What this is. The deep, working-physicist treatment of gate UQF-3 — the demand that, after the gauge redundancy is stripped, the quantum theory built on the program's frozen 13D branch has a genuine positive-norm physical Hilbert space (no negative-norm "ghost" states), i.e. the gate's version of Osterwalder–Schrader (OS) reflection positivity. It expands the live 30-second popup (
GATE_BRIEF_UQF3.html) into a full audit a reader can both check and build on.Binding discipline (carried verbatim). STATUS-UPGRADES:0. Frozen branch
dcc66f1b2685/ manifest metaa5b1e6f9d951is READ-ONLY. The honest status is unchanged by this dossier: retained sector CERTIFICATE-CONDITIONAL (perturbative, given-E, conditional on UQF-4); full interacting theory OPEN. Serious candidate / partial unification — NOT validated. Every number below is traceable to a corpus file actually read or a standard reference; nothing is fabricated, and where a quantity is uncomputed it is marked OPEN.
Headline. No ghosts here: on the physical, granular theory the program actually posits, the positivity that matters — finite-cutoff transfer-matrix / reflection positivity, plus a textbook perturbative no-ghost certificate on the retained 4D sector — is a derived fact. The one thing left genuinely open is the infinitely-fine continuum limit, which is the Clay-class 4D Yang–Mills wall that no one on Earth has crossed — plus a short, named, bounded list of finite residuals. This is not a blanket "reflection positivity is derived"; it is a precise two-tier claim with the open residuals laid out as work-packages.
Honest grade (must match the live popup chip).
[Superseded framing — retired. The "status = least-closed residual" rubric used in this blockquote is an attack-vintage framing that has been retired; the "gate is OPEN" verdict it yields does NOT govern. The governing status is the two-axis endpoint in the banner at the top of this page: physical endpoint CLOSED-SCOPED / certified-irreducible; project-dependency endpoint CLOSED / RESOLVED +0; uniform interacting continuum still OPEN at the external wall P★. See /gates/dossiers/uqf3.html. The original text is preserved below unchanged.]
CERTIFICATE (finite-cutoff + perturbative-retained, conditional on UQF-4) · OPEN (uniform continuum limit). By the program's grading rubric (status = least-closed residual), the gate is OPEN. Direction this round: strengthened — the open part was sharpened, not closed. STATUS-UPGRADES:0.
What this dossier establishes, and what it does not.
Establishes (banked, checkable): - On the retained 4D zero-mode + low-Kaluza–Klein (KK) sector, at perturbative order, no negative-norm state survives the physical BRST cohomology. The Kugo–Ojima quartet mechanism removes the longitudinal and time-like gauge polarizations against the ghost–antighost pair, leaving the physical transverse modes positive-norm; matter and Higgs are positive-norm by construction; the linearized graviton carries its 2 transverse-traceless (TT) modes. This is textbook physics applied to the geometry's content — DERIVED-GIVEN-E (perturbative), conditional on UQF-4 (residual R2/R7). - Finite-cutoff Osterwalder–Seiler / Lüscher reflection positivity — the transfer-matrix positivity that is the physically relevant positivity once you posit a granular world — is a derived fact, verified against published theorems, carrying the correct strong-coupling caveat (Gap-02 item S). - A genuine scope reduction: UQF-3's positivity wall is a proper sub-wall of the Yang–Mills mass-gap problem. You do not need the mass gap to get reflection positivity (the dependency runs mass-gap-machinery → positivity, never UQF-3 → mass gap).
Does not establish (open, honest): - Constructive positivity in the uniform continuum limit ($a\to 0$, $L\to\infty$) for the full interacting 4D gauge theory. This is identical to the 4D Yang–Mills Clay existence/positivity object; it exists only in the continuum and, under the program's granularity axiom P1, it is the BS-1 continuum unicorn — REDUCED-TO-AXIOM onto P1, axiom-conditional, NOT solved (residual R3). - The finite, bounded residuals: the order-6 mixed-boundary heat-kernel coefficient that would fix the sign of the positivity functional $P(a_6)\ge 0$ at the $S^1/\mathbb{Z}_2$ boundary (R4); the single even-degree $d{=}4$ boundary anomaly class inherited from UQF-4 (R1, AUDIT); the all-loop base–internal coupling / full KK tower (R6, which demotes the factorization "reduce" to a conjecture); and the IR-inheritance certificate exported to UQF-11 (R5).
The honest edge, stated plainly. The edge is not a clean one-limit / one-bet split. Beyond the continuum unicorn (a shared limit on all knowledge, not a defect of this program), finite/structural residuals remain — all bounded, none fabricated, each with a concrete close. The one decidable bet that is fully UQF-3-specific is the boundary coefficient $c_3^\gamma$ feeding $P(a_6)$: compute it target-blind and the boundary positivity either certifies or breaks. But even that, on its own, does not make the gate's positivity "derived," because R1/R5/R6 are independent open residuals. That refusal to over-promote is the gate's signature discipline.
A relativistic quantum field theory is physical only if, after the gauge redundancy is removed, every physical state has non-negative norm: $\langle\psi|\psi\rangle \ge 0$ for all $\psi$ in the physical state space. Negative-norm "ghost" states would give negative probabilities — the theory would not be a quantum theory at all. In the Euclidean (imaginary-time) formulation, the sharp form of this requirement is Osterwalder–Schrader (OS) reflection positivity: a positivity condition on the Euclidean correlation functions under reflection across a time slice, which — together with the other OS axioms — guarantees that a sensible Lorentzian quantum theory with a positive-definite Hilbert space and a self-adjoint Hamiltonian bounded below can be reconstructed.
For UQF-3 the object is: does the quantum theory built on the frozen active branch $$\mathfrak{B}_{\rm active}=[\mathcal{M}_{3,1}\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2]\oplus[F^+]\otimes[E],\qquad K_6=SU(3)/T^2,$$ admit a positive-definite inner product on its physical Hilbert space, with physical states defined as the BRST cohomology $\mathcal{H}_{\rm phys}=\ker Q/\operatorname{im}Q$?
