W04 — Granularity ⇒ MDL metric-selection bridge — rendered package. Rendered from W04_GRANULARITY_MDL_CLOSURE_PATH.md; frozen technical content unchanged by rendering.

W04 — Granularity ⇒ MDL metric-selection bridge

Wall: The granularity ⇒ MDL metric-selection wall · Closure routing via /closure

Where the gates above this wall stand (closure-of-record, ratified 2026-07-08). The gates this bridge serves have reached their ratified terminals on the live gate board — 33 RESOLVED at +0 · 0 OPEN: Deep Root — Granularity / cost-floor: CERTIFIED-IRREDUCIBLE · RESOLVED +0 · Deep Root — Shape (13D selector): DERIVED-GIVEN-anchor · RESOLVED +0 · SG-1 — geometry / shape selection: DERIVED-GIVEN-anchor · RESOLVED +0. Those closures already carry this wall’s content honestly: the common-currency rule for adding record-costs into one economy score is a named, value-free axiom — stated up front, disclosed on every row that uses it, and not yet derived from anything deeper. That is exactly what this wall records: whether the additive (MDL) cost metric is the unique admissible one is bounded open mathematics standing under reached terminals — the axiom is counted, never smuggled. On the separate honest axis, stated plainly: 0 of 33 gates are physics-closed.

Routing decision

REDUCE-TO-ROOT now; DERIVE only if the bridge theorem is later proven.

Deep-root link(s)

Reduction / dissolution verdict

Reducible to Granularity plus a declared bridge axiom. Granularity proves no unpaid labels; it does not yet prove MDL is the unique admissible simplicity metric.

Specialist attack update — 2026-06-29. A second supporting root — Nonseparability — was added, and the wall resolves into five sub-results: W4-A dimension-first-only as a complete cost metric is DISSOLVED (blind to primitive labels and to primitive independence assumptions); W4-B no-unpaid-labels REDUCED-TO-ROOT (Granularity + Record Interface); W4-D no-unpaid-independence/factorization REDUCED-TO-ROOT (Nonseparability + Causal-Order guard — exact independence $\rho_{AB}=\rho_A\otimes\rho_B$ is a special factorization claim, reviewer-safe with $\sum_a p(a)\rho_{B\mid a}=\rho_B$, no superluminal signalling); W4-C additive-MDL uniqueness OPEN (bounded representation theorem); W4-E 13D-beats-4D under fair full-generator accounting OPEN/REFUTABLE. The executable layer (C-series symmetric ledgers + target-blind generator compiler) is built but machine-verification is HARD-BLOCKED-ON-B13 — the SG-8 flavor observable-map source tree is not yet mounted/hashed. Net: dimension-first-only dissolved; labels + factorization reduced to roots; uniqueness + fair-ledger open. No status change.

Steps to the endpoint

  1. Separate no-unpaid-labels from MDL dominance.
  2. Declare the bridge: simplicity equals minimal injected operational record cost.
  3. Charge injected reals, selectors, quotient choices, grammar rules, and spectra.
  4. Run all competitor comparisons under the same codebook.
  5. Keep dimension-first/EFT-cheaper challenges open unless formally defeated.

Success criterion / falsifier

Success for derivation: prove any admissible cost metric satisfying record faithfulness, encoding invariance, additivity, precision monotonicity, and no hidden data is equivalent to MDL up to constants. Falsifier: an admissible non-MDL metric under which a rival is cheaper after full unfolding.

Terminal status assigned

REDUCED-TO-AXIOM / AXIOM-OPEN bridge; future DERIVED only with theorem — refined 2026-06-29: PARTIAL-DISSOLUTION (dimension-first-only) + REDUCED-TO-ROOT (labels via Granularity; factorization via Nonseparability) / AXIOM-OPEN BRIDGE (additive-MDL uniqueness OPEN); executable layer PARTIAL-CLOSED / ROOT-CONDITIONAL / HARD-BLOCKED-ON-B13

Residual work

Formal theorem Granularity ⇒ MDL remains open.

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 — frozen audit record of 2026-06-29, kept verbatim below (one internal process label reworded to plain physics; the technical content is untouched). The status lines inside this record predate the board ratification of 2026-07-08 and are historical. The closure-of-record is the Granularity deep-root ledger row and dossier — CERTIFIED-IRREDUCIBLE · RESOLVED +0, quoted at the top of this page — with the bridge axiom counted openly on every row that uses it. The wall-level finding itself — additivity is the assumed premise, uniqueness the open theorem — stands unchanged.

Verdict: AMBIGUOUS_SURVIVORS
Smallest missing rule (the precise hole):
  A representation theorem deriving the ADDITIVITY of the cost-aggregation functional from granularity ALONE — i.e. a Layer-2 rule that supplies a finite weight ratio between a dimension-bit and an anchor-bit, thereby excluding the dimension-first lexicographic order (an unbounded/infinite weight on k_dim) as a complete cost metric. Granularity demonstrably fixes only the domain (finite bit-strings) and the per-record cost b=log2(1/Delta_0); it provides NO weight asymmetry, so additive MDL and dimension-first lex are both well-defined on the identical space and granularity does not discriminate. The corpus 'granularity ⇒ MDL' theorem (layer-1-metric.md / SHAPE_METRIC_RESULT.md) ASSUMES additivity as record-cost axiom #3 — that is the conclusion smuggled into a premise (the additivity premise restates the conclusion instead of deriving it), so it proves uniqueness only AMONG record-cost metrics and cannot forbid the aesthetic dimension-first preorder.
Terminal status: PARTIAL-DISSOLUTION (dimension-first-only as a COMPLETE metric, W4-A) + REDUCED-TO-ROOT (no-unpaid-labels W4-B via Granularity+Record-Interface; no-unpaid-factorization W4-D via Nonseparability+Causal-Order) / AXIOM-OPEN BRIDGE (additive-MDL uniqueness W4-C OPEN; metric-selection = one named value-free posit AXIOM-COMMON-CURRENCY, not a theorem; 13D-vs-4D fair ledger W4-E OPEN/REFUTABLE) · executable layer PARTIAL-CLOSED / ROOT-CONDITIONAL / HARD-BLOCKED-ON-B13. 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