Main argument and missing coloring theorem

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Target and scope

The target is Problem 774, whose dated assessment records the integer question as open; this research does not change that status. The problem asks whether every proportionately dissociated set of positive integers is a finite union of dissociated sets. The attack seeks a counterexample: an infinite set $A\subset\mathbb N_{>0}$ with

1. one $\delta>0$ such that every finite $F\subseteq A$ contains a dissociated subset of size at least $\delta|F|$, and
2. no finite partition of $A$ into dissociated sets.

A finite family is dissociated if its only zero relation with coefficients in $\{-1,0,1\}$ is trivial. Write $r(F)$ for its largest dissociated subset size, $\eta(E)=\min_{\varnothing\ne F\subseteq E}r(F)/|F|$ for the hereditary extraction constant, $\chi(F)=\chi_1(F)$ for the dissociated coloring number, $\chi_h$ for the coloring number forbidding integer relations of coefficient height at most $h$ with unrestricted finite support, $\chi_{\mathbb Q}$ for the rational-independence coloring number, and $\rho$ for the positive-real Sidon parameter. Vector configurations are treated through their signed relation supports; the integer realization below makes that exact.

Standing

E0774 is not solved. Uniform extraction is proved for several explicit finite families, and the integer encoding and separated-block assembly are exact, so the whole problem reduces to one finite statement, (F) below. No family has a proved unbounded actual dissociated coloring number, and that is the single gap. Every result on these pages is author-recorded: separate reviewers checked most arguments, but no verification record is filed, so no result here carries a tier, and a composition of author-recorded proofs remains author-recorded. The roots-of-unity analytic Sidon inequality and the extraction-denominator statement are machine-checked as the tier-2 claims L20 and L21 (each with a fidelity audit filed and graded pass for the Lean sources and statement as they stood on 2026-09-28, the default branch at 13:50:48Z, and the warrant bound by a grade decision to the Lean v4.34.1 tree that the non-author clean gate of 2026-10-13 checked). The angular coloring bound for the prime grids is the tier-0 claim L123, and its decisive open question is the open claim L124; every other result is author-recorded without a claim card. No counterexample is claimed. The finite computations in the evidence folder check certificates and record diagnostics; none proves an unbounded statement.

The reduction to finite blocks

The finite-block theorem is proved with an elementary argument: if finite nonempty $E_t\subset\mathbb N_{>0}$ satisfy $r(F)\geq\delta|F|$ for every $F\subseteq E_t$ and every $t$, with one $\delta>0$, and $p_1=1$, $p_{t+1}>\sum_{i\le t}p_i\sum_{x\in E_i}x$, then $A=\bigcup_tp_tE_t$ is infinite, every signed relation in $A$ separates by blocks, $A$ is proportionately dissociated with the same $\delta$, and

$$\chi(A)=\sup_t\chi(E_t).$$

Restriction proves the lower bound; if the supremum is a finite $q$, coloring every block with that palette gives a proper global coloring by block separation. The exact rational-to-integer encoding realizes any finite configuration of nonzero, pairwise non-antipodal rational vectors by distinct positive integers whose ternary relation supports coincide with the original ones, up to column signs, so extraction and coloring are preserved. Both statements are author-recorded and use no analytic theorem.

Consequently the required theorem is:

(F) There are finite integer-vector configurations $F_j$ and one $\delta>0$ such that every subset of every $F_j$ contains a dissociated subset of relative size at least $\delta$, while $\sup_j\chi_1(F_j)=\infty$.

Uniform extraction is the first half of (F); unbounded actual coloring is the second. The families below each satisfy the first half. None is known to satisfy the second.

Uniformly extracting families

Distinct-prime tensors

Let $p_1=3,p_2=5,p_3=7,\ldots$ enumerate the odd primes, and set

$$S_p=\{e_1,\ldots,e_{p-1},-e_1-\cdots-e_{p-1}\},\qquad E_k=\bigotimes_{i=1}^k S_{p_i}.$$

Put $N_k=\prod_{i\leq k}p_i$, $d_k=\prod_{i\leq k}(p_i-1)$, and $B_k=N_k+2$. In any fixed order of the $d_k$ coordinates, define

$$F_k=\left\{\left|\sum_{j=0}^{d_k-1}v_jB_k^j\right|:v\in E_k\right\}. \tag{1}$$

These are distinct positive integers. Every column coordinate is $0,\pm1$, so a signed sum has digits bounded by $N_k<B_k-1$; its leading nonzero digit cannot cancel. The columns have no zero or antipodal pair. Therefore the encoding preserves all signed-relation supports, up to column sign changes, and preserves extraction and coloring.

The prime-tensor theorem gives one $\delta_0>0$ for all subsets of all $E_k$, hence of all $F_k$. Its proof:

1. Randomized lattice rounding gives $\rho(E\otimes S_p)\geq\rho(E)-Q(E)/(2p^2)$, with $Q(E)=\pi^2\prod_{q\text{ an existing side}}[(q-1+H_{q-1})/q]$; the target-dependent rounding distribution has mean zero and the stated second-moment bound, and no common distribution for all targets is asserted.
2. Elementary prime estimates and $H_{p-1}-1<\log p-\log(3/2)$ give, for $P=2^{40}$,

$$\sum_{p>P}\frac{Q(P,p^-)}{2p^2}<2^{-18}.$$

3. Apply Pisier's Theorem 2, the equivalence of Sidonicity with proportional quasi-independent extraction, consumed once as a claims-checked literature statement at its full strength, to the nested prime-tail union, obtaining one $\delta>0$. Restoring small-prime coordinates costs at most the fixed $M=\prod_{p\leq P}p$. Thus $\delta_0=\delta/M$ works uniformly, without blockwise decay. The Pisier constant is not numerically quantified.

The same argument, with cosetwise cyclotomic decomposition, proves that all roots of unity are proportionately dissociated in the additive complex plane; this is ordinary Sidonicity, not positive-real Sidonicity, and it does not imply an unbounded coloring number. The original tensor theorem, with sides $q_i=\lceil1000i\log_2(i+1)\rceil$ and positive-real constant at least $1-1/90000$, remains valid and uses Pisier once in the same way.

