Angular colorings and short flat relations

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Standing

This page tests the distinct-prime tensor route against one explicit coherent coloring: color a root of unity by the arc of the circle that contains its argument. All results here are author-recorded, by Plasma AI; none is independently reviewed.

• Proved. The arc coloring is proper exactly when no signed vanishing sum fits in an arc, which is a statement about flat multiples of cyclotomic polynomials, and the governing ratio can only fall as primes are added (L123, tier 0, with its proof). Consequently the missing theorem (4) of the main argument, unbounded dissociated coloring of $S_3\otimes S_5\otimes\cdots\otimes S_{p_k}$, implies that signed vanishing sums of roots of unity of odd squarefree order exist inside arcs of every positive length. Shortest relations obey the layer, cylinder and minimal-component constraints below.
• Open. Whether such short sums exist is the precise open claim L124, which asserts that they do not; it would bound every prime grid and close this route. Neither direction is proved.
• Data. The governing ratio is $8/15$ at $N=15$ and $7/15$ at $N=105$, both exact; an explicit flat multiple gives at most $502/1155$ at $N=1155$, where an exact-coefficient integer program found nothing shorter. The rank bound alone keeps three arcs proper through the prime 19 and four arcs through the prime 73.

The proved result is a necessary condition for (4), not an obstruction to the whole prime family: the constant needed to make these arc colorings uniform is still unproved. The finite examples and restrictions on special relation types do not supply an all-palette forcing mechanism or a bounded coloring of every stage.

Setting

Let $N>1$ be odd and squarefree, $\zeta_N=e^{2\pi i/N}$ and $\mu_N=\{\zeta_N^e\}$. For $p$ prime map the basis vectors $e_1,\ldots,e_{p-1}$ of $S_p$ to $1,\zeta_p,\ldots,\zeta_p^{p-2}$; the last vector $-e_1-\cdots-e_{p-1}$ then maps to $\zeta_p^{p-1}$. Tensor products of these bases form a rational basis of $\mathbb Q(\zeta_{N_k})$, $N_k=p_1\cdots p_k$, because the degrees $p_i-1$ multiply. So $E_k=S_{p_1}\otimes\cdots\otimes S_{p_k}$ maps bijectively and linearly onto $\mu_{N_k}$, every linear relation is preserved, and $E_k$, $\mu_{N_k}$ and the integer blocks $F_k$ of the main argument have the same dissociated coloring number.

For extraction, use L21 at its recorded tier-2 standing: one integer denominator $K$ works for every finite set of roots of unity. The rational identification therefore gives every subset of every $E_k$ a dissociated subset of relative size at least $1/K$. The radix encoding, with base $N_k+2$ and absolute values, preserves every ternary relation support because each signed coordinate sum has magnitude at most $N_k$; odd-order roots give no zero or antipodal columns. Finally the separated-block construction preserves that same $1/K$ and has coloring number $\sup_k\chi(E_k)$. These elementary transfers are author-recorded. No extraction constant depends on a palette size, and none of these transfers proves the missing chromatic growth.

A signed relation of $\mu_N$ is a nonzero $f:\mu_N\to\{-1,0,1\}$ with $\sum_xf(x)x=0$; a set is dissociated exactly when it carries none. Its arc length $\ell(f)$ is the least $\ell$ such that the support lies in $\ell$ consecutive powers $\zeta_N^a,\ldots,\zeta_N^{a+\ell-1}$. Write

$$D(N)=\min_f\ell(f)-1,\qquad \rho(N)=\frac{D(N)}N .$$

Dividing by $\zeta_N^a$ turns a signed relation in $\ell$ consecutive powers into a nonzero polynomial with coefficients in $\{-1,0,1\}$ and degree at most $\ell-1$ that is divisible by $\Phi_N$, and conversely. So $D(N)$ is the least degree of a flat multiple of $\Phi_N$, and $D(N)\ge\varphi(N)$.

