W09 — Moduli stabilization: saddle-vs-minimum and c_loop — rendered package. Rendered from W09_MODULI_STABILIZATION_CLOSURE_PATH.md; frozen technical content unchanged by rendering.

W09 — Moduli stabilization: saddle-vs-minimum and $c_{\rm loop}$

Wall: The moduli-stabilization wall (saddle-vs-minimum + c_loop) · Closure routing via /closure

Where the gates above this wall stand (ratified board, 2026-07-08). Two requirement-gates sit over this wall, and both are RESOLVED at +0 · DERIVED-GIVEN-anchor on the live gate ledger. SG-6 — moduli / vacuum stability: the vacuum sits still, and the geometry proves it — the sign that decides stability comes from a quantity built from the frozen shape that doesn't change under relabeling or reflection, an exact statement rather than a numerical guess, and it points to a genuine resting point, not a slide toward instability; the K₆ = SU(3)/T² three-way swap symmetry forces the resting point and splits the three size-parameters into one breathing size plus a two-part shape change, with the one outside number (the electroweak scale, 246 GeV) charged openly as the second ruler and the special resting point declared as a stated assumption. UQF-10 — compactification consistency: the curled-up shape holds together quantum-mechanically — the infinite tower of heavy modes cancels in exact boson–fermion pairs with no leftover knob, the swap symmetry collapses the stiffness matrix to a single number (Schur's lemma), and independent internal cross-checks (a curvature number of 5/2, a curvature eigenvalue of 5/12, a supersymmetric trace of −4, a curvature stiffness of +1/3) all agree. The board stands at 33 gates RESOLVED at +0 · 0 ANCHORED at +1 · 0 OPEN — and, on the separate honest axis stated just as plainly, 0 of 33 are physics-closed: every terminal rests on declared measured anchors.

What this page records is the wall's bounded residual program standing under those reached terminals — the full admissible-rep multiplicity table, μ_cell, the c_loop sign certificate. Open strengthening work under closed gates — not open gates.

Routing decision

ANCHOR as an OPEN bounded stabilization wall.

Deep-root link(s)

Reduction / dissolution verdict

Cannot reduce/dissolve. The missing objects are concrete: admissible-rep multiplicity, saddle/minimum decision, $\mu_{\rm cell}$, and $c_{\rm loop}$ sign. The saddle-vs-minimum decision has since been made at gate level — the stability sign is an exact, relabeling-invariant statement of the frozen shape and points to a genuine resting point (SG-6) — while the multiplicity table, $\mu_{\rm cell}$, and the $c_{\rm loop}$ sign certificate remain this wall's bounded residuals, shown openly under the terminals above.

Steps to the endpoint

  1. Freeze moduli coordinates and potential terms.
  2. Define admissible representation multiplicities before evaluation.
  3. Compute Hessian and classify saddle/minimum.
  4. Compute or explicitly anchor $\mu_{\rm cell}$.
  5. Compute $c_{\rm loop}$ sign target-blind.
  6. Link SG-6 and UQF-10 so the wall is not double-counted.

Success criterion / falsifier

Success: stabilized positive sector or bounded allowed saddle; $\mu_{\rm cell}$ emitted/anchored; $c_{\rm loop}$ sign reproduced. Falsifier: negative mode that cannot be lifted, wrong sign, or uncharged tuning.

Outcome on the stability leg (ratified): the vacuum sits still, and the geometry proves it. The deciding sign comes from an exact invariant of the frozen shape and points to a genuine resting point; the K₆ swap symmetry forces that point and collapses the stiffness matrix to a single number, and the internal cross-checks (5/2, 5/12, −4, +1/3) agree. The negative-mode falsifier stays armed on the row. SG-6 dossier · UQF-10 dossier.

Terminal status assigned

OPEN-ANCHORED / MODULI-BOUNDED — the wall register's own row vocabulary for this bounded residual program. The gates above it, SG-6 and UQF-10, stand RESOLVED at +0 on the ratified board.

Residual work

Multiplicity, $\mu_{\rm cell}$, $c_{\rm loop}$, full stability certificate — bounded strengthening work under the ratified terminals of SG-6 and UQF-10. Delivering these strengthens the rows; not delivering them changes nothing on the board.

Anti-claims

Systematic multi-pass analysis — 2026-06-29 (historical audit snapshot)

The generic multi-pass analysis was run on this wall and independently cross-checked (verdict confirmed; no overstatement / target-dependence / misclassification).

Preserved verbatim as the dated working record. Its gate-status lines — “no minimum and no saddle asserted” and the route-inconsistency note — are the mid-audit reading and are superseded by the ratified board quoted at the top of this page: SG-6 and UQF-10 stand RESOLVED at +0 on the live /gates/ ledger, the resting point proven by an exact invariant of the frozen shape (SG-6 · UQF-10). The wall-scoped residuals (multiplicity table, $\mu_{\rm cell}$, $c_{\rm loop}$ certificate) remain this wall's live bounded work.

Verdict: NOT_EXHAUSTED_BLOCKED
Smallest missing rule (the precise hole):
  The admissible-rep (p,q) multiplicity table on K6=SU(3)/T^2 together with the regularization-stable s=-1 (zeta) continuation of the graded supertrace sum (-1)^F d n (w1^2+w2^2) on the ISOLATED trace-free shape doublet at a critical point. This single object is the generator the doublet-sign segment (R2->R3) needs to be enumerable; its absence blocks the Casimir net sign, which in turn blocks the 2x2 signature. (The closed-form per-shell multiplicity m0(p,q)=min(p,q)+1 if (p-q)=0 mod3 else 0 is verified, but the full table and the regularized supertrace are not assembled, so the segment is NOT_EXHAUSTED_BLOCKED, not merely uncomputed.)
Terminal status: OPEN-ANCHORED / MODULI-BOUNDED — criticality DERIVED-GIVEN-E and free; shape doublet LEANING STABLE (tree-level {+1,+1}, target-blind, cross-confirmed against a falsifiable single-slice -1); breathing singlet c_loop route-INCONSISTENT (Route A -43/504 anchor-falsified vs canonical -16/315); mu_cell firewalled (no v-independent value). No minimum and no saddle asserted; bounded falsifier armed but not yet runnable. Anchored != physics-closed; 0 gates physics-CLOSED.

Shared closure invariant

A gate is terminal when its wall has the correct endpoint: derived, reduced to a root, dissolved, anchored, or wall-recorded. It is not terminal merely because the strongest imaginable version of the problem has been solved.

Status-inflation guardrails: - DISSOLVED does not mean SOLVED. - SELECTED does not mean FORCED. - GIVEN-E does not mean DERIVED-E. - MEASURED-ANCHOR does not mean DERIVED VALUE. - WALL-RECORDED does not mean PHYSICS-CLOSED. - A bounded local certificate does not become global completion.

Completion tests for this wall file

This wall file is complete only if: - The wall is named exactly. - The routing decision is explicit. - At least one deep-root link is listed. - The endpoint status is explicit. - Local/banked wins are separated from the missing object. - The success criterion or falsifier is target-blind. - The anti-claims block prevents status inflation. - Any failure to reduce or dissolve is stated honestly. - Residual work is finite where possible; unicorn work is wall-recorded.

Completion report

Tests passed: - Wall named exactly. - Routing decision assigned. - Deep-root links listed. - Endpoint status assigned. - Honest failure to reduce/dissolve stated where applicable. - Success criterion/falsifier included. - Anti-claims included.

Tests failed: - None in this handoff draft.

Open items: - Execute the residual work listed above.

Assumptions made: - Uses the uploaded wall register and the /closure endpoint strategy.

See also