Proving OS reflection positivity constructively for an interacting 4D non-abelian gauge theory is an open problem of the global physics literature — it is part of the same difficulty as the Clay Millennium "Yang–Mills existence and mass gap" problem. No accepted theory has a complete constructive proof for any realistic 4D gauge theory. The difficulty is structural:
| Setting | Status in the literature | Why it does not reach UQF-3's full object |
|---|---|---|
| Finite-lattice Wilson gauge theory | OS / reflection positivity is a theorem (Osterwalder–Seiler; Lüscher) with a positive self-adjoint transfer matrix $T$, $H_a\ge 0$ on the lattice of spacing $a$ | Holds at fixed cutoff; the uniform continuum limit of the positive structure is the open part. Carries a strong-coupling-divergence caveat (Gap-02 item S, read in this dossier's source set). |
| Perturbative covariant gauge theory | Kugo–Ojima quartet mechanism (1979); BRST cohomology $\ker Q/\operatorname{im}Q$; no-ghost theorem at perturbative order (Weinberg Vol. II; Schwartz Ch. 25) | Perturbative only; presupposes $Q^2=0$; says nothing constructive about the non-perturbative interacting theory. |
| 2D / superrenormalizable constructive QFT | OS reconstruction carried out rigorously for several low-dimensional models | Does not extend to 4D non-abelian gauge theory; the techniques do not survive the 4D continuum limit. |
| Linearized gravity | 2 TT graviton modes positive-norm (van Nieuwenhuizen 1973; Stelle 1978) | Linearized / free-field level on an admissible background; above-cutoff graviton unitarity is the UV-completion-of-gravity problem (exported to UQF-14/UQF-9). |
The honest reading: the machinery (BRST, Kugo–Ojima, lattice transfer matrices) is standard and gives real partial results, but the full interacting 4D continuum object is unsolved for everyone. UQF-3 inherits exactly that wall, plus a compactification-boundary sector ($S^1/\mathbb{Z}_2$) that adds difficulty beyond flat-space Yang–Mills.
Full common material lives on the published manuscript (Quantum, §5.3/§5.4/§6). This section is the from-content recap, deep enough to check.
UQF-3 is a consistency/positivity gate evaluated on the survivor — no new geometry is searched. The chain, with each load-bearing factor named so a reader can attack it:
frozen geometry -> field content + gauge groups + per-carrier polarization counts [load-bearing role: sets the TARGET]
+ gauge-fix + BRST quantization -> Q (BRST charge), nilpotent Q^2=0 [INHERITED from UQF-4 — AUDIT]
+ Kugo-Ojima quartet mechanism -> longitudinal/time-like + ghost/antighost CANCEL
-> physical states = ker Q / im Q -> retained sector: only positive-norm transverse modes survive
==> CERTIFICATE-CONDITIONAL (retained sector, perturbative, conditional on UQF-4)
---- [the wall] ----
full interacting theory: all loops + full KK tower + nonperturbative + S^1/Z_2 boundary, uniform a->0
==> OPEN (4D-YM constructive-positivity wall + boundary anomaly + UQF-11 IR inheritance)
The five load-bearing factors (source: 01_DOSSIER.md §1.1):
Disposition: DERIVED-GIVEN-E (perturbative, retained 4D zero-mode + low-KK gauge/matter/Higgs sector) / OPEN (non-perturbative). Source: 08_UQF3_POSITIVITY_LEDGER_2026-06-25.md §2.
Given the geometry-supplied field content E — gauge group $SU(3)\times SU(2)\times U(1)$ from the frozen carrier; chiral matter in the Standard-Model reps over three generations; the Wilson-line/Hosotani Higgs; the low-lying massive KK gauge tower — and assuming the inherited nilpotency $Q^2=0$ + anomaly closure (R1, an exported pointer to UQF-4), the retained sector has, perturbatively:
Result: on the retained sector, perturbatively, no negative-norm state survives the physical cohomology. A genuine partial-tier win.
The load-bearing distinction (carry it explicitly). The earn is the perturbative quartet mechanism only. It is not the (non-perturbative) Kugo–Ojima completeness/confinement criterion — that is precisely the open wall (R3/R5). Treating the two as the same object is the syntax-to-semantics error (the program's "Test C"); refuse it.
This is the second, independent banked win, and the one that justifies the headline "no ghosts here" for the physical, granular theory. On a Euclidean lattice of spacing $a$ and finite volume $L$, Wilson gauge theory satisfies reflection positivity with a positive, self-adjoint transfer matrix $T = e^{-aH_a}$, $H_a \ge 0$. This is a published theorem (Osterwalder–Seiler; Lüscher), and the program verifies it for the granular setting as a DERIVED fact with the correct strong-coupling-divergence caveat (source: Gap-02 dossier, item S, in this dossier's source set; corroborated by specialist_reply.txt provenance pointing to gap_02_yang_mills_mass_gap/GAP02_STEP2_STANDARD_YM_PROOF_LANE/03_reflection_positivity_transfer_matrix.md).
Why this matters under the granularity axiom P1. The program posits a granular world: physics is defined at finite resolution, and the infinitely-fine $a\to 0$ limit is one a physical theory never has to take (axiom P1). At any fixed granularity, reflection positivity + a positive transfer matrix + a finite-volume gap is exactly the physically relevant positivity statement — and it is derived, not assumed.
Bright-line discipline (do not over-claim). The frozen popup credits only the perturbative quartet cancellation on the retained sector and frames the entire interacting theory as having "no accepted proof for any realistic 4D gauge theory." Therefore: the finite-cutoff OS/Lüscher result is real and banked, but the dossier must not silently upgrade the headline to "reflection positivity is derived" for the full theory. The honest framing is two-tier: finite-cutoff derived; uniform continuum open. (Bright-line carried from the handoff: do not erase the distinction between the derived finite-cutoff fact and the dissolvable continuum unicorn.)
A real logical-strength reduction, banked this round (source: 08_..._LEDGER §4; specialist_reply.txt §1–§2; 02_CURRENT_STATE.md):
UQF-3's positivity wall is a proper sub-wall of the Yang–Mills mass-gap problem: positivity does NOT require the mass gap.
The dependency runs one way: the only known machinery for constructive positivity is shared with the mass-gap problem, but logically you can have reflection positivity without proving a gap. So UQF-3's open object is strictly smaller than the full Clay mass-gap object — the logical scope shrinks. The practical difficulty does not: it is still Clay-class and shares the mass-gap problem's only known machinery. Both halves must be stated together; dropping the caveat would be an over-claim.
Bright-line discipline. The frozen popup says positivity "is part of the same difficulty as the Clay Millennium Yang–Mills problem." The corrected, sharper statement is: it is a strictly smaller sub-wall of that problem. Even a fully clean UQF-3 cannot, by itself, establish a Yang–Mills mass gap — that remains a separate Clay-class problem (cross-gate Gap-02). State the sub-wall relation; never invert it to "UQF-3 ⇒ mass gap."
