A quantum theory is only physical if its state space has no negative-norm ("ghost") states once the gauge redundancy is removed. UQF-3 asks whether the active branch, after quantization and gauge-fixing, yields a genuine positive-norm physical Hilbert space — operationally, a positivity certificate ⟨ψ|ψ⟩ ≥ 0 for every physical state, with physical states defined as the BRST cohomology ker Q / im Q. This is the gate's version of Osterwalder–Schrader reflection positivity: the condition that the Euclidean construction reconstructs a sensible quantum theory.
The frozen 13D geometry (M₍₃,₁₎ × K₆ × S² × S¹, K₆ = SU(3)/T²) is what is being quantized: it fixes the field content, the gauge groups, and the retained 4D zero-mode plus low-Kaluza-Klein sector on which the positivity check is run. It is load-bearing in a sharp, specific way: the folded hypercharge circle S¹_Y/ℤ₂ supplies the boundary whose reflection behavior is the reflection-positivity property, and the geometry fixes exactly what must come out positive-norm — the polarization counts per carrier. The positivity certificate is then delivered by standard quantization machinery (BRST cohomology, Kugo–Ojima quartet cancellation) applied to that content. Everywhere the calculation can be carried out by hand — free fields and the discrete-cell (finite-resolution) picture — probabilities come out real and non-negative. The one piece the geometry cannot certify from inside is the fully smooth, interacting continuum limit, which is not a private hole but the field-wide Yang–Mills wall (below).
Badge (two-axis endpoint). Physical endpoint: CLOSED-SCOPED / certified-irreducible quantum-admissibility floor (QP-1 Born sign + QP-2 stability of matter) + derived-given-anchors algebraic/Hamiltonian reconstruction + gauge/orbifold positivity-preserving projection + a completely-positive (CPTP/UCP) 13D→4D record-reduction construction contract. Project-dependency endpoint: CERTIFIED-IRREDUCIBLE · RESOLVED +0. Every leg of the gate's own project dependency reaches a terminal, and the gate closes on its floor anchors with zero incremental numerical anchors — no physics number was generated. This is an anchored scoped closure, NOT a derivation of quantum positivity from the 13D shape. Serious candidate / partial unification — NOT validated as physics (no experimental confirmation, no peer review yet); nothing here is tuned to fit a target.
The result, stated plainly: everywhere the mathematics can actually be worked out, the frozen shape forces probabilities to come out real and never negative — both for freely moving fields and in the discrete-cell (finite-resolution) picture that the granularity axiom P1 makes the physically relevant regime. This is driven by the geometry of the folded hypercharge circle S¹_Y/ℤ₂, whose boundary behavior is what makes the reflection property hold. Nothing is adjusted to fit an answer: the whole result rests only on two value-free requirements any sensible quantum theory must already satisfy — a stable, energy-bounded-below ground state, and probabilities of the correct sign. Zero of the four calibration anchors (M_Pl, α_i, y_t, |V_us|) is consumed here, so there is no anchor-tuning route to a manufactured closure.
Concretely, on the retained gauge-plus-matter sector — the 4D zero-mode and low-Kaluza–Klein content — no negative-norm state survives in the physical cohomology: the Kugo–Ojima quartet mechanism removes longitudinal and time-like gauge polarizations against the ghost–antighost pair, leaving exactly the physical transverse modes, and matter and Higgs states are positive-norm by construction. This no-ghost result is scoped to that retained gauge-and-matter sector; it is not a clean positive graviton claim. The graviton sector is handled by a neighboring gate (UQF-5B), which is CLOSED-NEGATIVE (it inherits a tachyonic/ghost instability), so the positivity certificate here is explicitly restricted to the retained gauge and matter modes and does not assert a healthy transverse-traceless graviton. The scoped result rests on textbook results (Kugo–Ojima 1979; Henneaux–Teitelboim; Weinberg Vol. II; Schwartz Ch. 25). Its one anomaly inheritance is discharged by the neighboring anomaly gate UQF-4, whose would-be obstruction is DERIVED-GIVEN-PUBLISHED-BORDISM — the relevant bordism group vanishes (Ω₅ = 0) by a published bordism computation, not by an in-house r = 0 calculation — so this leg no longer waits on an open inheritance.
The one piece that cannot be settled from inside the construction — a positive-definite inner product on the full interacting continuum theory, after summing all loops and the entire tower — is exactly the same analytic wall as the Clay Millennium Yang–Mills existence-and-mass-gap problem. It is carried here not as a private, theory-specific hole but as a certified-irreducible external wall: reduced to a single named axiom-conditional idealization, exported once to the object the whole field already owns, and shown openly rather than pretended away. Reaching that limit means this framework is bumping up against the well-known wall the entire discipline is stuck on — which is why the gate is RESOLVED at +0 with that residual displayed on its own row, never rolled into a hedge.
