# The Tail-Count Transport: an exact per-level evaluator of $G_2$, and why its truncated chains fail

**Status: Draft 2, 2026-09-19, revising the manuscript of return #23 (2026-09-11) after referee report 71 (reject pending scope, implication and attribution corrections; central evaluator preserved). What this revision changes: the no-chain conclusion is restricted to what is demonstrated, the failure of the specified truncated chains and of histogram closure, since the old word and $q$ do determine the folded word (§1, §6); the loose-certificate biconditional is corrected in direction and threshold (§4, §9); the run-cost cap is stated as the upper bound on the number of kills it is, and the claim that a chained index must diverge at its rate is withdrawn (§6); the finiteness of both $L$-sums is proved by the mean-gap argument and Theorem 2 no longer cites a count its own proof assumed (§3, §4); the Holt–Rudd $(q-2)$ multiplier is quoted with its divisibility condition and the one-class counterexample to the unrestricted form (§1, §8), and the constrained-coding citation names the zero-loop extension (§2); the tail inequality's exactness is stated for the maximum support endpoint only (§3). Added: the mechanism of the margin trend, which return #1154 supplies, with the sharp form of the inequality and its exact-identity regime (§5). The proven core the referee named, the operator identity, Lemmas 1 and 2, Theorem 1, Theorem 2 and the fold-41 run, is unchanged. Nothing here is integrated into a live document. External submission is governed by the house moratorium recorded in `paper/PAPERS.md`.**

*Parent records: `research/history/staging/attack-foldL-03-transport.md` (the origin record: the inequality, its proof for the loose form, the no-chain argument), `research/history/staging/verify-tailcount-transport.md` (the adversarial re-derivation and its three corrections), `research/history/staging/frontier37.md` (the eighth fold, the three certificates, the untruncated check), `research/history/staging/import-maxplus.md` (the two-semiring reading), `research/U-FRAME.md` §11 and `research/operator-and-pair-count.md` (the live layer and the operator's honest limit), `research/OUTCOMES.md` (chaining, closed), `research/a3-05-bound-L.md` and `research/kappa-not-L.md` (the Alternation Lemma and the run-cost theorem). Calibration ladder: proven, verified, measured, conjectured, refuted. A script output is a measurement or a verification, never a proof.*

## Abstract

Fix the tile $T_x$ of twin-admissible residues modulo $x\#$ (classically, the two-class primorial wheel) with cyclic gap word $g_0, \dots, g_{D-1}$, and fold it by the next prime $q$. Holt and Rudd's transfer operator writes every gap of the folded tile as a window of the old word, a left gap, $L$ dead interior slots and a right gap, counted with a multiplicity $\nu_q(i, L) \in \{0, \dots, q-2\}$ over the $q$ alignments of the fold. Two one-line bounds, $\nu_q(i, 0) \le q - 2$ and $\nu_q(i, L) \le 2$ for $L \ge 1$, turn the operator into an inequality on tail counts: with $N(\theta)$ the number of old gaps of size at least $\theta$ and $Q_L(\theta)$ the number of old windows of $L + 1$ gaps with sum at least $\theta$ whose $L - 1$ interior gaps admit a legal walk on a two-element set modulo $q$,

$$N_{\mathrm{new}}(\theta) \le (q - 2)\,N(\theta) + 2 \sum_{L \ge 1} Q_L(\theta) \qquad \text{for every } \theta.$$

The inequality is proven (§3), in the loose form the origin record proves and in the alternation-refined form the live layer states, whose relaxation step the record argues in one sentence and we write out. It is verified with zero violations at every threshold at the eight folds $q = 11$ to $37$, with the sum over $L$ untruncated on the record's frontier producer and on a parallel port written for this paper, and on two further implementations over shorter ranges (§5); the ninth fold, $q = 41$, was run for this paper on the parallel port in one hour of wall time on nine threads and gives zero violations at 91 thresholds with the refined certificate again equal to $G_2(41\#) = 546$ (§4). Its consequence for the largest gap is the certificate $G_2(T_q) \le M_{\mathrm{alt}}$, the largest sum of a window of the old word whose interior class word is alternation-legal, and we prove that this certificate is an identity: $G_2(T_q) = M_{\mathrm{alt}}$ at every fold, because every such window sits inside a realised kill run at some alignment (§4). The record had this as measured at eight folds and open. The loose certificate, with the alternation condition dropped, is a bound and not an identity; it is strict at fold 29 (270 against 258) and exact at the other seven. So $G_2$ of the folded tile is a restricted maxsum statistic of the old cyclic gap word, computable from the word and $q$ alone, and the tail-count inequality is the derivation of that identity rather than a separate instrument. What the identity does not give is a bound over several folds from a summary of the old word: the old word and $q$ determine the folded word (the fold algorithm itself), but a scalar evaluator does not retain the word its next application needs, the gap histogram is not closed under folding (0 of 50 shuffles of the $T_{19}$ word at fixed histogram return the true $G_2(T_{23})$; a statement about words, with the tile caveat), and the fixed-index truncations of the window family that the record tried certify a constant against a diverging truth (verified; §6). Those are the chains that fail; no impossibility theorem about every iteration or every coarser invariant is claimed. Chaining the specified truncations is closed in the programme's outcome register. Below the record gap the inequality's margin has a mechanism: its maximal ratio is the relaxation constant $(q-4)/(q-2)$ of the exact survival count plus a merge excess under $0.007$ from fold 29 on, and with the exact count restored the inequality is an identity as soon as no old gap is $\equiv 0, \pm 2 \pmod q$ (return #1154; §5). The identification of the operator over the counting semiring with the fold over the max-plus semiring is inferred from the two constructions and is not searched (§7). The operator, the closure theorem and the $(q - 2)$ driving-term multiplier for gap sums not divisible by $q$ are Holt and Rudd's (2014); the tail-count form, the run term, the alternation refinement and the exact evaluator are not found in print, at the calibration of a search in one owning convention (§8). The setting is the programme's: a new framework and vocabulary over classical sieve-theoretic objects, a new lens. Nothing here bears on the twin prime conjecture or on the exponent $4.26645$.

## 1. What this paper does not do

Nothing in this paper bears on the twin prime conjecture, on the exponent $\beta_2 = 4.26645$ of the two-dimensional sieve, or on the growth of $G_2(x\#)$ with $x$. The paper evaluates the largest gap of one folded tile from the gap word of the tile before it, exactly, and shows that the evaluation cannot be iterated through the summaries the record tried: a fixed-index truncation of the window family, or the gap histogram. The old word and $q$ do determine the folded word, by the fold algorithm of §2 applied once more, so what is lost in iteration is the word, not the information.

That is the shape of the result and the proposal asks that a paper lead with it. Chaining the Tail-Count Transport on the tile is CLOSED in `research/OUTCOMES.md` (row "chaining the Tail-Count Transport on the tile", 2026-08-19): a fixed window index certifies a constant against a diverging truth, and forcing the index to grow prices out at the coordinate-free cap of §6. The single-fold instrument stays exact because the fold sums over all $q$ alignments of the old tile, so every merge that can happen does happen in some copy; the same sum is what a chained bound would have to give up. This paper proves the exactness (§4) and reproduces the closure (§6).

The per-level identity of §4 is small. It says that the fold's new record gap is the largest sum of a window of the old word whose interior class word modulo $q$ is a walk on two states. The argument is the one by which the companion draft on the run length $L$ (written 2026-09-11 from `paper/proposals/prop-exact-fold-L.md`, under review) shows that $L$ is a word statistic, applied to sums rather than to lengths, and it is the $m = 1$ case of the copy theorem of `research/U-FRAME.md` §5a with straddling windows included in the maximum. It evaluates $G_2(T_q)$ from $T_x$ without folding, in one pass over the old word, at folds whose answers were in OEIS A144311 before this programme existed. What it sharpens is the negative, in the form the evidence supports: the record gap of the next level is determined by the old word together with the residues of the old word modulo $q$; the histogram of the word does not determine it (§6, Fact 3), and the truncated chains of §6 do not bound it. Whether some other summary or an arithmetic argument could is not decided here.

The framework is a new framework and vocabulary over classical sieve-theoretic objects, a new lens, and the objects here have owners. The cycle of gaps, the fold recursion, the closure theorem, the histogram transfer operator with its binomial eigenvectors, and the $(q - 2)$ driving-term multiplier for gap sums not divisible by $q$ are Holt and Rudd's, from 2014 (`research/PRIOR-ART.md`; §8); the two-class slot-count identity $D_{\mathrm{new}} = D(q-2)$ of §2 adapts their one-class framework and is not printed there. What is ours is the reading of the operator as an inequality on tail counts, the run-correction term $2\sum_L Q_L$, the alternation refinement, the exact evaluator and the certificate use, and the two-class arithmetic under all of them.