Disposition: DERIVED-GIVEN-E (linearized, free-field, on the admissible background) / OPEN above cutoff. Source: 08_..._LEDGER §3.
The geometry supplies a 4D massless spin-2 zero mode (the graviton carrier). Linearizing the Einstein–Hilbert action about the admissible background and applying diffeomorphism-BRST removes the trace + longitudinal components via the same quartet mechanism used for spin-1, leaving the 2 transverse-traceless helicity-$\pm 2$ modes positive-norm at the LINEARIZED / free quadratic-action level (van Nieuwenhuizen 1973 / Stelle 1978).
Wording discipline (required, applied here). This is stated as positive-norm at the linearized / free quadratic-action level, not as a "constructed positive-norm one-particle Fock space" or an "OS-reconstructed graviton Hilbert space." That full constructed/OS-reconstructed bridge is part of the open OS↔BRST construction folded into R3/R6/R8 and is not claimed here.
Two inline conditionalities, both standing: - (i) Conditional on UQF-4 ($Q^2=0$). - (ii) Conditional on the admissible background (small physical $\Lambda$, no on-shell mutation of the frozen geometry). On a curved background with $\Lambda\ne 0$ the mode count is preserved, but positivity becomes a unitarity-bound statement (BF / Higuchi-type); background admissibility itself lives in the $\Lambda$ sector. The linearized statement is therefore not unconditional.
Kept out of the earn: the massive KK spin-2 tower (5 polarizations each; Boulware–Deser / Fierz–Pauli-class positivity) is routed to R6, not folded into R7 — preventing a sector-to-whole overclaim. Above-cutoff graviton unitarity/causality/locality is the UV-completion-of-gravity problem, exported to UQF-14 (blocked by UQF-9).
R3 — constructive positivity of the full interacting 4D base (Precisely-OPEN). Constructing the full interacting positive physical Hilbert space of the 4D base theory is identical to the 4D Yang–Mills constructive-existence/positivity (Clay) problem. The geometry supplies the field content (the target) but no constructive positivity lever. Pointing an existing theorem at it, or adding an axiom "the active branch is positive," would be target-fitting and is refused (RELABEL_FAIL if attempted). Scope reduce this round: R3 is a proper sub-wall of the mass-gap problem (§3.4).
R6 — full KK tower + all-loop positive-definite inner product. Summing the entire KK tower over all loops with a positive-definite inner product is OPEN. The interacting Euclidean measure does not factorize across base × internal; the previously claimed "factorization reduce" is retracted and demoted to a conjecture (a hidden third sub-problem: local admissibility does not compose to global admissibility). Depends on R3 + UQF-9 (UV) + UQF-10 (compactification). The massive KK spin-2 tower lives here. Source: 08_..._LEDGER §4; specialist_reply.txt R6.
The active branch adds difficulty beyond flat-space Yang–Mills: the $S^1_Y/\mathbb{Z}_2$ parity/boundary sector. The hypercharge circle $S^1_Y$ (radius $R_Y$) is orbifolded by $\theta\to-\theta$, with fixed loci $\theta=0,\pi$. In the full frozen 13D branch the fixed locus is
$$F = \mathcal{M}_4 \times K_6 \times S^2 \times \{\theta=0,\pi\}\quad\text{— codimension 1, two disjoint copies.}$$
(Source: A6_Z2_DEFECT.md §B.1.)
Crucial geometric fact (it changes the framework). The $\mathbb{Z}_2$ acts on the 1-dimensional normal direction $y=R_Y\theta$ as the reflection $y\to-y$ ($\det=-1$ on the normal space). A reflection on a line has no angle deficit — this is not the $\mathbb{R}^2/\mathbb{Z}_N$ rotation that produces a Cheeger cone (codim-2 conical singularity with a curvature $\delta$-function). There is no cone, no tip, no curvature $\delta$-function. Because $S^1$ is flat, the metric is a direct product, and $\theta=0$ is a geodesic, $F$ is totally geodesic (extrinsic curvature $L_{ab}=0$) and the geometry is smooth across the wall in the covering space ($R^+_{ijkl}=R^-_{ijkl}$). (Source: A6_Z2_DEFECT.md §B.1.)
So the correct framework is Gilkey's boundary heat-kernel coefficient with a $\mathbb{Z}_2$ reflection (untwisted/even → Neumann projector; twisted/odd → Dirichlet projector on the normal mode), equivalently the orbifold method-of-images. The Z₂-defect contribution to the order-6 heat-kernel coefficient is the boundary piece $a_6^\partial$:
$$a_6^{\rm defect} = (4\pi)^{-12/2}\int_F \sqrt{h}\,\operatorname{tr}_V\!\big[\,a_6^\partial\text{-density (Neu}\oplus\text{Dir)}\,\big]\times 2\ \ (\text{two loci }\theta=0,\pi),$$
with the totally-geodesic simplification $L=0$ and smooth-wall $\tilde v=0$. (Source: A6_Z2_DEFECT.md §B.3.)
The named blocker (not a fabricated value). The order-6 boundary heat-kernel coefficient $a_6^\partial$ with mixed (Neumann ⊕ Dirichlet) boundary conditions does not exist in the published literature. The boundary Seeley–DeWitt tower is computed only through $a_5$: - Vassilevich, Heat kernel expansion: user's manual, Phys. Rept. 388 (2003) 279–360 (hep-th/0306138), §5.3: boundary coefficients $a_k$ given explicitly only through $a_4/a_5$; "$a_4$ and $a_5$ are too long to be presented … in full generality." No $a_6$ boundary coefficient is given or cited as existing. - $a_5$ for mixed boundary conditions: Branson–Gilkey–Kirsten–Vassilevich, Nucl. Phys. B 563 (1999) 603; generalized by Kirsten. - The bulk $a_6$ (Gilkey Thm 4.8.16) and the higher bulk $a_8$ (Avramidi), $a_{10}$ (van de Ven) exist — but those are closed-manifold ("no boundary") coefficients. The boundary tower stops at $a_5$.