Set $s_1=1$ and recursively

$$s_{k+1}=1+\sum_{i\leq k}s_i\sum_{a\in F_i}a,\qquad A_{\rm prime}=\bigcup_{k\geq1}s_kF_k. \tag{2}$$

By the finite-block theorem, $A_{\rm prime}$ is infinite and proportionately dissociated with constant $\delta_0$, and

$$\chi(A_{\rm prime})=\sup_k\chi(F_k)=\sup_k\chi(E_k). \tag{3}$$

It remains to prove

$$\boxed{\sup_k\chi_{\rm diss} (S_3\otimes S_5\otimes\cdots\otimes S_{p_k})=\infty.} \tag{4}$$

This is unproved. The coloring numbers are nondecreasing in $k$, since each grid embeds as a slice of the next. The checked three-prime example has coloring number three, and the same value holds through the prime prefix ending at 19; at prime 23 only $3\le\chi_{\rm diss}\le4$ is established. The vanishing rational rank fraction $d_k/N_k$ does not establish (4). Every fixed finite coloring of the squarefree odd-root grid is equivalent, by the root-transversal equivalence, to one prime-fiber transversal; the rotation-averaging identity there gives $\eta(\mu_n)=\Psi(n)/n$ and $\chi_f(\mu_n)=n/\Psi(n)$, so the remaining problem is exactly integral coloring, not another local extraction condition.

Coloring the roots by arcs of argument gives a necessary condition for (4). Let $D(N)$ be the least degree of a nonzero polynomial with coefficients in $\{-1,0,1\}$ divisible by $\Phi_N$. By L123 (tier 0), coloring $\zeta_N^e$ by $\lfloor qe/N\rfloor$ is proper exactly when $qD(N)\ge N$, and $D(N)/N$ does not increase along divisibility, so

$$\chi_{\rm diss}(E_k)\le\lceil N_k/D(N_k)\rceil,\qquad c:=\lim_kD(N_k)/N_k=\inf_{M>1\text{ odd squarefree}}D(M)/M. \tag{4a}$$

Hence (4) implies $c=0$: signed vanishing sums of roots of unity of odd squarefree order must exist inside arcs of every positive length. The open claim L124 asserts $c>0$, which would bound every prime grid by $\lceil1/c\rceil$ and close this candidate. Known values are $D(15)/15=8/15$, $D(105)/105=7/15$ and $D(1155)/1155\le502/1155$; the angular-coloring page gives the structure that short relations must have and the evidence in both directions. Neither $c=0$ nor $c>0$ is established, so this is a necessary-condition reduction, not a uniform obstruction to the prime family. Refuting L124 alone would still leave the all-palette conclusion (4) unproved.

Linear-side axis lines and their covers

For $d_i=i+2$, let $L_t$ consist of all full axis-parallel line indicators on $X_t=\prod_{i\le t}[i+2]$. Write $N_t=|X_t|=(t+2)!/2$, $s_t=\sum_{i\le t}1/(i+2)$, and $M_t=|L_t|=N_ts_t$. The directional phase theorem proves

$$\rho(L_t)\ge \min_{i\le t}\prod_{j=i+1}^t\cos(\pi/(j+2))^{i+2} \ge e^{-\pi^2/2}. \tag{5}$$

A regular polygon of the $d$-th roots of any prescribed unit complex number contains the disk of radius $\cos(\pi/d)$. Independent root choices with that same positive real expectation align all new-axis terms simultaneously, while an old direction-$i$ line has expected multiplier $\cos(\pi/d)^{d_i}$. Optimizing the old coefficients after multiplying by this attenuation gives the displayed induction; for the linear schedule the logarithmic losses telescope to $\pi^2/2$. Primitive slice embeddings make the $L_t$ nested in one free abelian group; one application of Pisier's Theorem 2 to that union gives one $\delta_{\rm line}>0$ for every subset of every $L_t$. This is full proportional dissociation, not bounded-order freeness, and it does not inherit from the prime-tensor theorem.

Enumerate $X_t$ lexicographically by $j=0,\ldots,N_t-1$, put $B_t=M_t+2$, and set

$$F_t^{\rm line} =\left\{\sum_{x\in\ell}B_t^{j(x)}:\ell\text{ is a full axis line}\right\}, \qquad a_1=1,\quad a_{t+1}=1+\sum_{u\le t}a_u\sum_{v\in F_u^{\rm line}}v, \qquad A_{\rm line}=\bigcup_{t\ge1}a_tF_t^{\rm line}. \tag{6}$$

The carry-free realization and the separated-block lemma give, unconditionally, an infinite proportionately dissociated integer set with

$$\chi_1(A_{\rm line})=\sup_t\chi_1(L_t), \tag{7}$$

whose right side is not known to be infinite.

The cover theorem extends (5) to all finite incidence covers: for every cover degree and every incidence permutation, $\rho(\widetilde L_t)\ge e^{-\pi^2/2}$, and with every row duplicated $R$ times before covering, $\rho(\widetilde L_t^{[R]})\ge e^{-2\pi^2}$, uniformly in $t$, $R$, the cover degree and the permutations; a finite sine-kernel root distribution supplies the target-independent moments $c_k\ge e^{-2\pi^2k^2/D^2}$ needed when an old column meets a new block up to $R$ times. Placing all countably many covered configurations in independent free abelian summands and applying Pisier once gives one extraction constant for the entire collection; the corresponding explicit integer set $A_{\rm cov}$ satisfies

$$\chi_1(A_{\rm cov})= \sup_{t,R,m,\,\text{incidence permutations}} \chi_1(\widetilde L_t^{[R]}). \tag{8}$$

Row projection shows that deleting all duplicated rows except the first at each base point maps a cover of a duplicated line matrix onto an ordinary cover on the same distinct columns with a larger kernel, so common row duplication is chromatically redundant: with $\Gamma_t=\sup_{m,\pi}\chi_1(\widetilde L_t)$ over ordinary covers,

$$\sup_{R,m,\pi}\chi_1(\widetilde L_t^{[R]})=\Gamma_t, \qquad \chi_1(A_{\rm cov})=\sup_t\Gamma_t, \tag{9}$$

and the duplicated family inherits the stronger bound $e^{-\pi^2/2}$ directly. Infinitude of (9) is unproved.

