# Mathematics: Argentina vs Cape Verde This note is the public mathematical proof for the Turtle pregame score cloud. It proves a finite selection theorem from stated inputs; it does not certify the future result. The purpose is to make the final three score claims change mechanically when a data value, threshold, or rule changes. ## Abstract The final disclosed set is `2-0`, `1-0`, `3-0`. The low-event comparison is `1-0` against `1-1`; the pressure-extension comparison is `2-0` against `3-0`. Turtle keeps `1-0` because it dominates `1-1` in exact probability, selection index, and the Argentina clean-sheet ledger. Turtle then admits `3-0` because the predeclared knockout-pressure index is large enough to move the best B2 extension above the nonzero-draw branch. ## Definitions Let $$G=\{0,1,\ldots,7\}\times\{0,1,\ldots,7\}.$$ For a score $s=(a,c)$, define the independent Poisson score kernel $$P(s)=P(a,c)=\operatorname{Pois}(a;\lambda_{ARG})\operatorname{Pois}(c;\lambda_{CPV})=e^{-(\lambda_{ARG}+\lambda_{CPV})}\frac{\lambda_{ARG}^a}{a!}\frac{\lambda_{CPV}^c}{c!}.$$ Here $\lambda_{ARG}=1.5055$ and $\lambda_{CPV}=0.5978$. Let $S_{10}$ be the first ten scores in $G$ after sorting by descending $P(s)$, with exact ties broken lexicographically by $(a,c)$: $$S_{10}=\{1-0, 2-0, 0-0, 1-1, 2-1, 0-1, 3-0, 3-1, 1-2, 4-0\}.$$ Let $H(s)$ be historical bucket support and $B(s)$ be regulation result-branch mass. Turtle's selection index is $$I(s)=0.50\frac{P(s)}{P_{\max}}+0.30H(s)+0.20\frac{B(s)}{B_{\max}}.$$ This index is not a betting probability. It is a deterministic ranking functional used to choose public score claims. ## Evidence Ledger Argentina scored in 48/50 last-50 rows, 10/10 last-10 rows, and 3/3 current World Cup rows. The only Argentina zero-goal rows in the last-50 ledger are `0-2 Uruguay` and `0-1 Ecuador`. Thus the low-event question is not whether Argentina scores; the realistic low-event question is whether Cape Verde score once. Argentina win scoreline modes in the last-50 ledger are `2-0` twelve times, `1-0` ten times, and `3-0` seven times. In the current World Cup sample Argentina's scores are `3-0`, `2-0`, and `3-1`; Lionel Messi appears as a scorer in all three source rows, including late scoring entries. This does not prove a third goal, but it is evidence that Argentina's pressure can persist after the match is already controlled. Cape Verde's current World Cup path is `0-0 Spain`, `2-2 Uruguay`, `0-0 Saudi Arabia`: two clean sheets and one scoring draw. Cape Verde draw modes are `0-0` seven times and `1-1` five times, while Cape Verde loss modes are `0-1` four times, `0-2` three times, and `0-3` once. This evidence keeps the low-event branch alive, but it supports `1-0` more directly than `1-1` because Cape Verde have blanked in two of their three current World Cup rows. ## Candidate Table | Rank | Score | P(s) | Bucket | Branch | Turtle index I(s) | Status | Rule | |---:|---|---:|---|---|---:|---|---| | 1 | 1-0 | 0.183758 | Argentina win | B1: Argentina 1-0 control | 0.787500 | DISCLOSED | R4 | | 2 | 2-0 | 0.138326 | Argentina win | B2: Argentina margin >= 2 | 0.658880 | DISCLOSED | R4 | | 3 | 0-0 | 0.122056 | Draw | B0: 0-0 blank draw | 0.489380 | EXCLUDED | R1 | | 4 | 1-1 | 0.109842 | Draw | BD: nonzero draw | 0.526145 | EXCLUDED | R5 | | 5 | 2-1 | 0.082685 | Argentina win | BAC: Argentina win, CPV scores | 0.444982 | EXCLUDED | R2/R4 | | 6 | 0-1 | 0.072959 | Cape Verde win | BC: Cape Verde win | 0.316976 | EXCLUDED | R1 | | 7 | 3-0 | 0.069418 | Argentina win | B2: Argentina margin >= 2 | 0.427633 | DISCLOSED | R5 | | 8 | 3-1 | 0.041495 | Argentina win | B2: Argentina margin >= 2 | 0.320405 | EXCLUDED | R3 | | 9 | 1-2 | 0.032829 | Cape Verde win | BC: Cape Verde win | 0.248217 | EXCLUDED | R2c | | 10 | 4-0 | 0.026127 | Argentina win | B2: Argentina margin >= 2 | 0.271092 | EXCLUDED | R3 | ## Comparison Rules The candidate branches are $$B_2=\{a-c\ge2\},\quad B_1=\{a-c=1,c=0\},\quad B_{AC}=\{a>c,c>0\},\quad B_0=\{(0,0)\},\quad B_D=\{a=c,a\ge1\},\quad B_C=\{c>a\}.$$ The low-event gate compares `1-0` against `1-1`. It asks whether Cape Verde's single transition goal has enough support to break Argentina control. The pressure-extension gate compares `2-0` against `3-0`. It asks whether Argentina's favorite pressure should include a second B2 score after the primary B2 representative is already disclosed. Define the knockout-pressure index $$\Pi=0.06r_{B2}^{ARG,wc}+0.04r_{CS}^{ARG,wc}+0.02r_{2+}^{ARG,wc}+0.02r_{blank}^{CPV,wc}.$$ For this match, $\Pi=0.120000$ because Argentina's current World Cup margin-by-2 rate is 1.000, clean-sheet rate is 0.667, two-plus-goal rate is 1.000, and Cape Verde's current World Cup blank rate is 0.667. ## Proof The exact-score mode is `1-0`, since $P(1,0)=0.183758$ is the largest probability in $S_{10}$. It also beats `1-1` by the low-event gate: $P(1,0)=0.183758>P(1,1)=0.109842$ and $I(1,0)=0.787500>I(1,1)=0.526145$. Thus the low-event branch is represented by `1-0`, not `1-1`. The two-goal branch is generated because 31.4%>28.0%, 31.4%>25.9%, and 31.4%>14.7%. In $S_{10}\cap B_2=\{2-0,3-0,3-1,4-0\}$, the primary representative is `2-0` because $I(2,0)=0.658880$ exceeds $I(3,0)=0.427633$, $I(3,1)=0.320405$, and $I(4,0)=0.271092$. The remaining comparison is `3-0` against `1-1` for the third public slot after `1-0` has already won the low-event gate. Without pressure, `1-1` has the larger index: $I(1,1)-I(3,0)=0.098512$. With the declared pressure index, $I(3,0)+\Pi=0.547633>I(1,1)=0.526145$. Therefore `3-0` is disclosed as the pressure-extension score and `1-1` is retained only as the named draw-risk branch. Exhaustion is now immediate. `0-0` and `0-1` fail the Argentina no-score gate. `2-1` lies in $B_{AC}$ and is weaker than `1-0` by both probability and index. `1-2` is in the Cape Verde-win branch, and $P(\mathrm{CPV\ win})