W14 — Inflation spectrum derivation — rendered package. Rendered from W14_INFLATION_SPECTRUM_CLOSURE_PATH.md; frozen technical content unchanged by rendering.

W14 — Inflation spectrum derivation

Wall: The inflation-spectrum derivation wall · Closure routing via /closure

Where the gate stands — ratified board, 2026-07-08. The requirement-gate this wall serves, Gap-08 — inflation spectrum (“Does the framework owe an early-inflation prediction?”), is RESOLVED at +0 · DISSOLVED-GIVEN-root on the live gate board. The ledger reads: “The answer is no — and that ‘no’ is itself the meaningful result, not a dodge.” None of the framework’s structural ingredients requires an early-inflation episode, so none is added — while the finite CMB records (scalar tilt, amplitude, tensor bound) remain real measured anchors. As a consistency check only, never a derivation: the candidate scalar tilt n_s ≈ 0.964–0.968 sits in the Planck neighborhood (n_s = 0.9649 ± 0.0042), and the candidate tensor ratio r ≈ 0.0035–0.010 lies below current bounds (BICEP/Keck r₀.₀₅ < 0.036, 95%) — a sharp, refute-only falsifier window that LiteBIRD-era measurements can test. Provenance: anchor ledger · dossier · the residuals.

The board stands at 33 RESOLVED at +0 · 0 ANCHORED at +1 · 0 OPEN (ratified 2026-07-08); on the separate honest axis, 0 of 33 gates are physics-closed — the measured-anchor floor is permanent, and it is stated plainly. This page is the wall-level closure-path register for the optional candidate-derivation lane: the bounded residual work recorded below stands under that reached terminal. None of it is terminal-blocking, and the wall-register row vocabulary (OPEN-ANCHORED, TESTABLE-CANDIDATE, …) grades the wall’s residual lanes, never the gate.

Routing decision

DERIVE if theorem and normalization source-trace are supplied; otherwise OPEN-ANCHORED / TESTABLE.

Deep-root link(s)

Shape
Scale
Record Interface
Causal Order
Granularity

Reduction / dissolution verdict

Cannot reduce/dissolve. The inflation spectrum is a real observational target. Candidate values require theorem and normalization trace.

Steps to the endpoint

State candidate theorem $\lambda^2 = 1/6$.
Derive it or demote to candidate.
Trace the $81/121$ normalization source.
Compute spectral observables with frozen inputs.
Compare against CMB constraints with predeclared tolerance.
Mark outcome as derived, candidate, or refuted.

Success criterion / falsifier

Success: theorem for $\lambda^2=1/6$, source-traced $81/121$ normalization, predictions in observational window. Falsifier: theorem fails, normalization is fitted, or CMB predictions miss.

Outcome on this lane (ratified): the derive-or-demote step completed to a real, banked negative. The kinetic normalization K_σσ = 24 is geometry-forced; the candidate theorem λ² = 1/6 failed and is retired CLOSED-NEGATIVE on the ledger; the surviving geometry-forced slope family {8/3, 4, 22/9} carries the pre-registered, refute-only tensor window r ∈ [3.5, 36]×10⁻³ into LiteBIRD-era tests (~2030). The 81/121 source-trace remains bounded shown work under the closed gate. Gap-08 dossier.

Terminal status assigned

OPEN-ANCHORED / TESTABLE-CANDIDATE

That is the wall register’s row vocabulary for the candidate-derivation lane below — it is not a gate grade. The requirement-gate standing over this wall, Gap-08 — inflation spectrum, reached its ratified terminal: DISSOLVED-GIVEN-root · RESOLVED at +0 — no early-inflation episode is owed, and the finite CMB observables remain intact as measured anchors.

Residual work

Theorem, normalization trace, CMB prediction ledger.

Anti-claims

Candidate spectral value is not derivation.
CMB fit is not source-traced normalization.
Inflation is not closed until theorem exists — this binds the wall’s optional derivation lane only; the requirement-gate (Gap-08) owes no inflation episode and is RESOLVED at +0 by dissolution.

Systematic multi-pass analysis — 2026-06-29

Historical record. The block below is the 2026-06-29 audit output, preserved as published history; it was independently cross-checked at the time (verdict confirmed; no overstatement / target-dependence / misclassification). It predates the 2026-07-08 board ratification: Gap-08 — inflation spectrum has since reached its ratified terminal, DISSOLVED-GIVEN-root · RESOLVED at +0 — the framework owes no early-inflation episode, and the audit’s named holes have since been executed to their honest endpoints: the kinetic normalization K_σσ = 24 has been shown geometry-forced, the λ² = 1/6 candidate theorem was retired CLOSED-NEGATIVE — the register’s own derive-or-demote step completed to a real, banked negative — and the surviving geometry-forced slope family {8/3, 4, 22/9} carries the pre-registered, refute-only tensor window r ∈ [3.5, 36]×10⁻³ into LiteBIRD-era tests, a falsifier that stands as a strength (Gap-08 dossier). What bookkeeping remains is listed above, and none of it is terminal-blocking.

Verdict: NOT_EXHAUSTED_BLOCKED
Smallest missing rule (the precise hole):
  The extraction rule R2 fixing K_sigmasigma=24 from the split n=(6,2,1): the frozen sigma-kinetic dimensional-reduction datum (D.1 region, ~corpus L10268) that would show whether K_sigmasigma=24 is FORCED by the geometry (vs a recorded convention) has not been located/read at the kinetic-metric level, so the split->lambda^2=1/6 forcing segment cannot be fixed. Its companion hole: the (9/11)^2=81/121 normalization factor (III-b) has no source anywhere in the corpus (independently confirmed) and the 24/9=8/3 slope-vs-normalization bridge is unbanked.
Terminal status: OPEN-ANCHORED / TESTABLE-CANDIDATE — NOT_EXHAUSTED_BLOCKED on the K_sigmasigma=24 forcing source (A1-GEOM, frozen sigma-kinetic reduction at ~L10268 un-read) and the (9/11)^2=81/121 normalization source-trace (III-b, no corpus source on record), inside a declared EXCLUDED cosmology sector; convention-independent refute-valid LiteBIRD (~2030) falsifier pre-registered; no status changes; no gate 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.