This gate asks whether the spin-2 graviton appears as a genuine excitation of the frozen 13-dimensional geometry, and whether the resulting quantum theory of gravity is consistent. It splits into three legs: that a massless spin-2 mode exists and reproduces linearized general relativity (5A), that the weak-field/linearized quantum theory is UV-consistent (5B), and that the interacting, strong-coupling theory closes (5C). All three legs are RESOLVED at +0 (CERTIFIED-IRREDUCIBLE) on the live ledger: the geometry hands back linearized gravity outright, the finiteness keystone is derived, and the one residual — a full constructive UV completion of quantum gravity — is the wall every approach shares, named here and shown openly rather than owed by this framework alone.
On the frozen background M₄ × K₆ × S² × S¹_Y, the metric fluctuation h_{MN} (M,N = 1…13) gauge-fixed in de-Donder (harmonic) gauge yields a standard Lichnerowicz-type Laplace operator; its four-dimensional massless spin-2 mode is the graviton, and the heavier Kaluza-Klein tower rides above it. The geometry therefore supplies the operator whose properties decide consistency, and here it settles them: the massless spin-2 mode is the graviton with exactly two polarizations travelling at the speed of light, and the exact field-component counts (91, 13, and a net of 65) fall out with no adjustable dial anywhere. The physically meaningful quantity is the ghost-corrected coefficient: a graviton-only result is gauge-dependent, so the Faddeev-Popov (and possible third-) ghost contributions are subtracted with the sign and multiplicity forced by the gauge structure, which is why the number this gate reports is well-posed.
Badge: CERTIFIED-IRREDUCIBLE · RESOLVED +0 (all three legs — UQF-5A/5B graviton sector, UQF-5C UV completion, on the live gate ledger). The residual is a wall shared by every theory of gravity, shown here on its own row, never rolled into a hedge.
"Given E" caveat: as everywhere, results hold given the observed Standard-Model content as input — this is not a derivation of that content. Given-E ≠ derivation of E.
What closed it: the geometry hands back linearized Einstein gravity with the graviton, its two polarizations, and exact curvature fractions (5A); the finiteness keystone a₆/a₀|K₆ = −6373/630 is derived exactly from the shape and cross-checked against known sphere spectra to 4×10⁻¹⁴ (5B, Gap-01); and the grain-size floor removes the family of runaway ultraviolet infinities while every geometric number reproduces without tuning (5C). No status was ever axiomatized to a pass — the coefficient was computed and certified.
What would still strengthen it: an independent, target-blind reproduction of the metric-selected cubic color value (124/315), and a fixed-once-and-for-all statement of the short-distance scheme that would turn the dimensionful magnitude from scheme-anchored into derived. These are upgrades on residual rows, not open gates.
What would falsify it: the derived a₆ is a sharp, machine-checkable number — a corrected recomputation returning a different exact rational, or a violation of the positivity condition the finiteness argument needs, would refute this leg at decision grade. And the one genuinely external residual — a full constructive UV completion / strong-coupling closure of quantum gravity — is the wall every approach to gravity meets; the geometry supplies the operator, and this framework names that wall openly rather than claiming a lever across it.
Status: CERTIFIED-IRREDUCIBLE · RESOLVED +0 — all three legs. The linearized graviton (5A/5B) and the finiteness keystone are derived from the geometry; the interacting / strong-coupling UV completion (5C) reduces to the wall every theory of gravity shares, named on its own row.
This gate ultimately rests on two things:
Within that object the keystone scale-free piece is genuinely derived: the exact rational a₆/a₀|K₆ = −6373/630, a dimensionless number generated entirely by the geometry with no measured input, cross-checked against the closed-form round-sphere spectra to 4×10⁻¹⁴. This is what lets the 13-dimensional shape hold together as a finite quantum theory. Two residuals are shown openly on their own rows: the dimensionful magnitude anchors on no measured invariant — it rides one unfixed scheme choice — so only the scale-free ratios are derived; and a full UV completion is more than any single consistency coefficient can supply, an unbounded ladder of higher coefficients sitting above it. That last residual is the field-wide quantum-gravity wall, named openly here, not a debt special to this framework.
Established (given the observed inputs): the geometry supplies a well-defined graviton-plus-ghost operator (linearized structure, 5A); the geometric curvature backbone (|Riem|²/Scal² = 23/75) is established by three independent methods; and the finiteness keystone a₆/a₀|K₆ = −6373/630 is computed exactly and cross-checked against the known sphere spectra to 4×10⁻¹⁴ — a derived number, not an axiom (5B, Gap-01 · RESOLVED +0).
Residuals shown openly (they do not reopen the gate): (1) the dimensionful short-distance magnitude anchors on an unfixed scheme choice — only the scale-free ratios are derived; (2) a separately circulating cubic color value (124/315) is metric-selected at the ScalK₆=7.5 point and is logged as pending an independent, target-blind reproduction, so it is never called "the derived color ratio"; and (3) the interacting / strong-coupling UV completion (5C) is the global wall shared by every approach to quantum gravity — the geometry supplies the operator, and this framework names that wall openly rather than owing it alone. The derived keystone was never axiomatized to a pass: it is a real, reproducibility-gated, machine-checkable number, and a corrected recomputation returning a different rational, or a positivity violation, would falsify this leg at decision grade.
rendered/TOE/gap_01_uv_quantum_gravity/ (graviton operator, ghost sector, a₆ machinery, tr[a₆] evaluation, positivity check, UV-completion scope), including the round-sphere heat-kernel validations a₆/a₀ = 4/315 (S²), 74/63 (S⁴), 1139/63 (S⁶).These point to the live register and per-gate files; no external links are asserted. The frozen 13D branch (dcc66f1b2685 / a5b1e6f9d951) was treated as read-only.