# Turtle Mathematical Proof: Mexico vs England

Kickoff: `2026-07-05 8:00 PM ET`

This proof is intentionally mechanical. If a probability, threshold, or rule changes, the disclosed set changes by recomputation, not by prose.

## 1. Definition 1

Let G={0,...,7} x {0,...,7}. S10 is the ten scores with largest P_count(s), tie-broken lexicographically by (home goals, away goals): {1-1, 0-1, 1-0, 0-0, 1-2, 0-2, 2-1, 2-0, 2-2, 1-3}. No exact score outside S10 can enter C4; it may enter only a named post-lock/live-state audit.

## 2. Definition 2

For s=(h,a), P(s)=Pois(h;lambda_MEX) Pois(a;lambda_ENG), where lambda_MEX=1.0523 and lambda_ENG=1.1960.

## 3. Definition 3

Let r_MEX=7.40 and r_ENG=8.90. MEX evidence is filtered to opponents within epsilon of r_ENG; ENG evidence is filtered to opponents within epsilon of r_MEX.

## 4. Definition 4

Each historical row receives weight recency * competition * squad_overlap * exp(-distance^2/(2 sigma^2)), sigma=0.75. If the realm has fewer than five usable rows, epsilon widens to 1.25. If effective n still stays below five, the realm statistic is shrunk toward the last-50 statistic and readiness is marked weak.

## 5. Definition 5

LL(s)=log P_count(s)+0.70 log P_result(s)+0.85 log P_spread(s)+0.45 log P_total(s)+0.45 log P_BTTS_clean(s). TurtleIndex normalizes exp(LL) and adds bucket, branch, form, readiness, and pressure support.

## 6. Definition 6

C4 is the ordered set of four scores selected by R2-R6 and the score-four gate. The fourth score is a normal member of C4, selected by R5 when it protects an unrepresented branch. Four-score accuracy uses all four scores.

## 7. Data 1

Mexico last-50 W-D-L is 27-12-11; GF/M=1.500, GA/M=0.900.

## 8. Data 2

England last-50 W-D-L is 32-11-7; GF/M=2.300, GA/M=0.680.

## 9. Data 3

Opponent-threshold realm: Mexico vs England-realm has 8 raw rows, effective n=2.924, readiness=usable, GF/M=1.2849, GA/M=1.2006, scored=0.6581, clean-sheet=0.4956.

## 10. Data 4

Opponent-threshold realm: England vs Mexico-realm has 14 raw rows, effective n=9.228, readiness=strong, GF/M=1.6182, GA/M=0.9271, scored=0.8490, clean-sheet=0.3631.

## 11. Data 5

Result thread probabilities are MEX win=0.304, draw=0.323, ENG win=0.373.

## 12. Data 6

Spread thread order is draw=0.390; away_by_2_plus=0.231; away_by_1=0.150; home_by_1=0.146; home_by_2_plus=0.083. Therefore the strongest spread before exact-score selection is draw.

## 13. Data 7

Total and clean-sheet threads are over1.5=0.695, over2.5=0.409, over3.5=0.225, BTTS=0.457, MEX clean sheet=0.312, ENG clean sheet=0.351.

## 14. Data 8

Recent scoring remains visible but cannot delete clean sheets by itself: Mexico scored in 90.0% of last ten; England scored in 80.0% of last ten. Current World Cup samples: 4 and 4.

## 15. Data 9

Public pre-match context reports Mexico as home side at Azteca and England managing fitness questions. Those facts are bounded modifiers; exact score selection still comes from the thread model.

## 16. Data 10

Manual 0-10 strength seeds: England 8.9, Mexico 7.4. Venue and tournament form narrow the effective match gap.

## 17. Rule R1

The only hard pre-selection removal is 0-0 when P(over 1.5) >= 0.640000. Raw scoring rate alone cannot delete a clean-sheet score.

## 18. Rule R2

Let B* be the spread state with maximum spread-thread probability. The first MAIN score is argmax TurtleIndex among eligible S10 scores with spread_key=B*.

