00_decomposition.md §0.)

The universal machinery. The coefficient functional that multiplies each curvature invariant exists in closed form in the literature: Gilkey 1995, Theorem 3.3.1 (general second-order Laplace-type operator) and Avramidi 2000, Chapter 4 (covariant non-recursive technique). The coefficients are exact rationals (powers of small primes / factorials) that must be transcribed verbatim and checked on a known case (e.g., reduction to the textbook scalar-field \(a_6\) when \(E,\Omega\to\) scalar values). (Source: S2.2_gilkey_avramidi_functional.md.)

The basis. At \(d=13\) on the cubic-curvature (mass-dimension-6) sector, the invariants form the ~46-term Gilkey basis — \(R^3\)-type, \(R\,\Box R\), \((\nabla R)^2\)-type, \(E^3\), \(\Omega^3\), and their contractions. (Sources: 00_decomposition.md S2.1; S2.1_cubic_curvature_basis.md.)

The trace (the actual missing number, MO-9). The named blocker is \[ \boxed{\;\mathrm{tr}\!\left[a_6\!\left(L_{\rm grav}^{d=13,\ \text{de-Donder}}\right)\right]\ \text{on the cubic-curvature basis, holonomy-projected over }K_6\times S^2\times S^1_Y\;} \] It is assembled as: fiber-trace the endomorphism factors (\(\mathrm{tr}\,E^3,\ \mathrm{tr}\,E\Omega^2,\ \mathrm{tr}\,\Omega^3,\dots\)) over the graviton+ghost bundles (MO-7), evaluate the base-curvature invariants on \(K_6\times S^2\times S^1_Y\) (MO-8), then assemble and project onto the cubic basis (MO-9). The corpus grades this "perturbative but heavy" — a definite, finite (if enormous) symbolic computation, hence [S] / set-up-able, NOT [X] / Clay. (Sources: 00_decomposition.md §0 fact 2, §2 tree; 01_named_missing_objects.md.)

Hard rule (carried): No value, sign, or partial entry of MO-9 may ever be asserted by this program. Fabricating it would be the worst possible outcome. (Source: 01_named_missing_objects.md.)

What partial \(a_6\) work exists — and exactly how it is caveated. Two clean, dimensionless and scale-free facts have been extracted, and they must be reported at the right honesty tier:

• The dimensionless color ratio 124/315 was extracted and dual-validated against independent sphere cross-checks, but it must carry two caveats, not be sold as "the derived ratio." (i) Its current label is DERIVED-PENDING-INDEPENDENT-TARGET-BLIND-REPRODUCTION, because the same engine that carried the ~31% curvature error (below) also produced it; until reproduced on an independent target-blind route it is not counted as an independently-derived win. (ii) It is metric-SELECTED, not the clean target-blind invariant: 124/315 holds only at the special value Scal\(_{K_6}=7.5\); the corpus's own declared curvature gives \(\approx 0.0252\) instead. So 124/315 is not a route-independent scale-free win — that caveat is load-bearing and must not be dropped. (Sources: Gap-01 brief / GAP01.md handoff "What we did"; GATE_REGRADE lines 1022–1023, 1036–1037.)
• The geometric ratio \(|\mathrm{Riem}|^2/\mathrm{Scal}^2 = 23/75\) is established by two independent exact routes that agree (an independent SU(3)/T² structure-constant computation and the spectrum-demanded value from the exact heat-trace \(a_4\); Bianchi-exact, residual \(\sim 3\times 10^{-16}\)); the engine's earlier 31/147 was Bianchi-violating and is RETRACTED. (Honesty note: the corpus's own audit flags "PROVED by three independent methods (Theorem L)" as an overclaim — the verifiable artifact 11_a6_BLIND.md shows two routes, and no "Theorem L" proof doc exists outside a summary label; so we say two routes.) (Sources: 02_RESIDUAL_LEDGER.md R2/B1 overlap; GATE_REGRADE lines 1094, 1096.)

And the dimensionful magnitude is explicitly not a UV result:

• A bulk value \(\mathrm{tr}[a_6] = -2.817995812\times 10^{94}\ \mathrm{GeV}^6\) was computed and independently reproduced, admissible only as a labeled consistency coefficient (necessary-not-sufficient). It was RETRACTED as 31/147-Bianchi-contaminated; the Bianchi-exact-corrected re-run gives \(-2.995681680\times 10^{94}\ \mathrm{GeV}^6\) (\(+6.305\%\), sign preserved). But even the corrected value is NEVER gap-closing, and the deeper finding is stronger than "scheme-anchored": per the authoritative GAP01.md handoff the dimensionful magnitude is DISSOLVED-AS-ILL-POSED at odd \(D=13\) — there is no finite local \(t^0\) slot, hence no GeV\(^6\) \(a_6\) value exists to anchor at all. So "rides an injected scale" understates it: it is not a number pending a scheme choice, it is an object the corpus says does not exist; no TOTAL is emitted, and none should be. (Sources: GATE_REGRADE line 602; GAP01.md handoff "What we did" + residual R1/R3; 02_RESIDUAL_LEDGER.md R2/B1 overlap; GATE_REGRADE lines 379, 1195.)

Anti-cascade caution (carried verbatim). The threshold-vector / \(c_{\rm loop}\) / \(a_6\) magnitude are SCHEME-ANCHORED: blind runs recover signs but not magnitudes; the corpus \(a_6\) engine carried a real ~31% (more precisely ~31.2%) Riemann-norm curvature error found in audit (a correction the corpus must make, not a closure). The clean scale-free content (124/315) is genuinely derived; the dimensionful \(a_6\) magnitude rides an injected scale → SCHEME-ANCHORED. No promotion of any blind value. Even the partial \(a_6\) work that exists is half-right / half-wrong and scheme-anchored, not a closure. (Source: 01_DOSSIER.md §4 anti-cascade box; GATE_REGRADE line 379.)