The same phase induction gives $\rho\ge e^{-\pi^2/2}$ for arbitrary signed linear direction matchings of sizes $i+2$ with cross-direction intersections at most one, each old coefficient prescaled by its actual number of new blocks met, and for finite field towers, denser linear flags, and nested polynomial systems, with $\rho\ge e^{-\pi^2}$ for $p\ge3$ and $\rho\ge\tfrac16e^{-\pi^2}$ for the nested binary polynomial model. Matching-decorated transversals keep $\rho\ge\tfrac12e^{-\pi^2}$. Each is a larger uniformly extracting family with the same missing coloring theorem; chromatic lower bounds are not known to pass between these families and the covers.

Difference towers

For a finite integer-vector family $A=(a_v)_{v\in V}$, $G=(\mathbb Z/N)^V$, and $D_N(A)=\{a_v\otimes(e_g-e_{g+e_v})\}$, the difference-tower theorem proves $\eta(D_N(A))\ge(1-1/N)\eta(A)$ for $N\ge4$, by cutting one global seam per coordinate, extracting incoming types at each head, and excluding relations at a maximal vertex; a phase induction on the same order gives $\rho(D_N(A))\ge(1-1/N)\rho(A)-1/N$ without a new Pisier application. For $A^{(j)}=D_{N_j}(A^{(j-1)})$ with $N_j=2^{j+3}$,

$$\eta(A^{(k)})\geq\eta(A^{(0)})\prod_{j=1}^k(1-1/N_j)\ge\tfrac78\eta(A^{(0)}).$$

Every integer relation of height $h$ lifts, for every modulus $N\ge4$, to a nonzero relation of height $\lceil h/2\rceil$ by a centered discrete-cube primitive, and the minimum height outside the cyclewise-constant subgroup is exactly $\lceil\lambda(A)/2\rceil$. All difference operations must precede exact scalar encoding, since an intermediate ternary-preserving encoding may alter the height-two input relations of a later stage.

Critical Boolean moments

For $1\le k<d$ let $E_{d,k}$ consist of the integer columns of all squarefree degree-at-most-$k$ moments of the points of $\{0,1\}^d$; there are $2^d$ distinct nonzero columns and $r=\sum_{s\le k}\binom ds$ rows. Coordinate flips act by unimodular row changes, so $\eta(E_{d,k})=\alpha(d,k)/2^d$ exactly, with $\alpha$ the maximum dissociated size. Along even $d$ with $k=\lfloor d/2-a\sqrt{d\ln d}\rfloor$, the congruence-sensitive entropy bound forces $\eta\to0$ for fixed $a>1/2$, and the Gaussian counting theorem gives $\chi_1(E_{d,k})=2$ eventually for every fixed $0<a<1/2$. The rounded boundary

$$k_d=\left\lfloor\frac d2-\frac12\sqrt{d\log d}\right\rfloor$$

is the critical family. Its extraction is proved: the integer relation lattice $K_d$ has rank $m=n-r$, $n=2^d$, determinant $2^{D/2}$ with $D=(d-k_d)\binom d{k_d}$, and is isometric to the integer Boolean polynomial lattice of degree at most $d-k_d-1$; the compression theorem proves that every nonzero sublattice has normalized covolume at least that of the full lattice (semistability); the weak stable-lattice free-energy theorem gives $\log\theta_L(t)\le\epsilon(t)\,\mathrm{rank}(L)$ with one $\epsilon(t)\to0$ uniformly over stable lattices, hence by Poisson duality $\theta_L(u)\le u^{-m}e^{\epsilon(1/u)m}$; since the normalized covolume tends to $\sigma_*=2^{1/(2\sqrt{2\pi})}>1$, fixing $1<\sigma_0<\sigma_*$ and $T$ with $\epsilon(T)\le\tfrac12\log\sigma_0$ and $s=1/(\sigma_0T)$ gives

$$\log s+n^{-1}\log\theta_{K_d}(s)\le-\tfrac14\log\sigma_0<0$$

eventually; and the finite phase-packing and Gaussian-width criterion, with flip averaging, turns that fixed-scale theta estimate into one hereditary full-dissociation extraction constant. The integer-graph and modular-depth reductions are exact alternative descriptions of the same target, not additional premises. The critical family's coloring is open: the entropy calculation gives only $\chi_1\ge6$ eventually at the boundary.

Simplicial boundaries

For the oriented boundaries of all $k$-subsets of $[n]$, the simplicial theorem identifies the integer relation lattice and its determinant exactly:

$$K_{n,k}=\bigwedge^k A_{n-1},\quad \operatorname{rank}K_{n,k}=\binom{n-1}{k},\quad \det(K_{n,k})^2=n^{\binom{n-2}{k-1}}.$$

A two-dimensional signed-permutation commutant proves irreducibility on its real span; uniqueness of the minimum-normalized-covolume sublattice then proves semistability. At $n/k\sim C\log n$ for one fixed $C>0$, the normalized covolume tends to $e^{1/(2C)}>1$ and the relation rank is asymptotic to the number of columns, so the same theta argument and signed permutation averaging give one hereditary extraction constant. Letting $C$ grow to force an entropy color bound would lose uniform extraction. The matroid partition theorem of Edmonds gives $\chi_1\le\chi_{\mathbb Q}=\lceil n/k\rceil$, so bounded $n/k$ cannot give unbounded coloring, while $n/(k\log n)\to\infty$ loses extraction by entropy; see the scoped simplicial checks.