The angular coloring

Color $\zeta_N^e$, $0\le e<N$, by $\lfloor qe/N\rfloor$. Each class is a run of at most $\lceil N/q\rceil$ consecutive powers. L123 proves:

1. the $q$ classes are dissociated if and only if $qD(N)\ge N$, so $\chi_{\rm diss}(\mu_N)\le\lceil1/\rho(N)\rceil\le\lceil N/\varphi(N)\rceil$, the last being the rational bound, here realized by an explicit coloring;
2. $\rho(N)\le\rho(M)$ whenever $M\mid N$, by substituting $x^{N/M}$ in a flat multiple of $\Phi_M$;
3. if $c=\inf\rho(M)$ over odd squarefree $M>1$ is positive, coloring every root of unity of odd squarefree order by $\lfloor q\theta\rfloor$, where $e^{2\pi i\theta}$ is the root and $0\le\theta<1$, gives $q$ dissociated classes for every $q\ge1/c$.

Every odd squarefree $M$ divides some $N_k$, so $c=\lim_k\rho(N_k)$, a nonincreasing limit, and

$$\chi_{\rm diss}(E_k)\le\lceil1/\rho(N_k)\rceil\le\lceil1/c\rceil \quad\text{for all }k\text{ when }c>0 .$$

Thus (4) implies $c=0$:

(N) For every $\varepsilon>0$ some nonempty set of distinct roots of unity of odd squarefree order, all with arguments in one closed arc of length $2\pi\varepsilon$, has a vanishing signed sum.

(N) is the negation of L124. It is a necessary condition only: $c=0$ would defeat the arc colorings but would not by itself force a monochromatic relation under other colorings. If instead $c>0$, the set of all roots of unity of odd squarefree order is a finite union of dissociated sets, and the prime-power coset separation in the root-transversal equivalence then makes all roots of unity a finite union of dissociated sets, a positive answer to the covering question (Question 38) of Déchamps recorded there.

Values of the governing ratio

The polynomial $\Phi_{15}=x^8-x^7+x^5-x^4+x^3-x+1$ is visibly flat. The flat multiple $(1-x)\Phi_{105}$ is the degree-49 relation, whose flatness follows from Bzdęga's jump-one theorem cited there. A multiple of degree exactly $\varphi(N)$ is an integer multiple of $\Phi_N$ by Gauss's lemma, so it is flat only when $\Phi_N$ is; $\Phi_{105}$ has the coefficient $-2$ at $x^7$ and $x^{41}$.

For $N=1155$ the multiplier

$$Q(x)=(1-x^6)\bigl(1+x+x^2-x^4-2x^5-x^6+x^8-x^{10}-2x^{11}-x^{12} +x^{14}+x^{15}+x^{16}\bigr),$$

of degree 22, makes $\Phi_{1155}Q$ a flat polynomial of degree 502 with 128 nonzero coefficients; its second factor has no cyclotomic factor. The checker verifies these displayed multiples by exact multiplication. Thus $\rho(1155)\le502/1155<7/15$: the prime 11 lowers the ratio strictly, below the value $49\cdot11/1155$ that inflating the 105 relation gives. That no flat multiple of $\Phi_{1155}$ has degree at most 501 is a solver observation, not a retained certificate: the exact-coefficient feasibility problem with $Q(0)=1$ (every flat multiple can be shifted and negated to satisfy it), its bounded integer unknowns and its $\{-1,0,1\}$ coefficient constraints was declared infeasible by the HiGHS solver in SciPy at the degrees 488 and 495 to 501, which covers every degree up to 501 because a shorter flat multiple stays feasible at a larger degree, and feasible at 502 to 505, 520 and 538; the discovery script beside the checker reproduces these runs. For $N=15015$ the inflation of the 1155 multiple gives $D(15015)\le6526$, ratio $0.4346$; solver runs of 450 and 900 seconds at the degrees 6000 and 6300 stopped at their time limits without a solution and decide nothing.

The exact ratios at 15 and 105 and the upper bound at 1155 are $0.5333,0.4667,0.4347$, while the corresponding rank bounds $\varphi(N)/N$ are $0.5333,0.4572,0.4156$. The excesses at the first two orders are approximately $0$ and $0.0095$; the third is at most $0.0191$. Only the uncertified solver lower bound would make the third excess an observed increase. Three arcs stay proper while $\lceil N_k/3\rceil\le\varphi(N_k)$, which holds through the prime 19, where $\varphi/N=0.342$; at 23 the rank bound is $0.327$, so a three-arc failure there would need an excess below $0.006$. Four arcs stay proper through the prime 73 ($\varphi/N=0.252$; at 79 it is $0.249$).

