# The u-frame: the Zone Postulate as one inequality in one variable *(Adopted 2026-08-16, from Chris's proposal to step forward on u rather than on p. This note holds the frame, its proofs, the measurements that go with it, and an honest account of what the reframing does and does not buy. Calibration marked throughout: PROVEN, VERIFIED by exact computation, MEASURED, REFUTED.)* ## 1. The problem with stepping on p The Gap Reformulation compares G₂(p#), the largest gap between twin slots, with the zone (p,p′²). It can be useful to index the process by all integers instead of successive primes. The u-frame makes that change of coordinates without changing the survivor condition. Dependence on the next prime is not itself an impossibility result for a formula or a proof. The earlier claim that no one can write p′ as a function of p was too strong; the issue is obtaining a useful estimate, not defining the function. The current arithmetic campaign is in `chen-fold-benchmark.md` and `chen-signed-target.md`; the u-frame remains a separate sufficient reformulation. ## 2. Fold on the odd numbers Fold the twin-slot comb by every odd number in turn, 3, 5, 7, 9, 11, 13, 15, 17, and so on. A fold by U lays U copies of the tile and strikes the residues 0 and −2 modulo U. **Inertness Lemma (PROVEN, one line).** If U is composite then the fold by U kills nothing. *Proof.* Let q be the least prime factor of U. Then q < U, so q was already stacked at an earlier fold. If U divides r then q divides r, so r was already dead; likewise if U divides r+2. Hence no slot is removed. ∎ The lemma needs no primality test and no table. It says every composite fold is a no-op by construction, which is what lets the fold index be closed-form while the productive folds remain exactly the primes. **VERIFIED.** Folding 3, 5, 7, 9, 11, 13, 15, 17 from the mod-2 comb: | fold U | composite | kills | G₂ after | |---|---|---|---| | 3 | no | 2 | 6 | | 5 | no | 2 | 12 | | 7 | no | 6 | 30 | | 9 | **yes** | **0** | 30 | | 11 | no | 270 | 42 | | 13 | no | 2,430 | 66 | | 15 | **yes** | **0** | 66 | | 17 | no | 400,950 | 108 | The ladder 6, 12, 30, 42, 66, 108 reproduces the prime-fold ladder exactly, with flat steps at the composites. Also verified for folds by 4, 6, 8, 9, 25, which kill nothing and leave G₂ at 12: the tile width balloons from 30 to 1,296,000 with no change to the hole pattern, because repeating a pattern cannot change its maximum gap. The waste is real and entirely harmless. ## 3. The frame G₂(u) is a step function of u, constant between primes and jumping only where a new prime enters. The Zone Postulate route becomes > **G₂(u) < u² for every integer u**, binding immediately after each jump, that is at u = p. By the Inertness Lemma only the prime folds contribute, so with c(p) the multiplier at the fold by p the telescope is > **G₂(u) ≤ 12 · Π_{p ≤ u} c(p)**, and π(u), the number of folds that are not no-ops, is closed-form by the prime number theorem. **The requirement is on the partial sums, not on a uniform constant (PROVEN, VERIFIED in `research/gate-multiplies.md` §7).** Read as a uniform c the statement is vacuous, since 12·c^{π(u)} < u² at every u forces c = 1: the permitted per-fold budget 2 ln u/π(u) tends to 0. Read correctly the condition is > **Σ_{p ≤ u} ln c(p) ≤ 2 ln u − ln 12 at every u**, and differentiating it along the primes, one fold at a time, gives the sharp per-fold rate > **ln c(p) ≤ 2 ln p / p**. Everything is now expressed in u alone. No prime locations appear anywhere. **That rate is a factor ln u tighter than dividing the budget by π(u).** The softer form ln c ≲ 2 ln²u/u does not telescope: Σ_{p≤x} ln²p/p ~ ln²x/2 against a budget of 2 ln x, so it overspends by ln x/4 per unit and certifies x^{ln x − 2}. VERIFIED numerically, the bound it licenses exceeds u² by 4.6× at u = 37, 149× at u = 100 and 1.6e10 at u = 1000. **And the sharp rate is where the truth already sits (MEASURED).** If G₂ ≍ (ln W)^α with ln W = θ(x), then c_true = (θ(p′)/θ(p))^α ~ 1 + α·ln p/p, so the entire asymptotic headroom is the factor α/2 and nothing else. §4 measures both ladders against 2 ln p/p and finds them spending at about the full rate. There is no room to be had here unless α is genuinely below 2, which is the exponent question of §4 and is not settled. **The Inertness Lemma is load-bearing, not decorative.** If we paid the multiplier at every odd fold rather than only at the primes, the same budget would be spread over u/2 folds instead of π(u) and the rate would tighten to ln c ≲ 4/u, harder by a factor of ln u / 2. The lemma is what buys that factor back. ## 4. What is measured **Both ladders spend their budget at about the allowed rate (MEASURED).** Per-fold multipliers of G₂ at @5→@7 through @31→@37: 2.50, 1.40, 1.571, 1.636, 1.389, 1.360, 1.265, 1.349, 1.517 geometric mean 1.43 over the last eight, with no trend across nine folds. Read against §3's sharp rate, that is ln c(p) as a fraction of 2 ln p/p: 1.65, 0.77, 1.15, 1.48, 1.06, 1.13, 1.01, 1.35, 2.14 so our multiplier uses 77 to 214 percent of its budget, and the fraction is not falling. Ziller and Morack's adversarial multiplier (§6a), which dominates ours at every level, extends the reading nine folds further. At p = 41 to 73 it reads 1.39, 0.97, 1.36, 0.75, 1.15, 1.10, 1.18, 1.00, 0.61, mean 1.06 so their sequence is spending at exactly the allowed rate. Their raw multipliers do fall, geometric mean 1.611 over p ≤ 37 against 1.157 over p = 41 to 73, but the budget falls with them and the ratio does not improve. **Neither ladder shows the decay the route needs.** **The lifetime slack, which is the sharpest way to say it (MEASURED).** Any chain of upper bounds ending in G₂(x#) < x² passes through quantities that are all at least G₂(x#), so its total multiplicative overshoot over the truth is capped by x²/G₂(x#). As ln(x²/G₂(x#)) that is 1.058, 0.940, 0.984, 0.878, 0.953, 1.182, 1.016, 0.953 nats at x = 11 to 37: flat across the whole reachable ladder, with no growing margin anywhere. **About one nat, spread over an infinite ladder, is the entire budget for every approximation a proof would make** (`research/gate-multiplies.md` §5). **As a single exponent (MEASURED, and the estimator is biased).** Writing G₂ ≍ (ln W)^α, the route needs α < 2. Fitted on our twelve levels: | fit window | α | |---|---| | all 12 levels | 1.476 | | from x = 7 | 1.674 | | from x = 13 | 1.861 | | from x = 19 | 1.983 | | last four (23 to 37) | 2.170 | Those numbers cannot be read at face value. `research/exponent-control.md` runs the same estimator on the one-class Jacobsthal h(p#) = A048670, where the answer is known: **58 terms, true exponent 1, measured 1.282 ± 0.008, positive bias in all 40 sliding windows, no drift toward the truth anywhere.