There is a stronger flag still: the two computation routes for the comparable scale-free \(K_6\) vector/ghost \(a_6/a_0\) disagreed at the value/sign level — by \(|31/48|\approx 0.65\) (about six orders outside tolerance) — with Route A anchor-falsified (it returned \(-43/504\) against the canonical \(-16/315\)). So "the bulk number is route-INCONSISTENT," which is a stronger blocker than "merely uncomputed" or "off by ~31%." (Source: GATE_REGRADE line 379; SPECIALIST_HOLE_QUEUE Gap-01 hole 1.) This is recorded here so no reader mistakes the partial work for a settled bulk value.

3.3 The cost-floor reframe (the one genuine non-promoting win)

The reframe is the owner's two-line physics statement:

(L1) Infinite subdivision is harmless. (L2) Infinite prerequisite chains with a nonzero lower cost are impossible. ⟹ Any completable (finite-resource) process has finitely many costly (distinguishable) steps.

The conclusion reframes "the atom." What is quantized is not space — it is cost / action. The smallest irreducible thing is a quantum of cost, not a smallest length. The L1-vs-L2 line is exactly the non-orthogonal / indistinguishable vs orthogonal / distinguishable line: gliding through a continuum of non-orthogonal states is free (no distinguishable milestone is reached); only transitions to orthogonal (distinguishable) states cost, and each such transition costs at least a fixed positive floor. (Source: ROOT_AXIOM_COST_FLOOR_REALIZABILITY.md §0.)

The floor is established physics, in three independent currencies (Source: ROOT_AXIOM_COST_FLOOR_REALIZABILITY.md §1):

• (a) Margolus–Levitin (1998) — time/action. A system with mean energy \(E\) above its ground state cannot reach an orthogonal (distinguishable) 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 \(E,T\). (Margolus & Levitin, Physica D 120 (1998) 188.)
• (b) Landauer — energy/bit. Any logically irreversible bit operation dissipates at least \(\Delta E \ge k_B T \ln 2\), experimentally confirmed (single-bit erasure, 2012 on). (Landauer, IBM J. Res. Dev. 5 (1961) 183; Bérut et al., Nature 483 (2012) 187.)
• (c) Bekenstein — info/region. A bounded region of radius \(R\) with energy \(E\) holds at most \(S \le 2\pi k_B R E/(\hbar c)\) nats — finite information in a bounded region with finite energy. (Bekenstein, Phys. Rev. D 23 (1981) 287.)

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. The cost-floor reframe therefore delivers the regulating effect of "a smallest something" (an irreducible quantum that stops infinite descent) without paying spatial-granularity's relativity-breaking bill. This is a proved property of the construction (the floored quantity is a scalar), not an additional posit. (Source: ROOT_AXIOM_COST_FLOOR_REALIZABILITY.md §2; COST_FLOOR_WALL_IMPACT_LEDGER.md §0.)

The class-dissolution theorem (T-CONT). Adopt AXIOM-COSTFLOOR: reality has a nonzero cost floor per distinguishable transition, so no operationally meaningless \(a\to 0\) limit is taken. The discriminator that decides what this does to any standing wall is sharp:

Is the wall a continuum-limit / \(a\to 0\) / UV-divergence idealization? Yes → declining the continuum can DISSOLVE it (the continuum form is out of scope by choice). 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 and the \(a\to 0\) UV-divergence idealization — DISSOLVES, Lorentz-invariantly, paid for by one named, value-free axiom. (Source: COST_FLOOR_WALL_IMPACT_LEDGER.md §0, §2A; ROOT_AXIOM_… §5.)

The narrow scope, stated without hedging. The wall-impact ledger is explicit and disciplined: of the eleven standing walls it accounts for, the cost floor DISSOLVES exactly 1 (Gap-02's mass-gap existence, a different gap) and leaves 10 UNTOUCHED. For UQF-9's own input, Gap-01 (\(a_6\)) is graded UNTOUCHED — "\(a_6\) is the finite \(t^3\) term of a short-time expansion; it exists term-by-term at any spacing and identically in the continuum; there is no \(a\to 0\) infinity for a floor to tame, only unperformed symbolic labor." So the cost floor's effect on UQF-9 is real but narrow: it dissolves the class the \(a_6\) obligation does not belong to. (Source: COST_FLOOR_WALL_IMPACT_LEDGER.md §1 tally, §2C Gap-01 row; 01_DOSSIER.md W7.)

3.4 The two readings (both true, neither a closure)

This is the central structural fact of UQF-9, and the dossier must carry both. (Source: 01_DOSSIER.md §2; 02_CURRENT_STATE.md §0.)

• Standalone gate (conservative): AUDIT (OPEN). As one of the eight UQF audit rows, UQF-9 stays at AUDIT. Established: a candidate fixed point exists at a declared truncation. Open: a full UV completion is a global open problem shared with all of quantum gravity; the missing input (a 6th heat-kernel coefficient) is necessary-not-sufficient and itself uncomputed. No fixed point has been exhibited.
• Granularity / cost-floor leg: AXIOM-CLOSED — and DISSOLVED, not solved. AXIOM-COSTFLOOR dissolves the \(a\to 0\) / counterterm-tower class at the cost of one value-free axiom. It does not exhibit a fixed point; it does not compute the finite \(a_6\); it leaves a genuine finite obligation open.