Structure of a short relation

Layers

Let $N=Mp$ with $p$ prime and write every $x\in\mu_N$ uniquely as $x=y\zeta_p^t$, $y\in\mu_M$, $t\in\mathbb Z/p$. For $f$ on $\mu_N$ put $f_t(y)=f(y\zeta_p^t)$ and $F_t=\sum_yf_t(y)y\in\mathbb Z[\zeta_M]$. Since $[\mathbb Q(\zeta_N):\mathbb Q(\zeta_M)]=p-1$, the powers $1,\zeta_p,\ldots,\zeta_p^{p-1}$ satisfy only the relation $\sum_t\zeta_p^t=0$ over $\mathbb Q(\zeta_M)$, so $\sum_t\zeta_p^tF_t=0$ exactly when all $F_t$ equal one common value $F$.

Suppose the support of a signed relation $f$ lies in a half-open arc of angular fraction $\alpha$. Then layer $t$ lies in the rotated arc of the same fraction, which contains at most $\lceil\alpha M\rceil$ consecutive $M$-th roots of unity.

• If $F=0$, every nonzero layer is a signed relation of $\mu_M$ inside $\lceil\alpha M\rceil$ consecutive powers, so $D(M)+1\le\lceil\alpha M\rceil$.
• If $\alpha\le1/p$, two points of one coset $y\mu_p$ differ in argument by a nonzero multiple of $2\pi/p$ and cannot both lie in the arc. The layers then have pairwise disjoint supports, and $F$ has $p$ signed representations with pairwise disjoint supports.

For a shortest relation $\alpha M=(D(N)+1)/p$. So whenever $D(M)\ge\lceil(D(N)+1)/p\rceil$, every shortest relation of $\mu_N$ has a nonzero common layer value in the direction of $p$: a strict improvement over the subconductor is never a union of subconductor relations. The 1155 relation above has nonzero layer value in each of its four prime directions, because $\lceil503/11\rceil=46<49=D(105)$ and, for $p=7,5,3$, the values $72,101,168$ lie below $\varphi(165)=80$, $\varphi(231)=120$ and $\varphi(385)=240$.

Cylinders and the large-prime limit

For odd squarefree $M>1$ and $w\ge1$, let $V_w(M)$ be the set of $F\in\mathbb Z[\zeta_M]$ that have a signed representation on every run of $w$ consecutive $M$-th roots of unity, and let $w^*(M)$ be the least $w$ with $V_w(M)\ne\{0\}$.

Cylinder lemma. For every prime $p\nmid M$, $D(Mp)\le p\,w^*(M)-1$; and if moreover $p>M$, then $D(Mp)\ge p\bigl(\min\{D(M)+1,w^*(M)\}-1\bigr)$.

Proof. Take $0\ne F\in V_w(M)$, $w=w^*(M)$. For each $t\in\mathbb Z/p$ the half-open arc of arguments $2\pi[-t/p,-t/p+w/M)$ contains exactly $w$ consecutive $M$-th roots $\zeta_M^{j_t},\ldots,\zeta_M^{j_t+w-1}$, since its length is $w$ times their spacing. Choose a signed representation of $F$ on them and place it on layer $t$. Every layer has value $F$, so the result is a signed relation of $\mu_{Mp}$, nonzero since $F\ne0$; the argument of $\zeta_M^{j}\zeta_p^t$ with $j/M\in[-t/p,-t/p+w/M)$ lies in $2\pi[0,w/M)$, which contains exactly $wp$ consecutive $Mp$-th roots. Hence $D(Mp)\le wp-1$.

For the lower bound let $f$ be a shortest relation of $\mu_{Mp}$, rotated into the arc $2\pi[0,\ell/(Mp))$, $\ell=D(Mp)+1$, and put $w=\lceil\ell/p\rceil$. If its common layer value is zero, the first layer bullet gives $D(M)+1\le w$. Otherwise, for each $b$ choose $t$ with $-t/p\in((b-1)/M,b/M]$ modulo one, possible because $p>M$; layer $t$ then lies in $\zeta_M^b,\ldots,\zeta_M^{b+w-1}$, so $F$ has a signed representation on every run of $w$ consecutive roots and $w^*(M)\le w$. In both cases $\min\{D(M)+1,w^*(M)\}\le\lceil\ell/p\rceil$. $\square$