The missing theorem

For each family above, the missing statement is the second half of (F). In its sharpest available forms:

• for the prime grids, equation (4), which requires the short signed vanishing sums of (4a), the negation of L124;
• for the line family and its covers, infinitude of (7), (8) or (9);
• for the simplicial family, fix one $C>0$ and $k=\lfloor n/(C\log n)\rfloor$:

$$\forall q<\infty\ \exists n:\quad \text{every }q\text{-coloring of the oriented }k\text{-subset boundary columns contains a nonzero monochromatic ternary cycle};$$

• for the critical moments, unbounded $\chi_1(E_{d,k_d})$: a nonzero monochromatic ternary function with the required vanishing moments under every finite palette.

E0774 remains open; the missing theorem is (4), or any of these equivalents for another uniformly extracting family. A proof of any one, followed by the exact integer encoding and separated assembly, would give the required counterexample. Nothing weaker suffices: rational rank, rational homology, or $\chi_{\mathbb Q}$ growth; a monochromatic relation of height depending on the palette ($\forall q\,\exists h(q)$ rather than $\exists h\,\forall q$); fixed-order $h$-freeness; the failure of restricted palettes; extraction constants that decay with the stage; or any bounded computation.

Routes toward the coloring theorem, by subject

Each route below is author-recorded with its exact outcome. Retired routes are kept so that their obstructions are not rediscovered; the condition that would reopen each is stated on its page.

Restrictions on colorings of the prime grids

Angular colorings. The arc colorings are one coherent coloring of all roots of unity of odd squarefree order that every forcing mechanism for (4) must defeat. Three arcs stay proper through the prime 19 and four arcs through 73 by the rank bound $D(N)\ge\varphi(N)$ alone. A relation monochromatic for $q$ arcs has span below $2\pi/q$; with Conway–Jones' length bound for minimal vanishing sums this means that for $q=3$ every minimal component has at least 84 terms and a reduced exponent with at least eight prime factors. So the three-arc coloring defeats every mechanism whose relations have a minimal component of at most 83 terms or of reduced exponent dividing $3\cdot5\cdots19$, including menus with at most 83 terms, two-prime crosses and checkerboards, every relation inside a slice over at most seven primes, and their tensor products with signed functions on coprime factors; determinant flattening and adaptive nondecomposable constructions are not covered. A shortest relation that improves on a subconductor has nonzero common layer value in each improving prime direction. For a new prime $p>M$, the cylinder lemma places $D(Mp)/(Mp)$ within $1/M$ below $\min\{D(M)/M,w^*(M)/M\}$, with $w^*(M)\ge\varphi(M)$. Whether $D(N_k)/N_k$ falls below $1/3$ at some stage, first possible at the prime 23, is open.

Profile compression. For any proper $q$-coloring of the squarefree odd-root grid, let $m_p$ be the number of distinct coordinate color profiles in the prime-$p$ coordinate, a profile recording the color in every context of the other coordinates. The coefficient-safe profile compression theorem proves

$$\sum_{p\text{ odd}}\left(\frac1{m_p}-\frac1p\right)\leq2\log(48q), \tag{10}$$

with a finite-prefix version of the same bound. Monochromatic relations in the compressed indexed tensor lift across disjoint boxes with coefficients in $\{-1,0,1\}$. This excludes uniformly bounded profile counts and fixed-order bounded-state scans, but not arbitrary colorings: the case $m_p=p$ for every prime is not ruled out, and a finite palette does not bound the number of profiles. The partial-box theorem extends compression to exactly monochromatic cells and proves that every fixed disjoint-box complexity is finitely colorable (16 colors for two boxes); chain circuits show that unbounded box complexity alone forces nothing, since chains needing $k$ boxes all admit one 30-color proper coloring.

Bounded height and determinant flattening. For every $q$, with $h_q=\lceil e^{3q}\rceil$ and $P\ge e^{8(q+1)}$, every $q$-coloring of the tensor over primes $p\le P$ excluding any one prime $\ell$ has a monochromatic nonzero integer relation of height at most $h_q$; when $\ell\ne2$ the grid includes $S_2$. Grouping a factor relation by coefficient magnitude gives disjoint signed sums $w_{i,m}$ with $\sum_mmw_{i,m}=0$; a minimal Hall-tight subfamily (equally many factors and distinct magnitudes) gives a square matrix with vanishing determinant and a nonempty expansion term, and coprime conductors with disjoint group supports keep the expanded coefficients in $\{-1,0,1\}$. Thus $h$ conductor-disjoint height-$h$ dependent factors suffice if the union of the nonempty permutation-product blocks has one color. Forcing that monochromatic determinant support under an arbitrary palette is the unproved step; whole-context focusing raises the palette to $q^{|X|}$, and excluding used primes reduces the harmonic density to $\alpha_S=\prod_{p\in S}(1-1/p)$, so the eventual height condition becomes $\log(h+1)>q/\alpha_S$. See bounded height and flattening and the height-descent templates, which separate genuine one-step algebraic descent from the unproved matching of actual color classes.

Fixed-height focusing. Write $G=\bigoplus_{p\text{ odd prime}}\mu_p$. The fixed-height focusing theorem proves

$$\boxed{\chi_h(G)=\infty\text{ for one fixed finite }h \quad\Longrightarrow\quad\chi_1(G)=\infty,} \tag{11}$$

and in fact, for every fixed $h$, finiteness of the two coloring numbers is equivalent. The proof pays both focusing costs: infinite $\chi_h$ survives deleting finitely many conductor primes, a finite witness defeating $q$ colors gives a full-profile palette $q^{|F|}$ on the remaining tail, the same height $h$ defeats that palette, and induction produces $h$ conductor-disjoint height-$h$ dependent factors with an actually monochromatic full product, to which determinant flattening applies. An idempotent-ultrafilter formulation is proved conditionally on one common height for every member. The hypothesis of (11) is unproved: what is known is $\forall q\,\exists h(q):\chi_{h(q)}(G)>q$, not $\exists h\,\forall q:\chi_h(G)>q$.