** The bias grows with ladder length and then sticks at +0.28, and the pure power law wins the model comparison by 47 AIC units with white residuals while getting the exponent wrong by 0.28. So a clean fit here carries no evidence about the asymptote, and lengthening the ladder does not repair it. Corrected by the control's bias at matched width, the readings are **α = 1.57 ± 0.06 for the adversarial h₂ on 19 terms and 1.50 ± 0.05 for our own G₂ on the 22 trusted terms** (the 22-term refit, `research/exponent-control.md` §5), with a hard floor α ≥ 1 from h₂ ≥ h pointwise. **Central estimate 1.50 for G₂ (h₂'s figure stays 1.57), practical bracket 1.3 to 1.8, floor 1**, and α = 2 disfavoured by the one-sidedness of the bias rather than excluded by the data. Caveat, and it matters: our local slopes are 1.16, 1.08, 2.03, 0.91, 1.58, 2.03, 1.63, 1.73, 1.45, 2.11, 3.21. The final 3.21 at x = 37, where G₂ jumped 348 to 528, drags every recent fit upward. Drop it and the fit from x = 19 is 1.716. One data point is doing most of the work, which is the same small-sample disease the control diagnoses. **What the exponent question is for, and what it is not for.** The proven bound is G₂ ≪ (log q)^{4.2665+ε}, from the two-dimensional sieve limit β₂ = 4.26645; the Zone Postulate needs exponent 2. **The job is an exponent reduction, 4.2665 → 2, not a fight over a constant.** Measuring our own α more precisely says whether the target is true. It does nothing about the 2.27 of exponent between what is proven and what is needed. **So G₂(41#) is not the decisive measurement.** Measured on 48 control cases, adding an eleventh term to a ten-term fit moves the exponent by 0.022 on average and 0.078 at worst, against a bias of +0.262. Its real value is as a falsifiable test of the Poisson law, which predicts **476 to 633 with centre 513** (`research/G2-STATE.md` §6.2), against Ziller and Morack's free-choice ceiling of 894. ## 5. What makes the maximum gap (MEASURED) For each fold, the stretch that became the new record, and what filled it: | fold | new G₂ | kills that made it | merged sub-gaps | old G₂ | |---|---|---|---|---| | 7 | 30 | 2 | 6+12+**12** | 12 | | 11 | 42 | 1 | 12+**30** | 30 | | 13 | 66 | 1 | 36+30 | 42 | | 17 | 108 | 2 | 30+**66**+12 | 66 | | 19 | 150 | 1 | 42+**108** | 108 | | 23 | 204 | 3 | 24+48+90+42 | 150 | | 29 | 258 | 2 | 60+60+138 | 204 | In four of seven folds the winning stretch contains the old maximum gap itself, so the record is made by killing a slot at the end of the current worst gap and absorbing its neighbour. Chris's original mechanism was right. What was wrong was the arithmetic: the absorbed piece is a neighbouring gap of one or two mean gaps, not another copy of the maximum, so the multiplier is 1 + absorbed/G₂ and not 2. **REFUTED on the way in:** the guess that the maximum doubles fails at the first fold, 12 to 30 rather than 24. **Adjacent kills, and why they are not the lever.** Two slots at distance g can both be killed by p only if g ≡ 0 or ±2 (mod p), which is PROVEN in one line from the strike classes {0, −2}. Because grain gaps are multiples of 6 bounded by G₂, only a few gap values qualify, and they are the large rare ones: | fold p | qualifying gaps | share of all gaps | longest adjacent-kill run | |---|---|---|---| | 7 | {12} | 66.7% | 2 | | 11 | **none** | 0% | **1** | | 13 | {24} | 4.4% | 2 | | 17 | {36, 66} | 4.85% | 2 | | 19 | {36, 78} | 4.9% | 2 | | 23 | {48, 90, 138} | 3.1% | 3 | | 29 | {60, 114, 174} | 3.1% | **2** | | 31 | | | 4 | | 37 | | | 4 | The last column is **L, the longest adjacent-kill run**, whose diagonal is L = 2, 1, 2, 2, 2, 3, 2, 4, 4 at folds 7 to 37. §10 holds its definition, its proven bounds, and the reason it is not the quantity to aim at; §8 holds its custody. At p = 11 no gap qualifies, so that fold provably cannot make a single adjacent kill, which is why the run series dips. But the kills column of the previous table never exceeds 3 even where longer runs were available, so **bounding the adjacent-kill run bounds the wrong quantity.** The record gap is assembled from a few large gaps, not from a long run of typical ones. ## 5a. The copy-theorem reformulation, and where the decay comes from *(2026-08-16, from Chris's proposal to attack the multiplier through the copy structure. This is the session's main result and every step below is either proven or verified; the final extrapolation is flagged as such.)* **Step 1, the kill law (PROVEN).** Fold T_x by p. Each old slot s sits at p positions s + kW, and gcd(W,p) = 1 makes those p residues mod p distinct, so **every old slot loses exactly 2 of its p copies**. Read by copy instead of by slot, with w = W mod p and r_i = s_i mod p, the slot at index i in copy k dies iff r_i + kw ≡ 0 or −2 (mod p). Hence: > **Copy k deletes exactly the slots whose residue mod p lies in the 2-set > {−kw, −kw−2}.** The fold is not p mysterious operations. It is p residue-class deletions applied to one fixed pattern. **Step 2, the exact reformulation (VERIFIED at six folds).** > G₂(new) = max over the p 2-sets {a, a−2} of the largest gap left after > deleting those residue classes from the old tile. Checked against the actual fold at p = 7, 11, 13, 17, 19, 23: exact equality every time. **And it holds for the whole maxsum family, exactly (VERIFIED 40 of 40, over five folds and m ≤ 8):** > maxsum_m(new) = max over the p 2-sets {a, a−2} of maxsum_m(old minus those > classes). No straddling window ever beats a single-copy one, and the reformulation above is the m = 1 case. This is the real closure, and it lives at the residue level rather than at the level of the numbers maxsum_j(old). It is also **the one object in the branch whose index cost is zero**, since m stays m under folding, which is why `research/gate-multiplies.md` §9 finds it immune both to the no-fixed-point argument and to the Overshoot Budget. What it needs is a way to bound a residue-deleted maxsum without going through a kill count, which is TODO item 0c and which nobody has tried directly. That "nobody" is this repository and not the literature: no script in `research/` bounds a residue-deleted maxsum — the thirty-one that compute maxsum, every one checked, go through a kill count. It is an internal absence, and the literature form of the claim would have to meet `SEARCH-CONVENTIONS.md`. **The one identity the maxsum family does have is PRIOR ART, and it must not be presented as new.** On a cyclic gap word of D gaps summing to W, > **maxsum_m + minsum_{D−m} = W**, and hence `sd_m = sd_{D−m}`, verified EXACT at all 1484 m on T₁₃. It is the complement identity of the circular scan statistic (Cressie, *J. Appl. Probab.