## 2. Setting and the operator

Fix $x \ge 5$, let $W = x\# = \prod_{p \le x} p$ and let $T_x$, the tile, be the set of residues $s$ modulo $W$ with $\gcd(s(s+2), W) = 1$, listed as $s_0 < \dots < s_{D-1}$ inside $[0, W)$ and read cyclically, $D = \prod_{3 \le p \le x}(p - 2)$. Every slot is $5$ modulo 6, so every gap $g_i = s_{i+1} - s_i$ (with $g_{D-1} = W + s_0 - s_{D-1}$) is a positive multiple of 6. Write $G_2(T_x) = \max_i g_i$ and $\bar m = W/D$ for the mean gap. Partial sums from slot $i$ are $G_j(i) = g_i + \dots + g_{i+j-1}$, $G_0(i) = 0$, with indices modulo $D$ and one full turn adding $W$. Write $N(\theta) = \#\{i : g_i \ge \theta\}$ for the tail count of the old word.

Fold by a prime $q > x$. The big tile is the set of $qD$ integers $s_i + kW$, $0 \le k < q$, in $\mathbb{Z}/qW$, and the fold deletes the positions $v$ with $q \mid v$ or $q \mid v + 2$; what survives is $T_q$ when $q$ is the next prime after $x$ (`research/a3-05-bound-L.md` §1). A gap of $T_q$ is a maximal run of deleted positions together with the two live positions bounding it: the lift of an old slot $i$ survives, the lifts of the next $L$ slots die, the lift of slot $i + L + 1$ survives, and the new gap is $G_{L+1}(i)$.

**The operator.** For the lift $s_i + kW$ of slot $i$ in copy $k$, put $a = -(s_i + kW) \bmod q$. The lift of slot $i + j$ dies exactly when $s_i + kW + G_j(i) \equiv 0$ or $-2 \pmod q$, that is, when $G_j(i) \equiv a$ or $a - 2 \pmod q$; the lift of slot $i$ itself dies exactly when $a \in \{0, 2\}$. Since $\gcd(W, q) = 1$, as $k$ runs over the $q$ copies the alignment $a$ runs over $\mathbb{Z}/q$ exactly once. Hence (`research/a3-09-histogram-operator.md`, "The operator"; `research/U-FRAME.md` §11)

$$\mathrm{count}_{\mathrm{new}}(d) = \sum_{i=0}^{D-1} \sum_{L \ge 0} [G_{L+1}(i) = d]\; \nu_q(i, L), \qquad \nu_q(i, L) = \#\{a \in \mathbb{Z}/q : G_j(i) \in \{a, a-2\} \text{ for } 1 \le j \le L;\ a \notin \{0, 2\};\ G_{L+1}(i) \notin \{a, a-2\}\},$$

all congruences modulo $q$. The three conditions are, in order: the $L$ interior slots die, the left endpoint lives, the right endpoint lives. This is Holt and Rudd's discrete dynamic system for the populations of gaps across one step of the sieve (arXiv:1408.6002 §5, pp. 17–19), written on the gap word; the corpus records it as prior art and instructs that nothing in it be presented as new structure (`research/PRIOR-ART.md`, "A9 is demoted"). Its honest limit, recorded before any of the present work, is that it is closed on the gap word and not on the gap histogram, so it is an exact simulator and not a source of bounds (`research/operator-and-pair-count.md`, "The honest limit").

**The exact identity underneath it (proven).** For fixed $i$ and fixed $a \notin \{0, 2\}$, exactly one $L \ge 0$ satisfies the conditions of $\nu_q(i, L)$: $L$ is the number of consecutive $j \ge 1$ with $G_j(i) \in \{a, a - 2\}$, which is finite because the big tile has live positions. Hence $\sum_{L \ge 0} \nu_q(i, L) = q - 2$ for every $i$, and summing over $i$ gives the census recursion $D_{\mathrm{new}} = D(q - 2)$ (`research/history/staging/verify-tailcount-transport.md` (a), "No double counting"). The operator is an equality; everything below is a relaxation of it term by term.

**Classes and the two-state walk.** Write $\eta = +1$ when $q \equiv 1 \pmod 6$ and $\eta = -1$ when $q \equiv 5 \pmod 6$. A gap $g$ is of class $0$, $+2$ or $-2$ when $g \equiv 0$, $+2$ or $-2 \pmod q$, and qualifies when it is of one of these three classes. The classes are single arithmetic progressions of modulus $6q$ with least members $6q$, $(3 + \eta)q + 2$ and $(3 - \eta)q - 2$, and the two non-zero least members sum to exactly $6q$ (`research/a3-05-bound-L.md` §3 Lemma 2, proven). A word $c_1, \dots, c_m$ of classes is *alternation-legal* when it is the label sequence of a walk on two states $\mathrm{A}, \mathrm{B}$ in which class $0$ keeps the state, class $-2$ is the step $\mathrm{A} \to \mathrm{B}$, and class $+2$ is the step $\mathrm{B} \to \mathrm{A}$: equivalently, every letter qualifies and the non-zero letters strictly alternate in sign. This is the Alternation Lemma of `research/kappa-not-L.md` (proven), whose language family extends the $B = 1$ charge constraint of constrained coding by a zero loop at each state: Marcus, Roth and Siegel's charge graph (§2.3) is binary, with steps $\pm 1$ only and no self-loops, and at $B = 1$ its capacity is $0$ bits; the ternary language here keeps class $0$ as a self-loop at both states, so its graph has adjacency matrix $\begin{pmatrix}1&1\\1&1\end{pmatrix}$ and capacity $\log_2 2 = 1$ bit per symbol (a two-line computation, no novelty claimed; the lemma is the corpus's, `research/kappa-not-L.md` "ATTRIBUTION"; referee report 71 §5 and review 69 identified the mismatch with the printed binary language).

**Two window families.** For $L \ge 1$ and a threshold $\theta$, let

$$Q_L^{\mathrm{loose}}(\theta) = \#\{i : G_{L+1}(i) \ge \theta,\ g_{i+1}, \dots, g_{i+L-1} \text{ all qualify}\}, \qquad Q_L^{\mathrm{alt}}(\theta) = \#\{i : G_{L+1}(i) \ge \theta,\ \text{the class word of } g_{i+1}, \dots, g_{i+L-1} \text{ is alternation-legal}\}.$$

For $L = 1$ there is no interior and both count every window of two adjacent gaps with sum at least $\theta$; for $L = 2$ the interior is one gap and the two coincide; they can differ from $L = 3$ on. Always $Q_L^{\mathrm{alt}} \le Q_L^{\mathrm{loose}}$. The origin record's $Q_L$ is the loose one (`research/history/staging/attack-foldL-03-transport.md` §1); the live layer's is the refined one (`research/U-FRAME.md` §11).

## 3. The transport inequality

**Lemma 1 (multiplicity bounds; proven).** $\nu_q(i, 0) \le q - 2$, with equality exactly when $q \mid g_i$; and $\nu_q(i, L) \le 2$ for every $L \ge 1$.

*Proof.* For $L = 0$ the conditions are $a \notin \{0, 2\}$ and $a \notin \{g_i, g_i + 2\}$, so $\nu_q(i, 0) = q - |\{0, 2, g_i, g_i + 2\}|$, which is $q - 2$ when $g_i \equiv 0$, $q - 3$ when $g_i \equiv \pm 2$, and $q - 4$ otherwise. For $L \ge 1$ the condition at $j = 1$ alone forces $a \in \{G_1(i), G_1(i) + 2\}$, a set of two elements, and the remaining conditions only shrink it. $\square$

(`research/history/staging/verify-tailcount-transport.md` (a); the $L = 0$ case is the adapted pair correlation $\rho_q(g)$ of `research/a3-09-histogram-operator.md`, "The diagonal part".)

**Lemma 2 (the interior of a counted window is alternation-legal; proven).** Let $L \ge 1$ and put $A_L(i) = \{a \in \mathbb{Z}/q : G_j(i) \in \{a, a-2\} \text{ for } 1 \le j \le L\}$, the set of alignments at which the $L$ interior slots of the window $(g_i, \dots, g_{i+L})$ all die, with no condition on the endpoints. Then $A_L(i) \ne \emptyset$ if and only if the class word of $g_{i+1}, \dots, g_{i+L-1}$ is alternation-legal, and $|A_L(i)| \le 2$. In particular $\nu_q(i, L) > 0$ implies that the interior class word is alternation-legal, and a fortiori that every interior gap qualifies.

