The Geometry
of Chance.

Monte Carlo counts qualifying draws in a finite random sample and divides by the sample size. The 320,000 comparison portfolios share 4,096 evaluation draws, which supports paired comparison but adds common finite-sample uncertainty. A reported standard error across generated portfolios alone does not include all of that uncertainty.

Exact enumeration visits every main-number combination. Exact inclusion–exclusion instead computes the same count analytically when its preconditions hold. It is not an estimate just because it avoids visiting every draw.

The 116 named fixed configurations include MC-selected best, representative and worst examples, plus coverage curves. A sample-selected “best” is not certified as the exact best among every generated portfolio. Separate P003 formula validation covered 22 named entries representing 20 unique portfolios, with agreement of integer counts and an additional seed check.

P003-linear-1.1 requires every pair of games to share at most one number. For n games, the recorded exact 3+ count is:

n × 194,130 − 400n₀ − 3,700n₁ + 64T − Q

Here n₀ counts disjoint pairs, n₁ pairs sharing one number, T triangles with three distinct pair-intersection numbers, and Q four-game groups with six distinct pair-intersection numbers. Divide by 8,145,060 for probability. This formula is not valid for arbitrary overlapping portfolios.

Within this restricted family, inclusion–exclusion ends at four games: five pairwise-linear triples cannot fit on six drawn numbers. The separate family-specific upper bound is not a bound for arbitrary portfolios, and its attainability is not established.

For a single game, P(exactly k main matches) = C(6,k) × C(39,6−k) / C(45,6). Every fixed game has that same marginal probability. For 25 games, E[number with k matches] = 25 × P(exactly k matches), whether or not the games overlap.

The prior four fixed examples all have expected counts of approximately 0.5610148974 games with exactly three matches and 0.0341157708 games with exactly four. Multiplying these expectations by fixed tier payouts preserves the same expected subtotal. A subtotal is not total winnings, net profit or an investment return.

1. Hypothesis: less redundant coverage might increase the probability of at least one low-tier hit at the same game count.
2. Fixed comparisons: four 25-game configurations were enumerated exactly. Equal expected winning-game counts separated coverage from expected payout.
3. Scope correction: the broader 32-series study used a shared 4,096-draw sample. Its per-portfolio distribution and selected extrema must be described as estimates.
4. Analytic revision: P003-linear-1.1 imposed a pairwise-overlap condition, enabling exact inclusion–exclusion for each of 10,000 outputs. The older P003 / structure-1.0 series remains part of the MC comparison set.
5. Deterministic verification: recorded full-enumeration cross-checks agreed with the analytic integer totals. Global optimality and future predictive performance remain unproved.

This public account was checked against the retained structure-study summary, calculation-method record, candidate-scope inventory, final distribution and final verification, plus the earlier prize-study exact results. A separate read-only review checked the numerical interpretation. No new research runs were performed for this page.

The retained local evidence records fixed seeds, versioned methods and exact-count comparisons. This is an internally audited engineering case, not peer-reviewed research or independent public reproduction. Raw portfolios, purchased numbers, operational databases and source code are not released here; any release needs a separate publication review.

All claims concern the declared uniform 6/45 model and the recorded experiments. The 10,000 P003 outputs sample one construction family; they are not 10,000 independent model classes, nor proof that every output is structurally unique.