# Mathematics: Colombia vs Ghana

This document is the pregame Turtle proof. It proves a finite selection statement from visible data and fixed rules. It does not use market price as the prediction engine.

## Definitions

$$G=\{0,1,\ldots,7\}\times\{0,1,\ldots,7\}.$$

$$P(a,g)=\operatorname{Pois}(a;\lambda_{COL})\operatorname{Pois}(g;\lambda_{GHA}).$$

Here $\lambda_{COL}=1.3838$ and $\lambda_{GHA}=0.6833$. The candidate universe is

$$S_{10}=\{1-0, 0-0, 2-0, 1-1, 0-1, 2-1, 3-0, 1-2, 3-1, 0-2\}.$$

Turtle ranks candidates using

$$I(s)=0.45E(s)+0.25H(s)+0.20B(s)+0.10F(s)+0.04M(s),$$

where $E$ is the normalized exact-cell probability, $H$ is historical bucket support, $B$ is branch mass, $F$ is current-form fit, and $M$ is the bounded emotional/danger index.

## Candidate Table

| Rank | Score | Branch | P(s) | I(s) | Status | Rule |
|---:|---|---|---:|---:|---|---|
| 1 | 1-0 | B_home_win | 0.175122 | 0.673938 | MAIN | R2 exact-cell leader |
| 2 | 0-0 | B_draw_low | 0.126549 | 0.604272 | EXCLUDED | R1 hard gate |
| 3 | 2-0 | B_home_cushion | 0.121170 | 0.525311 | MAIN | R3 favorite-cushion representative |
| 4 | 1-1 | B_draw_low | 0.119660 | 0.563192 | SCORE_4 | R5 fourth-score |
| 5 | 0-1 | B_away_win | 0.086471 | 0.456191 | EXCLUDED | R2-R6 exhaustion |
| 6 | 2-1 | B_home_btts | 0.082795 | 0.523646 | MAIN | R4 complementary scoring branch |
| 7 | 3-0 | B_home_cushion | 0.055893 | 0.313272 | EXCLUDED | R2-R6 exhaustion |
| 8 | 1-2 | B_away_win | 0.040882 | 0.278204 | EXCLUDED | R2-R6 exhaustion |
| 9 | 3-1 | B_home_cushion | 0.038191 | 0.264521 | EXCLUDED | R2-R6 exhaustion |
| 10 | 0-2 | B_away_win | 0.029543 | 0.217273 | EXCLUDED | R2-R6 exhaustion |

## Proof Ledger

**Definition 1.** Let G={0,...,7} x {0,...,7}. S10 is the first ten scores after sorting by P(s): {1-0, 0-0, 2-0, 1-1, 0-1, 2-1, 3-0, 1-2, 3-1, 0-2}.

**Definition 2.** For s=(a,g), P(s)=Pois(a; lambda_COL) Pois(g; lambda_GHA), lambda_COL=1.3838, lambda_GHA=0.6833.

**Definition 3.** Branches are deterministic: B_home_win, B_home_cushion, B_draw_low, B_draw_high, B_away_win, B_home_btts.

**Definition 4.** I(s)=0.45 exact probability index + 0.25 bucket support + 0.20 branch mass index + 0.10 current form fit + 0.04 emotional index.

**Data 1.** Colombia last-50 W-D-L is 31-12-7; GF/M=1.860, GA/M=0.800.

**Data 2.** Ghana last-50 W-D-L is 16-14-20; GF/M=1.220, GA/M=1.320.

**Data 3.** Recent form: Colombia scored in 8/10 last-10; Ghana scored in 6/10 last-10.

**Data 4.** Current World Cup rows: Colombia n=3 with GF/M=1.333; Ghana n=0 with GF/M=0.000.

**Rule R1.** Hard scoring gates remove only scores contradicted by near-perfect recent and current-tournament scoring evidence.

**Rule R2.** The eligible exact-cell leader, i.e. the score with maximum P(s), must enter C4.

**Rule R3.** If P(Colombia win)>P(draw), choose the best I(s) representative from B_home_cushion.

**Rule R4.** If BTTS probability is at least 0.30 and Colombia win exceeds Ghana win, choose the best Colombia-win score with one Ghana goal.

**Rule R5.** Choose the fourth score by the best nonzero draw/danger branch; it is part of C4.

**Rule R6.** If fewer than three score representatives exist, fill remaining score slots by highest eligible I(s). Scores with total 0 are removed if P(over 1.5)>=0.55.

**Inclusion Lemma 1.** 1-0 is disclosed because it lies in B_home_win, passes R1, and is selected by R2 exact-cell leader with I=0.673938 and P=0.175122.

**Inclusion Lemma 2.** 2-0 is disclosed because it lies in B_home_cushion, passes R1, and is selected by R3 favorite-cushion representative with I=0.525311 and P=0.121170.

**Inclusion Lemma 3.** 2-1 is disclosed because it lies in B_home_btts, passes R1, and is selected by R4 complementary scoring branch with I=0.523646 and P=0.082795.

**Fourth Score Lemma.** 1-1 is the fourth C4 score because it is the best excluded danger branch with I=0.563192.

**Exclusion 0-0.** 0-0 is excluded by R1 hard gate: EXCLUDED_BY_R1_TOTAL_ACTIVITY_GATE

**Exclusion 0-1.** 0-1 is excluded by R2-R6 exhaustion: 0-1 loses branch or index comparison to the disclosed representatives.

**Exclusion 3-0.** 3-0 is excluded by R2-R6 exhaustion: 3-0 loses branch or index comparison to the disclosed representatives.

**Exclusion 1-2.** 1-2 is excluded by R2-R6 exhaustion: 1-2 loses branch or index comparison to the disclosed representatives.

**Exclusion 3-1.** 3-1 is excluded by R2-R6 exhaustion: 3-1 loses branch or index comparison to the disclosed representatives.

**Exclusion 0-2.** 0-2 is excluded by R2-R6 exhaustion: 0-2 loses branch or index comparison to the disclosed representatives.

**Theorem.** Under Definitions 1-4, Data 1-4, and Rules R1-R6, C4={1-0, 2-0, 2-1, 1-1}.

Final Answer: $\boxed{T_3=\{1-0, 2-0, 2-1\},\quad T_4=\{1-1\}}$.