*Proof.* Suppose $a \in A_L(i)$. Assign to each $j \in \{1, \dots, L\}$ the state $\mathrm{A}$ if $G_j(i) \equiv a$ and $\mathrm{B}$ if $G_j(i) \equiv a - 2$; the two are exclusive because $q > 2$. Since $g_{i+j} = G_{j+1}(i) - G_j(i)$, the class of $g_{i+j}$ for $1 \le j \le L - 1$ is $0$ when the state is kept, $-2$ on $\mathrm{A} \to \mathrm{B}$ and $+2$ on $\mathrm{B} \to \mathrm{A}$, and no other value is possible; so the interior class word is the label sequence of a walk, which is the definition of alternation-legal. Conversely, given a legal walk with starting state $S_1$ for the interior word, choose $a \equiv G_1(i)$ if $S_1 = \mathrm{A}$ and $a \equiv G_1(i) + 2$ if $S_1 = \mathrm{B}$; by induction on $j$ the state of $G_j(i)$ follows the walk, so every $G_j(i)$, $1 \le j \le L$, lies in $\{a, a - 2\}$ and $a \in A_L(i)$. The bound $|A_L(i)| \le 2$ is the $j = 1$ condition. Finally $\nu_q(i, L)$ counts a subset of $A_L(i)$. $\square$

Lemma 2 is the step the proposal names as "argued in one sentence as a valid relaxation and never written out" (`paper/proposals/prop-tailcount-transport.md` §2), which the record states as "restricting $Q_L$ to interiors whose class word admits a legal walk on a 2-set is still a valid relaxation of $\nu_q(i, L) > 0$ (it *is* the exact non-emptiness of $A_L$ minus the endpoint conditions)" (`research/history/staging/verify-tailcount-transport.md` (c)). The proof above is that sentence with its induction supplied, and it carries slightly more: non-emptiness of $A_L(i)$ is *equivalent* to legality, not only implied by it, which is what §4 uses.

**Theorem 1 (the Tail-Count Transport; proven).** Folding $T_x$ by a prime $q > x$, for every threshold $\theta$,

$$N_{\mathrm{new}}(\theta) \;\le\; (q - 2)\,N(\theta) + 2 \sum_{L \ge 1} Q_L^{\mathrm{alt}}(\theta) \;\le\; (q - 2)\,N(\theta) + 2 \sum_{L \ge 1} Q_L^{\mathrm{loose}}(\theta),$$

where $N_{\mathrm{new}}(\theta)$ is the number of gaps of the folded tile, over one period $qW$, of size at least $\theta$. The sums over $L$ are finite: the loose sum terminates at $L = R + 1$, where $R$ is the length of the longest run of consecutive qualifying gaps in the old word, and on a primorial tile $R \le D - 1$; the refined sum vanishes for $L > 2D$.

*Proof.* By the operator, $N_{\mathrm{new}}(\theta) = \sum_i \sum_{L \ge 0} [G_{L+1}(i) \ge \theta]\,\nu_q(i, L)$. The $L = 0$ terms are at most $(q - 2)\,\#\{i : g_i \ge \theta\} = (q - 2) N(\theta)$ by Lemma 1. For $L \ge 1$ each term is at most $2\,[G_{L+1}(i) \ge \theta]\,[\nu_q(i, L) > 0]$ by Lemma 1, and $[\nu_q(i, L) > 0] \le [\text{interior word alternation-legal}] \le [\text{interior gaps all qualify}]$ by Lemma 2; summing over $i$ gives $2 Q_L^{\mathrm{alt}}(\theta) \le 2 Q_L^{\mathrm{loose}}(\theta)$. A window with $L - 1 > R$ interior gaps has a non-qualifying interior gap, so $Q_L^{\mathrm{loose}}(\theta) = 0$ for $L > R + 1$. That $R$ is finite on a primorial tile is the mean-gap argument (referee report 71 §4): with $r$ the largest prime at most $x$, $\bar m(T_5) = 10$ and $\bar m(T_p) = \bar m(T_{p^-})\,p/(p-2)$ at each prime level, so by induction $\bar m(T_r) \le 2r$ since the preceding prime is at most $p - 2$; for $q > r$ the smallest qualifying gap is $\theta_0 = 2q - 2\eta > 2r \ge \bar m$, so some old gap lies below $\theta_0$ and does not qualify, and $R \le D - 1$. (For an arbitrary periodic word in which every gap qualifies the loose sum need not terminate; the first draft claimed the $2D$ cutoff for both sums and proved it for neither.) For the refined family: a window with alternation-legal interior has, by Lemma 2, an alignment at which its $L$ interior slots all die in one copy; a run of $L \ge qD$ consecutive dead lifts would contain a full period of the big tile and hence a live position, which is impossible, so the $L$ dead lifts are $L < qD$ distinct positions, and since the big tile has exactly $qD - D(q-2) = 2D$ dead positions, $Q_L^{\mathrm{alt}}(\theta) = 0$ for $L > 2D$. $\square$

(Loose form: `research/history/staging/attack-foldL-03-transport.md` §1, PROVEN, re-derived independently in `research/history/staging/verify-tailcount-transport.md` (a), verdict STANDS. Refined form: `research/U-FRAME.md` §11, via Lemma 2 here.)

Two remarks on what the theorem is. No arithmetic enters beyond Lemma 1: the copy boundary costs nothing because the sum over alignments is the sum over copies, and a window straddling the boundary of the old period is the same term as any other, the turn adding $W$ to the position and $W \bmod q$ to the residue (`research/history/staging/verify-tailcount-transport.md` (a), "The copy boundary wraps correctly"). And the inequality is a statement about the whole tail profile, at every $\theta$ at once, which is where it has content: at thresholds below $G_2(T_x)$ the first term is live and the relaxations of Lemma 1 cost something (§5), while above $G_2(T_x)$ the first term vanishes. The tail inequality stays an inequality there: at fold 17 and $\theta = 108 > 66 = G_2(T_{13})$... the table of §5 has $N_{\mathrm{new}} = 20$ against a refined right side of $2 \cdot 20 = 40$, so the multiplicity slack of Lemma 1 is still spent on the tail counts. What §4 sharpens to an identity is the support endpoint, the largest $\theta$ with a nonzero right side, not every tail count.

**Corollary 1 (the certificate; proven).** Let $M_{\mathrm{alt}} = \max\{G_{L+1}(i) : L \ge 1,\ \text{interior class word alternation-legal}\}$ and $M_{\mathrm{loose}}$ the same maximum over windows whose interior gaps all qualify, so $G_2(T_x) < M_{\mathrm{alt}} \le M_{\mathrm{loose}}$ (the window $(g_i, g_{i+1})$ with $g_i = G_2(T_x)$ has $L = 1$ and sum above $G_2(T_x)$). Then $G_2(T_q) \le M_{\mathrm{alt}} \le M_{\mathrm{loose}}$.

*Proof.* Take $\theta = M_{\mathrm{alt}} + 6$. Then $N(\theta) = 0$ and $Q_L^{\mathrm{alt}}(\theta) = 0$ for every $L$, so $N_{\mathrm{new}}(\theta) = 0$ by Theorem 1; gaps are multiples of 6. $\square$

This is the origin record's "Certified bound: $G_2(\mathrm{new}) < \min\{\theta : \mathrm{RHS}(\theta) < 1\}$" and its "equivalent reading": the certificate is the largest sum of consecutive old gaps whose interior gaps all qualify modulo $q$ (`research/history/staging/attack-foldL-03-transport.md` §1). The adversarial record observed that at the certificate the first term is inert at every fold in reach, so the certificate is a restricted-maxsum quantity and the tail-count framing is the derivation rather than the instrument (`research/history/staging/verify-tailcount-transport.md` (b)). Corollary 1 makes that observation a proof: the first term is inert above $G_2(T_x)$ at every fold, by definition; the run terms are not inert there, only their support is exact (§4).

## 4. The certificate is exact

**Theorem 2 (the per-level evaluator; proven here).** For every $x \ge 5$ and every prime $q > x$,

$$G_2(T_x \text{ folded by } q) = M_{\mathrm{alt}} = \max\{\,G_{L+1}(i) : L \ge 1,\ \text{the class word of } g_{i+1}, \dots, g_{i+L-1} \text{ modulo } q \text{ is alternation-legal}\,\}.$$

*Proof.* The inequality $\le$ is Corollary 1. For $\ge$, let $(g_i, \dots, g_{i+L})$ be a window with alternation-legal interior. By Lemma 2, $A_L(i) \ne \emptyset$; pick $a \in A_L(i)$ and the copy $k$ with $-(s_i + kW) \equiv a \pmod q$, which exists and is unique because $W$ is invertible modulo $q$ (the Chinese-remainder step of Holt and Rudd's Theorem 2.3). In that copy the lifts of slots $i + 1, \dots, i + L$ all die. They are distinct positions of the big tile: a run of $qD$ or more consecutive dead lifts would contain a full period and hence one of the $D(q-2) > 0$ live positions. The maximal run of dead positions of the big tile containing them is finite for the same reason, and it is bounded on the left by a live position at or before the lift of slot $i$ and on the right by a live position at or after the lift of slot $i + L + 1$. The gap of the folded tile between those two live positions contains the window, so it is at least $G_{L+1}(i)$. Hence $G_2(T_q) \ge G_{L+1}(i)$ for every such window, and $G_2(T_q) \ge M_{\mathrm{alt}}$. $\square$

The argument is the one by which the run length $L$ is shown to be the longest alternation-legal window of the old word: the alignment is recoverable from any slot of a run, and a legal window is therefore realised as dead slots in some copy (`research/a3-05-bound-L.md` §1; `research/history/staging/attack-foldL-01-census.md`). What Theorem 2 adds is that dropping the two endpoint-live conditions of $\nu_q(i, L)$ can only produce windows that are sub-windows of realised runs, whose sums are smaller, so the maximum is unchanged.