Binary circuit lifting. In the standard integral tensor matrix of an odd-prime grid, every minimal binary dependence on at most eighteen columns lifts to a full-support ternary relation, nineteen is sharp, and on a tail whose first three permitted primes are $r_1<r_2<r_3$ every binary dependence of support smaller than $2r_3$ contains a ternary dependence (binary lifting). For a dissociated $n$-point set $C$ the full binary kernel has distance at least nineteen, so

$$\operatorname{rank}_{\mathbb Q}A_C\geq \operatorname{rank}_{\mathbb F_2}\overline A_C \geq\log_2\sum_{j=0}^{9}\binom nj. \tag{12}$$

This is a support-size threshold, not a height bound; (12) still permits nullity density tending to one; projected circuits need not keep one color; and arbitrary scalar encodings do not preserve the modular statement.

Coefficient growth and the staircase. For conductor-disjoint relation supports $X_i$ whose full product has one color, with nonzero coefficient-magnitude sets $I_i$, $m_i=|I_i|\ge2$ and $b_i=\min I_i$, the weighted-simplex counting theorem forces a nonzero ternary relation whenever

$$\prod_i\frac{m_i}{m_i-1} \ge1+\sum_i\frac1{m_i-1} \sum_{a\in I_i\setminus\{b_i\}}\log_2(b_i+a), \tag{13}$$

together with a sharper exact integer subset-counting inequality, a palette version for $K$ individually monochromatic blocks ($2^K\le\prod_t(1+W_t/q)^q$), and the at-most-$q$-per-rank rule for constant-ratio-two blocks. If every $m_i\le M$, a dissociated full product must have $\sum_{i\le r}\log_2(2h_i-1)>(M/(M-1))^r-1$ with $h_i=\max I_i$, a doubly exponential height escape that no focusing argument controls. The root height-escape construction shows such escape is possible with actual cyclotomic units: fresh prime triples give circuits with magnitude pairs $\{1,2^{2^{i-1}}\}$, every finite full product is dissociated, and the least relation height of the $j$-th tail is exactly $2^{2^{j-1}}$, so height-free root flattening is false. The prime staircase constructs actual-root dissociated sets $C_t$ with saturated column image, an integral kernel basis of primitive height-two circuits with connected support graph, full kernel rank of density tending to one, and a mandatory coordinate in every nonzero signed-sum class; consecutive prime triples $p_i<q_i<r_i$ give coefficient-two circuits of rank $d_i=3p_i+2q_i+2r_i-14\le7r_i$, and index-two basis replacements one factor at a time give

$$R=\prod_i d_i,\qquad |C_t|=R\Bigl(1+\sum_i\frac1{d_i}\Bigr),\qquad k_t=R\sum_i\frac1{d_i}, \tag{14}$$

with every ternary relation in $C_t\otimes S_p$ containing a full fiber and $\chi_1(C_t\otimes S_p)=2$; also $\chi_h(C_t)\le\lceil\log_2(h+1)\rceil$. This retires density-only descent: no rank-density threshold below one, even with integrality, unit norms, saturation and connectedness, rescues a single-base descent. The two-factor products $C_d\otimes C_e$ likewise have two rationally independent color classes. The earlier prime-pivot family (nullity density $1/20$, half-density ceiling for its $2I$ invariant) and the equal-modulus example (powers of $(1+i\sqrt{15})/4$, nonintegral, with full height-two kernel rank of density tending to one and mandatory coordinates) are the weaker precursors; root integrality is an essential distinction but does not by itself supply descent.

Recurrence and piecewise syndeticity. Using the Hare–Yang extraction theorem (Section 3, Proposition 2, Lemma 3 and Theorem 2, for the torsion-free additive group with constants uniform at each fixed height) on fresh root grids, recurrent high-height roots produce nested sets $D_j$ that are height-$j$ independent, have $\chi_{\mathbb Q}(D_j)=\infty$, and satisfy $xD_k\subset D_j$ for $x\in D_j$ and late $k$; an ordinary idempotent ultrafilter contains all $D_j$, every member having infinite rational coloring number, yet dissociated $D_1$ shows this supplies no common height. Calling $A$ piecewise syndetic when $FA$ is thick for some finite $F$, an elementary partition argument proves

$$\chi_h(G)<\infty \quad\Longleftrightarrow\quad \text{some height-}h\text{ independent }A\subset G \text{ is piecewise syndetic}, \tag{15}$$

an exact reformulation of the unproved premise of (11); the $D_j$ are not piecewise syndetic, positive-pattern return sets can need arbitrarily large finite covers, and a syndetic translate-limit normalization gives uniformly recurrent minimal-shift points without a quantitative cover bound.

Other prime-grid restrictions. The Galois color-slice criterion makes differences of equal-value supports with one invariant sign function ternary after collisions and gives a composition-count sufficient condition without monochromatic whole blocks; its coordinatewise count test is excluded for the linear line family by an all-target generating bound. Galois squaring turns a coefficient-two root relation $2w=\sum_{b\in B}\epsilon_bb$ into the integral pair-product relation $w^2=\sum_{\epsilon_b=-1}b^2+\sum_{b<c}\epsilon_b\epsilon_cbc$, nonzero for $|B|>4$ by character dynamics, but with uncontrolled collisions and colors. Fixed-base defects stabilize: $r(B\otimes S_p)=pa-d(B)$ for $p\ge(3^a+1)/2$, $a=r(B)$, with no control over changing bases. Private conductor lifts preserve all integer relations and product injectivity yet always admit two rationally independent colors. Prescribed relation menus can have every determinant support split by two colors, even with common-color anchored axes. Parameter-word encodings of one fixed alphabet into conductor-disjoint odd-root groups admit a finite coloring whose monochromatic parameter images are rationally independent, so the Ramsey theorems of Graham–Rothschild, Prömel–Voigt, Voigt and Karagiannis supply no relation-preserving amplification by themselves. The prime reset and plateau give $\alpha_3(S_3\otimes S_5\otimes S_7)=2/3$, $3\le\tau_3(S_3\otimes S_5\otimes S_7)\le4$, and ordinary coloring exactly three through prime 19; tensor holes prove a root-coverage recurrence that amplifies the signed-span separation threshold while its input invariant is small, but the three-prime jump resets it maximally, and no restart hypothesis after arbitrarily many jumps is established. Factor sizes bounded by $Cj$ for one fixed $C$ are incompatible with uniform extraction even as retained subsequences, which does not cover the $j\log j$ scale; separation lifts bound thresholds after large factors. The exact twenty-point circuit in $S_3\otimes S_5\otimes S_7$, of rank nineteen with saturated column lattice and one coefficient two, refutes any bare rank-or-saturation criterion.