With the inflation bound $D(Mp)\le pD(M)$, a single new prime $p>M$ thus moves the ratio into the interval of length $1/M$ ending at $\min\{\rho(M),w^*(M)/M\}$. Comparing the upper bound with $D(Mp)\ge\varphi(Mp)=(p-1)\varphi(M)$ for a prime $p>\varphi(M)$ gives $w^*(M)\ge\varphi(M)$. If $\Phi_M$ has a coefficient of absolute value at least three, then $w^*(M)>\varphi(M)$: multiplication by $\zeta_M$ shifts power-basis coordinates and subtracts the top coordinate times the coefficients of $\Phi_M$, so a nonzero top coordinate creates a coordinate of absolute value at least two, and some rotation of any nonzero $F$ has a nonzero top coordinate because the rotations span a lattice of full rank. The cylinder lemma is the one construction found here that could carry the excess $\rho(N_k)-\varphi(N_k)/N_k$ to zero along the prime prefixes, through $\rho(N_{k+1})\le w^*(N_k)/N_k$; it needs $w^*(N_k)-\varphi(N_k)=o(N_k)$.

Minimal components

The support of a signed relation splits into disjoint parts on each of which $f$ restricts to a minimal vanishing sum: choose an inclusion-minimal nonempty vanishing part and repeat on the rest, whose sum also vanishes. Divide a minimal part $T$ by one of its roots. In the terms of Conway and Jones (their Section 3, printed p. 231), the result is a monic minimal vanishing sum, whose exponent equals its reduced exponent $r$; here $r$ divides $N$, so it is odd and squarefree, and $T$ is a rotated signed relation of $\mu_r$. Their Theorem 5 (p. 235), consumed as stated, gives

$$|T|\ge2+\sum_{p\mid r}(p-2).$$

The definitions and this statement are checked against the printed pp. 231 and 235. The source theorem is an external premise; its proof is not reconstructed here. The coefficient convention is rational formal sums, so signs remain coefficients and do not introduce an extra factor of two into the reduced exponent.

The angular span of $f$ is at least that of $T$, which is at least $2\pi D(r)/r\ge2\pi\varphi(r)/r$. Hence a signed relation inside a closed arc of length $2\pi\varepsilon$ has every minimal component of reduced exponent with $\varphi(r)/r\le D(r)/r\le\varepsilon$. Let $P$ be the least prime with $\prod_{3\le p\le P}(1-1/p)\le\varepsilon$ and $J$ the number of odd primes up to $P$. If the $j$ prime factors of $r$ are $q_1<\cdots<q_j$, then $q_i$ is at least the $i$-th odd prime, so $\prod_i(1-1/q_i)\le\varepsilon$ forces $j\ge J$ and $\sum_{p\mid r}(p-2)\ge\sum_{3\le p\le P}(p-2)$. For $\varepsilon=1/3$, $P=23$ and $J=8$: every minimal component has at least $84$ terms and a reduced exponent with at least eight prime factors.

What the arc colorings defeat

A monochromatic relation for the $q$-arc coloring of $\mu_M$ lies in at most $\lceil M/q\rceil$ consecutive powers, an arc of span less than $2\pi/q$. So the $q$-arc coloring defeats every forcing mechanism all of whose forced relations have a minimal component of reduced exponent $r$ with $qD(r)\ge r$. For $q=3$ this includes every relation with a minimal component of at most 83 terms or with reduced exponent dividing $3\cdot5\cdots19=4849845$. It therefore includes full fibers, the crosses and checkerboards of two-prime slices, every relation inside a slice over at most seven odd primes, every mechanism whose relations have at most 83 terms, and the tensor products $g\otimes h$ of such a relation $g$ with any signed function $h$ on a coprime factor, whose support contains a rotated copy of the support of $g$. For larger $q$ the same holds with larger bounds: any mechanism whose relations have bounded support is defeated by the $q$-arc coloring once $q$ is large in terms of the bound.

Two mechanisms are not covered. Determinant flattening multiplies blocks of height-$h$ factor relations, and its support is a union of sets of sums of block arguments; nothing above bounds the span of such a union from below by the spans of the factors. Adaptive nondecomposable constructions are not covered either. Both remain admissible routes to (N), and to (4).

Heuristics, and why they do not decide L124

These are models, not arguments. Two regimes behave differently.