**Relation to the copy theorem.** The copy theorem of `research/U-FRAME.md` §5a step 2 states, at $m = 1$, that $G_2(T_q)$ is the maximum over the $q$ two-sets $\{a, a - 2\}$ of the largest gap of the old tile with those residue classes deleted, read as a single copy of period $W$; the record grades it verified (40 of 40 cells over five folds, $m \le 8$) and notes that it rests on a measured clause the corpus has not proved, that no window straddling a copy boundary of the big tile ever beats a single-copy one (`research/gate-multiplies.md` §6 and §9). Theorem 2 is that statement with the straddling windows kept in the maximum: a window of the cyclic word that crosses the end of the old period has its residues shifted by $W \bmod q$ on the far side, so it is legal in the sense of Lemma 2 with the shift and is realised in the big tile, and it is in general not a window of any single residue-deleted copy. Theorem 2 therefore proves the copy theorem's $m = 1$ case in the form that includes straddling windows, and leaves the record's clause, that they never win, exactly where the record has it: measured. The record's open question, "whether $M_{\mathrm{alt}} = G_2$ ALWAYS" (`research/history/staging/verify-tailcount-transport.md` (c); `research/history/staging/frontier37.md` §9, "measured at eight folds and proven at none"; `paper/proposals/prop-tailcount-transport.md` §1 and the third upgrade trigger of its §5), is answered: yes, and the reason is the containment above, which the record's "one-line structural argument that makes $M_{\mathrm{full}}$ exact" did not include.

**What the theorem says and does not say.** It says that the record gap of the folded tile is a statistic of the old cyclic gap word together with the residues of its gaps modulo $q$: a restricted maxsum, in which the restriction is the two-state walk. It says that the biconditional the origin record stated with a hedge, "$G_2(x'\#) < x'^2$ if and only if no window of the old word with qualifying interiors sums to $x'^2$ or more, to within the 0 to 12 units measured" (`research/history/staging/attack-foldL-03-transport.md` §1), holds without a hedge for the alternation-legal family at every threshold, by Theorem 2. For the loose family the general-threshold biconditional fails in its forward direction: at fold 29, $G_2 = 258$ and $M_{\mathrm{loose}} = 270$, so at $\theta = 264$ the statement $G_2 < \theta$ is true while "no loose window has sum $\ge \theta$" is false. The reverse direction, "no loose window of sum $\ge \theta$ implies $G_2 < \theta$", is Corollary 1 and cannot fail. At the record's own threshold $\theta = q^2 = 841$ both sides are true at fold 29, so that example does not refute the $q^2$ statement; the first draft said otherwise and is corrected (§9 item 1). It does not say anything about the folded word's own gap word, which is what the next fold needs; that is §6. And it does not make the exactness at eight folds a discovery: with Theorem 2 in hand, the agreement $M_{\mathrm{alt}} = G_2$ in the tables below is a check on the implementations, not evidence about tails, and the instrument is an identity rather than a bound.

**Verified.** The table gives the three certificates of the record, the two of Corollary 1 and $M_{\mathrm{full}} = \max\{G_{L+1}(i) : \nu_q(i, L) \ge 1\}$, which is $G_2(T_q)$ by construction, at the eight folds in reach, against the exact ladder (`research/exact-g2-ladder.js`, which is A144311 + 1 term by term).

| fold $q$ | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 |
|---|---|---|---|---|---|---|---|---|---|
| $M_{\mathrm{loose}}$ | 42 | 66 | 108 | 150 | 204 | **270** | 348 | 528 | 546 |
| $M_{\mathrm{alt}}$ | 42 | 66 | 108 | 150 | 204 | 258 | 348 | 528 | 546 |
| $M_{\mathrm{full}} = G_2(T_q)$ | 42 | 66 | 108 | 150 | 204 | 258 | 348 | 528 | 546 |
| $L(T_x, q)$ | 1 | 2 | 2 | 2 | 3 | 2 | 4 | 4 | 3 |
| longest qualifying run $R$ of the old word | 0 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 |

The fold-41 column is the run for this paper, described below.

Sources: folds 11 to 29, `research/attack-foldL-03-transport.js` (loose certificate and truth; `research/history/staging/attack-foldL-03-transport.md` §1), re-run for this paper in 14 s with all self-tests passing; folds 11 to 31 on an independent implementation written from the operator definition, `research/history/staging/verify-tailcount-transport.md` (b), (c); folds 11 to 37 on a third implementation, `research/attack-frontier37-02-transport.js`, embedded OUTPUT, 661.9 s, whose calibration leg reproduces the first two cell for cell (`research/history/staging/frontier37.md` §3, §5). Fold 29 is the only fold at which the loose and the refined certificates differ, and the loose certificate's window there, of sum 270 with $L = 3$, has an interior that qualifies gap by gap and is not a legal walk (`research/history/staging/verify-tailcount-transport.md` (c), (d)). The row $R$ is from the frontier producer's "longest qualifying gap run" and the row $L$ from its "L(direct)". A fourth check, written for this paper and run at folds 7 to 23 on the stored tiles (`job67-alt-exact-check.js`, under 3 s here), enumerates every window with an alternation-legal interior, computes $A_L(i)$ directly from its definition, confirms it non-empty with at most two elements, extends the dead run containing the interior in the copy of each $a \in A_L(i)$ on the big tile, and asserts that the realised gap contains the window: 769,282 containments at fold 23 with zero violations, the window counts 5, 15, 141, 1557, 23363, 390521 agreeing with the frontier producer's alternation-legal window counts at the five shared folds, and its $\nu$-identity counter, which increments once for every $a \notin \{0, 2\}$ after the run scan terminates, closes at $q - 2$ for every $i$; that counter is a termination and mechanism check, not an independent enumeration of the $\nu$ distribution, whose identity is proved in §2. None of this is a proof.

**Fold 41 (verified, run for this paper).** The record prices the ninth fold at 2.4 h plus 5.4 h single-threaded and names it as the cheap check (`research/history/staging/frontier37.md` §9; `research/G2-STATE.md`, 5.05 h for the streaming leg). We ported the frontier producer's engine to worker threads (`job67-transport-parallel.js`: the stream of $T_{37}$ out of $T_{23}$ is cut into 1,147 blocks, each scored by an engine warmed by a pre-roll of 256 slots and closed by a post-roll of 256, with runs attributed to the block where they start, gaps and windows to the block of their first index; histograms and maxima merged at the end) and validated it against the served producer at folds 11 to 37, where every printed figure agrees, including $D(T_{37})$, the 12,338,231,614 enumerated runs and the two window counts. At fold 41, on 9 threads, 3,607.8 s wall and 28,346 s of CPU:

| $q = 41$ | |
|---|---|
| $D(T_{37})$ streamed | 217,929,355,875 (exact) |
| $D(T_{41})$ counted | 8,499,244,879,125 $= 39 \cdot D(T_{37})$ (exact) |
| $G_2(T_{41}) = M_{\mathrm{full}}$ | 546, against the ladder's 546 |
| $M_{\mathrm{loose}}$, $M_{\mathrm{alt}}$ | 546, 546 |
| $L(T_{37}, 41)$ | 3; best merged gap by run length 1, 2, 3: 540, 540, 546 |
| longest qualifying run of $T_{37}$ at $q = 41$ | 3 |
| maximal kill runs enumerated | 434,169,935,510 |
| windows scored, loose and alternation-legal | 219,618,074,383 and 219,618,073,707 |
| transport inequality, 91 thresholds, untruncated | 0 violations, loose and refined |
| $N_{\mathrm{new}}(546)$, $\sum_L Q_L(546)$ | 4 and 4 (the ladder's `nmax` at $x = 41$ is 4) |

So the alternation-refined certificate is exact at a ninth consecutive fold, as Theorem 2 requires, and the loose one is exact for the eighth time in nine. The captured run reports no unclosed runs and no histogram overflow, with the longest qualifying and direct runs at 3 against a 256-slot overlap and a 1,024-slot ring, and all counts below $2^{53}$; those conditions make the stored overlap adequate for this run and are not a guarantee for deeper folds. Referee report 71 exercised the port's three-fold branch serially on $T_7$ folded by 11, 13, 17 and scored at 19 across 221 blocks with the overlap reduced to 8, matching directly built histograms and window counts (`port-fixture.cjs`, its artifact). The proposal's first upgrade trigger ("fold 41 keeps zero violations at every threshold and the alternation-refined certificate again returns $G_2$ exactly") fires and its first downgrade trigger does not (`paper/proposals/prop-tailcount-transport.md` §5). After Theorem 2 the certificate half of that trigger was not in doubt; the inequality half, at the 91 thresholds below the record, was.