Anchor and gluing constructions

The small-loss anchor gluing of the Grow source fails at every arity $m\ge3$: the large-arity obstruction on that page and the exact boundary certificate show that the deletion $A$ of the point $3$ from the fifteen-point Grow–Whicher example has $\chi(A\otimes S_3)=3$ with universal signed span $U_3=\{0\}$, before and after every exact scalar encoding. Every fixed number of disjoint constant-sign boxes being finitely colorable also blocks the scalar-source simplex gluing at arities $m\ge q+2$. The quadratic analytic gluing lemma remains available for a different source but does not propagate anchors; mobile buffers above a source-dependent length prevent chromatic increase for finite-threshold sources; and saturated Hadamard gluing gives cheap $q\to q+1$ jumps from a fixed anchored source of order $r=2^k\ge2$ but loses every next-color anchor at sufficiently large order and under diagonal padding with $s\ge2q$, the threshold being source-dependent. No replacement source with a surviving anchor is known. The 3-by-5 tensor layers show that adding a larger simplex can lower the coloring number ($\chi(X\otimes S_7)=3$ but $\chi(X\otimes S_{11})=2$ for $X=S_3\otimes S_5$), so monotone growth in the side sizes must not be assumed.

Line covers: restricted colorings, amplifiers and automata

The line algebra gives $\operatorname{rank}L_t=N_t-\prod_i(d_i-1)$, a saturated column image and an integer kernel generated by elementary full-plane trades; prime-minus-one sides embed in prime grids and coefficient-two circuit slices give a second realization. On three-direction slices, rational dependence gives a ternary subrelation (the slices are cographic, so each color complement must be connected); in four directions the exact height-two circuit of 27 lines in $[3]^4$ shows rational dependence need not. A proved entropy estimate gives $q\ge2\log(h+1)$ for any global height-$h$ coloring of the linear family, hence eventually $\chi_2\ge3$, and a spectral refinement retaining the averaged row covariance still yields only palette-dependent heights. Full products of two growing linear-side line families have

$$\frac{r_{\rm diss}(L_s\otimes L_t)}{|L_s\otimes L_t|} \le\frac{\log_2(1+st)}{s_ss_t}\longrightarrow0, \tag{16}$$

so full-product flattening is not a uniformly extracting construction, and trimming every alphabet to a proper subset gives each retained line a private point, destroying all line dependencies.

Quotienting the point lattice by all lines gives point columns $\bigotimes_iS_{i+2}$ with

$$\chi_1\Bigl(\bigotimes_{i\le t}S_{i+2}\Bigr) >\frac{t+2}{8\bigl(1+(2/\log2)\log^2(t+2)\bigr)}, \tag{17}$$

so every fixed point coloring forces a monochromatic flat vector in the integer line image; the missing step is a same-color bounded line preimage. Integer height-preserving preimages exist in at most three directions by an explicit graph gauge, without color control, and the real version survives adjoining binary sides; twenty stored four-direction samples are finite examples only. Colored preimages from common real spans exist in two directions, but an explicit rationally independent twelve-line class in three directions has a nonzero flat image on five points and no nonzero flat lift on that support, so independence of the individual classes and real-span membership do not suffice.

Every fixed nonzero integer kernel vector of a $0/1$ matrix has a ternary lift in some finite incidence cover (selecting $|a_v|$ signed copies and pairing incidences; degree $h\lfloor t/2\rfloor$ suffices for height $h$ and row degree $t$), and every finite list lifts simultaneously in a product abelian cover (cyclic lifts), but these covers are tailored to the relation and their sheets are not forced to share a subsequently assigned color. A fixed connected partial lift containing two copies of one base column type is always splittable by three colors, and all canonical cylindrical lifts and deck translates of any finite prescribed relation list are split by one $3\chi_1(A)$-color palette, so fixed-template fiber focusing is refuted. The sufficient cover comparison that would settle (8), between the supremum of covered ternary coloring numbers and the rational coloring number of the base, is neither proved nor assumed.

Genuine color-unavoidable results exist but do not transfer: the explicit fourteen-column two-sheet cover amplifier raises $\chi_1$ from two to three, and circular-interval bases with $n$ columns on $r$ rows have prechosen cyclic unrolling covers with supremal coloring at least $\lceil n/r\rceil$, which proves the cover inequality for all distinct nonzero binary matrices of rank at most three; but the growing circular-interval family fails uniform extraction, and an ordinary line cover on $mN$ rows retains at most $(mN-1)/2$ interval columns, so bounded-exception intervalization cannot handle line bases of growing density. Generic voltages give a prechosen finite cover defeating every affine homomorphic $q$-palette $c(v,g)=\phi(g)+c_v$ into groups of order at most $q$ when $n>qr$, on every modulus $N\ge2$, together with the canonical period-$d$ formulas when $N\equiv1\pmod d$; arbitrary nonlinear palettes are not covered, and the K_{3,3} witness refutes a proposed nonzero-XOR certificate. Cofactor templates on bases with three universal rows and unbounded $n/r$ are split by seven colors for odd moduli (two for even), by the alternating sum $-1$ of the integer inversion statistic on the six matchings of $K_{3,3}$, so no density lower bound holds for that template hypergraph. Signed-cographic gluing has at most three colors, since its full relation space is a signed cut space and every subset has rational rank at least a third of its size. Free noncommuting incidence voltages produce a forest and hence no finite-support relations. Fixed-symbol automata describe all ternary relations of one integer translation symbol by finite residual states: infinite proper colorings periodicize, cyclic admissible degrees are ultimately periodic, and for large cyclic degree $\chi_\infty\le\chi_{\rm cyc}\le2\chi_\infty$, so unbounded $\chi_\infty$ for a growing family of fixed symbols would give the required cover coloring, but no density-based lower bound for $\chi_\infty$ is proved. Canonical one-circuit covers of a primitive nonternary rational circuit have actual sheet-parity two-colorings at the stated large prime degrees $2P$, so an amplifier must couple different relation spaces. Periodic masks retain the rational/ternary distinction even with connected support (a thirteen-column full circuit of height two), and binary-power voltages give the arbitrary bounded-period exclusion $n\le qr\ell$ for proper periodic $q$-colorings of period $\ell$, with no comparison against the automaton's possible periods. Pure-transversal resets survive arbitrary first-step twists and need compatible pulled-back row partitions afterwards; a size-conditioned reset holds under $nd<d^D$.

