W15 — Born-rule weight / measure — rendered package. Rendered from W15_BORN_WEIGHT_CLOSURE_PATH.md; frozen technical content unchanged by rendering.

W15 — Born-rule weight / measure

Wall: The Born-rule weight / measure wall · Closure routing via /closure

Where this wall’s gate stands — board of record, ratified 2026-07-08. The requirement-gate this wall serves, Born — probability weight, is RESOLVED at +0 · CERTIFIED-IRREDUCIBLE on the live gate board (33 RESOLVED at +0 · 0 ANCHORED at +1 · 0 OPEN). The ledger row, verbatim: “The exponent in the ‘square the amplitude’ rule for quantum odds comes out to exactly two — not approximately two, but forced to be two, with no room to nudge it.” The closure is exactly the honest reduction this wall demanded: one named, value-free posit — BORN-A1, non-contextuality — is declared openly, and Gleason’s theorem then forces the exponent p = 2 above the measured quantum-kinematics floor, with no numbers tuned or fit. This page records the bounded residual work standing under that reached terminal; the wall register’s OPEN-ANCHORED vocabulary below grades the wall’s remaining work items, never the gate. And on the board’s separate honest axis, 0 of 33 gates are physics-closed — nothing is solved from nothing; the measured floor is ≥ 1 by design, and the board says so plainly.

Routing decision

REDUCE-TO-ROOT if measure assumptions are declared; DERIVE only if a non-circular proof is supplied.

Deep-root link(s)

Reduction / dissolution verdict

Can be reduced only honestly: if noncontextuality, additivity, and measure uniqueness are declared as paid assumptions. Without that, the wall remains open.

Leg A1 reconciliation — 2026-06-29. Leg A1 (non-contextuality / cross-context gluing $a=a'$) is REDUCED-TO-AXIOM onto a declared non-contextuality posit — the honest endpoint — carrying the standing anti-claim that this posit is import-forbidden as a derivation: it is NOT forced by the $K_6$ admissibility-chamber algebra (invariance is provably silent on cross-context gluing). These are complementary, not contradictory: reduced-to-a-declared-axiom, with the explicit guard that the axiom is not claimed to be derived. Convertible path: computing the von Neumann algebra type on the chamber algebra (no type-$I_2$ central summand $\Rightarrow$ Gleason forces non-contextuality) would upgrade A1 to ROOT-FORCED-GIVEN-the-algebra; until then it is reduced-to-axiom, not derived.

How the gate actually closed (2026-07-08): by the honest branch of this routing decision. The measure assumption is declared — the named, value-free posit BORN-A1 (non-contextuality) — and with it granted, Gleason’s theorem forces the exponent to exactly 2; the gate rests as CERTIFIED-IRREDUCIBLE at +0 with the posit shown, not smuggled (Born dossier). The convertible path above — deriving the von Neumann algebra type from the chamber algebra — remains the named upgrade that would turn the declared posit into a forced one: bounded work, listed below, standing under a closed gate.

Steps to the endpoint

  1. Separate probability normalization, additivity, noncontextuality, and $|\psi|^2$ weight.
  2. Attempt derivation from framework roots only.
  3. If noncontextuality is imported, declare it.
  4. Prove A2 measure uniqueness or leave it open.
  5. Check no-signalling compatibility.

Success criterion / falsifier

Derivation success: non-circular $|\psi|^2$ proof from roots, not imported Born rule. Reduced endpoint success: all measure assumptions named and minimal. Falsifier: alternate admissible measure consistent with declared roots/records.

Terminal status assigned

REDUCED-TO-AXIOM / OPEN-ANCHORED; DERIVED only with proof

That is the wall register’s grade for the wall’s own remaining work — register vocabulary, kept deliberately. It grades the residual derivation debt, not the requirement-gate: on the board of record the gate is closed — Born — probability weight · RESOLVED +0 · CERTIFIED-IRREDUCIBLE (ledger · dossier, ratified 2026-07-08).

Residual work

Noncontextuality, A2 uniqueness, non-circular weight derivation.

Anti-claims

Systematic multi-pass analysis — 2026-06-29

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

Published history — dated snapshot, kept verbatim. The block below is the frozen output of the 2026-06-29 wall-analysis run, preserved unedited: publishing dated working notes is part of this site’s discipline. The gate statuses quoted inside it (“ANCHOR-CLOSED but physics-status OPEN,” the leg-level grades) are mid-analysis snapshots that predate the board’s ratification. The current status of this wall’s gate is the one at the top of this page — Born · RESOLVED +0 · CERTIFIED-IRREDUCIBLE (ratified 2026-07-08; the live ledger wins every conflict). The wall-work items it names remain the live residual list for this wall.

Verdict: WALL_RECORDED
Smallest missing rule (the precise hole):
  A framework-internal rule that forces the cross-context gluing a=a' (i.e. non-contextuality of the frame function p on P(A)) from the frozen K6 admissibility-chamber operator algebra ALONE — with the von Neumann algebra TYPE (no type-I2 central summand; Z(A) and the mixed anomaly xi_R4 uncomputed, I2-DANGER-OPEN) DERIVED from the geometry rather than imported. Given that single rule, Gleason/Busch/Bunce–Wright deliver Tr(rho E); without it the geometry->non-contextuality->Tr(rho E) chain is broken at the first link. (The A2 sibling-rule: a framework-FORCED compact symmetry acting transitively/ergodically on C, which finite W=S3 provably cannot supply.)
Terminal status: on the ratified board, Born — probability weight (Gap-14/Gap-15) stands CERTIFIED-IRREDUCIBLE · RESOLVED +0 (see /gates/); register axis: OPEN-ANCHORED / WALL-RECORDED — the bounded open work standing under that terminal: Genuine wins: A1 separation countermodel (DERIVED, terminal as a no-go) and the K6=SU(3)/T2 Weyl-chamber A2 existence-localization (DERIVED-GIVEN-E); (G1)/(G2) forcing and the non-contextuality=>Tr(rho E) conditional theorem are DERIVED-GIVEN-E. Leg A1 functional form is AXIOM-OPEN / import-forbidden (NOT REDUCED-TO-AXIOM, NOT closed); vN algebra type / xi_R4 is OPEN / import-dependent (I2-DANGER-OPEN); Leg A2 selector measure is OPEN (uniqueness provably fails, point-mass disclosed, inter-sector weights silent); chamber uniqueness OPEN; CH-2 honest-halts; measured floor ANCHOR-BORN-QUANTUM-KINEMATICS is a MEASURED-ANCHOR (legitimate >=1 floor). No gate physics-CLOSED. no status changes.

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