## 5. The inequality below the record gap

Above $G_2(T_x)$ Theorem 1 is the identity of §4 relaxed by the endpoint conditions, and it costs nothing. Below $G_2(T_x)$ it is an inequality on the whole profile and the relaxations of Lemma 1 are visible. The record checks it at every threshold on the value grid, not only at the endpoint, with the $L$-sum untruncated.

| fold $q$ | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 |
|---|---|---|---|---|---|---|---|---|---|
| thresholds tested | 7 | 11 | 18 | 25 | 34 | 45 | 58 | 88 | 91 |
| violations, loose and refined | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\max_\theta N_{\mathrm{new}}(\theta)/\mathrm{RHS}(\theta)$, loose | 1.0000 | 1.0000 | 0.8881 | 0.8975 | 0.9180 | 0.9324 | 0.9348 | 0.9477 | 0.9551 |
| at $\theta =$ | 36 | 54 | 36 | 36 | 42 | 42 | 42 | 48 | 72 |
| $N_{\mathrm{new}}(G_2(T_q))$ | 4 | 12 | 20 | 20 | 4 | 2 | 4 | 2 | 4 |
| $\sum_L Q_L^{\mathrm{loose}}(G_2(T_q))$ | 2 | 6 | 20 | 16 | 4 | 4 | 4 | 2 | 4 |
| $\sum_L Q_L^{\mathrm{alt}}(G_2(T_q))$ | 2 | 6 | 20 | 16 | 4 | 2 | 4 | 2 | 4 |

Source: `research/attack-frontier37-02-transport.js`, embedded OUTPUT (`research/history/staging/frontier37.md` §4), and the fold-41 column from the run for this paper (§4); folds 11 to 29 agree to four decimals with `research/history/staging/verify-tailcount-transport.md` (b), which ran the sum untruncated and thereby discharged the origin record's honest limit about its $L \le 8$ truncation (`research/history/staging/attack-foldL-03-transport.md` §7); the origin producer, re-run for this paper, prints the same five ratios at folds 11 to 23 with 0 violations. The last three rows are the tail-count reading at $\theta = G_2(T_q)$, where $N(\theta) = 0$ and the inequality reads $N_{\mathrm{new}} \le 2 \sum_L Q_L$; the frontier producer prints the loose sum, and the refined sum is from the parallel port's logs (§10), which print both. The row $N_{\mathrm{new}}(G_2(T_q))$ equals the number of positions attaining the record gap in `research/exact-g2-ladder.js` (`nmax` 4, 12, 20, 20, 4, 2, 4, 2, 4), recovered by a route that never visits a position. At folds 11 and 13 the inequality is attained with equality at some threshold.

**The trend, and its mechanism (proven form; measured excess).** The maximal ratio rises at every fold from 17 upward, 0.8881 to 0.9551 over six steps. The record called this a trend with no mechanism (`research/history/staging/frontier37.md` §9) and the first draft of this paper left it so. Return #1154 (job 2458) supplies the mechanism from Lemma 1 itself. The survival count is exact: $\nu_q(i, 0) = q - 4 + c(g_i)$ with $c(g) = 2, 1, 0$ for $g \equiv 0$, $g \equiv \pm 2$, else, so Theorem 1's first term relaxes $q - 4 + c(g_i)$ to $q - 2$, and at any threshold where that term dominates the right side the ratio is close to $(q-4)/(q-2)$: $13/15 = 0.8667$, $15/17 = 0.8824$, $19/21 = 0.9048$, $25/27 = 0.9259$, $27/29 = 0.9310$, $33/35 = 0.9429$, $37/39 = 0.9487$ at $q = 17, 19, 23, 29, 31, 37, 41$, against the measured $0.8881, 0.8975, 0.9180, 0.9324, 0.9348, 0.9477, 0.9551$: an excess of $+0.021, +0.015, +0.013, +0.007, +0.004, +0.005, +0.006$, which is the merge term (20 to 39 per cent of the right side at the maximizing threshold) counted at 2 per window against fewer realised merges. The deceleration the first draft measured is the baseline's own, $d/dq\,(q-4)/(q-2) = 2/(q-2)^2$, and the sequence tends to 1 like $1 - 2/(q-2)$; the extrapolation to fold 53 in the first draft is withdrawn. The **sharp form** $N_{\mathrm{new}}(\theta) \le \sum_{g_i \ge \theta}(q - 4 + c(g_i)) + 2\sum_L Q_L(\theta)$ is proven by the same decomposition (Lemma 1 used exactly); its maximal ratio is $0.990$ to $0.9999$ at every fold tested, attained at $\theta = 1$, and it becomes an identity for every $\theta$ as soon as no old gap is $\equiv 0, \pm 2 \pmod q$, because then $\nu_q(i,0) = q-4$ for all $i$, every adjacent pair merges in exactly two copies, and $Q_L = 0$ for $L \ge 2$ (the isolated-deletion lemma of return #1072, whose proof uses only that hypothesis): measured at $T_{19}$ folded by $q = 71, 73, 79, 83, 89$ and at $T_{23}$ by $107, 109$ with identity gap $0$, and at $T_{19}$ by $41, 43, 67$ the loose ratio already reaches $1.0000$ above $G_2(T_{19})$ where every potential merge is realised (`tct2458.py`, 23 folds, zero violations of either form; return #1154). So the referee's remark that endpoint and alignment information can suppress non-realised merges is exactly right, and the price of the $L = 1$ term is measured: it is the difference between the loose and the sharp ratios. Return #1154 pre-registers the next consecutive step, $41 \to 43$: maximal loose ratio in $[0.955, 0.960]$, baseline $39/41 = 0.9512$; a value outside $[0.9512, 0.965]$ refutes the reading. The maximizing threshold, $\theta = 36$ to $72$, is far below the record gap and has no bearing on the certificate.

**The standing route, for comparison (measured).** The copy-theorem chain of `research/U-FRAME.md` §5a bounds $G_2(T_q)$ by $\mathrm{maxsum}_{L+1}(T_x)$, the largest sum of $L + 1$ consecutive old gaps, which needs $L$ and takes no account of which windows can be realised. At folds 11 to 29 that bound reads 42, 96, 138, 168, 228, 300 against the truth 42, 66, 108, 150, 204, 258, a loss of 0, 0.375, 0.245, 0.113, 0.111, 0.151 nats per fold, where the alternation-legal certificate loses 0 at every fold by Theorem 2 and the loose one loses 0.045 nats once, at fold 29 (`research/attack-foldL-03-transport.js`, "certified bound" table, re-run; `research/history/staging/attack-foldL-03-transport.md` §1). Against $q^2$, the certificate's margin $\ln(q^2/M_{\mathrm{alt}})$ equals the truth's own slack $\ln(q^2/G_2(T_q))$ at every fold, 1.058, 0.940, 0.984, 0.878, 0.953, 1.182, 1.016, 0.953 at folds 11 to 37 (`research/gate-multiplies.md` §5, the Overshoot Budget's slack column): the evaluator certifies $G_2(T_q) < q^2$ at each of the eight folds from the old word alone and gives away none of the lifetime budget doing it, which is the origin record's reading and follows from Theorem 2.

## 6. Why it does not chain

The evaluator takes the old gap word and $q$ and returns $G_2$ of the folded tile. To evaluate the next level it needs the folded tile's gap word and its residues modulo the next prime $q'$. Three facts, each at its own calibration, say what the record tried and what failed; the closure of the route in `research/OUTCOMES.md` rests on the first. None of them is an impossibility theorem: dropping positive terms from the right side of an upper bound is not an admissible upper-bound step, so a truncated chain's failure says nothing about chains that treat the omitted terms, and the histogram shuffle is a statement about words.

**Fact 1 (a fixed index certifies a constant; verified).** To iterate Theorem 1 one must carry the whole window family $S_m(\theta) = \#\{i : G_m(i) \ge \theta\}$, $m \ge 1$, of which $N = S_1$ is the first member, since $Q_L$ is a restricted $S_{L+1}$. The origin record transports the family by the same alignment count, $S'_m(\theta) \le q\,S_m(\theta) + \sum_{K \ge 1} c(m, K)\,S_{m+K}(\theta)$ with $c(1, K) = 2$ and $c(m, K) = 2(m + K - 1)$ for $m \ge 2$ (`research/history/staging/attack-foldL-03-transport.md` §4; stated there, not re-derived here), and truncates it at a fixed index $M$, dropping every term with $m + K > M$, which is optimistic. Starting from the exact profiles of $T_{11}$:

| $M$ | 4 | 8 | 12 | 16 |
|---|---|---|---|---|
| $\mathrm{maxsum}_M(T_{11})$, the ceiling | 108 | 180 | 240 | 330 |
| the chain's certificate at folds 13 to 29 | 108 at every fold | 180 at every fold | 240 at every fold | 330 at every fold |
| truth 66, 108, 150, 204, 258 | false from fold 19 | false from fold 23 | false from fold 29 | above $q^2$ at folds 13 and 17 |

(`research/attack-foldL-03-transport.js`, part 4B, re-run for this paper.) Nothing feeds $S_M$, so its support never grows and the chain prints a constant against a diverging truth. The verdict is a negative reached from an optimistic truncation, which only strengthens it. The adversarial record's correction is that this fixed-index argument carries the no-chain verdict on its own, and the cap below is not what carries it (`research/history/staging/verify-tailcount-transport.md` (d), "the stated reason is mis-attributed"); the outcome register's row credits both (`research/OUTCOMES.md`, "chaining the Tail-Count Transport on the tile", CLOSED 2026-08-19).