For the concrete field, flag and polynomial models, full-line Fourier/Radon equations determine every nonconstant new coefficient, integer coefficients require congruences modulo the newest field size, ternary coefficients require bounded oscillation, and off-color columns pin the remaining constants; the initial $p$-plane image contains $p\mathbb Z^X$ but has no color-safe ternary preimage property. The checksum evidence shows an explicit eight-column monochromatic relation against one binary Catalan-parity palette at polynomial stage one, the analytic four/six/eight-column trade defeating every zero-offset binary linear-intercept checksum, and an explicit proper two-coloring of the whole same stage into two rational bases of determinant $2^8$; the offset theorem then defeats every common-linear-intercept binary palette with arbitrary direction offsets by stage two. None of these is a lower bound for arbitrary colors. Direction and profile limits show that bounded reciprocal direction mass gives bounded rational colors and that sixteen initial-plane labels defeat full-profile homogenization along every later multiscale line. Long shifts prove that uniform full extraction plus every prescribed $k$-edge increasing-path relation on pair labels forces $n\le k+2(k-1)\log_2(2(k-1))/\delta$, so those prescribed supports have bounded coloring at fixed $\delta$, and an elementary binary rank theorem constrains their realizations; the earlier shift-hypergraph bound $n\le\ell+k+2^{k-1}-2$ gives a six-coloring of the prescribed supports in the high-fractional regime.

Difference towers: color limits

The color limits are exact. Hereditary field-rank extraction has the same product bound, so Edmonds' theorem bounds every summable-loss tower from a fixed finite seed; with each seed column normalized by its lowest $p$-valuation, if every valuation layer has hereditary residue-rank density $\delta$ the tower has at most $\lceil1/(\delta P)\rceil$ dissociated colors at retention $P$, and powers-of-$p$ seeds have exactly two colors at every positive depth when $P\ge1/2$ despite seed rational coloring number $n$. Every canonical centered-cube lift of input height at least two contains a complete coordinate cycle, and one two-coloring splits all such templates. Typed directed difference families with at most one incoming arc of each type have $\eta(B)\ge\eta(A)/2$ and $\chi_1(B)\le2\chi_1(A)$. Local kernel vectors integrate to a ternary array exactly when the coordinate-line sums vanish and cumulative sums on every line have oscillation at most two; a prescribed color support additionally pins those sums to a common zero, which rational dependence does not supply. Small-loss towers on varying line seeds remain an unproved target.

Boolean moments: coloring side

The entropy theorem gives, for a dissociated subfamily of $m$ columns, $m\le\frac12\sum_{s\le k}\binom ds\log_2(2\pi e[m/4^{s+1}+1/12])$, hence $\alpha(d,k)\le\frac{d-k}2\binom dk+\frac12\log_2(2\pi e/3)\sum_{s\le k}\binom ds$ for $k\le d/2$; all nonzero ternary relations have support at least $2^{k+1}$; a random two-coloring splits every affine subspace large enough to support a relation in the critical range, but genuinely non-affine flat circuits exist at every order (the equal-moment triples $\{1000,0101,0011\}$ and $\{0001,1101,0010\}$ tensored with parity on $k-1$ fresh bits give a unique flat circuit of size $6\cdot2^{k-1}$ with no affine $(k+1)$-flat in its support); and a fixed-density first-moment alteration bound $T(p)=O(2^d)$ on minimal trade supports would give a bounded proper coloring, so that particular certificate cannot establish (F). The phase-rounding criterion proves $\rho(E_{d,k})\ge\exp(-2\pi^2T/n)$ from a rational basis of the projected lattice $\Lambda=P_W\mathbb Z^n$ (covolume $2^{-D/2}$, dual to the relation lattice) with Gram–Schmidt coordinates at most $1/4$ and energy $T$; the needed basis with $T=O(n)$ is not constructed, and mere reordering of projected coordinate generators cannot meet the cap at the boundary. The integer-graph reduction writes $K=\{(z,Cz):z\in\mathbb Z^m\}$ with explicit Möbius coefficients $C_{B,T}=(-1)^{k-|B|+1}1_{B\subseteq T}\binom{|T|-|B|-1}{k-|B|}$, surjective over the integers for $2k<d$, and proves $\vartheta_H(s)\le\vartheta_K(s)\le\vartheta_{\mathbb Z}(s)^r\vartheta_H(s)$ and $s_0^{m-r}\le\det H\le s_0^m$ for the hard kernel $H=\ker_{\mathbb Z}C$, with a rank-one countermodel showing that full semistability, not rank or puncturing alone, is what the bridge needs. The affine-class theorem shows an affine syndrome class is dissociated exactly when it is rationally independent, equivalently when $B_k+H=\mathbb F_2^d$ for its dual code, a missing frequency coset giving the explicit ternary relation $\chi_w1_C$, and a product of coordinate-pair factors defeats every fixed finite-abelian checksum palette at the boundary. The middle-layer determinant, by the checked diagonal form of inclusion matrices (Ghorbani et al., Corollary 3), satisfies $\log_2\det K_{\rm sl}/\binom d{d/2}\sim\tfrac12\sqrt{\ln d}$ at the rounded critical scale, so it grows and supplies no vanishing obstruction. The palette-height bound proves that every fixed $q$-coloring of sufficiently large critical moments has a monochromatic integer relation of height at most $\lfloor\sigma_*^q\rfloor$, arbitrary-palette but not ternary control. Full tensors of two critical moment blocks, two critical simplicial blocks, or one of each lose uniform extraction when both indices grow, with the exact obstruction