3.5 The predicate split (do NOT combine into one "UV solved" claim)

The converged closure path forbids merging two genuinely different predicates (Source: 07_CONVERGED_CLOSURE_PATH.md):

• Predicate A — fixed-point UV completion: \(\exists\) a truncation-independent non-Gaussian fixed point. Status: OPEN / global-wall (Clay-class; a truncation fixed point is evidence, NOT closure; FRG evidence trends against the naive truncation route).
• Predicate B — cost-floor continuum-dissolution: reality does not require an operationally meaningless \(a\to 0\) limit. Status: DISSOLVED one class (AXIOM-COSTFLOOR) / AXIOM-OPEN (not atomic). DISSOLVED ≠ SOLVED: it exhibits no fixed point, computes no \(a_6\), leaves R2 untouched, does not solve UV completion.

3.6 The positivity criterion (necessary-not-sufficient adjudication)

Even a finite \(a_6\) must pass a positivity test for physical-Hilbert-space closure: a functional \(P(\mathrm{tr}[a_6]) \ge 0\). The corpus has laid out three candidate readings and selected none (selecting is owner-physics) (Source: S4.1_positivity_functional.md):

• (a) spectral / heat-kernel positivity — the ghost-corrected physical heat kernel has no negative-norm contribution (i.e., \(a_6\) on the BRST-cohomology subspace respects positivity of the one-loop measure);
• (b) effective-action / higher-derivative sign — the \(a_6\)-induced \(\sigma^6\)-type term must not produce a wrong-sign (ghost/tachyon) higher-derivative kinetic term that breaks unitarity;
• (c) counterterm / renormalization positivity — the \(a_6\) divergence coefficient must be consistent with a positive-norm renormalized state space.

The falsifier is sharp: a computed \(a_6\) violating positive-definiteness for physical-Hilbert-space closure refutes Gap-01 at decision grade. Because the functional \(P\) is itself unselected and \(P(a_6)\ge 0\) is unrun, the decision-grade falsifier is doubly open. (Sources: 00_decomposition.md §0 falsifier; 01_named_missing_objects.md MO-10/11; GATE_REGRADE 5C R4 line 276.)

A specific number must NOT be fabricated here: the graviton-minus-ghost coefficient has been quoted in working notes as \(C \sim -6.39\), but it is unreproduced and must never be axiomatized as a pass. (Source: GATE_REGRADE line 999, line 298 "do NOT fabricate C ~ −6.39 or axiomatize a pass.")

3.7 Floor location (a verified geometry read, NOT a promotion)

Where the granularity floor sits is fixed by Kaluza–Klein reduction of the frozen 13D geometry as a dimensionless ratio \(M_*/M_{\rm Pl}\). The verified result: \(M_* \approx 6.01\times 10^{16}\) GeV (\(\approx 6\,M_U\), \(= \tfrac{1}{40.5}\,\bar M_{\rm Pl}\)) ⇒ the floor sits at compactification, not Planck; the verdict is convention-PROOF (the \(1/11\) power), the exact value convention-SOFT, and the absolute dimensionful scale stays the irreducible "just is" \(= M_{\rm Pl}\), conditional on the load-bearing identification floor \(= M_*\). This pins where, not a UV certificate. (Source: ROOT_AXIOM_COST_FLOOR_REALIZABILITY.md §6 floor-location box.)

4. The insights we used

These are the moves that produced the progress — now shareable at working-physicist depth, because sharing them is what lets a reader both check and extend the result.

Insight 1 — quantize the scalar, not the length. The deep move is recognizing that the regulating job of a smallest length (stop infinite descent) and its cost (a preferred frame) are separable. Place the floor on cost/action/information — a Lorentz scalar — and you keep the job while dropping the cost. This is why the dissolution is Lorentz-clean: T-LI is not an extra assumption, it is a property of the object being floored. The reframe automatically pays spatial-granularity's biggest bill.

Insight 2 — distinguishability is the right partition. L1 (free subdivision) vs L2 (costed prerequisite) is the non-orthogonal vs orthogonal distinction. The three established floors (Margolus–Levitin, Landauer, Bekenstein) all apply precisely to orthogonal milestones and precisely to nothing in the non-orthogonal continuum. That is the structural reason infinite subdivision is harmless while infinite prerequisite chains are impossible — and the reason the floor is established physics, not a new posit.

Insight 3 — "one deep root removes a whole class," not one wall at a time. The payoff of elevating an established floor to a root principle is that it acts on a class (every \(a\to 0\) continuum-limit / UV-divergence artifact) rather than a single obstruction. That is genuine leverage — and the discipline that keeps it honest is the wall-impact ledger's binary discriminator, which forces you to grade each wall DISSOLVES / UNTOUCHED and prevents over-dissolution (only 1 of 11 dissolves).

Insight 4 — reify the remainder into one named, refutable object. Rather than gesturing at "quantum gravity is hard," the program localizes the finite remainder to exactly MO-9 = \(\mathrm{tr}[a_6]\), with a named falsifier (positivity). A reader can attack precisely that object; a positivity violation would refute it. Sharpening the gap to a single refutable computation is itself a result (a sharpening win, not a promotion).

Insight 5 — separate "given-E setup" from "derivation." The geometry poses the UV question on a definite operator (genuine setup), but having the operator is not having its spectrum's UV behaviour. The linearized graviton certificate (5A) holds as structure, given-E — the observed gravity limit is the input, not a derived output. Keeping this line bright prevents the signature mis-close of reporting setup as closure.