**Fact 2 (a cap on the kills a window of given span can carry; proven cap, measured spend).** A window of sum $\theta$ that carries $K \ge 2$ dead slots has $K - 1$ interior gaps forming an alternation-legal word, whose sum is at least the run-cost floor $c_{\min}(K - 1) = 3q(K - 1) - (q + 2\eta)[K - 1 \text{ odd}]$ (`research/a3-05-bound-L.md` §4 Theorem A and Corollary A1, proven), so

$$K \le 1 + \frac{\theta + q + 2}{3q}.$$

The record states this as $K \le 1 + \theta/(3q)$ (`research/history/staging/attack-foldL-03-transport.md` §4; `research/history/staging/verify-tailcount-transport.md` (d)), which is the even case; for odd $K - 1$ the single unpaired interior gap can be as small as $2q - 2\eta$ rather than $3q$, and the displayed form covers both parities. At $\theta = q^2$ the difference is one third of a unit of index. This is an upper bound on $K$ for a window of fixed span: it says a window of sum $\theta$ carries at most about $\theta/(3q)$ dead slots. It does not say that a window of sum about $q^2$ needs about $q/3$ qualifying gaps, and it does not show that a chained certificate's index must diverge at that rate: the two unconstrained outer gaps can carry a large sum with $L = 1$ and an empty interior (referee report 71 §3 exhibits the abstract window $(24, 30)$ at $q = 7$, sum $54 \ge q^2$, empty interior; not a tile window, and enough to show that the converse needs control of the outer gaps). The first draft drew that converse and it is withdrawn. The index actually spent by the exact certificate, the $L$ of its attaining window, is 1, 1, 2, 1 or 2 (a tie), 3, 2, 2 at folds 11 to 31 (`research/history/staging/verify-tailcount-transport.md` (d), which corrects the origin record's 1, 1, 2, 2, 3, 3), and the truncation index the untruncated sum needs, $R + 1$, is 1, 2, 2, 2, 3, 3, 4, 4 at folds 11 to 37 (§4 table). The cap and the spends are kept separate: at fold 37 and $\theta = 528$ the cap reads $1 + (528 + 39)/111 = 6.11$ against $R + 1 = 4$, so the first draft's "within one of the cap at every fold" was false.

**Fact 3 (no statistic coarser than the word is closed; measured, and the caveat travels with it).** The merge condition is a condition modulo $q$, and $q$ changes at every level, so an invariant read at a fixed modulus or a fixed moment order cannot see it; to see it at every level an invariant would have to carry the residue of every gap modulo every prime, which by the Chinese remainder theorem is the gap word (`research/history/staging/attack-foldL-03-transport.md` §5, item 2). And the histogram of the word is not enough: at fold 23 the true $G_2(T_{23}) = 204$, and 50 random shuffles of the $T_{19}$ gap word at fixed histogram, seed 12345, give a folded $G_2$ with minimum 264, median 288 and maximum 354, none returning 204, while the shuffled words' folded $\sum g^2$ and $N(\ge 120)$ move by under 0.2% and 8% (`research/history/staging/verify-tailcount-transport.md` (f), strengthening the origin record's five shuffles). The caveat the record attaches is that a shuffled word is not a tile, so this refutes histogram closure at the level of words, which is the level every candidate statistic is defined at; it does not exclude an argument that uses arithmetic structure a shuffle destroys (`research/history/staging/attack-foldL-03-transport.md` §5, caveat).

**The window frame does not rescue it (measured).** In a window of fixed length at height $x$ the multiplier on the first term drops from $q - 2$ to 1, because the window does not grow, and the $L \ge 2$ terms vanish identically once the localized word has no qualifying gap at all, which is measured to happen at $p = 701$ in $[0, 2 \cdot 10^9)$ and never to return (the last fold with a kill run of length 2 or more is $p = 421$, and the last with any gap reaching $2p - 2$ is $p = 1021$); the transport there is exact and one line. The chain of the displayed relaxation still fails, on the $L = 1$ term, which has no interior condition and which the relaxation always contains; endpoint and alignment information can suppress non-realised merges (§5, the sharp form), and the failure below is that of the displayed bound: the bound $\mathrm{maxsum}_2$ multiplies the record by about 1.20 per fold against a truth that multiplies it by 1.013, about 39 nats over 233 folds against a lifetime budget of about 0.6 (`research/history/staging/verify-tailcount-transport.md` (e)). The record's reading, which we adopt, is that the tile frame's exactness and the window frame's chainability are the same property with opposite signs: on the tile every adjacent pair that can merge does merge in some copy, which is Theorem 2; in a window the alignment is fixed and almost no pair merges, and a bound that maximises over alignments throws that away at every fold.

**Where this leaves the object.** By Theorem 2 the per-level evaluator is an identity, and the no-fixed-point argument of `research/gate-multiplies.md` §2 lists exactness as one of its three escapes: an identity returns the truth, not a floor, and there is no gate to stay under. The escape is not free. The identity's input at level $x$ is the full word of $T_x$ with its residues modulo $q$, and the Overshoot Budget of the same note caps the total overshoot of any chain of upper bounds ending in $G_2(x\#) < x^2$ at $\ln(x^2/G_2(x\#))$, measured 0.88 to 1.19 nats on the fourteen-term ladder $x \le 43$, predicted to tend to $\ln(1/0.55) = 0.598$ (`research/gate-multiplies.md` §5, conditional on the measured law $G_2 \sim 0.55(\ln W)^2$), and rising to 1.2946 at $x = 79$ on the A144311 terms (`research/history/staging/import-maxplus.md` §5). A chain built from Theorem 2 spends none of it at any single fold and has nothing to carry to the next. What remains is the decay hypothesis on runs of qualifying gaps that `research/kappa-not-L.md` names as what remains: a window with alternation-legal interior of sum $x'^2$ carries at most about $x'^2/(3q)$ dead slots (Fact 2's cap), and how many it needs is not determined by the cap; no polynomial moment of the gap distribution can supply the decay of long qualifying runs (`research/kappa-not-L.md`, "The wall, located precisely"). This paper does not touch it.

## 7. One operator, two semirings (inferred)

The record reads the fold in two algebras. Over the counting semiring it is the operator of §2 on the gap word, whose population form is Holt and Rudd's bidiagonal transfer matrix on driving-term populations, with binomial left and right eigenvectors and the closed-form spectrum $a_j = \prod_{17 \le q \le p}(q - j - 1)/(q - 2)$ (arXiv:1408.6002 §5.1 and Table 1; `research/PRIOR-ART.md`, "A9 is demoted"); it is an exact simulator with no bounds, closed on the word and not on the histogram (§2; §9 item 11). Over the max-plus semiring $(\mathbb{R} \cup \{-\infty\}, \max, +)$ the same fold is $q$-fold cyclic duplication of the tile's circuit followed by elimination of the dead vertices, which replaces two consecutive arc weights by their sum; the word that comes out is the true fold, gap for gap, at 6 of 6 folds from $3 \to 5$ to $17 \to 19$, so the copy theorem of `research/U-FRAME.md` §5a is a semiring identity (`research/history/staging/import-maxplus.md` §1b, VERIFIED). There $G_2 = \|A\|_{\max}$ is the operator norm and not an eigenvalue: the max-plus Perron root of the tile is the mean gap $\bar m = W/D$, Mertens, and the whole growth sits in $G_2/\bar m$, which runs 1.2000 to 7.2716 over $x = 5$ to 23 (`research/history/staging/import-maxplus.md` §2a, VERIFIED). The tropical operator would give bounds directly and is not closed on any finite maxsum vector: all $q$ alignments of a fold share one maxsum vector and their folded $G_2$ values spread by 0.09 nats at $x = 29$ (`research/history/staging/import-maxplus.md` §1c). Tropicalization buys the bound and loses the closure.

