# The localized gap: the merge structure, and why the chain does not close *(Calibration marked throughout: PROVEN, VERIFIED by exact computation, MEASURED, PRIOR ART. Scripts: `research/localized-01-ladder.js`, `-02-fixed-window.js`, `-03-merge-lemma.js`, `-04-maxsum.js`. All run in under a minute. Change history in `research/history/CHANGELOG.md`.)* Read `research/PRIOR-ART.md` §"Holt and Rudd" first. The fusion mechanism below is theirs, published 2014. Ours is the two-class form and the application to a maximum gap, and that application is on Holt's own list of open problems. ## 1. The verdict The chain does not close, and it fails on a one-line averaging argument rather than on any sieve question. What survives is the growth law of §6, which answers a question U-FRAME §9 had left open since 2026-08-16. ## 2. Two elementary facts **Fact A (PROVEN).** For x ≥ 3, no two twin slots of T_x are 2 apart. If s and s+2 were both twin slots then s, s+2 and s+4 would all be x-rough, and those three cover every residue class mod 3. **Fact B (PROVEN).** Folding T_x by p deletes exactly the slots in classes {0, −2} mod p, which are 2 apart. In an interval shorter than p − 2, two kills would have to be either p apart, impossible inside the interval, or exactly 2 apart, which Fact A forbids. **An interval of length below p − 2 contains at most one kill.** ## 3. The Localized Merge Lemma Write M(T_x, Y) for the largest twin-slot gap of T_x among gaps starting below Y, and maxsum_m for the largest sum of m consecutive such gaps. > **Lemma (PROVEN).** If M(T_x, Y) ≤ (p − 2)/4 then M(T_p, Y) ≤ maxsum₂(T_x, Y). *Proof.* Write M = M(T_x, Y). A new gap G fuses j+1 consecutive old gaps and so contains j kills, giving G ≤ (j+1)M. Suppose G ≥ p − 2. Partition G into ⌈G/(p−2)⌉ subintervals each shorter than p − 2; by Fact B each holds at most one kill, so j ≤ ⌈G/(p−2)⌉ ≤ G/(p−2) + 1. Then G ≤ (G/(p−2) + 2)M, so G(1 − M/(p−2)) ≤ 2M, and with M ≤ (p−2)/4 this gives G ≤ (8/3)M ≤ (2/3)(p−2), contradicting G ≥ p − 2. So every new gap is shorter than p − 2, holds at most one kill, and fuses at most two old gaps. ∎ The boundary question, whether a fused gap starting below Y can end above it, is settled and empty: see §8. **Why it must be localized.** On the full tile M = G₂, and the computed ladder puts G₂ between x′²/4.5 and x′²/3.2 at every level, against a hypothesis that asks for G₂ ≤ (x′−2)/4. The hypothesis fails at every level and by a growing margin. In the head M ≈ c·k·ln³x against p ≈ x, so it holds from some point on. **Where it actually turns on.** At Y = 10⁹, that is k = 2.17, the hypothesis first holds at **x = 13933** (VERIFIED), against 13,630 predicted by solving x* = 9.6·ln²x*·ln(Y/m̄). At k = 3 that is x* ≈ 2.4·10⁴ with Y ≈ 1.4·10¹³, **out of computational reach by four decades.** The lemma is true and the regime where it applies is not one we can survey. ## 4. Why the chain does not close The telescope is M(x_n, Y) ≤ maxsum_{n+1}(T_{x₀}, Y), one index per fold, against A5 Theorem B's proven L ≈ 0.18x on the full tile. The per-fold index cost of 1 is real, and it is the right way to read Chris's localization payoff. The chain fails on the number of folds it can traverse. > **Deficit Lemma (PROVEN).** For 1 ≤ m ≤ D/2, maxsum_m ≥ m·m̄·(1 − O(m/D)), by > averaging and nothing else. > VERIFIED: min R = 1.187, 1.280, 1.296 across the scan, never below 1. > **Traverse Bound (PROVEN).** Given the Deficit Lemma, the telescope's own > hypothesis caps a block at j ≤ x/(9.6 ln²x) folds **for any constant C, > including C = 1**, against the π(x) ≈ x/ln x folds needed. Short by 9.6·ln x, > which is 93 at x = 16001 and grows. **MEASURED chain survival: 0 folds for every x ≤ 12143 at Y = 10⁹, and exactly 1 fold at x = 13933 and x = 16001, against the 1863 that π(16001) requires.** **The obvious repair is closed (PROVEN).** Weakening the gate to M ≤ α·p fails at every α, because the gate feeds back. If the chain proves M ≤ B(x), Fact B permits B/x kills per gap, the telescope index is B/ln x, and the bound returned is about 2.4·R·B·ln x. **The map B ↦ 2.4·R·B·ln x is expanding for every x ≥ 2**, so there is no fixed point. **The obstruction is not the size of the gate. It is that the gate multiplies wherever the index accumulates against a fixed base.