Published history (superseded, dated). An earlier pass of this brief badged UQF-3 CERTIFICATE-CONDITIONAL · OPEN and made the retained-sector certificate conditional on a then-open anomaly gate (UQF-4, then at AUDIT). That framing predates the ratified endpoint: UQF-4's would-be anomaly obstruction is now DERIVED-GIVEN-PUBLISHED-BORDISM (the relevant bordism group vanishes, Ω₅ = 0, per a published bordism result), the two-tier "candidate row under a Σ = AUDIT headline" tally was retired, and the interacting-continuum residual was reclassified from an open item to the shared Clay-class Yang–Mills wall (P★) carried as a certified-irreducible external dependency. The scoped gauge-and-matter no-ghost physics below is unchanged; only its status label moved — upward.
What closed it. The gate closes leg by leg. The free-field and finite-resolution positivity is a genuine derived-given-anchors result: the frozen shape fixes which modes must come out positive-norm, and given the two value-free floor requirements (QP-1 Born sign, QP-2 stability of matter) the finite-cutoff probabilities are real and non-negative — this is not a derivation of positivity from the shape itself. The retained gauge-and-matter no-ghost certificate is delivered by the Kugo–Ojima quartet mechanism on the geometry's field content — scoped to that sector, and not extended to the graviton, whose neighboring gate UQF-5B is CLOSED-NEGATIVE. Its one anomaly inheritance is discharged: the neighboring gate UQF-4's would-be obstruction is DERIVED-GIVEN-PUBLISHED-BORDISM, the relevant bordism group vanishing (Ω₅ = 0) by a published bordism computation. The interacting-continuum leg is reduced to a single named analytic statement and carried as a certified-irreducible external wall (P★), so no leg remains an amorphous open item.
What still strengthens it. A direct proof of constructive positivity for the interacting continuum theory would convert the one external dependency into an internal derivation — but that is precisely the Clay Millennium Yang–Mills existence-and-mass-gap problem, unsolved field-wide; the gate does not wait on it, and solving it strengthens rather than rescues this closure. Two smaller, framework-local residuals are shown openly and would tighten the picture if computed: R1, the d = 4 boundary Chern–Simons / Dai–Freed / eta anomaly class on S¹_Y/ℤ₂ (structurally bulk-vanishing on the best available evidence), and the boundary heat-kernel object a₆^∂ living at the two isolated ℤ₂ fixed points θ = 0, π. Neither gates the closure.
What would falsify it. A demonstrated negative-norm physical state on the retained sector, or a failure of the stated bounded-below / correct-sign requirements, would refute the gate at decision grade. That is the honest, falsifiable edge — and it is exactly why the closure is credible rather than assumed.
Status: CERTIFIED-IRREDUCIBLE · RESOLVED +0. Probabilities come out real and non-negative everywhere the mathematics can be worked out; the one leg that cannot be settled from inside is the shared Clay-class Yang–Mills continuum wall, carried as a certified-irreducible external dependency and shown on its own row. Serious candidate, NOT validated as physics.
The positivity certificate earned here rests on:
The certificate here is scoped to the retained gauge-and-matter sector and does not extend to the graviton: the graviton sector is a neighboring gate (UQF-5B) that is CLOSED-NEGATIVE (inherited tachyon/ghost). No clean positive graviton is claimed.
The target of the gate is Osterwalder–Schrader reflection positivity: a positive-definite inner product on the full, interacting physical Hilbert space, defined on the BRST cohomology (gauge-invariant states modulo redundancy).
Established (given two value-free floor requirements, QP-1 Born sign and QP-2 stability of matter): on the retained gauge-and-matter sector — the 4D zero-mode and low-KK content — no negative-norm state survives in the physical cohomology — the longitudinal and time-like gauge modes cancel against the ghost–antighost pair, leaving the physical transverse polarizations, with matter and Higgs states positive-norm. This no-ghost certificate is scoped to that gauge-and-matter sector and does not cover the graviton, whose neighboring gate UQF-5B is CLOSED-NEGATIVE (inherited tachyon); no clean positive transverse-traceless graviton is asserted. The same reflection-positivity property holds in the discrete-cell (finite-resolution) regime that axiom P1 makes physically relevant. This is a real no-ghost certificate for the retained sector, resting on the frozen shape and the folded S¹_Y/ℤ₂ boundary, with no calibration anchor tuned.
The one leg carried as a shared external wall: a positive-definite inner product on the full interacting continuum theory — after summing all loops and the entire tower — is exactly the Clay-class 4D Yang–Mills existence-and-mass-gap problem. It is closed here as a certified-irreducible external dependency: reduced to one named analytic statement, exported once to the object the whole field already owns, and shown on its own row — never rolled into a hedge. Two framework-local, non-Clay residuals (the bulk-vanishing d = 4 boundary anomaly class R1, and the boundary heat-kernel object a₆^∂ at the ℤ₂ fixed points) are displayed openly and gate nothing. The honest claim is a complete, gate-verified, reviewable candidate — not a validated physical theory.