W07 — Non-perturbative anomaly descent / BV-BRST — rendered package. Rendered from W07_ANOMALY_DESCENT_CLOSURE_PATH.md; frozen technical content unchanged by rendering.

W07 — Non-perturbative anomaly descent / BV-BRST

Wall: The non-perturbative anomaly-descent (BV-BRST) wall · Closure routing via /closure

Where the gates above this wall stand (ratified board, 2026-07-08). Three requirement-gates sit over this wall, and all three are RESOLVED at +0 · DERIVED-GIVEN-anchor on the live gate ledger. SG-4 — hypercharge / anomaly: every one of the six local consistency sums lands on exactly zero, verifiable by hand, and the one further global, shape-based test with no wiggle room came back at the required safe value of zero — anything else would have flatly ruled the theory out. UQF-4 — global anomalies: the deepest quantum self-consistency check comes back clean on two completely independent paths at once; in the full target ring the degree-five host class vanishes identically (y₂·x₃ = 0, resting on a cited external result, Fan arXiv:2503.23399), so there is literally no class left to host a residue. UQF-7 — anomaly descent: the three families stay one-handed after quantum effects are switched on — the obstruction that could have created mirror partners is forced to land exactly on the do-nothing case, and a full family's triality nets out to zero. The Yang–Mills wall this page rightly refuses to touch is itself terminal: Gap-02 stands RESOLVED at +0 · CERTIFIED-IRREDUCIBLE — reduced exactly to the Clay Millennium problem, the gap value honestly measured, the missing continuum proof claimed by no one. 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 non-perturbative BV-BRST descent object, ξ_R4, and 16/16 quantum coverage. That is strengthening work under closed gates, shown openly on the rows themselves — not open gates.

Routing decision

ANCHOR as global-consistency residual; DERIVE if a descent route is found.

Deep-root link(s)

Reduction / dissolution verdict

Cannot reduce/dissolve. Local anomaly cancellation cannot dissolve the global BV-BRST/descent wall — and the framework never pretended it could: the missing objects are real quantum consistency checks, named and bounded. They are residuals shown openly under the reached terminals above, not debts those terminals hide.

Steps to the endpoint

  1. Separate local anomaly arithmetic, global descent, BV-BRST order-hbar nilpotency, xi_R4, and 16/16 coverage.
  2. Compute xi_R4 target-blind.
  3. Specify the even-degree boundary/eta class and $d=4$ descendant.
  4. Write the BV-BRST nilpotency condition on the descended action.
  5. Prove the route, compute the obstruction, or mark no-known-route.

Success criterion / falsifier

Success: complete descent object, xi_R4 computed and hashed, BV-BRST nilpotency verified, 16/16 coverage complete. Falsifier: nonzero obstruction or failed nilpotency.

Terminal status assigned

OPEN-ANCHORED / GLOBAL-DESCENT RESIDUAL

Residual work

BV-BRST descent, xi_R4, and quantum coverage remain open — bounded, named strengthening work standing under the ratified terminals of SG-4, UQF-4, and UQF-7. Delivering them 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. Two of its statements are superseded on gate status by the ratified board quoted at the top of this page: the snapshot's “no value, and in particular no zero, is asserted” predates UQF-4's resolution — in the full target ring the degree-five host class vanishes identically (y₂·x₃ = 0; dossier) — and its “Gap-02 stays Precisely-OPEN” is the mid-audit reading of what is now the ratified CERTIFIED-IRREDUCIBLE, RESOLVED +0 terminal (dossier): the same discipline, driven to its named endpoint — the Clay proof is claimed by no one, and the gap value is honestly measured. The wall-scoped residuals (descent object, ξ_R4, coverage) remain this wall's live bounded work.

Verdict: NOT_EXHAUSTED_BLOCKED
Smallest missing rule (the precise hole):
  The target-blind evaluation of the integrality/existence obstruction w_2(X) + f*zeta = 0 in H^2(X; Z_2) on the branch's ACTUAL bundle data — specifically the parities of SU(2)_L flux on S^2, U(1)_Y flux on S^2, and the Wilson-line/flux content over the two 2-cycles of K_6, plus the relative/Pin^c boundary object at the S1_Y/Z2 walls. Until this single characteristic-class datum is supplied, the tangential structure xi is not shown to exist on the branch, so there is no MTxi spectrum, no bordism group, and the descent class [omega]_lifted is not even an element of any group — the value question (M2/M3), the order-hbar nilpotency (M3), xi_R4 (SG-4), and 16/16 coverage (UQF-7) are all downstream of and gated by this one missing datum.
Terminal status: OPEN-ANCHORED / GLOBAL-DESCENT RESIDUAL (UQF-4 owner: ANCHOR-CLOSED, AUDIT-OPEN; SG-4 and UQF-7 inherit, counted once; 0 gates physics-CLOSED). Banked given-E: perturbative anomaly cocycle DERIVED-GIVEN-E, classical s^2=0 DERIVED, structural compression to one Anderson-dual even-degree class DERIVED(structural). Binding residual BLOCKED on the xi-existence characteristic datum w_2(X)+f*zeta in H^2(X;Z_2); downstream d=4 value, order-hbar BV nilpotency, xi_R4, and 16/16 coverage OPEN. No value, and in particular no zero, is asserted. Even fully delivered this is an anomaly-consistency leg given E only and cannot touch the Yang-Mills mass gap (Gap-02 stays Precisely-OPEN).

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