Exact checks for the Schur-growth notes

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From the repository root run:

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

The program uses exact integer and rational arithmetic and checks, for the refutation of L261, that every relation among the differences of $\{4^0,\ldots,4^{n-1}\}$ is an endpoint triangle for $n\le8$; for the refutation of L266, that the digit and digit-sum constraints select exactly the powers of $b$ for $1\le n\le5$, $q\in\{3,4,8\}$ and $b=\max(4,n+2,q+1)$, by examining every integer in $[0,b^n-1]$ (at most 59049 candidates per case); and that the three-coloring of $K_{16}$ by the cubic-residue classes of $\mathrm{GF}(16)$ has no monochromatic triangle and every six-vertex subset has every color. Its transfer to differences of sixteen powers of 18 has 120 differences and 560 Schur triples, none monochromatic. The three-box argument of L266 makes this a counterexample at $\rho=1/3$; the huge ambient interval is handled by the written cell and extension proofs, not enumerated. For the refutation of L260, that the block composition is Schur-free with both palettes for every pair of tripling colorings with at most four colors and for the retained Bohr coloring, and that one composite of $[1,63239]$ has a longest color-omitting dilated prefix below the bound of Lemma C; for the proof of L262, the multiplier of its Lemma 2 on every sum-free arc with endpoints in $\tfrac1{120}\mathbb Z$, and the sum-free arcs of the tripling colorings for $k\le7$; and for the signed-cube reduction, the base-$b$ embedding for $r\le5$, the digit-cube cells for $r\le4$, the binary-value and support maps for $r\le6$, that $Q_3^+$ has no Schur-free two-coloring, and the colorings behind $\chi_4\le3$, $\chi_5\le4$, $\chi_6\le5$ and $\chi_7\le5$; and for the palette-rule page, that a retained four-coloring of $Q_4^+$ is Schur-free and, with the four twisted blocks $\{(x_1,x_2,y_1,y_2):x_1+y_1=s_1,\ x_2+y_2=s_2\}$ (sums mod $2$) and its retained palettes, obeys rule (R), while one block uses all four colors on its internal pairs and the sum of $2^{|U_B|}$ over the blocks is $28>2^4$; that coloring is a function neither of the support nor of the binary value. Controls confirm that the Schur test counts equal summands, that the prefix statistic sees the late color of a tripling coloring, and that the rule-(R) test rejects a palette containing an internal color.

Three inputs are retained. bohr_psi_m20.json is a 20-color coloring of $[1,800]$ whose classes lie in Bohr sets $\{x:\{\theta_ix\}\in(1/3,2/3)\}$, found by produce.py bohr 20 800 4 (seed 4) as in the construction of Lemma B of the refutation of L260. The twenty frequencies are retained as exact numerator-denominator pairs, and the checker verifies each element's arc membership using integers. The discovery sampler uses floating arithmetic, but the retained witness and its verification do not rely on rounding. schur_44.json is a coloring of $[1,44]$ by four colors with no monochromatic $x+y=z$, found by the depth-first search produce.py schur 44 4; composed by Lemma A with one color on $[1,1]$, it gives the five-coloring of $[1,133]$ behind $\chi_7\le5$. The checker verifies both inputs exactly and does not rely on either search; rerunning produce.py reproduces both files byte for byte. twisted_cube_r4.json holds the coloring of the forty vectors of $Q_4^+$ and the four palettes. It was found by a SAT search (kissat) over colorings and palettes with the twisted blocks fixed, block $(0,0)$ required to use every color inside and each other block allowed at most two colors missing from its interior. That search is not retained; the checker verifies the witness directly.

The largest interval coloring checked has length 63239; the largest signed cube checked has dimension seven. The program uses the repository's Python environment, including NumPy, on one thread and below 1 GiB. Expected runtime is a few seconds. The checks illustrate proved lemmas or verify finite witnesses, and prove no asymptotic statement. There is no independent review. The program exits nonzero on any failed check, also under optimized Python.