* 14 (1977); Naus; Wallenstein–Naus; Glaz–Naus–Wallenstein 2001 chs. 8–10, 17), so it is OWNED and no sentence here may claim it (`IMPORT-MAP.md` row 1, `history/staging/identifications-prior-art.md`). What it buys is a constraint rather than a bound: any growth law for maxsum_m must be consistent with its own mirror at D − m, which is what killed the `√m` fluctuation factor's reading at large m. **Step 3, a two-sided bound in the old gap word alone — the LOWER half is now PROVEN (2026-08-19), the upper half VERIFIED at 329 (tile, prime) cells, folds 7 to 37.** A maximal run of L deleted slots merges L+1 consecutive gaps, and every single slot dies in exactly 2 of its p copies (CRT), so the record adjacent pair merges somewhere in the big tile — that one line proves the lower half (`history/staging/attack-l1-residue.md`, Lemma C). Hence > **maxsum₂(old) ≤ G₂(new) ≤ maxsum_{L+1}(old)** where maxsum_m is the largest sum of m consecutive gaps and L is the longest adjacent-kill run. Both halves hold at every cell tested (`research/a3-10-lower-tightness.js`): the lower bound is exact at 3 of the 9 ladder folds, the upper at 2. The upper bound beats the naive (L+1)·G₂ everywhere, overshooting by 0 to 45% against 20 to 216%. **The lower half does not become exact with depth.** Its excess runs 6, 0, 0, 12, 0, 18, 24, 18, 120 at folds 7 to 37, so the recursion is not determined by maxsum₂ and cannot be made so; §10 gives the two mechanisms that break it, one of them arithmetic and permanent. **A lower recursion that chains, for the whole family (VERIFIED 40/40).** maxsum_{m+1}(old) ≤ maxsum_m(new), with ratios 1.000 to 1.354, so G₂(T₂₃) = 204 ≥ maxsum₂(T₁₉) = 186 ≥ … ≥ maxsum₆(T₇) = 108, one index per fold. It is a lower chain, so it bounds G₂ from below where the Zone Postulate needs an upper bound (`research/gate-multiplies.md` §6). **The upper half is LOOSE, and the strategic verdict follows: aim at κ(m), not at L.** Write j\*(m) for the sharp shift, the smallest j with maxsum_m(new) ≤ maxsum_{m+j}(old). Then 1 + j\*(1) is the EFFECTIVE run length, the smallest window of old gaps that covers the new record, and it measures 3, 2, 2, 3, 2, 3, 3, 3, 4 at folds 7 to 37 against L + 1 = 3, 2, 3, 3, 3, 4, 3, 5, 5: never larger, strictly smaller at three folds, and at fold 31 it is 3 where L + 1 = 5. **A proof aimed at L is therefore aiming past the target.** The object to aim at is κ(m), the maximum kill count in a window of m consecutive new gaps, which step 3a defines and which generalises L to the whole family. The effective run length is the tighter coordinate and a strictly weaker thing to bound than L, but it is not a shortcut: §10 shows that a constant value for it is impossible, and fold 37 is where 3 stops holding in any case. **Step 3a, the family closes and L generalises to κ(m) (VERIFIED 40 of 40, over five folds and m ≤ 8).** Any m consecutive new gaps are exactly m + (kills inside) consecutive old gaps, so > **maxsum_m(new) ≤ maxsum_{m+κ(m)}(old)**, with κ(1) = L. Closure is real, and the price is that the parameter generalises from the number L to the sequence κ(m). **The strong forms are REFUTED, and they are left visible.** maxsum_m(new) ≤ maxsum_{m+L}(old) fails at every fold from m = 4 to 6 (T₁₉→T₂₃, L = 3, m = 6: 462 against maxsum₉ = 390), so κ(m) ≤ L is refuted. κ(m) ≤ L + 2 holds for all m ≤ 8 at those five folds but is not universal either: it fails at fold 11, where L = 1 and κ(6) = 4, precisely the fold where no gap value qualifies, so L collapses to its floor while κ does not, and it fails by 2 at fold 7, where L = 2 and κ(7) = κ(8) = 6. *(The fold-7 counter-example was added 2026-08-18. It had been hidden by a truncated cyclic replay in `research/a3-05-bound-L.js`, which printed the fold-7 κ row as 2, 2, 2, 1, 0, 0, 0, 0; the corrected row is 2, 2, 4, 4, 5, 5, 6, 6.)* The L-free form maxsum_m(new) ≤ maxsum_{m+1}(old) + G₂(old) holds for m ≤ 8 and dies at m = 10 (maxsum₁₀(T₁₃) = 282 against 210 + 42). Empirically maxsum_m(new) ≤ maxsum_{m+4}(old) covers every measured case with no L in it at all, and equality with maxsum_{m+j\*} occurs in 20 of 40 cases. maxsum_m/m is not monotone, though maxsum is subadditive with zero violations, so Fekete gives convergence without monotonicity. **What κ(m) buys, and what it does not.** It has far better tail structure than a longest run: `research/gate-multiplies.md` §4 measures κ(m)/m falling monotonically from 2 or 3 at m = 1 to 0.25 to 0.28 at m = 32, at every fold. And it is the exact statement of what has to be bounded, with about L + 2 of headroom to spend. What it does not do is remove the dependence on a run parameter: the L dependence moves into κ(m) and stays, and §10's Theorem C carries A5's structural cap over to κ, so redirecting L to κ does not move the wall either. **Step 4, the additive form of the recursion, is FALSE.** The tempting move is to write maxsum_{L+1} ≈ G₂ + L·m̄ with m̄ the mean gap, giving > G₂(new) ≤ G₂(old) + L·m̄, hence G₂(x#) ≲ 12 + Σ_{p≤x} L(p)·m̄(p), so that the recursion becomes a sum rather than a product. Checked fold by fold in `research/a3-02-diagonal-f.js` with the measured L, it fails at two of the seven folds: | fold | G₂(old) + L·m̄ | true G₂(new) | verdict | |---|---|---|---| | 13 | 76 | 66 | holds | | 17 | 106 | 108 | **fails** | | 19 | 154 | 150 | holds | | 23 | 227 | 204 | holds | | 29 | 260 | 258 | holds | | 31 | 379 | 348 | holds | | 37 | 477 | 528 | **fails** | At T₁₃ folded by 17 the approximation gives 106.4 against the true 108, and at T₃₁ folded by 37 it gives 476.8 against the true 528. Both are clean failures, not rounding, and step 3's proven two-sided bound holds at both of them. The reason is the one §5 already recorded and this step ignored: **the absorbed neighbour is not a mean gap.** It is drawn from the same fat tail as the record itself. At fold 29 the merged sub-gaps were 60 + 60 + 138 against a mean of 28.1. Substituting m̄ for a tail draw understates the merge. **The coefficient is wrong as well as the tail draw.** The step is written with an implicit slope of 1 in ρ = (maxsum_j − G₂)/((j−1)·m̄). Taken as the median over j ≤ 8 that slope measures 1.33 at T₁₁ and 1.71 at T₂₃, and over individual (tile, j) cells it ranges 1.02 to 1.84 across that span, reaching 2.39 at T₂₉. It is larger at the top of the reachable ladder than at the bottom but not monotone in the level: the median reads 1.33, 1.26, 1.15, 1.27, 1.71 at T₁₁ to T₂₃. Whichever reading is taken it is not 1 (`research/gate-multiplies.md` §8). **Consequence, and it is the important one.