Exact small cases for twisted layers

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Purpose

The twisted-layer note uses a handful of finite facts: the structure of the proper 2-colorings of $\mu_{15}$, a family of nine layer colorings with trivial signed-span intersections, the root-span fraction of arc classes below the rank threshold, the survival of signed pairs and triples of roots under coordinate twists, the least flat-multiple degrees $D(165)$ and $D(195)$, the numbers behind Lemma 7, the lower bound on the first moment of common values, and survival sets of signed sums under all coordinate twists at 105. This probe checks them exactly. It proves no statement about larger conductors; the race model that these numbers inform is a heuristic.

What is checked

Run, from the repository root,

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

The checker uses exact integer arithmetic (NumPy int64 arrays with bounds checked before arithmetic), exact rational logarithm enclosures, the standard library and the shared root tools checker. The full run takes about twenty minutes, most of it at the conductor $255255$; --quick omits that conductor and the seeded parts of items 8 and 9 and takes under a minute. Before each coordinate recurrence, the bound $\|v\|_\infty+|v_{d-1}|\max_{j<d}|\Phi_M[j]|$ is evaluated with Python integers and required to be at most $2^{60}$. It bounds every product and sum in the update. The stored coordinate tables also satisfy $2M\max|v_{i,j}|\le2^{60}$, bounding every later signed sum and difference in the fixed input domain. Its named checks are:

1. $\mu_{15}$. Enumerating all $3^{15}$ signed vectors gives 446 nonzero signed relations with 127 supports, 98 of them minimal, of sizes 3, 5, 6, 7, 8 and 9; the largest dissociated subsets have 8 elements; there are 1440 proper 2-colorings, all with classes of sizes 7 and 8; the minimal supports are invariant under the 720 coordinate permutations of $\mathbb Z/3\times\mathbb Z/5$, and the colorings form four orbits of 360. In every coloring the 8-element class signed-spans all 15 roots, and the 7-element class spans 9 roots in 360 colorings and 11 roots in the other 1080.
2. Nine layers. The nine labeled colorings in assets/nine_layers.json are proper, and the successive signed-span intersections of their classes shrink to $\{0\}$ for both colors, with sizes 2187, 447, 117, 87, 29, 21, 7, 1, 1 for color 0 and 6561, 447, 227, 67, 49, 13, 9, 7, 1 for color 1. By the layer criterion this gives $\tau_2(S_3\otimes S_5)\le9$.
3. Arc classes. For $M\in\{105,165,195,1155,15015,255255\}$ and $q\in\{3,4\}$, the arc of $\lceil M/q\rceil$ consecutive roots lies below the rank $\varphi(M)$ and no root outside it has power-basis coordinates in $\{-1,0,1\}$ supported on the arc, so the arc spans exactly its own roots.
4. Signed pairs. For the $q=3$ arc class at $M=105$ and $M=1155$, every ordered pair of distinct roots and each relative sign is tested; per agreement pattern of grid coordinates, the fraction inside the arc's signed span is $1/3$ for pairs on one coset of $\mu_3$ with sign $+1$, and at most $2/15$ for every other pattern and sign, the maximum being attained.
5. Two arcs at 165 and 195. $\Phi_{165}$ and $\Phi_{195}$ have height two and $(1-x)\Phi_{165}$, $(1-x)\Phi_{195}$ are flat, so $D(165)=81$ and $D(195)=97$ and two arcs are not proper there.
6. Signed triples at 105. For the $q=3$ arc class at $M=105$, every ordered triple of distinct roots and each sign pattern is tested, and the orbits under coordinate twists are read off from the equality types of the three residues at each prime. Leaving out the orbits whose values visibly have lower complexity (two equally signed points on one coset of $\mu_3$, a whole coset of $\mu_3$, or three equally signed points on one coset of $\mu_5$), every orbit survives with probability at most $4/45$, and every orbit with positive survival needs no more layers than the roots, that is $\lambda\le|O|^{-\log3/\log105}$, with $|O|$ counted as ordered triples (an overcount, which only strengthens the check). The logarithmic comparison uses rational intervals: after reducing an argument to $y\in[1,2)$ by powers of two, set $z=(y-1)/(y+1)$ and bound $\log y$ by the first 32 terms of $2\sum_{j\ge0}z^{2j+1}/(2j+1)$. The omitted tail is at most $2z^{65}/(65(1-z^2))$; the same series at $z=1/3$ bounds $\log2$. Upper bounds on $\log|O|\log(1/\kappa)$ are compared with lower bounds on $\log M\log(1/\lambda)$. An inconclusive interval fails the check; floating-point output is used only for display.
7. The first moment of Lemma 7. With $p$ the next prime and $|G_M|$ the product of the factorials of the primes of $M$, the lower bound $(2^{|C|}-1)/|G_M|^{p-1}$ exceeds $2^{12}$ for a class of $\lceil M/2\rceil$ roots at $M=1155$ and exceeds $2^{2479}$ for classes of $\lceil M/q\rceil$ roots at $M=15015$ and $q\in\{2,3,4\}$, by exact integer comparison; for $q=3$ at $M=1155$ it is below one, so the lemma says nothing there.
8. Survival sets at 105. For the arc class of 35 roots at $M=105$, and for a signed sum $F$ of some of its roots, the checker computes, over all $3!\,5!\,7!=3628800$ coordinate twists $g$, whether $\hat gF$ lies in the class's signed span; below the rank this means that its power-basis coordinates are in $\{-1,0,1\}$ and vanish outside the arc. The sums of the whole class, with all signs $+1$ and with alternating signs, survive exactly the identity and the reflection $e\mapsto34-e$, as Lemma 10 of the note predicts, with no other survivor. The full run adds seeded sums: twelve on 20, 22 and 24 points meet fewer than 11 of the 720 left cosets of the affine group (at most eight), each of four on 10 points meets at least 11 (29 to 415), and among ten on 17 points one meets at least 11 cosets and one survives twists that move its support off the arc; the relation between each such twisted image and its representation inside the arc lies on at most three grid lines (cosets of some $\mu_p$). The threshold 11 is the next prime, the number of layers that Lemma 9 can use for the step to 1155.
9. Survival sets of layered classes at 105. Seven twists of $\mu_{15}$ drawn from a fixed seed layer the three arcs of $\mu_{15}$ into a 3-coloring of $\mu_{105}$; the checker confirms by the layer criterion, with exact signed spans at 15, that it is proper and stays proper after deleting any one layer. Each class has 35 roots, linearly independent (rank 35 modulo the prime $2^{31}-1$). A separately checked right inverse modulo 65521 recovers the only possible ternary coefficient vector, and multiplication by the original integer matrix confirms it. This certifies rejection as well as acceptance without floating-point arithmetic. The sum of each whole class survives only the identity among all twists; in the full run, six seeded signed sums on 20 class points survive at most two twists each (fewer than 11), and six on 10 class points survive 52 to 165 twists each.

Inputs

assets/nine_layers.json lists the color-0 classes of the nine layers as exponents $e$ of $\zeta_{15}^e$; color 1 is the complement. The family was found by a greedy search over all proper 2-colorings of $\mu_{15}$; the checker validates the retained family and does not repeat the search.

Observations that are not certificates

For signed triples at $M=1155$ the note quotes a Monte Carlo estimate, 1500 random triples for each orbit of ordered triples and each sign pattern, with a fixed seed; it is an estimate, not a check. Two further findings come from an exact-coefficient integer program (HiGHS through SciPy, one thread) and are solver verdicts only. The layering of the nine colorings, padded by repetition to 11 and 13 layers, gives 2-colorings of $\mu_{165}$ and $\mu_{195}$ in which the program found no signed relation in either class, while it found one in an arc of $\lceil N/2\rceil$ roots and in the layering with all layers equal; the layer criterion and item 2 already prove these colorings proper. In the coloring of $\mu_{165}$ the program found and checked representations of 71 of the 83 roots outside the 82-element class and of 70 of the 82 roots outside the 83-element class, and declared the remaining 12 roots of each unrepresentable.