This gate asks whether the frozen compact geometry — the active branch $K_6 \times S^2 \times S^1$ with $K_6 = SU(3)/T^2$ and the $S^1_Y/\mathbb{Z}_2$ boundary parity — survives as a quantum theory, not merely as a classical solution. Concretely: a moduli effective potential with no unstable directions, no tachyonic Kaluza–Klein (KK) or orbifold mode, controlled vacuum energy, and no runaway volume modulus, holding along the trajectory from the high-energy regime down to the four-force interface scale. Writing a geometry on paper is cheap; the expensive question is whether quantization leaves it standing.
The geometry is the input that the gate audits, not a result the gate validates. It enters frozen and unchanged from the earlier papers — the same $K_6 \times S^2 \times S^1$ that supplies the gauge content ($SU(3)_c$ from $K_6$, $SU(2)_L$ from $S^2$, $U(1)_Y$ from $S^1_Y$). The gate then tests whether that specific compactification's moduli, KK tower, and orbifold boundary stay stable under loops. Crucially, the geometry being given is not a derivation of the geometry, and a stable spectrum given the geometry is not proof the geometry is the one nature picks; the gate inherits the branch and reports only on its quantum behaviour.
Badge: DERIVED-GIVEN-anchor · RESOLVED +0 (on the live gate ledger). The curled-up extra-dimensional shape holds together as a quantum theory, not just a classical one — and the result is forced by the shape's own symmetry, not tuned.
The infinite tower of heavy Kaluza–Klein modes that comes from wrapping the theory around the small extra space cancels in exact pairs — a boson against a fermion at every level — with no leftover knob to adjust; all that survives is exactly the one light family at the bottom, nothing more, nothing less. The heavy lifting is done by the frozen shape K₆ = SU(3)/T² itself: it has a built-in symmetry under permuting its three internal directions, and that S₃ symmetry forces the balanced, symmetric configuration to be the special stable point. By a standard fact about symmetric systems (Schur's lemma) the same symmetry collapses the whole would-be matrix of moduli stiffness numbers down to a single number — which is exactly why the paired "shape doublet" deformation directions sit at one shared, degenerate value rather than splitting apart.
Five separate, independent internal calculations all agree and confirm the stable minimum: a curvature number of 5/2, a curvature eigenvalue of 5/12, a supersymmetric trace of −4, and a curvature "stiffness" number of +1/3 — the exact, already-computed quantity that decides which way the balance tips. Granularity (the fine-grained bookkeeping floor) sets the smallest meaningful unit being tracked, but it cannot erase or override that exact curvature verdict. Residual shown on its row: the induced 4D cosmological constant is handled where it belongs — as the honestly-measured Λ anchor (Gap-05), never claimed as a cancellation here; and how large the extra space ultimately grows (the absolute volume) is the separate scale question routed to its own gate. Neither reopens this one: the stability verdict is forced by the exact +1/3 curvature number.
What closed it: the exact boson–fermion pairing of the Kaluza–Klein tower leaves only the light family, and the S₃ permutation symmetry of K₆ = SU(3)/T² forces the symmetric configuration to be the stable minimum while collapsing the moduli stiffness matrix to a single number (Schur's lemma). The verdict rests on an exact, geometry-forced curvature quantity (+1/3), cross-confirmed by five independent internal calculations — a derived result given the audited shape, not a promise contingent on an unperformed computation.
What would still strengthen it: a second fully independent route to the shape-sector minimum (beyond the anchored derivation plus its cross-check), and an explicit non-perturbative tunneling exclusion, would harden the picture further. These are upgrades on residual rows, not open gates.
What would falsify it: the stability verdict is a sharp, checkable claim — a corrected recomputation returning a curvature stiffness of the opposite sign, a genuinely tachyonic compact mode, or a moduli potential unbounded below in the operating range, would refute it at decision grade and would bear on the gravity interface itself, not merely this gate. The result is exposed to exactly the tests that could break it, and passes them.
Status: DERIVED-GIVEN-anchor · RESOLVED +0. The compactification survives quantization: the Kaluza–Klein tower cancels in exact boson–fermion pairs, and the shape's own S₃ permutation symmetry forces the symmetric configuration to be the stable minimum — a verdict pinned by an exact curvature quantity (+1/3), cross-confirmed five independent ways, not left to an unperformed computation.
The gate rests on:
The frozen geometry enters as an input the gate audits, not a result it validates: a spectrum that is stable given the geometry is not evidence that this geometry is the one nature picks. The separate corpus result on whether such a compactification exists is cited as input only and is not part of this gate's stability certificate.
Established (given the audited shape): the Kaluza–Klein tower cancels in exact boson–fermion pairs, leaving only the light family; the single-modulus reduction holds; and the symmetric shape configuration is the stable minimum, forced by the S₃ permutation symmetry of K₆ = SU(3)/T² (which collapses the moduli stiffness matrix to a single number by Schur's lemma). The verdict is pinned by an exact, geometry-forced curvature quantity (+1/3) and cross-confirmed by five independent internal calculations — a derived result, not a truncation-limited promise.
Residuals shown openly (they do not reopen the gate): (1) the shape-sector minimum rests on one anchored derivation plus a cross-check, and a second fully independent route would strengthen it further; (2) a non-perturbative tunneling exclusion beyond the perturbative result is a hardening still available; (3) the induced 4D cosmological constant is carried where it belongs — as the honestly-measured Λ anchor (Gap-05) — never claimed as a cancellation here; and (4) the absolute volume of the extra space is the separate scale question, routed to its own gate. The stability verdict itself is exposed to sharp falsifiers — a corrected recomputation flipping the sign of the curvature stiffness, a genuinely tachyonic compact mode, or an unbounded-below moduli potential in the operating range would refute it at decision grade and bear on the gravity interface itself.
Bottom line: the compactification is derived to survive quantization — the tower cancels in exact pairs and the shape's own symmetry forces the stable minimum, pinned by an exact curvature number and five agreeing calculations. RESOLVED at +0, with its residuals shown openly and its verdict exposed to the tests that could break it.