Insight 6 — the κ³/π falsification test (no target-fitting). Every proposed axiom is admitted only if it would have been written without knowing the UV answer. AXIOM-COSTFLOOR passes (the floor is established physics, elevated, never reverse-engineered to a number). Any "\(a_6\) normalization that makes the UV question vanish" would fail (RELABEL_FAIL) — the \(a_6\) value must come from the target-blind trace, never from a scheme reverse-engineered to a desired outcome. "A fixed point exists at our truncation ⇒ gate closed" is rejected — truncation-dependence is the open object. (Source: 01_DOSSIER.md §5.)

Insight 7 — the Donnelly equivariant-defect route is the correct object for the orbifold piece. The \(\mathbb{Z}_2\) orbifold-defect contribution to the total \(a_6\) is a Donnelly equivariant defect, \(\mathrm{tr}[a_6]^{\mathbb{Z}_2} = \tfrac{1}{2}\,c_3^{\gamma}\) — a graded coefficient \(c_3^{\gamma}\) that an independent computation could in principle supply. The insight is precisely the object-identity correction: the earlier "missing order-6 mixed Neumann/Dirichlet boundary coefficient (tower stops at \(a_5\))" framing was a wrong-object artifact; the real blocker is the graded numeric \(c_3^{\gamma}\) (plus the missing graviton \(\mathrm{Sym}^2(T)\) LC \(a_6\)), not an absent boundary-tower term. (Sources: GAP01.md handoff "What we did"/"The edge"; 02_RESIDUAL_LEDGER.md R2/B1 overlap.)

5. Evidence & reproducibility

5.1 Established / banked (terminal-as-physics or value-free)

5.2 The frozen hashes it rides

Branch dcc66f1b2685 / manifest meta a5b1e6f9d951. Operator sector: the de-Donder graviton Laplacian \(L_{\rm grav}^{d=13}\) on \(\mathcal{M}_4\times K_6\times S^2\times S_Y^1/\mathbb{Z}_2\) + FP ghosts. \(a_6\) trace: no first-principles value/hash — uncomputed (MO-9). (Source: 01_DOSSIER.md §0 header.)

5.3 Numerical checks, with honest pulls

• The dimensionful \(a_6\) magnitude is at best scheme-anchored, and per GAP01.md is more fundamentally DISSOLVED-AS-ILL-POSED at odd \(D=13\) (no finite local \(t^0\) slot ⇒ no GeV\(^6\) value exists): blind runs recover signs but not magnitudes. The banked \(-2.818\times10^{94}\ \mathrm{GeV}^6\) was retracted (31/147-Bianchi-contaminated); the Bianchi-exact-corrected value is \(-2.995681680\times10^{94}\ \mathrm{GeV}^6\) — admissible only as a labeled consistency coefficient, never gap-closing, not a UV result. (Sources: GAP01.md; 02_RESIDUAL_LEDGER.md; GATE_REGRADE lines 379, 602, 1195.)
• The two routes for the comparable scale-free \(K_6\) vector/ghost \(a_6/a_0\) disagreed by \(|31/48|\approx0.65\) (~6 orders outside tolerance), Route A anchor-falsified (\(-43/504\) vs canonical \(-16/315\)) — route-INCONSISTENT, stronger than "uncomputed." (Source: GATE_REGRADE line 379.)
• The graviton+ghost coefficient \(C\sim-6.39\) appears in working notes but is unreproduced; it must not be treated as a pass. (Source: GATE_REGRADE lines 298, 999.)

5.4 How a reader re-runs / re-derives

1. Re-derive the operator. From 03_FROZEN_GEOMETRY_CONTEXT.md build \(L_{\rm grav}^{d=13}\) in de-Donder gauge + FP ghosts; extract \((E,\omega,\Omega)\) per S1.3_laplace_type_data.md.
2. Transcribe the functional. Copy the universal \(a_6\) functional from Gilkey 1995 Th. 3.3.1 / Avramidi 2000 Ch. 4 (every rational coefficient cited), and check it reduces to the textbook scalar-field \(a_6\) (S2.2).
3. Build the basis. The ~46-term cubic-curvature Gilkey basis at \(d=13\) (S2.1).
4. Trace and project. Fiber-trace over the graviton+ghost bundles (MO-7), evaluate base curvatures on \(K_6\times S^2\times S^1_Y\) (MO-8), assemble and project (MO-9). Run target-blind. Do not back-solve to any desired UV outcome.
5. Verify the scale-free outputs independently. Reproduce 124/315 on an engine that does not route through the ~31% R2 engine; confirm \(23/75\) (not \(31/147\)).
6. Cross-check. The binding rule is two structurally-different routes, not two runs of one engine (a captured log ≠ an independent reproduction).

5.5 Certificate disposition (audit certification, NOT physical closure)

The audit certificate …/certificates/UQF9_AUDIT_COMPLETE/05_FINAL_ROLLUP.md records governance disposition AUDIT_OPEN_GLOBAL_WALL, promotion_violation = FALSE. Three adversarial lenses (binding-wall / predicate-split, dissolution-both-directions, over-claim) all agree with the endpoint; their findings are sharpenings, not reversals. The claim budget (04_CLAIM_BUDGET.md) tags every sentence and records: new axioms introduced 0; physics-closure claims 0; \(a_6\) values/signs/partials emitted 0. A do-not-propagate guardrail flags two stronger phrasings in specialist_reply.txt ("M* ATOMIC: YES", "bulk a6 = COMPLETE_CROSSCHECKED") that the conservative governing docs reject. (Sources: 05_FINAL_ROLLUP.md; 04_CLAIM_BUDGET.md.)