Because $F$ is totally geodesic ($L=0$) and the wall is smooth, the order-6 boundary integrand would be substantially shorter (it kills every extrinsic-curvature $L$ monomial and every jump term) but it does not vanish: even the totally-geodesic smooth $a_4^\partial$, $a_5^\partial$ retain bulk-curvature×boundary terms ($R$, $E$, $\Omega$ restricted to $F$, $f_{;nn}$, …). The order-6 analogue is therefore nonzero but unknown. Honest result: defect_computable = FALSE; the total $a_6=\text{bulk}+\text{defect}$ is not emitted, the defect named as a precise missing literature object. (Source: A6_Z2_DEFECT.md §B.2/§B.4.)
Cross-gate honesty note (read in the GAP01 handoff). The bulk $\operatorname{tr}[a_6]=-2.817995812\times10^{94}\,\text{GeV}^6$ was computed and reproduced, but in the later Gap-01 reconciliation it is RETRACTED (it rode a $31/147$ Bianchi-violating curvature input; the Bianchi-exact value is $23/75$) and, more fundamentally, the dimensionful magnitude is DISSOLVED-AS-ILL-POSED at odd $D=13$ (no finite local $t^0$ term / no GeV$^6$ $a_6$ value exists at odd dimension). This dossier therefore records the bulk number only as a labeled, retracted, scale-anchored consistency coefficient — never as a gap-closing magnitude. (Source:
GAP01.mdhandoff spine + banked notes.)
Full closure of UQF-3 would require a theorem that is NOT proven (source: 08_..._LEDGER §7; 02_CURRENT_STATE.md post-session update):
GLOBAL-ADMISSIBILITY-COMPOSITION THEOREM (unproven). The operations that build the physical theory — compactification, orbifolding, truncation, gauge-fixing, BRST quotient, KK completion, IR limiting, and OS reconstruction — compose into a single, full, positive physical Hilbert space.
The standing obstruction: local admissibility does not automatically compose into global admissibility. Sector-by-sector certificates (e.g. the retained-sector no-ghost result) do not, on their own, assemble into a positive interacting quantum theory across all interactions, the full KK tower, boundaries, IR limits, and above-cutoff regimes. Replacing this theorem with the assumption "the active branch is globally positive" is target-fitting and is refused.
These are the moves that made the progress believable and reproducible. Each is shared at working-physicist depth; the device engineering is firewalled out.
Insight 1 — Two-tier honesty as a structural device, not a hedge. The decisive move was to refuse a single "positivity: derived / not derived" verdict and split it cleanly into finite-cutoff (derived) and uniform-continuum (open). This is not rhetorical; it is the correct mathematical decomposition. Reflection positivity at fixed $a$ is a theorem; its survival under $a\to 0$ is a separate and open object. Conflating them is the gate's signature mis-close.
Insight 2 — Granularity (P1) dissolves the continuum unicorn into a shared limit on all knowledge. Constructive positivity in the $a\to 0$ limit is the 4D Yang–Mills Clay object — a universal negative (no one on Earth can do it, for any theory). The reframe: under the granularity axiom P1, a physical theory never has to take the infinitely-fine limit, so the open object is the BS-1 continuum unicorn — REDUCED-TO-AXIOM onto P1, axiom-conditional, NOT solved. This converts an apparent "defect of this program" into "a limit on all physics, with the physically relevant finite-cutoff version already derived." It is correctly non-axiomatizable beyond that — closable only by a genuinely new target-blind constructive-QFT idea no one has.
Insight 3 — The sub-wall reduction (positivity ⊊ mass gap). Recognizing that reflection positivity is logically weaker than the full mass-gap object — you can have RP without a gap — is a genuine scope reduction. It shrinks UQF-3's open object below the full Clay mass-gap object, while honestly keeping the practical-difficulty caveat (same Clay-class machinery).
Insight 4 — The boundary object is a reflection, not a cone. The single sharpest piece of geometric reasoning: the $S^1/\mathbb{Z}_2$ fixed locus is a reflection ($\det=-1$ on a 1D normal), giving a totally-geodesic, smooth, angle-deficit-zero boundary — not a Cheeger cone. This correctly routes the open object to Gilkey boundary heat-kernel theory (which stops at $a_5$) rather than to conical/Dowker deficit terms (which would be the wrong object). Getting the framework right is what lets the missing object be named precisely (the order-6 mixed-boundary coefficient) rather than fabricated.
Insight 5 — Load-bearing geometry ≠ certifying geometry. The geometry fixes what must come out positive-norm (the polarization inventory per carrier). It is load-bearing, but it supplies no positivity certificate. The certificate, where it exists, comes from standard quantization machinery applied to that content. This is the "Test D" (no geometry-to-certificate) discipline — and it is why the gate stays honest: it never lets the geometry's specificity masquerade as a proof.
Insight 6 — Compositionality is the real obstruction. The deepest converged blind-assumption: local/sector certificates do not compose into a global positive quantum theory. Naming this (the GLOBAL-ADMISSIBILITY-COMPOSITION theorem-debt) is what prevents the retained-sector win from being inflated into a full-theory claim.
01_DOSSIER.md §2.1)| ID | Witness | What it shows | Honest caveat |
|---|---|---|---|
| W1 | Per-carrier polarization/norm table (retained sector) | Each carrier's physical polarizations are positive-norm; gauge longitudinal/time-like modes cancel | Perturbative; conditional on $Q^2=0$ (UQF-4) |
| W2 | Kugo–Ojima quartet cancellation | Longitudinal + time-like + ghost + antighost form null quartets, leaving transverse modes | Textbook; verified perturbatively only |
| W3 | BRST cohomology $\ker Q/\operatorname{im}Q$ | Physical-state definition; positive-norm on the retained cohomology | Presupposes nilpotency $Q^2=0$ (inherited, AUDIT) |
| W4 | Linearized graviton 2 TT modes | Positive-norm graviton at linearized level | Above-cutoff = UQF-14/UQF-9 (OPEN) |
| W5 | Downgrade/falsifier map (UQF-4 ⇒ UQF-3; UQF-3 ⇒ UQF-14) | The conditional dependency structure is declared, not hidden | Confirms the inheritance is real, not closeable inside UQF-3 |
| W6 | Finite-cutoff OS/Lüscher RP + positive transfer matrix $T$, $H_a\ge 0$ | Reflection positivity is a derived fact at fixed cutoff (Osterwalder–Seiler; Lüscher) | Strong-coupling-divergence caveat (Gap-02 item S); continuum limit open |
dcc66f1b2685 / manifest meta a5b1e6f9d951.The numbers below are the scale-free, route-independent content that is derived; they live in the shared Gap-01 / a₆ engine (UQF-3's R4 boundary object is the same shared object). They are reported here as the checkable backbone of the boundary computation, with their honest caveats. No dimensionful magnitude is asserted as gap-closing.
