Evidence

../

The entry point checks the finite evidence for the sieve lead with the shared failure harness. It uses the root Python environment with NumPy and SciPy, reads the two JSON assets below, and writes no output files. Run it from the repository root:

uv run --no-sync python wiki/research/leads/property_p_sieve_multiplicity/evidence/main.py

Expected runtime is about 90 seconds. Solver calls are restricted to one thread and must report completion; their optimality remains a floating-point report. There is no reduced mode. The checks cover:

• direct property-P tests of the truncated gcd-one example, the self-inverse-residue witness, and the congruence blocks;
• exhaustive residue occupancy for six coprime pairs and for $\{15,21\}$;
• greedy integer and prime property-P sets and the stated finite bounds;
• the first-prime normalization, exact ownership, inherited exclusions, strict descent and the mass and Euler-weight telescoping identities, including children containing 1 or 2;
• the sibling-discount family for $(k,M)=(3,105),(3,315),(4,1155),(5,15015)$; the largest uses the proved automatic-interval reduction for property P, and the others also receive direct checks;
• the semiprime family for $y=7,13,31$, including the greedy matching, nonempty small-prime branches and discarded-large-prime bounds;
• the direct-toll obstruction with $M=625,1225$, including its fixed early element, exact root list and nonempty prime branches;
• nine retained above-product witnesses, checking every element's domain, admissible residues, property P and exact rational mass;
• the retained two-block witness, with exact mass above $0.0929$, and the separate exhaustive matching upper bound $0.101959737$ over all $2^{14}$ early subsets;
• the three further exhaustive matching bounds for later interval starts 625, 3125 and 78125, with their complete candidate counts and rounded separate masses. Each covers the same $2^{14}$ early choices.

The two-block upper bounds in the checker use integer weights rounded upward, disjoint edges representing forbidden distinct pairs, and every early subset. They are certified finite upper bounds when replayed; they do not assert exact optima. The lower witness was found by a time-limited mixed-integer search and is checked directly. The nine above-product witnesses were also found by the mixed-integer solver; replay of their feasibility does not certify their optimality. Earlier finite-window maxima in main.py retain that same distinction.

Arithmetic for witness properties, masses, ownership and the matching bound is exact. Numerical regional comparisons and solver optimality are explicitly numerical. The infinite semiprime obstruction uses the written prime-number-theorem argument; no finite grid replaces it. These checks prove neither (H), (U), nor E12(iii), and confer no independent review.