** Every argument of the form "Σ L·m̄ ~ x·ln³x ≪ x², so the route closes" runs through this step and therefore does not stand. That includes step 7's own dichotomy below and §11's tail argument. What survives is step 3, the PROVEN two-sided bound, verified to fold 37. **Any asymptotic must be carried by maxsum_{L+1} directly**, and `research/gate-multiplies.md` §8 prices exactly that: the sharp per-fold budget of §3 goes through step 3 iff L ≤ 0.19 to 0.31·p/ln p on average over the ladder, against §10 Theorem B's proven L ≤ 0.18 p. We have no valid asymptotic in either direction, and the gap is a factor 0.58 to 0.95 ln p. **Step 5, why the lower bound decays (MEASURED).** maxsum₂ = G₂ + (the gap adjacent to the maximum), and the neighbour is a typical gap while the maximum grows. Measured maxsum₂/G₂ = 2.00, 1.40, 1.571, 1.455, 1.389, 1.240, 1.147 across the seven folds. That is the decay mechanism, exactly stated. **Step 6, what controls L (PROVEN criterion, MEASURED behaviour).** Two adjacent slots at distance g can both be deleted only if g ≡ 0 or ±2 (mod p), so L is governed by f, the fraction of gaps meeting that condition, via roughly L ≈ ln D / ln(1/f). Since gaps are multiples of 6, the smallest qualifying value is about 2p. Sweeping L(T_x, p) over tiles T₇ to T₂₃ and primes 7 to 200 shows L falling to 1 as p grows at fixed tile, and rising with the tile at fixed p; the fold ladder rides the diagonal, where L = 1, 2, 2, 2, 3, 2, 4, 4 at folds 11 to 37, so **L is not monotone**: it steps back to 2 at fold 29 before reaching 4 at fold 31, and §10 gives the mechanism. Fold 37 is measured in research/a3-10-lower-tightness.js by streaming T₃₁, all 6,226,553,025 slots of it. **Step 7, the dichotomy, and which branch we are on.** L ≈ ln D/ln(1/f) with ln D ≈ x gives: - f roughly constant ⇒ L ~ x/3.2, which is linear; - f decaying ⇒ L polylog. The branch matters because step 3, priced against §3's per-fold budget, needs L ≤ 0.19 to 0.31·p/ln p (`research/gate-multiplies.md` §8). Linear L misses that by a log; polylog L clears it with room. **The old shortcut through Σ L·m̄ is not available**: that sum is step 4, and step 4 is false. **MEASURED, and it is the second branch.** The qualifying threshold 2p grows linearly in x while the mean gap grows only like ln²x, so the threshold recedes into the tail of the gap distribution: | tile | fold p | mean gap | threshold 2p | 2p/mean | f | ln(1/f) | L | |---|---|---|---|---|---|---|---| | T₁₁ | 13 | 17.1 | 26 | 1.52 | 4.44e−2 | 3.11 | 2 | | T₁₃ | 17 | 20.2 | 34 | 1.68 | 4.85e−2 | 3.03 | 2 | | T₁₇ | 19 | 22.9 | 38 | 1.66 | 4.88e−2 | 3.02 | 2 | | T₁₉ | 23 | 25.6 | 46 | 1.80 | 3.11e−2 | 3.47 | 3 | | T₂₃ | 29 | 28.1 | 58 | 2.07 | 3.07e−2 | 3.48 | 2 | | T₂₉ | 31 | 30.1 | 62 | 2.06 | 3.74e−2 | 3.29 | 4 | | T₃₁ | 37 | 32.2 | 74 | 2.30 | 1.84e−2 | 3.99 | 4 | The last two rows come from streaming T₂₉ and T₃₁ without ever storing either tile (`research/a3-02-diagonal-f.js`, attack A2); §8 holds the custody for all seven rows. Off the diagonal the collapse is unmistakable: for T₂₃, f runs 0.041, 0.031, 0.031, 0.012, 0.0034, 0.0010, 0.0003, then exactly 0 from p = 71. **MEASURED, on seven points: ln(1/f) tracks 2p/m̄, and seven points cannot pin the slope.** The raw diagonal is not monotone, since T₂₉ steps back up. It is not flat either. Regressing ln(1/f) on 2p/m̄ over the seven points gives slope 1.062 and intercept 1.357 with R² 0.730, **but SE 0.289, so the 95% interval is [0.32, 1.81]**. That contains 1, which is why the exponential-tail model is not contradicted here, and it equally contains the 1.4016 of the alias-free recomputation, since the published 42-point census fit was defective from x = 37 (`research/f-decays.md` header, `history/staging/fdecay-deep.md`). Two extra points move this slope to 1.881. Do not quote 1.06 as a measured exponent. The exponential-tail model predicts slope exactly 1 with intercept ln(m̄/6), which runs 1.05 to 1.68 over this range. As a parameter-free forward prediction rather than a fit, ln(1/f) ≈ 2p/m̄ + ln(m̄/6) lands within 11% at six of the seven points. So f is a tail probability at a receding threshold, as claimed, and that is now measured. The wobble has a cause. Almost all of f sits in one histogram bin, the smallest multiple of 6 that is ≡ 0, ±2 (mod p): 24, 36, 36, 48, 60, 60, 72 at p = 13 to 37. That value tracks 2p to within 8% but repeats across consecutive folds, and the bin also carries a Hardy-Littlewood comb weight, measured between 0.73 and 1.94 with no trend. Over this range the comb noise is the same size as the decay, which is why five points could not separate the branches. **And the repeats are not accidents: twin folds share a threshold, so the diagonal is a staircase whose treads are the twin pairs** (PROVEN, §12). Both members of a twin pair give the same minimal qualifying gap 2(p+1), so f cannot advance across a twin fold. That is the whole of the wobble at 17 and 19 and at 29 and 31. **Consequence for L.** Since 2p grows linearly and m̄/ln²x is measured flat at 2.7 to 3.1, the threshold recedes without limit, so ln(1/f) → ∞ and L ≈ ln D/ln(1/f) ≈ m̄/2 ≈ 1.4 ln²x, which is polylog. **The shape check is weaker than this section used to claim**, and the correction of L(T₂₃, 29) from 3 to 2 on 2026-08-17 is why. L/ln²x runs 0.348, 0.304, 0.249, 0.346, 0.203, 0.353, 0.339 across the seven folds: a factor of 1.7 between its ends and not flat. L/lnD runs 0.408, 0.274, 0.200, 0.234, 0.126, 0.208, 0.177, which falls overall but not monotonically, reversing twice. Neither ratio is steady enough to call the branch on its own, and **the two regressions no longer separate the branches at all**: L on lnD gives R² 0.683 and L on (ln lnD)² gives R² 0.627, where before the fix they read 0.904 and 0.867 (`research/a3-02-diagonal-f.js`). If L were linear in x the second ratio would be the flat one, and it is the less flat of the two, so the reading still points the same way. So the evidence favours the branch the route needs on the ratio comparison and the mechanism rather than on any fit, and it does so because **the prime grows linearly while the mean gap grows logarithmically**, which is the structural asymmetry the lens is built on. A polylog L clears the 0.19 to 0.31·p/ln p threshold with a margin growing like p/ln³p, crossing 1 near p ≈ 800. It is evidence for a branch, not a proof of one. **Honest limits.** 2p/m̄ still only moves from 1.52 to 2.30, so the lever arm is short and a true slope near 0.7 is not excluded by seven points. The independence model overestimates L by a factor of about four, predicting 8 to 16 where we measure 2 to 4, so the shape of L ~ c ln²x is supported but the constant is not. The exponential-density model is also better than it deserves to be, since the grain at T₂₃ is still markedly more regular than exponential. The mechanism is identified and every link is checkable; the asymptotic claim is not established. **Where this leaves the question.