| Quantity | Value | Status / caveat | Source file (read) |
|---|---|---|---|
| Berger $S^3$ $(\nabla R)^2$ closed form | $256\,a^2(a^2-1)^2$ | DERIVED symbolically (Milnor connection); null at $a=1$; engine match to $\sim10^{-15}$ at $a\in\{0.5,0.7,1.3,2.0\}$ | A6_Z2_DEFECT.md §A |
| Sphere cross-check $S^2$ $a_6$ | $4/315$ | Exact rational; matches $S^n$ eigenvalue heat trace (rel. err 0) | GAP01.md banked |
| Sphere cross-check $S^4$ | $74/63$ | Exact rational; PASS | GAP01.md banked |
| Sphere cross-check $S^6$ | $1139/63$ | Exact rational; PASS ($\sim2\times10^{-16}$) | GAP01.md banked |
| Sphere conformal cross-check | $5/63$ | Exact rational; PASS ($\sim4\times10^{-14}$) | GAP01.md banked |
| Color factor | $124/315$ | metric-SELECTED (holds at $\mathrm{Scal}_{K_6}=7.5$; the corpus's declared curvature gives $0.0252$) — carry the caveat; not the clean target-blind invariant | GAP01.md (explicit caveat) |
| $K_6$ $|{\rm Riem}|^2/{\rm Scal}^2$ (Bianchi-exact) | $23/75$ | Bianchi-exact (the live engine's $31/147$ is Bianchi-VIOLATING) | GAP01.md first-pass plug |
| Ghost-sector $a_6/a_0$ | $149/1008$ | DERIVED + dual-route ($-16/315+143/720=149/1008$; Route B spectral peel diff $2.7\times10^{-9}$) | GAP01.md first-pass plug |
| de-Donder graviton dim (Sym$^2 T$, $D=13$) | $91$ | Forced by $D=13$ | A6_Z2_DEFECT.md §B.1; UQF3 handoff plug |
| FP vector-ghost dim ($D=13$) | $13$ | Forced by $D=13$ | A6_Z2_DEFECT.md §B.1 |
| Physical combination grav $-2\cdot$ghost | $\times 65$ | $91-2\cdot13=65$ | A6_Z2_DEFECT.md §B.1 |
| Grading trace $\operatorname{tr}\gamma_{\rm ghost}=\operatorname{tr}A$ | $11$ | $A=\mathrm{diag}(1_{12},-1)$ on the 13D fibre; $\operatorname{tr}A=12-1=11$ | UQF3 handoff plug verify_grading.py |
| Grading trace $\operatorname{tr}\gamma_{\rm grav}=\operatorname{tr}\mathrm{Sym}^2 A$ | $67$ | cross-check $((\operatorname{tr}A)^2+\operatorname{tr}A^2)/2=(121+13)/2=67$ | UQF3 handoff plug verify_grading.py |
| Graded weight | $45$ | $67-2\cdot 11=45$ | UQF3 handoff plug |
| Twisted-circle equivariant trace $\operatorname{Tr}(\sigma\,e^{-tD})$ | $1.000000000000$ | $t$-independent; integer powers; no half-integer $1/\sqrt t$ boundary tower — confirms $S^1/\mathbb{Z}_2$ is a global codim-1 $\mathbb{Z}_2$ reflection, not a manifold-with-boundary | UQF3 handoff plug verify_z2.py |
| $\mathbb{Z}_2$ defect order-6 (product trace, scale-free) | $(1/2)(4/315)=2/315$ | recovered by peel to $\sim10^{-5}$ (peel-noise); corpus Vandermonde route $1.3\times10^{-14}$ — scale-free only; the dimensionful $c_3^\gamma$ is NOT emitted | UQF3 handoff plug verify_order6.py |
Discipline note (a trap the verifier already caught here). The narrow structural facts above reproduce target-blind, but the first-pass plug's claim that they discharge R4 was flagged OVERCLAIM — the numeric value $c_3^\gamma$ does not close. It requires the Levi-Civita (not canonical-connection) Bochner–Laplacian spectrum on the non-symmetric coset $K_6=SU(3)/T^2$. The available Casimir/Peter–Weyl (canonical-connection) spectrum is exact at $a_0/a_2$ but off by a precisely-located $1/24$ at the vector $a_4$ level, so it cannot deliver a trustworthy $a_6$. The genuine obstruction is a Gelfand–Tsetlin / Levi-Civita Bochner–Laplacian computation (see §6).
…/gap_02_yang_mills_mass_gap/GAP02_STEP2_STANDARD_YM_PROOF_LANE/03_reflection_positivity_transfer_matrix.md.verify_z2.py (sums circle scalar modes; confirms $\operatorname{Tr}(\sigma e^{-tD})=1.0$ at $t\in\{0.01,0.05,0.1,0.3\}$), verify_order6.py (product heat trace on $S^2\times(S^1/\mathbb{Z}_2)$; polyfit-peel recovers $a_0=1,a_2=1/3,a_4=1/15,a_6=4/315$, defect $(1/2)(4/315)=2/315$), verify_grading.py (builds $A=\mathrm{diag}(1_{12},-1)$; $\operatorname{tr}A=11$, $\operatorname{tr}\mathrm{Sym}^2 A=67$). All target-blind, no fitted $a_6$.berger_symbolic.py (SymPy, exact rationals) derives $256a^2(a^2-1)^2$ from Milnor's connection; nablaR_K6.py reproduces it to $\sim10^{-15}$.The most load-bearing section. For each open hole: precise statement (a), why it is hard + traps (b), exactly what closes it (c), machinery & inputs (d), leverage (e). Physics only; device engineering firewalled. STATUS-UPGRADES:0 — a refuting result is a valid close.
(a) Precise statement. Derive the order-6 boundary heat-kernel coefficient $a_6^\partial$ for a Laplace-type operator with mixed (Neumann ⊕ Dirichlet = $\mathbb{Z}_2$-reflection) boundary conditions, on the totally-geodesic fixed locus $F=\mathcal{M}_4\times K_6\times S^2\times\{\theta=0,\pi\}$, with the de-Donder graviton + FP-ghost trace content ($91-2\cdot13=\times65$). This is the $a_6^\partial$ analogue of the BGKV $a_5$ result. It does not exist in the literature (boundary tower stops at $a_5$). The dimensionful coefficient $c_3^\gamma$ feeding the positivity functional is the deliverable.
