Evidence

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The entry point builds the cover of Proposition 1 on the research page at $N=10^4$, $10^5$ and $10^6$ from its recipe: the class $1\bmod2$, residue $0$ for the primes in $(N^{1/u},N/2]$ with $u=4\log\log N$, and one prime of $(N/2,N]$ for each remaining integer. It checks that the cleanup primes exist, that the integers left before cleanup are exactly the even $N^{1/u}$-smooth integers, recomputed from largest prime factors, and that every integer in $[1,N]$ lies in a selected class. It then repeats the check with the largest tested prime cutoff at most $\sqrt N$ for which the cleanup primes still suffice, and checks by integer arithmetic that each resulting cost is less than $33/20$. These are finite checks of a recipe whose validity for all large $N$ is proved on the research page; they prove nothing asymptotic.

Two further clauses are seeded finite probes. In the window $(N^{1/3},N]$, with every class forced to have exactly $\lfloor N/p\rfloor$ points, the probe checks that the zero classes leave exactly the $N^{1/3}$-smooth integers and that no change of one prime's class lowers that number, at $N=3\cdot10^4$ and $10^5$; the same two checks run in the windows $(N^{1/4},N]$ and $(N^{1/5},N]$, with $N^{1/4}$- and $N^{1/5}$-smooth integers. It then runs a local search from a seeded random choice of classes and checks that it ends above the zero count. At $N=3\cdot10^4$ it also runs seeded Metropolis annealing by single-class moves: from the zero classes it never reaches fewer survivors than the zero classes, and from random classes it ends above them. The union probe chooses one full residue class modulo each prime in $(L^{1/2},L^{9/10}]$ inside $[0,L)$ for $L=3000$, $10^4$, $3\cdot10^4$, $10^5$ and $3\cdot10^5$, and reports the union sizes of the zero, random, greedy and locally improved choices after checking the maintained counts against a recount. The research page records the observed values and their limited bearing.

A separate exact witness at $N=105$ uses $0\bmod5$, $6\bmod7$ and $0\bmod11$. The checker verifies that the class sizes and all pair intersection sizes equal those for zero residues, while a triple intersection at $55$ increases the union from $40$ to $41$. The cost is exactly $167/385$. This is a finite improvement for a selected three-prime set; it is neither a cover nor an improvement for the full prime window.

A sixth clause probes the excess $X_S$ of display (26) of the research page: the union of one class modulo each prime in $(N^{1/3},\sqrt N]$, every class with exactly $\lfloor N/p\rfloor$ points, against its zero-class value, at $N=10^6$ and $10^7$. It checks that the zero-class union equals $\sum_p\lfloor N/p\rfloor-\sum_{p<q}\lfloor N/pq\rfloor$ exactly, since every pair of these primes has product at most $N$ and every triple more. From a seeded random start, coordinate ascent gives each prime in turn the class covering the most integers missed by the others; the probe checks the maintained counts against a recount and that the union never decreases, and reports the final excess and the number of integers in four or more classes. The comparison mean $N(1-\prod_{p\in S}(1-1/p))$ is for unrestricted uniform residues; the seeded searches use only classes of $\lfloor N/p\rfloor$ points. At $N=10^6$ a seeded annealing run by single-class moves is a second search, with its maintained union checked against a recount. The research page records the values and their limited bearing.

A seventh clause checks the centered construction (27). It chooses $b=\lfloor N/2\rfloor+1$ and uses $b\pmod p$ for the primes in $(z,N/2]$, with $z=N^{1/(4\log\log N)}$. It independently factors the distances $|n-b|$ to check that the survivors are exactly the integers with positive $z$-smooth distance from $b$. The center is already covered. The checker assigns distinct unused primes from $(N/2,N]$ to these survivors, then checks coverage from the complete list of residues. It also repeats the construction with the largest tested prime cutoff up to $\sqrt N$ fitting the cleanup supply, and checks the resulting cost is below $131/100$ by integer arithmetic. These are exact finite covers at $N=10^4$, $10^5$ and $10^6$, not a constant bound for all $N$.

Run from the repository root:

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

The full run takes about two minutes; --quick drops $N=10^6$, the second window order of each window and the three largest union sizes, shortens the annealing, and runs the sixth clause at $N=10^5$ only. The run is deterministic, reads no inputs and writes no files. Every failed check exits nonzero.