The statement that the two constructions are one operator in two semirings is INFERRED from their shapes (`research/history/staging/import-maxplus.md` §1d); no functor and no commuting diagram is written, and the record itself notes that the spectral object and the target object differ, $G_2$ being a norm while the tropical eigenvector's amplitude is several times $G_2$ and growing (its §2b: 7.4 times at $x = 23$). The proposals registry lists the semiring mapping among the owning conventions with no row in `research/SEARCH-CONVENTIONS.md` §1 (`paper/proposals/PROPOSALS.md`, "Recounted 2026-08-20"), so nothing in this section is claimed as new, and the referee's reading of it as a suggestive restatement (`paper/proposals/prop-tailcount-transport.md` §6) is one we cannot refute.

## 8. Prior art

The position is the registry's (`research/PRIOR-ART.md`, `research/SEARCH-CONVENTIONS.md` §1, `research/history/staging/proposals-prior-art.md` §2), not a hopeful one.

**Holt and Rudd own the operator, the closure theorem and the multiplier.** The cycle of gaps $\mathcal{G}(p\#)$, the recursion R1, R2, R3 (Lemma 2.1, §2.1 p. 5), the closure theorem "each possible closure of adjacent gaps in the cycle occurs exactly once", by the Chinese remainder theorem (Theorem 2.3, §2.2 p. 8), and the population model with its bidiagonal transfer matrix $M_J$ and binomial eigenvectors (§5 pp. 17–19, §5.1 p. 19) are in arXiv:1408.6002 (2014), read against the PDF of record and quoted verbatim in `research/history/staging/lit-pdf-holt-rudd.md`. The corpus demoted its own histogram operator in consequence, and both the live layer and the operator note instruct that nothing there be presented as new structure. One earlier citation in the corpus read "Holt and Rudd 2014, §6a", a section their paper does not have; it was this programme's own section and is corrected (`research/U-FRAME.md` §11, correction of 2026-08-18).

**The $(q - 2)$ multiplier is theirs too, as an identity.** Their §6.1, Corollary 6.3 with Figure 4 (pp. 25–26, read as page images 2026-08-19), carries the driving-term transport with no span hypothesis but with a divisibility condition: for a fixed gap sum $g$ with $q \nmid g$, "of the $q$ copies of $s$, two are eliminated as driving terms and $q - 2$ remain as driving terms of various lengths", giving $\sum_j n_{g,j}(qN) = (q - 2)\sum_j n_{g,j}(N)$, and printed page 26 explains why the two exterior closures then fall in different images (`research/history/staging/proposals-prior-art.md` §2; referee report 71 §5). The condition is not decorative: in the one-class cycle $\mathcal G(6)$ exactly one driving term has sum $10$, and folding by $q = 5$ gives four such driving terms in $\mathcal G(30)$, a factor $q - 1$ (the referee's exact enumeration). The first draft stated the multiplier for every gap size and is corrected. The two-class slot-count identity $D_{\mathrm{new}} = D(q-2)$ of §2 is a different object, proved independently, and unaffected. This corrected a sentence the corpus rested on, that their machinery is bounded by $|s| < 2p_1$ throughout; that sentence was false and is fixed in `research/PRIOR-ART.md`. Their object is the population of driving terms of a fixed sum, summed over lengths, in the one-class cycle, used for asymptotic ratios. They convert no population identity into a bound on a maximum gap; their corpus proves no non-trivial upper bound on a maximum gap (arXiv:1402.1970 §4, p. 10, reduces $g(Q) \le g(\bar q\#)$ and stops there, `research/history/staging/lit-pdf-holt-rudd.md` §3b) and does not study the spacing between occurrences of the gap 2, which is $G_2$ (`research/PRIOR-ART.md`, "the boundary").

**The multiplicity $\nu_p(s)$ and the fusion-spacing lemma.** The one-class statement that two fusions occur in the same image of the cycle exactly when $p$ divides their span is Holt's Lemma 2 in arXiv:2502.20470v3 §3, p. 5; the two-class form with its $\pm 2$ branch, and the alternation it forces, has no one-class counterpart there (`research/history/staging/proposals-prior-art.md` §1).

**What was searched and what was not.** The owning convention for "how a count of gaps moves from one fold to the next" has a row in `research/SEARCH-CONVENTIONS.md` §1 (written 2026-08-19), and the search in it returned the $(q - 2)$ transport without the inequality reading. The tail-count form, the inequality, the run term $2\sum_L Q_L$, the alternation refinement and the certificate use were searched with it and not found; the B-free and Toeplitz-word literature bounds gaps on a fixed set and nobody there iterates a level-to-level inequality on gap counts (`research/history/staging/proposals-prior-art.md` §2, verdict ADJACENT). The downgrade trigger's precondition, an owning-convention row written and searched, has fired, and the trigger itself did not (`paper/proposals/prop-tailcount-transport.md` §5, scored 2026-08-20). Not searched: Holt's GitHub notebooks; the covering-systems literature for a transport inequality; and the semiring mapping of §7, which has no owning-convention row at all. The standing assumption of the registry applies: prior art exists for more of this than has been found, and the burden is on us to look again. No claim of literature novelty is made for Theorem 2. It is a short consequence of the closure theorem and the Alternation Lemma, new to this corpus and not claimed beyond that.

## 9. Refuted and corrected claims on this line of work

Kept visible, as the house rules require.

1. **The loose certificate is not a general-threshold characterization, and the fold-29 example does not touch the $q^2$ statement.** `research/history/staging/attack-foldL-03-transport.md` §1 states "$G_2(x'\#) < x'^2 \iff$ no window of the old word with qualifying interiors sums to $x'^2$ or more", hedged "to within the 0 to 12 units measured". For the loose family the general-threshold biconditional fails in the forward direction (fold 29, $\theta = 264$: $G_2 = 258 < 264$ while a loose window sums to 270), not in the reverse direction as the first draft said; and at $\theta = q^2 = 841$ both sides hold at fold 29, so the record's $q^2$ statement is not refuted by that fold (referee report 71 §2). With the alternation-legal family the general-threshold biconditional is Theorem 2 and holds without a hedge.
2. **"Zero loss" was exactness by relaxation, and is now exactness by identity.** The origin record presented the certificate's exactness at five of six folds as the instrument's merit; the adversarial record reclassified it as the operator evaluated on its own support with two conditions dropped (`research/history/staging/verify-tailcount-transport.md` (c)); this paper proves that dropping those two conditions changes nothing (§4). The instrument is not a bound that happens to be tight. It is an identity.
3. **$M_{\mathrm{alt}} = G_2$ was carried as measured, at eight folds.** `research/U-FRAME.md` §11, `research/history/staging/frontier37.md` §3 and §9 and the proposal's §1 grade it measured and open. Theorem 2 proves it. The record's live-layer paragraph "Exact at folds 11 through 37 ... the exactness is structural" is right in its conclusion and should cite the containment argument rather than the eight data points.
4. **The stated reason for the no-chain verdict was mis-attributed.** The origin record's §4 says the rate is A5's cap; the fixed-index argument carries the verdict on its own (`research/history/staging/verify-tailcount-transport.md` (d)). The outcome register's row credits both; §6 follows the correction.
5. **The measured index spends were wrong at two folds.** 1, 1, 2, 2, 3, 3 in the origin record; 1, 1, 2, (1 or 2), 3, 2 on the alternation-refined certificate, fold 19 a tie and fold 29's 3 an artifact of the loose window that is not realisable (`research/history/staging/verify-tailcount-transport.md` (d)).
6. **The cap $K \le 1 + \theta/(3q)$ omits the odd-parity term.** The interiors of a window with $K - 1$ odd sum to at least $3q(K - 1) - q - 2\eta$, not $3q(K - 1)$; the corrected cap is $K \le 1 + (\theta + q + 2)/(3q)$ (§6, Fact 2). The adversarial record's re-derivation calls the record's form "sound, and slightly conservative", which is right in the regime it is used in and not as a statement for all $\theta$.
7. **The $L \le 8$ truncation of the origin producer** was an honest limit in `research/history/staging/attack-foldL-03-transport.md` §7 and is discharged: the sum terminates at $L = R + 1$ on its own (Theorem 1), and $R + 1 \le 4$ at every fold in reach.
8. **The adversarial re-derivation ran from scratchpad code.** The proposal's HELD trigger names this. The frontier producer, which is in the repository with its output embedded, recomputes folds 11 to 31 from an implementation written for that note and agrees with the adversarial table cell for cell, including the one cell where loose and refined differ (`research/history/staging/frontier37.md` §3, §5); the check script of §4 is a fourth implementation at folds 7 to 23. The adversarial numbers are reproducible from repository code even though the adversarial code is not.
9. **The corpus said Holt's machinery is bounded by $|s| < 2p_1$.** False; §8.
10. **A citation into "Holt and Rudd 2014, §6a"** pointed at a section that does not exist in their paper; §8.
11. **The record says both that the counting operator is closed on the histogram and that it is not.** `research/history/staging/import-maxplus.md` §1d: "The counting operator is closed on the histogram, which is why U-FRAME §11 calls it an exact simulator." `research/operator-and-pair-count.md`, "The honest limit": "the operator is closed on the gap word, not on the histogram, so it is an exact simulator rather than a source of bounds." The second is right, and the shuffle test of §6 (Fact 3) is the evidence: two words with the same histogram fold to different $G_2$. Holt and Rudd's $M_J$ acts on populations of driving terms indexed by length, which is a finer object than the single-gap histogram. §7 follows the second wording.
12. **First draft, title, abstract, §1 and §6: "the evaluation cannot be iterated", "nothing coarser can bound it", "no statistic of the old word supplies the next folded residues".** Withdrawn as impossibility claims: the old word and $q$ determine the folded word (referee report 71 §1, with an iterated tiny check $T_5 \to 7 \to 11 \to 13$ matching the directly built words); what fails is the specified truncated chains and histogram closure (§1, §6).
13. **First draft §4 and §9 item 1: the direction and threshold of the loose-family counterexample.** Corrected (item 1 above).
14. **First draft §6 Fact 2: "a window of sum about $q^2$ needs about $q/3$ qualifying gaps", "the index a chained certificate would have to carry diverges", "within one of the cap at every fold".** The cap is an upper bound on kills for a fixed span; the converse needs control of the outer gaps; at fold 37 the cap is 6.11 against $R + 1 = 4$ (referee report 71 §3). Withdrawn and corrected (§6).
15. **First draft Theorem 1: "both sums vanish for $L > 2D$".** Proved only for the refined sum; the loose sum's finiteness on a tile is the mean-gap argument, now in the proof, and Theorem 2's use of the $2D$ count is no longer circular (referee report 71 §4).
16. **First draft §3: "above $G_2(T_x)$ the theorem costs nothing" and "reduces to a statement about the largest gap".** Exactness is of the support endpoint only; at fold 17 and $\theta = 108$ the refined right side is 40 against $N_{\mathrm{new}} = 20$ (referee report 71 §4).
17. **First draft §8: the $(q-2)$ multiplier "for gaps of every size".** Holt–Rudd's Corollary 6.3 requires $q \nmid g$; the one-class $\mathcal G(6) \to \mathcal G(30)$ example gives $q - 1$ (referee report 71 §5). Corrected.
18. **First draft §2: the constrained-coding citation.** The printed $B = 1$ charge language is binary without zero loops and has capacity 0; ours has a zero loop at each state and capacity 1 bit; named as an extension (referee report 71 §5, review 69).
19. **First draft §5: "a linear extrapolation reaches 1 at about fold 53" and "one more point on a curve with no model".** The curve has a model, $(q-4)/(q-2)$ plus a shrinking merge excess (return #1154); withdrawn and replaced (§5).
20. **First draft §4: "$\sum_L \nu_q(i,L) = q-2$ at every $i$" as a check-script result.** The script's counter closes at $q - 2$ by construction; it is a termination check (referee report 71).