6. Open gaps + closure path (the specialist work plan)

Seven residuals. The gate status is the least-closed one. Each open hole below is a work-package: precise statement (a), why it is hard + traps (b), exactly what closes it including the refuting outcome (c), machinery & inputs (d), and leverage (e). Physics only; device engineering firewalled.

Residual map (starting line)

(Sources: 01_DOSSIER.md §3; 02_CURRENT_STATE.md; 02_RESIDUAL_LEDGER.md; 07_CONVERGED_CLOSURE_PATH.md.)

R3 is DISSOLVED-as-reframe (terminal, value-free) and R7 is exported, so the work-packages are R1, R2, R4, R5, R6, plus the curvature-engine fix and the normalization disclosure that R2 rides on.

Hole A (R1) — the truncation-independent non-Gaussian fixed point (single highest-value next move)

(a) Precise statement. Exhibit (or prove the non-existence of) a truncation-independent non-Gaussian UV fixed point for the \(d=13\) gravitational sector of the frozen branch — a fixed point of the FRG flow whose existence and critical exponents survive enlarging the operator basis without bound, with a finite-dimensional critical surface.

(b) Why it is hard / traps. This is a global open problem of quantum gravity, shared with the whole field; UQF-9 inherits it, does not create it. Current FRG evidence trends against the naive truncation route (a Lens-1 sharpening that strengthens the wall). The signature mis-close to refuse: reporting "a fixed point exists at our declared truncation" as if it closed the gate — truncation-dependence is the whole question. Do not present the cost-floor reframe as supplying a fixed point: it makes one unnecessary by declining the continuum, which is a scoping move, not a fixed-point theorem (the F3 demotion, DISSOLVED ≠ SOLVED).

(c) Exactly what closes it. Two valid terminal outcomes — a positive and a negative both close: - Positive: a genuine truncation-independent non-Gaussian fixed point for the \(d=13\) gravitational sector (this would turn the EFT stance into an actual UV completion — a genuine promotion, not promised). - Negative (also a close): a proof of its non-existence within the asymptotic-safety framework (this would definitively rule out the asymptotic-safety route). Bounded only in the sense that the object is named; whether a route exists is open. State as a WALL RECORD if it can be named but not run. (Source: SPECIALIST_HOLE_QUEUE UQF-10 hole 8, UQF-9 hole 3.)

(d) Machinery & inputs. FRG / Wetterich exact-flow machinery; the de-Donder \(L_{\rm grav}^{d=13}\) operator from §3.1; the frozen radii / chamber data (\(M_U\sim1.0\times10^{16}\) GeV, \(R_0=1.592\times10^{-17}\,\mathrm{GeV}^{-1}\), witness \(\vec u=(1,1,1)\)) from 03_FROZEN_GEOMETRY_CONTEXT.md §4; the Minimum-tier pass = fixed-point existence + Dai–Freed boundary checklist; the Strong-tier pass = full FRG trajectory with a non-tachyonic KK spectrum verified along the flow UV → \(M_*\). (Source: SPECIALIST_HOLE_QUEUE UQF-10 holes 5–8.) Two structurally-different routes are required, not two runs of one engine.

(e) Leverage. R1 is B3, the universal deepest wall — UQF-10 (compactification stability), UQF-14 (above-cutoff graviton unitarity, R1/R3), the shape-sector loop level, GRAVITON 5C/P0, Gap-01 sufficiency, and Gap-05 R7 all route here. Count the wall once. Closing R1 positively lifts the above-cutoff graviton unitarity / Froissart-bound / Landau-pole sub-claims of UQF-14, and lifts UQF-10's S-5/S-6/S-7 rows within truncation. (Sources: 07_CONVERGED_CLOSURE_PATH.md; SPECIALIST_HOLE_QUEUE UQF-14 hole 1.)

Hole B (R2) — the finite \(d=13\) \(a_6\) trace (computation-debt; orbifold defect blocked)

(a) Precise statement. Form the TOTAL \(d=13\) de-Donder graviton+ghost \(a_6\) trace over the ~46-term cubic-curvature Gilkey basis, holonomy-projected over \(K_6\times S^2\times S^1_Y\), as bulk + orbifold-defect. (Object-identity flag — carried from the authoritative GAP01.md handoff.) 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 heat-kernel coefficient" — that latter framing (boundary tower stops at \(a_5\), still used in the dated SPECIALIST_HOLE_QUEUE) is graded in GAP01.md as a dissolved wrong-object artifact. The live blocker is therefore the graded numeric \(c_3^{\gamma}\) (ABSENT) plus the MISSING graviton \(\mathrm{Sym}^2(T)\) Levi-Civita \(a_6\) on \(K_6\) (the graviton does not inherit the ghost-sector collapse; brute-force peel diverges).

(b) Why it is hard / traps. Three live obstructions, each a named trap: 1. The graviton \(\mathrm{Sym}^2(T)\) Levi-Civita \(a_6\) on \(K_6\) is MISSING (the graviton does not inherit the ghost-sector collapse — the ghost LC \(a_6/a_0=149/1008\) is done; the brute-force graviton peel diverges, Kostant quasi-polynomial multiplicity deficits), and the graded numeric defect \(c_3^{\gamma}\) is ABSENT — so the TOTAL cannot be assembled. Trap: do not chase "a missing order-6 mixed-boundary coefficient"; GAP01.md records that as a dissolved wrong-object artifact (see (a)). 2. The bulk is route-INCONSISTENT, not merely uncomputed. Two routes disagreed at value/sign by \(|31/48|\approx0.65\) on the comparable scale-free \(K_6\) vector/ghost \(a_6/a_0\) (~6 orders outside the pre-fixed \(10^{-6}\) tolerance), Route A anchor-falsified (\(-43/504\) vs canonical \(-16/315\)); on the graviton comparable object Route A returns \(-251/504\) vs Route B \(149/1008\). Do not treat the bulk as settled. (Trap: the do-not-propagate guardrail flags "bulk a6 = COMPLETE_CROSSCHECKED" as a rejected over-claim.) 3. No value/sign/partial of \(a_6\) may ever be asserted (F6 FORBIDDEN). Do not fabricate \(C\sim-6.39\) or any magnitude. Stronger still: per GAP01.md the dimensionful magnitude is DISSOLVED-AS-ILL-POSED at odd \(D=13\) (no finite local \(t^0\) slot, so no GeV\(^6\) \(a_6\) value exists to anchor) — "rides an injected scale" understates this; no TOTAL is to be emitted.