** It is no longer whether f decays, which is settled in the direction the route needs (§12 settles it on 42 exact points). The open question is the valid replacement for step 4, carried on step 3a's κ(m) rather than on L. Two suggestive coincidences pointed the way to the family: maxsum₄(T₂₃) = 348 = G₂(T₃₁) and maxsum₈(T₂₃) = 528 = G₂(37#), two and three folds ahead respectively. ## 6. What the reframing buys, stated honestly **It buys form.** The statement is now closed-form in one variable, the fold index is the odd numbers, which folds bite is settled by a one-line lemma rather than a lookup, and the count of productive folds comes from PNT. Any attack that would have needed to know where the next prime sits is now unnecessary, and a whole class of dead ends is closed off. **It does not buy difficulty, and it buys no slack either.** The requirement ln c(p) ≤ 2 ln p/p is exactly what the per-prime analysis already gave. The reframing moves no arithmetic, and the entire problem now sits in one place: **is the per-fold multiplier bounded by 1 + O(ln p/p), with the constant 2?** Nothing in this note bounds it, and §4's measurements put both our ladder and the adversarial one at about the full rate rather than under it. Note what that does to the win condition: a proof that c decays at *any* rate is worth nothing here. It has to decay at 2 ln p/p, which is the rate the truth already runs at. This is a change of coordinates rather than a change of terrain. That is worth having, since a proof has to be writable before it can be written, but it should not be mistaken for progress against the wall. ## 6a. PRIOR ART, and it is close: Ziller and Morack 2017 **Our G2 sequence is in OEIS, one offset away: it is A144311 + 1**, Andrew Carter, September 2008, 22 terms. The 2026-08-16 check recorded here read "not there — exact-term searches on 2,6,12,30,42,66,108,150,204,258,348,528 at three offsets all return nothing", and those searches were run in our own convention; A144311 carries `G2 - 1` under wording containing no "Jacobsthal", no "twin", no "primorial" and no "gap". `research/SEARCH-CONVENTIONS.md` §2 is the record of the miss. Consequences: the drafted submission is a **duplicate and must not be sent**, and G2(41#) = 546 is already published as A144311's a(13) + 1, so computing it buys verification rather than discovery. **But the programme of this note is published.** OEIS **A288815**, "Paired Jacobsthal function applied to the product of the first n primes" (Mario Ziller, 2017; Ziller and Morack, arXiv:1706.00317, *Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs"; keyword `hard,more`), carries this comment: > "If a(n) < p_n^2 - p_n holds for n>=3 then Goldbach's conjecture and the twin > prime conjecture hold as well." That is the Zone Postulate condition, stated in 2017, in a form STRICTLY STRONGER than ours. Their a(n) is the ADVERSARIAL paired Jacobsthal, the maximum over ALL choices of two residues per prime, while our G2 is the single arithmetic choice {0, -2}. So a(n) >= G2(p_n#) always, VERIFIED at all twelve shared terms. Their companion A072753 (maximum gap in two-stage prime sieves, 19 terms) is the same object in another normalisation, a(n) = 6*A072753(n) + 6, and is already known to this repo through the 2D-Rankin work. **Their 21 terms are free data, and they extend every reading we have.** The adversarial ratio a(n)/(lnW)^2 decays monotonically from n = 6: 1.411, 1.111, 0.997, 0.990, 0.882, 0.842, 0.806, 0.804, 0.758, 0.765, 0.704, 0.689, 0.674, 0.666, 0.649, 0.604. **The fitted exponent is 1.62 in the theta frame and 1.924 against x, and the Zone Postulate threshold is an x-frame quantity.** Fitting a(n) against ln W gives 1.653 on all 21 terms, 1.613 from n = 6, 1.623 on the last eight; the same 21 terms fitted against x give **1.924**. Quoting the theta-frame number against the critical exponent 2 compares two different frames. The control's bias is +0.282 in x and +0.220 in theta, so after correction the two frames land at **1.567 and 1.493** and most of the disagreement cancels (`research/exponent-control.md` §6). The corrected reading is section 4's: 1.57 for h₂ and 1.50 for G₂, bracket 1.3 to 1.8, floor 1, with 2 disfavoured rather than excluded. **What their data does settle.** Their 21 points with no outlier are a far better ladder than our twelve, and our sequence lives underneath theirs at every shared term. It also hands us G2(41#) <= 894 for free, which is a ceiling rather than a prediction; the Poisson law's 476 to 633 is the prediction (section 4). **And the verification of the target extends further than ours.** a(n) < p^2 - p holds at all 21 computed terms. The margin (p_n^2 - p_n)/a(n) reads 1.11, 1.40, 1.67, 1.04, 1.42, 1.33, 1.38, 1.80, 1.63, 1.88, 1.83, 1.73, 1.68, 1.94, 2.07, 1.92, 2.02, 2.02, 2.00, and against p_{n+1}^2 it reads 2.72, 4.03, 2.56, 1.93, 1.88, 2.05, 2.30, 2.14, 2.40, 2.37, 2.07, 2.12, 2.19, 2.45, 2.25, 2.36, 2.30, 2.17, 2.38. **The two conventions disagree on the trend and agree asymptotically**, because the OLS slope of log p_{n+1} on log p_n is 0.88 rather than 1 over this range: with p_{n+1}^2 the trend is -0.084 +- 0.045 and with p_n^2 it is +0.154 +- 0.035. Against x'^2, which is the Zone Postulate's own threshold, the margin is **flat at 2.2** with slope +0.018 +- 0.045 over [23, 73], minimum 1.880 at x = 17 and never revisited. Our G2 sits above that at 3.2 to 4.4. So the route is verified to the 21st prime in the adversarial form, hence a fortiori for our G2, and the margin is flat rather than growing. **Second prior art, and it is not close, it is the same object: Holt and Rudd 2014.** Fred B. Holt and Helgi Rudd, arXiv:1408.6002 §5, build the transfer matrix M_J on the cycle of gaps, eigenstructure and binomial eigenvectors included. §11's histogram transfer operator is a rediscovery of it in two-class vocabulary. Their "fusions" are our kills, their cycle of gaps G(p#) is our tile, and their R1/R2/R3 recursion (Lemma 2.1) is our fold. What is not in their corpus is the spacing between consecutive g = 2 occurrences, which is our G₂: their machinery is bounded by |s| < 2p throughout, which is exactly the regime a maximum gap leaves. See `research/PRIOR-ART.md` §"Holt and Rudd". **Nothing in §11 may be presented as new structure.** What is ours there is the exact head engine of the tail paragraph and the numbers it produces. **Positioning consequence.