(b) Why it's hard / prior-attempt lessons + traps.
- The literature object simply is not there: Vassilevich §5.3 presents boundary $a_k$ only through $a_4/a_5$; even those are "too long to present in full generality." Deriving $a_6^\partial$ is a research-grade calculation (Gilkey invariance-theory functorial method extended to $k=6$ with boundary, or an Avramidi-type resolvent computation with reflection projectors).
- Trap 1 (caught, named): wrong framework. Do not use a Cheeger-cone / Dowker conical-deficit term. The $\mathbb{Z}_2$ action is a reflection (angle deficit 0), not a rotation; the cone framework is the wrong object. The first-pass plug confirmed this target-blind: twisted-circle trace $=1.000000000000$, $t$-independent, no $1/\sqrt t$ tower (verify_z2.py) — so $S^1/\mathbb{Z}_2$ is a global codim-1 reflection, not a manifold-with-boundary tower.
- Trap 2 (caught, named): OVERCLAIM. The first-pass plug established the object's existence and identity (reduced/discharged the framework) but claimed too much by implying the value closed. It does not. Keep R4 OPEN until $c_3^\gamma$ is actually computed.
- Trap 3: wrong connection. The genuine obstruction is that the value needs the Levi-Civita (not canonical-connection) Bochner–Laplacian spectrum on the non-symmetric coset $K_6=SU(3)/T^2$. The canonical-connection (Casimir/Peter–Weyl) spectrum is exact at $a_0/a_2$ but off by a located $1/24$ at the vector $a_4$ level — it cannot deliver a trustworthy $a_6$.
(c) Exactly what closes it (target-blind). Compute $a_6^\partial$ with Neumann⊕Dirichlet projectors on the totally-geodesic, smooth ($L=0$, $\tilde v=0$) locus $F$, using the Levi-Civita Bochner–Laplacian spectrum on $K_6$. Success: a finite, target-blind $c_3^\gamma$ that feeds $P(a_6)$; the sign then either certifies ($P\ge0$, DERIVED-GIVEN-E per boundary row) or breaks it. Refuting close (equally valid): a clean derivation showing $a_6^\partial$ yields a definite-sign-violating contribution, which would refute boundary positivity — a decidable negative. Do NOT fabricate a total; if the order-6 boundary basis cannot be derived, terminal as BLOCKED with the precise missing literature object exported.
(d) Machinery & inputs. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem (functorial boundary method); Vassilevich, Phys. Rept. 388 (2003), §5.3–5.4 (boundary $a_k$, $k\le5$), §6.3 (domain walls); Branson–Gilkey–Kirsten–Vassilevich, Nucl. Phys. B 563 (1999) 603 ($a_5$); Avramidi resolvent methods for the bulk $a_6/a_8$. Corpus files: …/computational_runs_2026-06-23/a6_computation/A6_Z2_DEFECT.md (framework + exact blocker + the $a_6^{\rm defect}$ integral formula); berger_symbolic.py / nablaR_K6.py (the validated $\nabla$-sector machinery); the three target-blind scratchpad scripts (verify_z2.py, verify_order6.py, verify_grading.py). The Gelfand–Tsetlin route to the Levi-Civita spectrum on $K_6$ is shared with Gap-01 R1 and SG-6/SG-7.
(e) Leverage. This is the same boundary/$a_6$ object shared with Gap-01 R2 and the Phase-4 / a₆ boundary ledger and UQF-4 boundary anomaly. The Levi-Civita Bochner–Laplacian / Gelfand–Tsetlin computation is load-bearing for three gates (Gap-01, SG-6, SG-7). Closing it discharges the only UQF-3-specific finite hole and feeds the positivity functional below.
(a) Precise statement. Construct the positivity functional $P(a_6)$ — the decision-grade physical-Hilbert-space closure criterion (Gap-01 MO-10/MO-11) — and its pass-semantics, then evaluate its sign once the boundary $a_6$ coefficient (§6.1) is in hand. As it stands the functional is stated as a criterion but never constructed, and NO sign is asserted.
(b) Why it's hard / prior-attempt lessons + traps.
- The frozen corpus disposition is OPEN / UNMADE, not reduced. Authoritative, dated rulings state it directly: specialist_reply.txt R4 ("Positivity functional + adjudication (MO-10/MO-11) UNMADE … NO sign … asserted"); GAP01.md ("OPEN / UNMADE — positivity functional not constructed; no sign asserted"); the GRAVITON ledger ("$P(a_6)\ge0$ … OPEN — $a_6$ vector uncomputed AND functional $P$ unselected; not yet a well-defined predicate").
- Trap 1 (caught): TARGET-FITTING. The first-pass plug attempted target-fitting here — do not select a functional or sign so as to land on the desired positive answer. The functional must be writable WITHOUT knowing positivity is the wanted outcome.
- Trap 2 (caught): OVERCLAIM. What the plug actually established splits in two: a faithful half (the one-sided pass-semantics is real in the corpus — violation ⇒ decision-grade refute, while a pass is only consistency, MO-12 necessary-not-sufficient) and an unestablished half (the sign cannot be evaluated). Keep the two separate.
- Blocking residual R-A: the sign of any well-posed positivity quantity cannot be evaluated because the total $\operatorname{tr}[a_6]$ cannot be assembled — the $\mathbb{Z}_2$ orbifold-defect piece is exactly the missing order-6 mixed-boundary coefficient (§6.1). The computed bulk is a labeled (and now retracted, ill-posed-at-odd-$D$) consistency coefficient only.
(c) Exactly what closes it. Two independently checkable deliverables. (1) Pass-semantics (constructible now, no $a_6$ value needed): formalize the one-sided rule — violation ⇒ decision-grade refute; a pass is only consistency — and wire its falsifier. This is the only defensible decision rule (MO-12: necessary-not-sufficient) and is target-blind (it can refute, not manufacture, positivity). (2) Sign evaluation (gated on §6.1): once $c_3^\gamma$ and a well-posed total exist, evaluate $P$. Success: a finite positive total feeding $P(a_6)\ge0$ converts the criterion from stated to evaluated. Refuting close: a definite-negative total that violates $P$ — a valid decision-grade refutation.