## 19. Rule R3

After R2, the exact-cell anchor is the remaining eligible S10 score with maximum P_count(s).

## 20. Rule R4

After R2 and R3, scan spread states by descending spread-thread mass and skip already represented states. The first unrepresented spread whose representative has spread mass >=0.140000 or TurtleIndex >=0.280000 enters C4.

## 21. Rule R5

The fourth score is the remaining eligible score maximizing 0.35*P_spread + 0.25*form + 0.25*I + 0.15*P_BTTS_clean. It is part of C4, not a separate containment grade.

## 22. Rule R6

If fewer than three score scores survive, fill by highest remaining eligible I(s).

## 23. Inclusion Lemma 1

1-1 is in C4 by R2 spread-thread leader: P_count=0.132877, result=0.322765, spread=0.389749, total=0.285466, BTTS/clean=0.456967, I=0.811527. 1-1 maximizes I(s) inside the strongest spread thread draw=0.3897 and beats 2-2 by I=0.811527 > 0.413765.

## 24. Inclusion Lemma 2

0-1 is in C4 by R3 exact-cell anchor: P_count=0.126278, result=0.373147, spread=0.149981, total=0.305330, BTTS/clean=0.350650, I=0.426089. 0-1 is the exact-cell anchor after the spread leader and beats the next remaining count cell 1-0 by P_count=0.126278 > 0.111103.

## 25. Inclusion Lemma 3

0-2 is in C4 by R4 alternate-thread representative: P_count=0.075513, result=0.373147, spread=0.231324, total=0.285466, BTTS/clean=0.350650, I=0.452771. 0-2 represents the strongest unrepresented spread thread away_by_2_plus with mass 0.2313 >= 0.1400 and beats 1-3 by I=0.452771 > 0.304265.

## 26. Fourth Score Lemma

2-2 is score 4 by R5: spread=0.389749, form=0.527500, I=0.413765, BTTS/clean=0.456967. 2-2 is the best excluded thesis-compatible branch by spread/form/index/BTTS-clean comparator and beats 1-3 by score 4 0.440274 > 0.357450.

## 27. Exclusion Lemma

1-0 has status EXCLUDED_BY_SPREAD_THREAD by R4 spread-order comparison: 1-0 belongs to home_by_1, but R4 selected 0-2 from away_by_2_plus because P_spread(away_by_2_plus)=0.231324 > P_spread(home_by_1)=0.146244.

## 28. Exclusion Lemma

0-0 has status EXCLUDED_BY_TOTAL_THREAD by R1 total gate: 0-0 fails R1 because P(over 1.5)=0.694670 >= 0.640000, so the no-goal cell is removed before branch selection.

## 29. Exclusion Lemma

1-2 has status EXCLUDED_BY_SPREAD_THREAD by R2/R5 spread comparison: 1-2 shares away_by_1 with 0-1 and loses I=0.326606 < 0.426089.

## 30. Exclusion Lemma

2-1 has status EXCLUDED_BY_SPREAD_THREAD by R4 spread-order comparison: 2-1 belongs to home_by_1, but R4 selected 0-2 from away_by_2_plus because P_spread(away_by_2_plus)=0.231324 > P_spread(home_by_1)=0.146244.

## 31. Exclusion Lemma

2-0 has status EXCLUDED_BY_SPREAD_THREAD by R4 spread-order comparison: 2-0 belongs to home_by_2_plus, but R4 selected 0-2 from away_by_2_plus because P_spread(away_by_2_plus)=0.231324 > P_spread(home_by_2_plus)=0.082703.

## 32. Exclusion Lemma

1-3 has status EXCLUDED_BY_SPREAD_THREAD by R2/R5 spread comparison: 1-3 shares away_by_2_plus with 0-2 and loses I=0.304265 < 0.452771.

## 33. Theorem

Under Definitions 1-6, Data 1-10, and Rules R1-R6, the unique disclosed four-score card is C4={1-1, 0-1, 0-2, 2-2}.

## Final Card

- SCORE 1: `1-1`
- SCORE 2: `0-1`
- SCORE 3: `0-2`
- SCORE 4: `2-2`