## 10. Reproduction

Every number in this paper is read from an embedded OUTPUT block, a staging record that names its producer, or a run made for this paper whose log is listed here.

- `node research/attack-foldL-03-transport.js 23`: the loose certificates and truths at folds 11 to 29, the inequality at every threshold at folds 11 to 23, the chain table of §6, the index table; 14 s here (39 s recorded), all self-tests pass. Re-run for this paper; log `job67-foldL03-23.log`.
- `node --max-old-space-size=4096 research/attack-frontier37-02-transport.js full`: the three certificates, $L$, $D_{\mathrm{new}}$, the untruncated inequality and the window counts at folds 11 to 37; 661.9 s recorded, embedded 2026-08-19, re-verified with `node research/qc/embed.js --check --timeout 1800 --node-flag --max-old-space-size=4096 research/attack-frontier37-02-transport.js -- full` (`research/history/staging/frontier37.md` §8). Not re-run here.
- `node job67-alt-exact-check.js`: the mechanism check of Theorem 2 at folds 7 to 23, under 3 s; log `job67-alt-exact-check.log` (no timing line, so the log hashes reproducibly).
- `node --max-old-space-size=8192 job67-transport-parallel.js calib|fold31|fold37|fold41`: the parallel port; calibration at folds 11 to 29 in seconds, fold 31 in 4 s wall, fold 37 in about 2 min wall, fold 41 in 3,607.8 s wall on 9 threads (28,346 s CPU); logs `job67-calib.log`, `job67-fold31.log`, `job67-fold37.log`, `job67-fold41.log`. The logs carry timings, so compare figures rather than hashes.
- `research/import-maxplus-01-mapping.js` with argument 23 (`node research/qc/embed.js --check --streams both research/import-maxplus-01-mapping.js -- 23`): §7's semiring figures (`research/history/staging/import-maxplus.md` §8).
- `research/exact-g2-ladder.js`: the truths $G_2(q\#)$ and the attaining counts `nmax`.
- `python tct2458.py 13,17 17,19 19,23 19,29 ... 19,89` and `python tct2458.py 23,29 23,31 23,37 23,107 23,109` (return #1154, uploaded with this revision): the loose and sharp ratios, the $(q-4)/(q-2)$ baseline, the merge share and the identity gap at 23 folds; reproduces the loose maxima 0.8881, 0.8975, 0.9180, 0.9324 and the certificates 108, 150, 204, 270 as controls.
- Referee report 71's `small-checks.py` and `port-fixture.cjs` (its artifacts): the iterated-word check, the threshold example, the one-class divisibility exception, the abstract span example, and the serial three-fold branch test of the parallel port.

The independent adversarial implementation (`advtct-lib.js`, `advtct-cert.js`, `advtct-f31.js`) lives in a session scratchpad and is not in the repository; its numbers are reproduced by the frontier producer (item 8 of §9).

## References

Locators are those the research record carries; a locator identifies the evidence and does not mean the cited page was re-read for this draft unless the record says it was read as a page image.

- Referee report 71 on return #23, and return #1154 (job 2458; verifier `tct2458.py`) and return #1072 (the isolated-deletion lemma), platform records this revision builds on.
- F. B. Holt and H. Rudd, *Eratosthenes sieve and the gaps between primes*, arXiv:1408.6002v1 (2014): §2.1 Lemma 2.1 p. 5 (R1, R2, R3); §2.2 Theorem 2.3 p. 8 (each closure occurs exactly once); §5 pp. 17–19 and §5.1 p. 19, Table 1 (the transfer matrix $M_J$, its binomial eigenvectors and closed-form eigenvalues); §6.1 Corollary 6.3 and Figure 4 pp. 25–26 (the $(q - 2)$ driving-term transport for $q \nmid g$; the divisibility condition and p. 26's explanation checked by referee report 71). Lemma 2.1, Theorem 2.3 and §5 verified verbatim against the PDF of record in `research/history/staging/lit-pdf-holt-rudd.md`; Corollary 6.3 and Figure 4 read as page images in the officer pass of 2026-08-19, `research/history/staging/proposals-prior-art.md` §2.
- F. B. Holt, *Eratosthenes sieve supports the $k$-tuple conjecture*, arXiv:2502.20470v3 (2025), §3 Lemma 2, printed p. 5 (read as a page image).
- F. B. Holt, *Surviving Eratosthenes sieve I: quadratic density and Legendre's conjecture*, arXiv:2603.25915v1 (2026), section list checked for transport-shaped statements (`research/history/staging/proposals-prior-art.md` §2).
- B. H. Marcus, R. M. Roth and P. H. Siegel, *An Introduction to Coding for Constrained Systems*, lecture notes (edition and year not recorded in the registry), §2.3 p. 47 (the $B = 1$ charge constraint, binary, no zero loops), §3.2 p. 75 (capacity of the 2-charge example); the ternary zero-loop extension used here is not identified there (review 69, referee report 71).
- OEIS A144311 (A. Carter, 2008), the maximal gap between twin-admissible residues modulo primorials, 22 terms; the corpus's ladder is A144311 plus one, and its terms at $x \le 43$ are recomputations (`research/G2-STATE.md`; `paper/PAPERS.md`).
- R. A. Cuninghame-Green, *Minimax Algebra*, Springer, 1979; F. Baccelli, G. Cohen, G. J. Olsder and J.-P. Quadrat, *Synchronization and Linearity*, Wiley, 1992, ch. 3; R. M. Karp, A characterization of the minimum cycle mean in a digraph, Discrete Math. 23 (1978) 309–311 (the max-plus references of `research/history/staging/import-maxplus.md` §8, cited there without a novelty claim).
- H. Diamond and H. Halberstam, the dimension-2 sifting limit $\beta_2 = 4.26645$, as recorded and verified in `research/dhr-verification.md`.

## Authorship and AI disclosure

Sole author: Chris Benjaminsen. The framework, vocabulary, and driving questions are the author's, developed over six years of independent work. Formal derivations, literature audits, computations, and manuscript drafting were carried out using AI assistants under the author's direction. Computations have reproducible code and recorded outputs; asymptotic arguments require their stated mathematical inputs and are not proved by finite checks. Refuted intermediate claims are retained in the record.