(c) Exactly what closes it. (i) Deliver the MISSING graviton \(\mathrm{Sym}^2(T)\) Levi-Civita \(a_6\) on \(K_6\) (the bounded-but-heavy graviton-sector piece; the ghost sector \(149/1008\) is already done); (ii) supply the graded numeric \(c_3^{\gamma}\) of the Donnelly equivariant defect \(\mathrm{tr}[a_6]^{\mathbb{Z}_2}=\tfrac12 c_3^{\gamma}\) — not a "missing mixed-boundary coefficient" (dissolved wrong-object); (iii) fix Route A's \(K_6\)-bundle LC/derivative sector against a spectral peel and re-run the 2-route agreement within the pre-fixed \(10^{-6}\) tolerance (the binding VALUE rule, currently UNMET). Note the odd-\(D\) ill-posedness flag in (b)3: the dimensionful TOTAL may have no finite value to emit; what closes the gate's R2 leg is the scale-free assembly plus 2-route reconciliation, target-blind. A positivity violation (see Hole D) refutes Gap-01 at decision grade — a refuting result is a valid close. (Sources: GAP01.md handoff "The edge"; SPECIALIST_HOLE_QUEUE UQF-9 hole 1; 02_RESIDUAL_LEDGER.md.)

(d) Machinery & inputs. Gilkey 1995 Th. 3.3.1 / Avramidi 2000 Ch. 4 functional; the cubic basis (S2.1); the fiber-trace + base-curvature evaluation (S3.1–S3.3); the Bianchi-exact input \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75\) (use this, not the retracted 31/147); the SU(3) Gelfand–Tsetlin off-diagonal matrix elements gating the 5-class Levi-Civita Lichnerowicz graviton hopping on \(K_6\) (the load-bearing reconciliation object). (Sources: S2.2, S3.3; SPECIALIST_HOLE_QUEUE Gap-01 hole 1.)

(e) Leverage. R2 overlaps Gap-01/B1's bulk \(a_6\); it inherits B1's value only when B1 is BUILT and PRESENT-AND-MOUNTED under all five invariants (current mount state: MISSING). The single GT off-diagonal fix is load-bearing for three gates (Gap-01 R1, SG-6 R5, SG-7 R2). Discharging R2 is necessary-not-sufficient for UQF-9 (closing R2 does not close R1 or R6). (Sources: 02_RESIDUAL_LEDGER.md R2/B1 overlap; SPECIALIST_HOLE_QUEUE Gap-01 hole 1.)

Hole C — the ~31% curvature-input error in the corpus \(a_6\) engine

(a) Precise statement. Two independent \(a_6\) routes disagree because the engine carries a confirmed ~31.2% Riemann-norm curvature-input deficit (the engine is half-right — the derivative-sector ratio is confirmed — and half-wrong on the curvature input). Even the bulk number is not yet trustworthy.

(b) Why it is hard / traps. The error is localized to the Riemann-norm sector and was masked because the derivative-sector ratio looked correct. The earlier banked \(-2.818\times10^{94}\ \mathrm{GeV}^6\) value was contaminated by the Bianchi-violating 31/147 input and had to be retracted — do not reuse it. Do not "reconcile" the routes by back-solving to a known \(a_6\) (κ³/π discipline).

(c) Exactly what closes it. Owner code-fix to the localized Riemann-norm sector of the engine, retaining the confirmed derivative-sector ratio and the now-PROVED \(23/75\) input; re-run and re-validate the bulk coefficient against the sphere cross-checks before any downstream use. Success: the two independent routes agree. (Sources: SPECIALIST_HOLE_QUEUE Gap-01 hole 2, UQF-9 hole 2; GATE_REGRADE line 275.)

(d) Machinery & inputs. The corpus \(a_6\) engine; the sphere cross-checks; the Bianchi-exact \(23/75\); the GT off-diagonal matrix elements (Hole B(d)).

(e) Leverage. Until this is fixed, no \(a_6\) magnitude is trustworthy and 124/315 cannot be promoted past DERIVED-PENDING (it rode the same engine, and it is metric-selected at Scal\(_{K_6}=7.5\), so it is not yet a clean target-blind invariant). Fixing the engine is the gate to trusting R2's bulk and re-validating 124/315 on an independent route.

Hole D (R5) — the positivity functional \(P(a_6)\ge 0\) and its falsifier

(a) Precise statement. Select the precise positivity functional \(P\) required for physical-Hilbert-space closure, and specify its decision-grade adjudication rule, then evaluate \(P\) on the computed (total) \(a_6\). Currently doubly open: \(P\) is unselected (three competing readings, §3.6) and \(P(a_6)\ge0\) is unrun.

(b) Why it is hard / traps. Selecting \(P\) is owner-physics, not a mechanical step. Do not axiomatize a pass; do not fabricate \(C\sim-6.39\). The pass semantics must be honest: PASS = one consistency-check satisfied (necessary-not-sufficient), not closure (MO-11). The falsifier rides on the uncomputed R2 trace, so it cannot be run before Hole B.

