# C4 Proof V2.1

## Verdict

**Villanova CONDITIONAL PASS for the V2.1 pregame, regulation-time score-cloud procedure.**

This is a theorem about a declared selection rule, not a claim that the match
will finish in the four nodes. It is not yet an immutable public lock. Aster
may freeze only after hashing this proof and its ledger with a final card,
receiving Athena's compatible signature, and verifying that every public and
AI-readable surface says the same thing.

## Scope, chronology, and admission

- Fixture: England v Argentina, FIFA World Cup semifinal M102, Atlanta Stadium.
- England goals are first. Scope is regulation time only; extra time and
  penalties are excluded.
- No-leak cutoff: `2026-07-15T14:09:00Z` (`10:09 EDT`).
- `MODEL_PRECOMMIT_V2_1.md` and builder provenance were physically created
  before the V2.1 candidate files. The precommit fixes all coefficients,
  thresholds, constants, score universe, tie break, and builder SHA-256
  `DA5618F38340B94A5B909CC7D27166D3AE71D363D34E5E9CFB94895F2E2EE908`.
- The current builder hash matches that provenance. The V2.1 23-file manifest
  independently hash-matches at audit time.
- 100 of 100 rows are model-valid, 50 per team, with zero retrievals after the
  cutoff. England's Norway record is official regulation `1-1`; the extra-time
  winner is excluded.
- England-vs-Argentina: 5 selected realm rows, `n_eff=4.09341334`, conditional
  sparse pass with precommitted `kappa=5`. Argentina-vs-England: 7 rows,
  `n_eff=6.31718717`, pass.

The rejected `C4_LOCK_V2` root is historical and not used here. V2.1 is a
distinct, precommitted run.

## Declared index and C4 rule

For every `s=(e,a)` in `{0,...,7} x {0,...,7}`:

`L(s) = log P_count(s) + .35 log P_result(s) + .30 log P_margin(s) + .20 log P_total(s) + .12 log P_BTTS(s) + .18 log P_clean(s) + .10 log P_family(s) - penalty(s)`.

`P_family(s)` is the geometric mean of the pre-score result, margin, total,
BTTS, and clean components of the deterministic score family. `exp(L(s))` is
normalized over the 64-cell universe.

Enumerate four-score subsets of the top twelve cells. Require Argentina-win,
draw, clean-fork, and BTTS coverage only when their predeclared thresholds
activate. Maximize

`J(S) = mass(S) + .004*(distinct result classes - 1) + .003*(distinct total classes - 1) + .004*I(BTTS and no-BTTS) - .0035*(adjacent same-result pairs)`.

Break exact ties lexicographically by sorted score tuple. Every node must retain
at least `.50` across the eight declared stability forks.

## Reproduction, inclusion, and exclusion

Independent recomputation from `score_candidates.csv` agrees with the reported
top-twelve `L(s)` values to at most `1.65e-7` at displayed precision; posterior
mass is `1.00000001` due solely to eight-decimal display rounding.

| Node | Rank | Posterior | Stability | Status |
|---|---:|---:|---:|---|
| 1-1 | 1 | .17530478 | 1.00 | Included |
| 1-2 | 2 | .14142851 | 1.00 | Included |
| 0-1 | 3 | .11724027 | 1.00 | Included |
| 2-1 | 4 | .07345419 | .75 | Included, fragile |
| 0-0 | 5 | .07323734 | .25 | Excluded |
| 2-2 | 6 | .06732455 | .00 | Excluded |

The selected set `{1-1, 1-2, 0-1, 2-1}` has posterior mass `.50742775` and
objective `.52542775`. The closest eligible alternative
`{0-0,0-1,1-1,1-2}` has mass `.50721090` and objective `.52121090`.
The difference `.00421685` is mechanical: the selected set supplies England
result diversity (`.008` vs `.004`); its total and BTTS-fork bonuses tie.
Thus 2-1 defeats 0-0 through the declared rule, not narrative preference.

## Contradiction and stability lemmas

The complete ledger is `contradiction_ledger_v2_1.csv`.

1. BTTS is `.60144409`, no-BTTS `.39855591`; the C4 represents both forks.
2. Argentina-by-one is `.32515618`; 1-2 and 0-1 cover it.
3. England-win is `.23470553`; 2-1 survives only through declared diversity
   coverage and 6/8 retention. It must remain visibly **fragile**.
4. Clean sheets, high draws, ruthless cells, and chaos remain contrary live
   branches, but none beats the unique objective maximum.
5. 1-1, 1-2, and 0-1 survive 8/8 declared forks. 2-1 survives 6/8, above the
   `.50` threshold; 0-0 appears only in the negative-binomial and
   England-attack-minus-4 forks.
6. No NeohmX branch was supplied before cutoff, so Human Signal Preservation
   is not applicable rather than silently ignored.

## Theorem and limitations

Given the V2.1 precommitted data rule, shrinkage, score index, candidate grid,
objective, and tie break, the unique four-node selection is

`{1-1, 1-2, 0-1, 2-1}`.

This conditional pass remains limited by the sparse England realm, retrospective
strength proxy, unpromoted lineups, diffuse exact-score mass, fragile 2-1 node,
and normal match shocks. It makes no outcome-certainty or action-recommendation claim.

## Freeze conditions

1. Aster must include this proof, the ledger, the V2.1 builder provenance, and
   a final C4 card in a refreshed hash manifest.
2. The final card, lock document, and website must show exactly the four nodes
   above, an explicit regulation-time scope, and the conditional limitations.
3. Any post-freeze score change requires a new versioned run, never an edit.