** We must not claim the reduction; it is Ziller and Morack's, and we must not claim the operator; it is Holt and Rudd's. What remains ours, pending a proper read of all three papers, is the u-frame presentation of section 3, the Inertness Lemma, the mechanism results of section 5, and the specific arithmetic sequence G2, which is a smaller quantity than theirs and is **A144311 + 1** in OEIS, published by Carter in 2008 and extended twice since. Any writeup must cite them and Carter up front. **First move before any more work here: read arXiv:1706.00317 and arXiv:1706.03668.** They have thought about this exact inequality for nine years and their notes on the computation will say what is hard about extending the ladder. ## 7. Next 1. **The L question, priced — and re-scoped by the 2026-08-19 wave.** Step 3 goes through iff L ≤ 0.19 to 0.31·p/ln p on average; §10 Theorem B proves L ≤ 0.18 p. The wave settled what can and cannot close that factor of 0.58 to 0.95 ln p: the true per-fold L is EXACTLY the longest alternation-legal window of the old gap word (zero slack at all nine computable folds), but the maxsum route to proving it FLOORS at ≈ 0.183x (`maxsum_k ≥ G₂`, `a3-05-bound-L.md` §7), so the only object left is the window statistic's own growth — and even a perfect L cannot save the tile chain below p ≈ 800 (the true-L chain busts at fold 31, `gate-multiplies.md` §8). The live ground moved to the ZONE frame, where multi-kills extinguish (a measured law, five windows and four decades of Y, out-of-sample validated one decade blind at W = 2·10¹¹, where the last multi-kill fold measured 631 against the sealed band [571, 877], `history/staging/attack-foldL-06-scaling.md`, `history/staging/foldL-window5.md`; and the law's amplitude is no longer fitted — the Arratia–Goldstein–Gordon first moment DERIVES A = 2.2091e−2 against the record's fitted 2.4312e−2, with c at 0.37σ, beating the geometric null on both, `history/staging/import-stein.md`) and the transport chain's killer factor collapses (`verify-tailcount-transport.md`); what remains there was, briefly, one counting hypothesis at L = 1, stated on residues — and it is REFUTED AS A HYPOTHESIS: the residue condition is the kill condition with zero slack, the threshold above which the sum vanishes is provably ≤ G₂ (measured EQUAL at four of four windows), and the chain sum is zero iff G₂ < θ — the statement is the Zone Postulate in residue notation (`history/staging/attack-l1-residue.md`, Lemmas A/B/D). The relocation to the zone frame is real and its instruments stand; the destination is not a smaller problem. The open object remains H″ — the window statistic's growth. 2. **The copy theorem for the whole maxsum family (§5a step 2, VERIFIED 40 of 40 over five folds and m ≤ 8, not proven).** It is the one object in the branch whose index cost is zero, m stays m under folding, so it is immune both to the no-fixed-point argument and to the Overshoot Budget, for as long as the identity holds beyond the range it was checked in. What it needs is a way to bound a residue-deleted maxsum without going through a kill count. Nobody has tried it directly, meaning no branch here has; that is an internal absence, not a literature one (`SEARCH-CONVENTIONS.md`). 3. **DONE: G₂(41#) = 546 and G₂(43#) = 618 are exact terms** (the run cost 2m31s against the 37-hour pricing above — the cost-estimates-err-cheap rule's best exhibit), both inside their pre-registered Poisson windows. §4's fits are on the first twelve levels and predate them; a 14-term refit is pending (`research/exact-g2-ladder.js` holds the ladder). 4. The mechanism question is ANSWERED at the record level: the last FOUR record-setting folds (23, 29, 31, 37) assembled fresh maxima rather than extending the old one — read directly off the genealogy tree, which gives G₂(T₂₉) = 258 = 60 + 138 + 60 born @23/@19/@23 with the old record 204 absent from the ancestry (`history/staging/attack-foldL-04-amortized.md`). And the per-fold L question underneath it is settled exactly: L equals the longest alternation-legal window of the old gap word, zero slack at all NINE computable folds — eight twice independently, the ninth (L(T₃₁, 37) = 4, extremal word 150+72+150) by both routes in `history/staging/frontier37.md` — see also `attack-foldL-01-census.md`, `attack-foldL-02-bridge.md`. ## 8. Reproduction and custody The fold ladders and the max-gap forensics of §§2, 4 and 5 are regenerated from nothing by `research/uframe-repro-01-fold-ladder.js` and `research/uframe-repro-02-maxgap-forensics.js` (both embedded), which checked 150 printed figures and corrected three (§2's tile width 1,296,000; §4's drop-x37 fit 1.716; §5's fold-17 share 4.85%) — the record is `history/staging/consolidation-wave.md`. The tables are quotable. What follows is the part that had custody all along. **L(T₂₃, 29) = 2, and this is the value the whole fold-29 row rests on.** Confirmed independently five times: by A4, by A5, by A8 (`research/a3-08-adjacent-pairs.js`, itself three routes — brute force over the folded 215M-slot tile, the kill graph, and a direct window score), by A9's operator spectrum, and by A10 (`research/a3-10-lower-tightness.js`, three more routes, among them a direct triple test finding 243,816 deletable adjacent pairs and exactly zero deletable triples). That direct window score sits exactly 6 below §11's exact PAIRS(T₂₃, 29) = 243,822, and the 6 is the weight-2 term: d = 174 = 6p occurs 6 times and is realised by two copies rather than one. So the upper bound at fold 29 is maxsum₃ = 300 rather than maxsum₄ = 348, an overshoot of 16% over the true 258. **`research/Lgrowth.js` was corrected on 2026-08-16 and may be quoted.** Its run finder, shared with `research/killrun.js`, used to compare the incoming residue against the older member of the live pair rather than against the immediately preceding slot, so it accepted residue words like (r, r+2, r−2) whose last two entries differ by 4, and it reported 3 at fold 29. Both are fixed, and the corrected diagonal L = 2, 1, 2, 2, 2, 3, 2, 4 at folds 7 to 31 agrees with `research/a3-08-adjacent-pairs.js`, which found the same bug independently. The same routine generates its whole off-diagonal sweep. **The streaming custody, which is what the deep folds rest on.