# Turtle Mathematical Proof: Paraguay vs France

Kickoff: `2026-07-04 5:00 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): {0-1, 0-2, 1-1, 0-3, 1-2, 0-0, 1-3, 1-0, 0-4, 2-1}. 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_PAR) Pois(a;lambda_FRA), where lambda_PAR=0.6585 and lambda_FRA=1.9768.

## 3. Definition 3

Let r_PAR=6.60 and r_FRA=9.40. PAR evidence is filtered to opponents within epsilon of r_FRA; FRA evidence is filtered to opponents within epsilon of r_PAR.

## 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

Paraguay last-50 W-D-L is 16-14-20; GF/M=0.880, GA/M=1.120.

## 8. Data 2

France last-50 W-D-L is 34-8-8; GF/M=2.280, GA/M=0.880.

## 9. Data 3

Opponent-threshold realm: Paraguay vs France-realm has 6 raw rows, effective n=4.875, readiness=usable, GF/M=0.6896, GA/M=1.7396, scored=0.5158, clean-sheet=0.1773.

## 10. Data 4

Opponent-threshold realm: France vs Paraguay-realm has 22 raw rows, effective n=15.226, readiness=strong, GF/M=2.1417, GA/M=0.6883, scored=0.8717, clean-sheet=0.4689.

## 11. Data 5

Result thread probabilities are PAR win=0.161, draw=0.159, FRA win=0.680.

## 12. Data 6

Spread thread order is away_by_2_plus=0.414; away_by_1=0.265; draw=0.154; home_by_1=0.152; home_by_2_plus=0.015. Therefore the strongest spread before exact-score selection is away_by_2_plus.

## 13. Data 7

Total and clean-sheet threads are over1.5=0.688, over2.5=0.510, over3.5=0.305, BTTS=0.413, PAR clean sheet=0.145, FRA clean sheet=0.499.

## 14. Data 8

Recent scoring remains visible but cannot delete clean sheets by itself: Paraguay scored in 80.0% of last ten; France scored in 100.0% of last ten. Current World Cup samples: 4 and 4.

## 15. Data 9

Lineup conflict unresolved in public scout sources: Mbappe status, Diego Gomez return and Paraguay defensive availability are treated as bounded modifiers rather than hard score facts.

## 16. Data 10

Scout memo: France hold the strength edge through ranking, squad quality and tournament scoring; Paraguay carry low-block evidence.

## 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

0-2 is in C4 by R2 spread-thread leader: P_count=0.140093, result=0.679659, spread=0.414398, total=0.178118, BTTS/clean=0.498864, I=0.740957. 0-2 maximizes I(s) inside the strongest spread thread away_by_2_plus=0.4144 and beats 0-3 by I=0.740957 > 0.588489.

## 24. Inclusion Lemma 2

0-1 is in C4 by R3 exact-cell anchor: P_count=0.141735, result=0.679659, spread=0.264659, total=0.312297, BTTS/clean=0.498864, I=0.664791. 0-1 is the exact-cell anchor after the spread leader and beats the next remaining count cell 1-1 by P_count=0.141735 > 0.093326.

## 25. Inclusion Lemma 3

1-1 is in C4 by R4 alternate-thread representative: P_count=0.093326, result=0.159026, spread=0.154433, total=0.178118, BTTS/clean=0.412643, I=0.225854. 1-1 represents the strongest unrepresented spread thread draw with mass 0.1544 >= 0.1400 with I=0.225854.

## 26. Fourth Score Lemma

1-3 is score 4 by R5: spread=0.414398, form=0.727500, I=0.507267, BTTS/clean=0.412643. 1-3 is the best excluded thesis-compatible branch by spread/form/index/BTTS-clean comparator and beats 0-3 by score 4 0.515627 > 0.503866.

## 27. Exclusion Lemma

0-3 has status EXCLUDED_BY_SPREAD_THREAD by R2/R5 spread comparison: 0-3 shares away_by_2_plus with 0-2 and loses I=0.588489 < 0.740957.

## 28. 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=0.457659 < 0.664791.

## 29. Exclusion Lemma

0-0 has status EXCLUDED_BY_TOTAL_THREAD by R1 total gate: 0-0 fails R1 because P(over 1.5)=0.687703 >= 0.640000, so the no-goal cell is removed before branch selection.

## 30. 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 1-1 from draw because P_spread(draw)=0.154433 > P_spread(home_by_1)=0.152009.

## 31. Exclusion Lemma

0-4 has status EXCLUDED_BY_SPREAD_THREAD by R2/R5 spread comparison: 0-4 shares away_by_2_plus with 0-2 and loses I=0.437231 < 0.740957.

## 32. 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 1-1 from draw because P_spread(draw)=0.154433 > P_spread(home_by_1)=0.152009.

## 33. Theorem

Under Definitions 1-6, Data 1-10, and Rules R1-R6, the unique disclosed four-score card is C4={0-2, 0-1, 1-1, 1-3}.

## Final Card

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