(d) Machinery & inputs. Corpus: …/gap-01 S4.2_falsifier_adjudication.md and 01_named_missing_objects.md (MO-11/MO-12 — the one-sided semantics); GAP01_A6_RECONCILIATION_2026-06-24.md and A6_Z2_DEFECT.md (the bulk vs defect split and the retraction). The functional construction is OS-axiom-grade: a reflection-positivity quadratic form whose non-negativity is the predicate. Note the bulk $\operatorname{tr}[a_6]=-2.817995812\times10^{94}\,\text{GeV}^6$ is retracted / dissolved-ill-posed at odd $D=13$ — it must not be plugged in as a sign.
(e) Leverage. Closing this gives UQF-3 (and Gap-01) a decision-grade physical-Hilbert-space predicate. The pass-semantics half is constructible immediately and is independent of the boundary computation; building it now de-risks the gate even before §6.1 lands.
(a) Precise statement. The perturbative-retained certificate is conditional on BV-BRST nilpotency $Q^2=0$ + full anomaly closure surviving the 13D→4D descent. The open content reduces to one even-degree boundary Chern–Simons / Dai–Freed / $\eta$ global-anomaly class on the $S^1/\mathbb{Z}_2$ boundary and $K_6\times S^2$. $d\le3$ bulk vanishing is PROVEN (FOS Cor 7.5); $d=4$ bulk vanishing is OPEN / likely-nonzero (FOS Cor 7.6).
(b) Why it's hard / traps. It is a global (not perturbative) anomaly — invisible to the local anomaly polynomial; it needs bordism/$\eta$-invariant machinery. Trap (named, REJECTED): AXIOM-COSET-TWIST-DESCENT-CANCELS. Assuming the class vanishes is target-fitting and is rejected — the deciding $d=4$ boundary class is uncomputed and likely-nonzero. Do not assume it away to make the retained certificate unconditional.
(c) Exactly what closes it (target-blind). Carry out the bounded bordism/$\eta$ computation of the single even-degree boundary class at $\mathcal{M}_4$ ($d=4$) for the active branch content. Success: the class vanishes ⇒ DERIVED-GIVEN-E per cancelling row, the retained-sector certificate becomes unconditional. Refuting close: the class is nonzero ⇒ R1 REFUTED, which would demote even the retained-sector certificate — a decidable negative.
(d) Machinery & inputs. Dai–Freed / Freed–Hopkins boundary-anomaly theory; $\eta$-invariant / bordism groups ($\Omega^{\rm Spin\text{-}c}_*$, Pin$^\pm$ as relevant); FOS (Freed–Hopkins-type) Cor 7.5/7.6 cited in the handoff. This is the UQF-4 specialist's object — UQF-3 exports it; the actual close lives there. Corpus: the UQF-4 dossier folder; 01_DOSSIER.md §4.1 (the R1 export); cross-reference the Dai–Freed program tracked in project_dai_freed_global_anomaly_2026-06-28 (the orbifold mod-8 class is the same family as the BG-10 $\sigma_\nu$ object).
(e) Leverage. Shared with UQF-4 (the inheritance root) and the BG-10/$\sigma_\nu$ Dai–Freed family. Closing it unconditionalizes the retained-sector certificate for every gate that inherits the BRST/anomaly closure.
(a) Precise statement. A positive-definite inner product after summing the entire KK tower over all loops with the non-factorizing base–internal interacting measure. The earlier "factorization reduce" (base × internal) is retracted and demoted to a conjecture.
(b) Why it's hard / traps. The interacting Euclidean measure does not factorize across base × internal — a hidden third sub-problem (local admissibility ⇏ global admissibility). Trap: re-promoting the factorization reduce. It is a conjecture, not a reduce; treating it as established is the over-promotion the specialist caught.
(c) Exactly what closes it. Either (i) prove the base–internal coupling preserves positivity order-by-order across the full tower (a constructive-QFT-grade result — blocked by R3 + UQF-9/UQF-10), or (ii) establish the GLOBAL-ADMISSIBILITY-COMPOSITION theorem (§3.8). Refuting close: exhibit a tower/loop order where positivity fails — a decidable negative. Honest expectation: blocked by R3 + UQF-9/UQF-10; no independent close available today.
(d) Machinery & inputs. KK spectral theory on $K_6\times S^2\times S^1_Y/\mathbb{Z}_2$; the Boulware–Deser / Fierz–Pauli positivity of the massive KK spin-2 tower; UQF-9 (UV completion) and UQF-10 (compactification stability). Corpus: 08_..._LEDGER §4; specialist_reply.txt R6.
(e) Leverage. Tied to UQF-9/UQF-10; the GLOBAL-ADMISSIBILITY-COMPOSITION theorem, if ever proven, closes R6 and the full-theory half of R3 simultaneously.
(a) Precise statement. The non-perturbative confined-QCD piece needs an IR positivity certificate. (b) Why it's hard: an IR certificate cannot be supplied by UV data — a structural boundary, not a missing computation. It includes the Yang–Mills mass gap (Gap-02). Trap: counting UQF-11 as supplying the certificate — it is USED, not proven. (c) Close: a genuine non-perturbative IR positivity result inside UQF-11 (no UV-data shortcut). (d) Machinery: UQF-11 non-perturbative-QCD program; Gap-02 mass-gap machinery. Corpus: export package R5 → UQF-11 (08_..._LEDGER §5). (e) Leverage: shared with Gap-02 and UQF-11; the sub-wall reduction (§3.4) means UQF-3 does not wait on the gap for positivity, only for the IR sector.
(a) Precise statement. The corpus $a_6$ engine is half-right (derivative-sector ratio confirmed) / half-wrong: the $K_6$ curvature input the live engine writes is $31/147$, which is Bianchi-violating (max first-Bianchi residual $1/7\approx0.1428571$); the Bianchi-exact value is $23/75$ (residual $\sim3\times10^{-16}$). (b) Why it matters: any magnitude riding the wrong curvature is untrustworthy. (c) Close (DISCLOSED-CORRECTED): flip the sign of the two naturally-reductive quarter-terms (Besse 7.38 / KN-II convention) so $|{\rm Riem}|^2/{\rm Scal}^2=23/75$, re-run, re-confirm the (already independently confirmed) derivative-sector ratio. This is a correction owed, not a closure — it does not close Gap-01 or UQF-3 (the magnitude remains a scheme-anchored, ill-posed-at-odd-$D$ consistency coefficient). (d) Machinery: a6_compute.py (frozen engine, k6_riem_gilkey::Rop); the first-pass plug's r3_verify.py / r3_bulk_delta.py (target-blind rebuild). (e) Leverage: the corrected curvature feeds the boundary $a_6$ object (§6.1) and the same shared engine used by Gap-01/SG-6/SG-7. Trap: do not let the corrected bulk magnitude ($\sim-2.9957\times10^{94}$) be read as gap-closing — it never is.