$$\eta(E_1\otimes E_2)\le \frac{D_1r_2+D_2r_1}{2n_1n_2} +c_0\frac{r_1r_2}{n_1n_2},\qquad c_0=\tfrac12\log_2(2\pi e/3),$$

for rectangular moment tensors; a degree-$\ell$ height-two supported polynomial need not admit any nonzero ternary substitute on its support, and recoverable zero-preserving pointwise ternarization can require the full original degree. A low-degree binary annihilator does not itself provide a ternary trade: the four points $\{000,110,101,011\}$ support $1+x_1+x_2+x_3$ over $\mathbb F_2$, yet their degree-one real moment columns are independent (determinant $-2$ after the constant pivot), and tensoring shows $S^a\subseteq\{0,1\}^{3a}$ is rationally independent for moments through degree $a$ while its binary indicator has degree $a$. Polynomial fibers give actual-color flat representations on a focused progression by Prouhet identities (checked against Graham–Rothschild, Corollary 8), but that progression already has its own one-fiber relation, full fibers lose extraction, and the sparse signed-span target $\{w,2w\}\subset\Sigma(C)$ is exactly height-three dependence; they are not an extraction-preserving amplifier.

Simplicial boundaries: coloring side

Three colors avoid every decomposable flat cycle when $\log n=o(k)$: normalizing a flat Plücker coordinate gives $[I_k\mid D]$ with $D$ totally unimodular and $D\mathbf1=-\mathbf1$, for $a$ active extra vertices there are at most $2^{ka+a}$ normalized matrices and at least $ka$ supported bases for $k\ge4$, and the asymmetric local lemma of Moser–Tardos, Theorem 1.1 gives the coloring. One two-coloring avoids every disjoint join of simplex boundaries (smallest support $2^{a-1}(k-a+2)$ for $a$ nontrivial blocks with $\sum_i(|B_i|-1)=k$) and every flat cycle on at most $k+T$ vertices, $T=\min(n-k,\lfloor\sqrt{k/(16\log(2n))}\rfloor)$. In those particular colorings every monochromatic flat cycle is nondecomposable and exceeds the vertex bound; no arbitrary-color lower bound results. Elementary simplex boundaries alone are avoided by a local-lemma coloring whenever $e\,2^{-k}((k+1)(n-k-1)+1)\le1$, which holds for $n=O(k\log k)$. The small solver searches at $n=6,7$ are inconclusive. A proof of the simplicial form of (F) must therefore use color-adaptive nondecomposable cycles.

Retired mechanisms

Bounded-support representations with bounded entries force a uniform rational-independent partition, which retires the fixed-width route, including the actual fixed-order family of Nešetřil–Rödl–Sales, whose Theorem 1.1 gives, for each fixed $h$, a locally proportionally $h$-free set that is not a finite union of $h$-free sets, with an extraction constant $k(h)^{-k(h)}$ that depends on $h$; fixed-order $h$-freeness and decaying extraction constants do not meet the target, and the checked partial-Steiner Ramsey host theorem does not preserve cover degrees. Independently planted relations need correlated signs to admit distinct labels. An explicit saturated relation lattice disproves a uniform span-dimension gap for proportionately dissociated sets, so raw span dimension is not a viable surrogate. Multiplicativity of individual simplex Sidon constants is false; the proved replacement uses target-dependent randomized phase lifts and a summable second-moment bound. Short-arc flat root relations, such as the degree-49 flat multiple of $\Phi_{210}$, force unbounded angular color variation but no coloring contradiction, and no $o(n)$-degree flat family was found.

Held links, provisional premises, and gaps

Held links. Every native statement cited above is author-recorded on its own page, with the scope stated there, except the two roots-of-unity statements recorded as the tier-2 Lean claims L20 and L21; internal checks by separate reviewers are not a tier and are not independent verification of a proposed solution. The external theorems consumed at full strength are Pisier's Theorem 2 (claims checked in the held copy, constant unquantified), Edmonds' matroid partition theorem, the Moser–Tardos asymmetric local lemma, the Hare–Yang extraction theorem, Graham–Rothschild's Corollary 8, the Ghorbani–Mohammad-Noori–Pournaki diagonal form, Bzdęga's Theorem 1.5, Conway–Jones' Theorem 5 for minimal vanishing sums, Ramsey–Graham's Proposition 1.2 and Theorem 2.12, Regev–Stephens-Davidowitz's reverse Minkowski theorem (Theorem 1.2 of version 4 for the Gaussian coloring theorem, and the stable-lattice propositions of version 6 for the free-energy theorem), Vritsiou's regular-ellipsoid restatement of Pisier's 1989 theorem, and the single-set width-to-dissociation inequality of Lefèvre–Li–Queffélec–Rodríguez-Piazza, Theorem 3.4. Each consuming page states the exact passage used.

Provisional premises. No unconditional result above rests on an unproved premise. The conditional statements are labeled as such where they stand: the open claim L124 behind the closing direction of (4a), which would bound every prime grid; the fixed-height hypothesis of (11) and its piecewise-syndetic form (15); the bounded-energy small-coordinate basis of the phase-rounding criterion; the monochromatic determinant support of Hall-tight flattening and the growth inequality (13); the cover comparison inequality behind (8); and the color-adaptive nondecomposable cycle needed on the simplicial side. Any of these may be used provisionally with its qualification visible.

Gap. There is one gap, the second half of (F): an unbounded actual dissociated coloring theorem for a family with one uniform hereditary extraction constant. For the prime grids it requires, by (4a), signed vanishing sums of odd-squarefree-order roots of unity in arbitrarily short arcs, the negation of L124; a proof of L124 would instead close that family. Until the gap is filled, $A_{\rm prime}$, $A_{\rm line}$, $A_{\rm cov}$ and the moment and simplicial families are proved proportionately dissociated candidates, not counterexamples, and the target problem remains open.