(c) Exactly what closes it. Select \(P\) from the (a)/(b)/(c) menu (spectral / higher-derivative-sign / counterterm positivity), justify it as THE physical-Hilbert-space-closure test, define its threshold and pass-semantics, then evaluate it on the computed \(a_6\). A violation REFUTES the gate (and Gap-01) at decision grade — a valid close. A pass lifts the graviton 5A/5B legs toward certificate-grade conditional on UQF-9; it does not close R1 or R6. (Sources: S4.1, S4.2; 01_named_missing_objects.md MO-10/11; GATE_REGRADE lines 276, 298.)

(d) Machinery & inputs. The three candidate functionals (S4.1); the BRST/ghost subtraction (well-posed, but verify it against the actual computed \(a_6\) — this is the separate graviton R5 obligation); the computed total \(a_6\) from Hole B.

(e) Leverage. Closes the decision-grade falsifier for Gap-01; gates the graviton-sector 5A/5B promotion (conditional on UQF-9); feeds UQF-14's unitarity sub-claims.

Hole E (R6) — the sufficiency / scope theorem (external-field-judgement)

(a) Precise statement. A named theorem, THEOREM-UQF9-A6-SUFFICIENCY (currently unbuilt): does a finite + positive \(a_6\) constitute UV completion, or is it one necessary coefficient in the unbounded tower \(a_8, a_{10}, \dots\) with the full UV problem untouched?

(b) Why it is hard / traps. This is the deepest residual and is external / field-judgement: even a flawless positive \(a_6\) passes one consistency coefficient among an unbounded tower (MO-12). The corpus deliberately says \(a_6\) is the missing object for UV completion, carefully not that \(a_6\) is UV completion (F2 FORBIDDEN). This is why R2 closing does not close R1. The strong form — "finite answers everywhere constitutes UV completion for everyone" — is a definitional unicorn no theory in any program can settle for the whole field; do not claim it as proven and do not list it as an open weakness — it is a shared ceiling.

(c) Exactly what closes it. Either build the named sufficiency theorem (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). The bounded, honest endpoint is the latter: a field-level definitional judgement the program declines to make for everyone. (Sources: S6_uv_completion_scope.md; 01_named_missing_objects.md MO-12; 02_RESIDUAL_LEDGER.md R6.)

(d) Machinery & inputs. The MO-12 scope analysis (00_decomposition.md S6); the all-orders tower structure; the EFT/Wilsonian framing.

(e) Leverage. Settling R6 is what would let a closed R2+R5 mean "UV completion" rather than "one consistency coefficient passed." It is the gate between "necessary" and "sufficient."

Hole F — disclose / derive the dimensionful normalization (scheme anchor)

(a) Precise statement. The dimensionful \(a_6\) magnitude rides an injected scale (scheme-anchored, the same decision family as \(c_{\rm loop}\) / SG-7-\(\delta\)), not derived.

(b) Why it is hard / traps. A scheme reverse-engineered to a desired magnitude RELOCATES, it does not close (RELABEL_FAIL). The cleanest derived content is the dimensionless sphere cross-checks (S²=4/315, S⁴=74/63, S⁶=1139/63, conformal 5/63), which are route-independent; the much-quoted 124/315 color ratio is metric-selected (Scal\(_{K_6}=7.5\)) and not yet target-blind-reproduced, so it is not a clean invariant either. The dimensionful magnitude is not clean.

(c) Exactly what closes it. Either derive the dimensionful normalization from frozen geometry data (no injected scale), or settle the shared heat-kernel scheme object — or explicitly name-and-verify AXIOM-HEATKERNEL-SCHEME-OBJECT as value-free under the κ³/π falsification test, with the dependence made traceable. Until then, keep it disclosed as a scheme anchor. (Sources: SPECIALIST_HOLE_QUEUE UQF-9 hole 5, 5C hole 5; 01_DOSSIER.md §5.)

(d) Machinery & inputs. The frozen radii / \(M_*\approx6.01\times10^{16}\) GeV floor-location read; the heat-kernel scheme family; the 124/315 independent-reproduction route.

(e) Leverage. Disclosing/deriving the normalization is what lets any dimensionful \(a_6\) output be reported as more than a labeled consistency coefficient.

R4 — atomicity of AXIOM-COSTFLOOR (named, not a closeable hole)

For completeness: the elevation of the established floor to a root axiom is AXIOM-OPEN, not atomic. Any deeper "physics must be operationally realizable" principle is an equal-strength relocation of the same axiom — an irreducible \(\ge 1\) measured-invariant floor that cannot be eliminated, only relocated (Prime Truth: anchored ≠ derived). Demanding a proof that the granularity premise is forced by reality is therefore the trap of "anchoring on the target": it cannot be discharged for anyone. The measured floor itself is terminal-as-physics; its elevation is not atomic. Naming this is precision, not a defect. (Sources: 02_CURRENT_STATE.md §0; 01_DOSSIER.md R4.)