** T₂₉ was streamed out of T₂₃ in 34 s and T₃₁ out of T₂₉ in 126 s, and neither tile was ever stored. The streaming leg recovers G₂(31#) = 348 from T₂₉ in 10 s and G₂(37#) = 528 from T₃₁'s 6,226,553,025 slots in 519 s, both matching the published sequence, and the streamed 258 and 348 are the recorded 10th and 11th entries of a G₂ ladder built by an unrelated lattice walk. The generator reproduces T₂₃ bit for bit from T₁₉, and the five original diagonal rows of §5a step 7 reproduce exactly. **And G₂(37#) = 528 has a second, INDEPENDENT EXHAUSTIVE certificate** (2026-08-19, forty-third pass): `maxsum₁ = 528` over all 217,929,355,875 gaps of T₃₇, computed by the scan-statistic engine, which shares no code with either exact-ladder producer. That is a maximality certificate rather than an agreement — it is the maximum over the whole tile, not a match against a published term — and it is the strongest custody this corpus holds on its own central object (`history/staging/scanstat-t37.md`). ## 9. Honest status Asked directly whether this line is in a good spot. The calibrated answer is **better in understanding, not meaningfully closer to a proof.** **Solid.** Folding by p is exactly p residue-class deletions on one fixed pattern (PROVEN; VERIFIED at six folds). The two-sided bound maxsum₂ ≤ G₂(new) ≤ maxsum_{L+1} holds in the old gap word alone (VERIFIED at 329 cells over folds 7 to 37), and §5a step 2's copy theorem lifts it to the whole maxsum family exactly. The per-fold budget is sharp rather than approximate: §3's ln c(p) ≤ 2 ln p/p, with the whole route's lifetime slack measured at about one nat. The difficulty is localized to one named, cheaply measurable quantity. **Solid, and a hole that has since closed.** The growth law of maxsum_m is no longer missing. `research/localized-04-maxsum.md` measures **maxsum_m = m·m̄ + σ·√(2m·ln D)**, unfitted, holding to 12 percent over m ∈ [2 ln D, 1024] and over three decades of window size, with σ/m̄ ≈ 0.89 to 0.95. *(CORRECTED 2026-08-18 from "6 percent", which was the home's figure and which the home's own printed intervals refute: R/EV runs to 1.119 at x = 3499. This site is not independent corroboration, it cites the home in the same sentence.)* So R(m) = maxsum_m/(m·m̄) falls to 1 from above and never plateaus, and sup_m R(m) = R(1) ≈ ln D. That is measured in the head, on a segmented sieve of [0, Y); on the tile the related ratio ρ = (maxsum_m − G₂)/((m−1)·m̄) is larger and does not fall, reading 1.58, 1.78, 1.83, 1.41, 1.84, 2.39 at T₁₁ to T₂₉ as the maximum over m ≤ 8, which is not monotone in the level (see §5a step 4 for the central reading). The extreme-value model that predicts ρ → 1 assumes an exponential gap tail while the tile is measurably under-dispersed. Do not transfer the constant between the two settings, and do not extrapolate ρ in either direction. *(Settled 2026-08-20: these are TWO OBJECTS sharing a formula, not a contradiction with the seven-level refutation — that refutation is measured on the cyclic TILE family (`scanstat2.md`, `scanstat-t37.md`); the head/anchored window setting has its own constants, and this paragraph's no-transfer rule is exactly why the two sets of statements can both stand.)* **Not solid.** Nothing is proven about L. We have nine integers, 2, 1, 2, 2, 2, 3, 2, 4, 4, and a heuristic. The upper half of the bound is inert without an L bound, so the chain has a hole at exactly one point and everything rests there. Steps 1 and 2 of §5a are probably folklore: "folding a wheel by p deletes two residue classes" is what folding means, and a specialist would call it obvious. The clarity is useful to us; it is not a discovery. **The hole, priced.** `research/gate-multiplies.md` §8 runs step 3 with the true L at every fold, the most favourable reading the proven machinery allows, and chains it from G₂(T₁₁) = 42. The running slack ln(x²/bound) falls 1.058 → 0.186 over five folds, and **the chain dies at fold 31**: L(T₂₉, 31) = 4 forces maxsum₅(T₂₉) = 510, giving 1,380.5 against a target of 31² = 961, slack −0.363. VERIFIED by streaming all 214,708,725 slots of T₂₉. So the branch goes through iff **L ≤ 0.19 to 0.31·p/ln p** on average over the ladder, against §10 Theorem B's proven **L ≤ 0.18 p**. The gap is a factor **0.58 to 0.95 ln p** (0.18/0.31 and 0.18/0.19), and it is the only gap. The two ends are the two readings of ρ, 1.5 and 2.4, and since ρ is not monotone in the level neither end can be presented as the one the trend favours. **The structural warning, which should carry the most weight.** The chain, if completed, proves G₂ < x², which is the Gap Reformulation, which proves TPC. So by construction it must contain something at least TPC-hard. The only question is where, and the answer is L. Bounding L means showing that long runs of LARGE gaps do not occur in the grain at the scale ~2p, inside a tile of width p#. That is a statement about the tail of the gap distribution at a specific scale, which is an INTERVAL statement, not a residue one. Our own triage rule says that is the hard kind, and it is the same place the parity obstruction lives. The work here relocated the difficulty precisely; it did not dissolve it. Ziller and Morack have had the adversarial version since 2017 and have not closed it either. **Compute guidance.** Extending the diagonal f is minutes and discriminates the two branches, so it is worth doing on curiosity alone. It will show which branch the data favours, not which is true, since the branches separate only asymptotically. Do not budget serious compute against this line until L has a proof strategy that is not "measure it further." ## 10. L and κ(m): the proven bounds, and why L is not the target L is the longest adjacent-kill run, κ(m) its generalisation to the whole family, κ(1) = L. Both are governed by the criterion that two adjacent slots at distance g die together only if g ≡ 0, ±2 (mod p), whose closed form is three progressions of modulus 6p with weights 1, 1, 2, putting the smallest qualifying gap at exactly 2p ∓ 2 rather than "about 2p". **PROVEN:** the Alternation Lemma, which makes a run a two-state walk, raises the floor from 2p on a single gap to 3p on a window and forces the fold-29 dip; Theorem A, the Run Cost; **Theorem B, the first unconditional bound on L, L ≤ 0.18 p**; and Theorem C, which carries the argument to κ(m). **The wall:** Theorem B can never prove L below G₂/(3p), which is LINEAR, the failing branch, and Theorem C inherits the ceiling, so redirecting L to κ does not move it. **PROVEN, and it closes a route:** exactness of the lower bound can never hold from some level on, one of its two mechanisms being the permanent arithmetic event p | G₂, G₂ ± 2; and no constant can replace the effective run length either. The target is κ(m), and inside it the rarer ±2 class at 4p. **PRIOR ART on the Alternation Lemma's LANGUAGE, not on the lemma.