Exact checks of the displayed flat cyclotomic multiples

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Purpose

The angular-coloring page bounds the dissociated coloring number of the odd prime grids by $\lceil N/D(N)\rceil$, where $D(N)$ is the least degree of a flat multiple of $\Phi_N$, and quotes values of $D(N)$ together with the thresholds and one lattice covolume used in its discussion. The algebra on that page and on L123 is the durable justification; this probe checks the displayed polynomials and numbers exactly.

What is checked

Run, from the repository root,

uv run --no-sync python wiki/research/erdos_774/evidence/angular_flat_multiples/main.py

The checker uses exact integer arithmetic, the standard library and the shared root tools checker, performs no search and finishes in a few seconds. It builds $\Phi_{15}$, $\Phi_{105}$ and $\Phi_{1155}$ as Möbius quotients of the binomials $x^d-1$ and checks their degrees and that each divides $x^N-1$ but no $x^d-1$ with $d$ a proper divisor. Its named checks then establish:

• $\Phi_{15}$ is flat, so $D(15)=8$; $(1-x)\Phi_{105}$ is flat of degree 49 with 26 nonzero coefficients, and $\Phi_{105}$ has height two, attained at $x^7$ and $x^{41}$, so $D(105)=49$;
• the multiplier $(1-x^6)S$ displayed on the research page has degree 22 and constant term one, $\Phi_{1155}(1-x^6)S$ is flat of degree 502 with 128 nonzero coefficients, $\Phi_{1155}$ has height three, and $S$ has no cyclotomic factor;
• $(1-x^{11})\Phi_{105}(x^{11})$ is a flat multiple of $\Phi_{1155}$ of degree 539;
• the exact totient ratios of the odd prime prefixes cross $1/3$ between the primes 19 and 23 and $1/4$ between 73 and 79, the odd primes up to 23 have $\sum(p-2)=82$, and the ratio for 1155 is $480/1155$;
• the Gram determinant of $\Phi_{1155},x\Phi_{1155},\ldots,x^{22}\Phi_{1155}$ has half-logarithm $51.39$, and the section bound $2^{23}(503/23)^{23/2}$ has logarithm $51.42$.

These give $D(1155)\le502$ and the quoted values; a failed clause exits nonzero. The lower bound $D(1155)\ge502$ is not checked.

The solver searches

The discovery script poses, for given $N$ and degree $D$, the exact-coefficient feasibility problem of the research page: integer unknowns $q_0=1,\ldots,q_{D-\varphi(N)}$, bounded by the partial sums of the absolute coefficients of $1/\Phi_N$, with every coefficient of $\Phi_NQ$ in $\{-1,0,1\}$. It uses the HiGHS solver in SciPy on one thread, checks every reported solution by exact multiplication, and appends its verdicts to the ignored output/ folder. Its verdicts are observations, not certificates. For $N=1155$ the solver reported infeasibility at the degrees 488 and 495 to 501, hence at every degree up to 501, and solutions at 502 to 505, 520 and 538; the solution at 502 is the multiplier checked above. For $N=15015$, runs of 450 seconds at degree 6000 and of 900 seconds at degree 6300 stopped at the time limit without a solution, so they decide nothing about $D(15015)$.