(a) Precise statement. The gate is named for OS reflection positivity; the delivered object is a BRST no-ghost certificate. For free/perturbative fields the two are standard-equivalent (covers R2 spin-1 + R7 free-graviton). For the full interacting branch the OS↔BRST equivalence is part of the open construction. (c) Close: establish the OS↔BRST bridge for the interacting active branch — folded into R3 + R6. Terminal: DISCLOSED-CORRECTED (name precisely which positivity object is delivered) → full-branch equivalence stays OPEN (scoped). Trap: presenting the BRST no-ghost certificate as full OS positivity for the interacting branch. (e) Leverage: purely a scoping/disclosure correction; no new computation, but it keeps the headline honest.
Dissolved ≠ solved. The constructive positivity in the uniform continuum limit ($a\to0$, $L\to\infty$) for the full interacting 4D gauge theory is the 4D Yang–Mills Clay existence/positivity object. Under the granularity axiom P1 it is the BS-1 continuum unicorn — REDUCED-TO-AXIOM onto P1, axiom-conditional, NOT solved. Dissolving it onto P1 is a limit on all knowledge (no one on Earth has crossed this wall, for any theory), not a defect of this program — but "dissolved" is explicitly not "solved," and we never print the continuum result as proven. It is correctly non-axiomatizable beyond P1: closable only by a genuinely new, target-blind constructive-QFT idea no one currently has.
Selection ≠ derivation; given-E ≠ derivation-of-E. The frozen geometry is load-bearing — it fixes what must come out positive-norm (the per-carrier polarization counts) — but it supplies no positivity certificate. The retained-sector certificate terminates on E (geometry-supplied content) + textbook positivity machinery, conditional on UQF-4: it is DERIVED-GIVEN-E + standard-result-inherited, not geometry-validated positivity. Supplying the field content is not the same as proving the interacting theory is positive.
What is explicitly NOT claimed. - NOT a blanket "reflection positivity is derived." Finite-cutoff: derived. Uniform continuum: open (R3). Finite residuals: open (R1/R4/R5/R6). - NOT a positivity proof for the full interacting theory after summing all loops + the entire KK tower + the non-perturbative sector. That all-orders global-sufficiency demand is a quantum-gravity-wide open question; demanding it be proven complete is the same universal wall everyone faces (its finite inputs — the boundary $a_6$ coefficient and the UQF-4 $d=4$ boundary class — ARE bounded holes, listed in §6; only the all-orders global-sufficiency demand dissolves). - NOT "UQF-3 ⇒ Yang–Mills mass gap." Even a fully clean UQF-3 is an anomaly/positivity-consistency leg only; the dependency runs mass-gap-machinery → positivity, never the reverse. - NOT a dimensionful gap-closing $a_6$ magnitude. The bulk $\operatorname{tr}[a_6]=-2.817995812\times10^{94}\,\text{GeV}^6$ is a labeled, retracted, scale-anchored consistency coefficient, dissolved-as-ill-posed at odd $D=13$.
The anchors paid. The floor converges to two value-free measured-invariant anchors (source: specialist_reply.txt §3): AXIOM-STABILITY-OF-MATTER (the time-translation generator is self-adjoint and bounded below — a ground state exists) and AXIOM-BORN-SIGN (probabilities are real and non-negative). These are three representations of one contraction/unitarity fact (bounded-below-$H$ / positive-norm / Born-sign), not three independent invariants; both are ATOMIC and measured-invariant. Beyond the floor sits one explicitly-named, Precisely-OPEN Clay-class wall (AXIOM-RECONSTRUCTION-BRIDGE-OPEN: on $\mathfrak{B}_{\rm active}$ the full interacting Euclidean measure exists and reconstructs stability + Born-sign), with no constructive lever from the geometry, plus the exported pointer AXIOM-BRST-NILPOTENCY-INHERITED ($Q^2=0$, → UQF-4, not a UQF-3 anchor).
Honest endpoint. UQF-3 stays OPEN at its honest ceiling — serious candidate / partial unification, NOT validated. The retained 4D zero-mode + low-KK sector (gauge / matter / Higgs + linearized graviton) is CERTIFICATE-CONDITIONAL (perturbative Kugo–Ojima / BRST no-ghost, conditional on UQF-4) and finite-cutoff OS/Lüscher reflection positivity is a derived fact; while the full interacting positive physical Hilbert space remains OPEN behind the 4D Yang–Mills constructive-positivity wall (R3, now shown a proper sub-wall of the mass gap), the uncomputed $S^1/\mathbb{Z}_2$ boundary $a_6$ coefficient (R4), the inherited $d=4$ boundary anomaly (R1, AUDIT), the non-factorizing full-KK / all-loop coupling (R6), the IR inheritance (R5), and above-cutoff graviton / UV completion (R7) — because the GLOBAL-ADMISSIBILITY-COMPOSITION theorem is unproven and local admissibility does not compose into global admissibility.
STATUS-UPGRADES:0. Frozen branch dcc66f1b2685 / a5b1e6f9d951 READ-ONLY.
given-E ≠ derivation of E · dissolved ≠ solved · selection ≠ derivation
a textbook citation ≠ a from-content proof for the active branch · anchored ≠ derived
CERTIFICATE-CONDITIONAL ≠ unconditional · finite-cutoff derived ≠ continuum derived
Dossier built per DOSSIER_BUILD_PROTOCOL.md v1 (2026-06-29). Synthesizes: PER_GATE_DOSSIERS/UQF3_COMPLETION_HANDOFF/{01_DOSSIER, 02_CURRENT_STATE, 08_UQF3_POSITIVITY_LEDGER_2026-06-25, specialist_reply}; computational_runs_2026-06-23/a6_computation/A6_Z2_DEFECT.md; articles/GATE_BRIEF_UQF3.html; 30pagedoc/handoffs/{UQF3, GAP01}.md. Every number traced to a source actually read; uncomputed quantities marked OPEN. Physics shared; QC-chip / out-of-scope engineering firewalled.