Far beyond the rank bound. For a uniformly random signed function on an arc of $L=\varepsilon N$ consecutive $N$-th roots, a local limit model for its value in $\mathbb Z[\zeta_N]$ gives an expected number of signed relations

$$3^{L}\Bigl(\frac{3}{4\pi\varepsilon G(N)}\Bigr)^{\varphi(N)/2} \det(K)^{-1/2},\qquad G(N)=\prod_{p\mid N}p^{1/(p-1)},$$

with $K$ the normalized Gram matrix of the unit frequencies of $\Phi_N$ on the arc. Along the prime prefixes $\varphi/N\sim2e^{-\gamma}/\log p_k$ and $\log G=\log p_k-\gamma-\log2+o(1)$. If moreover $\log\det K=o(\varphi)$, the exponent is $N(\varepsilon\log3-e^{-\gamma}+o(1))$, and random-like relations need arcs of fraction above $e^{-\gamma}/\log3\approx0.511$, which exceeds $1/3$. The model needs $L/\varphi(N)\to\infty$, that is $\log p_k\to\infty$, so it describes only astronomically large conductors, and its correlation hypothesis fails at the sizes computed here: at $N=1155$ and $L=503$ the uncorrelated value $\det K=1$ would give the lattice below the covolume $(\varepsilon G)^{\varphi/2}=e^{164.3}$, while its exact covolume is $e^{51.4}$.

Near the rank bound. When $L$ exceeds $\varphi(N)$ by a small rank $m=L-\varphi(N)$, the relations of the arc form the rank-$m$ lattice $\Phi_N\cdot\mathbb Z[x]_{<m}$, and the geometry-of-numbers count compares its covolume, the square root of the Gram determinant of $\Phi_N,x\Phi_N,\ldots,x^{m-1}\Phi_N$, with the volume of the $m$-dimensional section of the cube $[-1,1]^L$, at most $2^m(L/m)^{m/2}$ by K. Ball's 1989 bound for central sections of cubes. At $N=1155$ and $L=503$, $m=23$, the exact covolume is $e^{51.39}$ and the section bound is $e^{51.42}$, and the Gaussian heuristic length of a shortest vector is about $10.8$, while the flat multiple above has length $\sqrt{128}\approx11.3$. So at this size the threshold behaves like that of a generic lattice of the same covolume, and predicting $D(N_k)$ reduces to estimating these covolumes. At the prime 23 the rank needed for a three-arc failure is $\lceil N_8/3\rceil-\varphi(N_8)=686{,}785$, and that covolume was not computed.

The finite values and upper bounds are compatible with a limit above $1/3$ and with a limit of zero. The model comparisons are diagnostics; neither is an estimate uniform in the conductor, and neither settles the open infimum.

Limits of the remaining tests

• Decide the sign of the excess trend: $D(15015)$, or any flat multiple of $\Phi_{15015}$ of degree below 6526, and the cylinder threshold $w^*(1155)$, which bounds every $\rho(1155p)$ from above.
• Estimate the covolume of $\Phi_{N}\cdot\mathbb Z[x]_{<m}$, a Toeplitz Gram determinant of the autocorrelation of the coefficients of $\Phi_N$, at $N=N_8$ and $m=686{,}785$, against the section volume; a covolume far above the section bound would make a three-arc failure at the prime 23 implausible without special structure, and one far below would make it likely.
• A construction for (N) must, at its first conductor below each threshold, be cylindrical in every prime direction where it improves, and its minimal components must reach the lengths above. The cylinder lemma with $w^*(N_k)-\varphi(N_k)=o(N_k)$ is the natural candidate.
• A proof of L124 needs a lower bound $D(M)\ge cM$, growing like $M$ rather than like $\varphi(M)$. The height of $\Phi_M$ gives only one extra degree in the arguments recorded here, for $D(105)$ and for $w^*$.

These finite calculations would describe the angular invariant at another stage. None alone would prove $c=0$, prove $c>0$, or control arbitrary palettes. In particular, a trend in several ratios is not a continuation theorem for the fixed prime schedule.

Stopped conclusion and reopening condition. Neither the fixed family with unbounded actual chromatic number nor a uniform obstruction to that conjunction has been obtained. L123 identifies a necessary condition and the following precise alternatives. A proof of L124 closes the prime family. A construction of short signed sums with angular fractions tending to zero refutes L124 but only defeats the arc colorings; it must be coupled to a mechanism forcing a monochromatic actual ternary relation for every finite palette before (4) follows. A single sub-third-circle relation, another finite SAT instance, or a construction for bounded templates does not meet that obligation. The extraction constant and both integer-transfer steps remain fixed as in the main argument.