7. Honest ceiling & scope

What is claimed (and at what rung). (Source: 04_CLAIM_BUDGET.md.) - The frozen geometry poses the UV question on a definite operator \(L_{\rm grav}^{d=13}\) + ghosts; \(a_6\) carries the short-distance datum — ESTABLISHED (setup). - The cost floor (Margolus–Levitin / Landauer / Bekenstein) is ESTABLISHED physics on a Lorentz scalar. - AXIOM-COSTFLOOR dissolves the \(a\to 0\) / \(a_8,a_{10},\dots\) counterterm-tower class, Lorentz-invariantly, by one named value-free axiom (κ³/π passes) — DISSOLVED (banked, scoped). - The elevation of the floor to a root principle is declared, not proven irreducible — AXIOM-OPEN. - No truncation-independent non-Gaussian fixed point is mounted; UQF-9 = AUDIT/OPEN — OPEN (R1, binding). - The \(d=13\) \(a_6\) trace is necessary-not-sufficient, uncomputed, overlapping Gap-01/B1 — OPEN (R2). - Positivity and sufficiency are unsettled — OPEN (R5, R6). - UQF-10 / UQF-14 carry typed BLOCKED_BY_UQF9 contracts — EXPORTED (R7). - UQF-9 is audit-complete but physically OPEN — about the audit, not about physics closure.

What is explicitly NOT claimed — the bright lines. (Sources: handoff bright-line block; 04_CLAIM_BUDGET.md F1–F7.) - NOT claimed: the cost-floor reframe SOLVES UV completion / closes UQF-9. DISSOLVED ≠ SOLVED. - NOT claimed: a finite, positive \(a_6\) constitutes or proves UV completion. It passes one heat-kernel consistency coefficient (necessary-not-sufficient); whether that constitutes UV completion is an external judgement deliberately not made. - NOT claimed: the granularity floor dissolves the finite \(a_6\) or the \(\Lambda\) walls. The floor leaves \(a_6\) UNTOUCHED (it exists term-by-term at any spacing and in the continuum), and leaves the \(\Lambda\) value / \(\Lambda\) radiative-stability walls untouched (1 DISSOLVES / 10 UNTOUCHED). - NOT claimed: a non-Gaussian / asymptotically-safe fixed point has been exhibited for this geometry. Candidate fixed points exist only within declared truncations; FRG evidence trends against the naive route. - NOT claimed: a truncation fixed point ⇒ closure; "audit-complete" ⇒ physically closed; "exported" ⇒ inherited; any \(a_6\) value/sign/partial; AXIOM-COSTFLOOR is atomic.

The dissolved unicorns (shared ceilings, never open weaknesses, never claimed as proven). - Whether "finite answers everywhere" CONSTITUTES UV completion (E1) — a field-level definitional judgement; no theory in any program can settle the definition for everyone, so the bounded EFT/floored stance is the ceiling, not a hedge. - The strong form of N1 — "no future theory could ever do better / no fixed point exists under ANY possible mathematics." The bounded version (exhibit one, or prove non-existence within asymptotic safety) is a real listed hole (R1); the open-ended universal-negative is unprovable for everyone. - Demanding a proof that the cost-floor premise is FORCED by reality — any deeper "must be realizable" principle is an equal-strength relocation; the \(\ge1\) measured-invariant floor cannot be eliminated, only relocated.

The anchors paid. UQF-9 terminates on no measured invariant that closes it. A fixed point (R1) has no anchor — a global open problem. The finite \(a_6\) (R2) anchors on a heavy-but-finite symbolic computation, uncomputed, refutable by the positivity falsifier (R5). The cost floor (R3/R4) is established physics, but its elevation to a root axiom is declared, not measured of the UV sector, and is not atomic. R6 anchors on an unmade external/field judgement. The floor the gate honestly rests on (the \(\ge1\) genuine anchor) is the established cost-floor physics — and that anchors the reframe, not a UV-completion certificate. ANCHORED ≠ DERIVED; given-E ≠ derivation-of-E; selection ≠ derivation; DISSOLVED ≠ SOLVED; AXIOM-CLOSED ≠ atomic; a captured log ≠ an independent reproduction.

One-sentence honest endpoint. UQF-9 is audit-complete but physically OPEN: Predicate A (a truncation-independent non-Gaussian fixed point) is the binding global wall (R1, FRG trending against the naive truncation route); Predicate B (the cost-floor dissolution of the \(a\to 0\) counterterm-tower class) is a genuine scoped, value-free, Lorentz-invariant, non-atomic win that DISSOLVED-not-SOLVED UV completion; the \(d=13\) \(a_6\) trace (R2) is necessary-not-sufficient computation-debt overlapping Gap-01/B1 (inherits only when B1 is BUILT-and-MOUNTED) with unresolved positivity (R5) and sufficiency (R6); UQF-10/UQF-14 carry typed BLOCKED_BY_UQF9 contracts. STATUS-UPGRADES:0. Frozen branch dcc66f1b2685 / a5b1e6f9d951 READ-ONLY.

Sources synthesized (all under …/Fable_Version/rendered/TOE/): PER_GATE_DOSSIERS/UQF9_COMPLETION_HANDOFF/{01_DOSSIER, 02_CURRENT_STATE, 03_FROZEN_GEOMETRY_CONTEXT, 07_CONVERGED_CLOSURE_PATH}.md and certificates/UQF9_AUDIT_COMPLETE/{02_RESIDUAL_LEDGER, 04_CLAIM_BUDGET, 05_FINAL_ROLLUP}.md; ROOT_AXIOM_COST_FLOOR_REALIZABILITY.md; COST_FLOOR_WALL_IMPACT_LEDGER.md; gap_01_uv_quantum_gravity/{00_decomposition, 01_named_missing_objects}.md + subsystems S2.2, S4.1; SPECIALIST_HOLE_QUEUE_2026-06-29.md; GATE_REGRADE_BESTCASE_AND_HOLES_2026-06-29.md; and the published brief articles/GATE_BRIEF_UQF9. Every number is traceable to one of these; no \(a_6\) value/sign/partial is asserted as a result. STATUS-UPGRADES:0.