** The two-state walk is the B = 1 charge constraint / alternate-mark-inversion of Marcus–Roth–Siegel §2.3 p. 47, its capacity in their §3.2 p. 75, so the constraint graph's strict soficity and its capacity ln 2 are reproductions and **nothing there may be presented as new structure**. Ours is the arithmetic: that the fold imposes this constraint, the 3/p → 2/p rate correction, the weight-(1,1,2) multiplicity, and the wall address (`IMPORT-MAP.md` row 2). Derivation, tables, the wall located precisely and the honest limits: [research/kappa-not-L.md](kappa-not-L.md). ## 11. The exact machinery: the histogram transfer operator and the pair count Two exact instruments, neither of which needs a tile in memory, are what make the f and L questions measurable at any level in reach. **PROVEN, VERIFIED exactly at six folds:** the new gap histogram is generated from the old gap word and q **and from nothing else**, which is the misalignment principle in its sharpest form. Its honest limit is that the operator is closed on the gap word rather than on the histogram, so it is an exact simulator and not a source of bounds. **PROVEN, VERIFIED at every fold to 37:** the exact pair count PAIRS(T, p), whose weight 2 on the p-divisible sizes is §10's multiplicity, and the alternation check at fold 37, the deepest the lemma has. The head engine gives the tail at any level and 31 exact diagonal points, carrying a FIT that must not be used past the range it was fitted on. The operator is PRIOR ART (Holt and Rudd 2014, **§5**, pp. 17-19, where M_J is built as a bidiagonal matrix; see this file's own **§6a**) and **nothing there may be presented as new structure**. *(Corrected 2026-08-18, PDF read: this said "Holt and Rudd 2014, §6a". Their paper has no §6a — its §6 is "Polignac's conjecture and Hardy & Littlewood's Conjecture B" and it carries §6.1 ("General recursion on cycles of gaps", p.23), §6.2 (p.26) and §6.3 (p.29), so the note that said it "is not subdivided" was wrong at the PDF (corrected 2026-09-13, job #598) — and the "§6a" was this document's own section all along, as `operator-and-pair-count.md`:13 makes explicit. As written it read as a citation into their paper and sent a reader nowhere.)* Statements, tables, the fit and its caution: [research/operator-and-pair-count.md](operator-and-pair-count.md). **The Tail-Count Transport [PROVEN, 2026-08-19].** Read as an inequality through its own multiplicity bounds (`ν_q(i,0) ≤ q−2`, `ν_q(i,L) ≤ 2`), the operator transports the whole tail-count profile: folding `T_x` by `q`, the count of new windows with sum ≥ θ satisfies `N_new(θ) ≤ (q−2)·N(θ) + 2·Σ_{L≥1} Q_L(θ)`, where `Q_L(θ)` counts old windows of L+1 gaps with sum ≥ θ whose L−1 interior gaps admit a legal walk on a 2-set mod q — so `G₂(new)` is at most the largest such window sum. Exact at folds 11 through 37 (eight folds; at 37 even the loose form is exact, `history/staging/frontier37.md`), and the exactness is structural: keeping the endpoint-live conditions as well turns the certificate into `G₂(new)` itself. The inequality's own margin narrows monotonically (max N_new/RHS 0.888 → 0.948 across folds 17..37, +0.012 per step) — it is proven, so the rise means the evaluator approaches tightness; fold 41 is the cheap check. A sharp per-level evaluator, not a source of a chained bound: on the tile the (q−2) factor forbids chaining, and the same alignment sum that forbids it is what makes the evaluator exact (`history/staging/attack-foldL-03-transport.md`, independently verified in `verify-tailcount-transport.md`). **PRIOR ART on the transport, and on the recursion it reads [corrected 2026-09-13, job #598].** The `(q−2)` transport above is Holt and Rudd's, with and without a span hypothesis, and it was owed here: **§3 Corollary 3.2, p.12** is the span-restricted form (`N_s(p_{k+1}#) = (p_{k+1} − j − 1)·N_s(p_k#) + 1·n_{s,j+1}(p_k#)` for `g < 2p_{k+1}`, coefficient `p − 2` at `j = 1`), and **§6.1 Corollary 6.3 with Figure 4, pp. 25–26** is the same transport **with no span hypothesis** at the level of the length-summed driving-term counts — *"of the `q` copies of `s`, two are eliminated as driving terms and `q − 2` remain as driving terms of various lengths"*, hence `Σ_j n_{g,j}(qN) = (q−2)·Σ_j n_{g,j}(N)` for `q ∤ g`. The recursion itself is **§2.1 and Theorem 2.3, p.8** on the primorial ladder and **Lemma 6.1, p.23 with Theorem 6.2, p.25** for general `G(N) → G(qN)`; the multiplicity counting `ν_q(i,0) ≤ q−2`, `ν_q(i,L) ≤ 2` is **§3 Lemma 3.1, p.11** and Figure 4's caption on p.25; and this section's own ledger `Σ_{L≥0} ν_q(i,L) = q−2` is that figure's sentence "two are eliminated … and `q − 2` remain", with the case split `ν_q(i,0) = q−2 ⟺ q | g_i` being the paper's. **Ours is the θ-tail restriction, the walk-qualified correction `Q_L`, the inequality in place of the length resolution, and the certificate form; nothing else in this paragraph may be presented as new structure.** Read at the PDF of record on 2026-09-13 (`sha256 672c4377…`, 34 pp., `pdftotext -layout`), with the page claims asserted by `verify-section61.mjs` (17 checks, exit 0); the measured correspondence on the ladder — 64 instances of Corollary 6.3 with `q ∤ g` exact, and the 8 instances with `q | g` where the factor is `q − 1` rather than `q − 2` — is in the note `theta-tail-transport` submitted with the same audit. ## 12. f, the qualifying-gap fraction: 42 exact points and the staircase f is the fraction of gaps that qualify, ≡ 0, ±2 (mod p), and it governs L through roughly L ≈ ln D/ln(1/f) (§5a step 6). **MEASURED on 42 exact points, x = 11 to 199, with no tile ever built:** f falls by a factor of 170 over that range (the census behind that figure is DEFECTIVE from x = 37 — shift-alias, `history/staging/fdecay-deep.md` — and the corrected factor is LARGER, so the reading strengthens; §5a step 7's seven-row table is x ≤ 31 and unaffected), which is the polylog branch, the one the route needs. It is the branch, not the route. The scatter is arithmetic, a singular-series comb on the minimal qualifying gap. **PROVEN:** both members of a twin pair share d_min = 2(p+1), so the qualifying threshold cannot advance across a twin fold and **the diagonal is a staircase whose treads are the twin pairs**, which is the whole of the apparent non-monotonicity; and Lemma B, the structural negative, which puts the cheap half of the grain census law out of reach exactly where f lives and caps the CENSUS at x ≈ 200; the deep-window instrument measures f to x = 829, and L_win = 2 at every level 97..829, so a window can measure f but never L. The corrected growth law is a fit over 51 window levels, not a theorem (`history/staging/fdecay-deep.md`). The census, both regressions, the comb and the caveats kept visible: [research/f-decays.md](f-decays.md). --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [history/CHANGELOG.md](history/CHANGELOG.md), indexed by document.*