** That last qualifier is the whole reach of the argument. `research/gate-multiplies.md` §2 states it as four hypotheses and tests every recursion in the repo against them: it closes this chain and A4's tile analogue, and it does not touch a chain that re-bases at every fold, because there the running bound cancels out of the cost instead of compounding. What closes the tile analogue is a different and simpler statement, the Overshoot Budget (`gate-multiplies.md` §5): any chain of upper bounds ending in G₂(x#) < x² has a total multiplicative overshoot capped at x²/G₂(x#), measured flat at 0.88 to 1.19 nats over the whole computed ladder. ## 5. MEASURED: the localized gap `research/localized-01-ladder.js`, k = 3, window Y = x³, 61 folds to x = 307, in 0.64 seconds. | quantity | reading | |---|---| | M(x, x³) / (k·ln³x) | flat in 1.2 to 1.6 across the ladder | | M(x, x³) / x² | falls monotonically, 0.4155 at x = 19 to **0.00923** at x = 307 | | folds moving M at all | 20 of 61 | | mean increment per fold | 14.16, max 144 | `research/localized-02-fixed-window.js` holds Y fixed, which isolates fold damage from window growth. Folds past x > √Y are frozen by the Impact Lemma. | Y | folds | folds moving M | mean increment | max | final M | |---|---|---|---|---|---| | 10⁴ | 44 | 10 | 4.64 | 42 | 210 | | 10⁵ | 113 | 18 | 5.52 | 132 | 630 | | 10⁶ | 301 | 25 | 4.80 | 234 | 1,452 | | 10⁷ | 781 | 28 | 2.20 | 210 | 1,722 | | 3·10⁷ | 781 | 41 | 2.80 | 246 | 2,190 | The increment is additive and small, and most folds do nothing. **The M/ln³x reading is confirmed independently and it is the one that governs the zone.** `research/maxgap-law.md` §8 measures M(x, x²) directly by a third engine out to x = 9973 and gets M/ln³x between 2.9 and 4.2 over a 47-fold range in x, which is this row read at k = 2. So the localized max gap is polylog in x, and the localized margin against the zone is **x²/(4.4 ln³x)** at the worst case across both engines (4.2 on this engine alone; 3.5 was the midpoint; `audit-cross-document-constants.md` A6), growing. Anything that projects the localized gap as linear in x is wrong by a factor x/ln²x. **Carry the coordinates with the constant.** The law behind the row is M(x, Y) ≈ c·m̄·ln(Y/m̄) and c is a surface c(x, lnD), not a constant (`maxgap-law.md` §4). Every c here is measured off the diagonal, at lnD ≈ 11 to 16 and x ≈ 10²·⁵ to 10⁴, where it runs 0.74 to 1.17. The same object read on the whole tile, where lnD = θ(x) − ln m̄, gives 0.46. The two are the same surface, not a disagreement, and neither number may be carried to the other's window. The merge lemma's conclusion holds well before its sufficient condition does. At k = 2, across all 820 folds to x = 6323, M(new) ≤ maxsum₂(old) never failed, and the kill count per fused gap measured 1 or 2 and never more. ## 6. The growth law of maxsum_m `research/localized-04-maxsum.js`, custody-checked against `localized-01` digit for digit at x = 307 by a different engine, ladder to x = 16001 in 18 seconds. > **maxsum_m = m·m̄ + σ·√(2m·ln D)**, so R(m) = 1 + (σ/m̄)·√(2 ln D / m), > with σ/m̄ = 0.892, 0.920, 0.949 at x = 997, 3499, 16001. Not fitted. Over m ≥ 2·ln D it holds to R/R_EV ∈ [0.981, 1.110], [1.038, 1.119], [0.990, 1.064]. R(m) → 1 and never plateaus. Below m ≈ 2 ln D the maximum is a single tail event, the same patch for m = 1 through 10; above it the argmax migrates and the behaviour is bulk √m concentration. So sup R = R(1), and **R(1) ≈ ln D is a property of the window rather than of the level**: R(1)/ln D ∈ [0.736, 1.170] across Y = 10⁷, 10⁸, 10⁹ and x from 89 to 16001. **This answers U-FRAME §9,** which records the hole as closed and points back here. The repo's named single hole was a proven upper bound on maxsum_m as a function of x that is not G₂ itself, one rung with no ladder. The ladder exists and is the law above. It is not where the difficulty lives. ## 7. Cross-checks against A5 and A9 **A5 agrees independently.** Theorem C transplanted gives κ(m) = 1 for all m ≤ 6 at every x ≥ 997, and Theorem B gives L ≤ 1 from x = 499 at Y = 10⁹. Localization repairs Theorem B by moving its ceiling rather than by a better argument: G₂/(3p) ≈ 0.18x on the tile becomes M(x,Y)/(3p) ≈ 0.8·ln²x·ln D/x in the head, dropping below 1 from x ≈ 470. **A9 departs by about 20% and should not be extrapolated.** Its tile fit rate 1.2992/m̄ predicts R(1) between 12.0 and 12.6 across the scan; measured is 12.6 to 18.7, mean ratio 1.23. The head's own rate is about 1.06/m̄. The far tail is better: A9 predicts 1.1e−3 gaps of size ≥ 2p′ at x = 1009, and there is one. ## 8. The boundary question is closed Three rules computed in one pass: gaps with left endpoint below Y, buffered, and fully contained. **(a)/(b) = 1.0000 and (a)/(c) = 1.0000 at every x, every m ≤ 1024, at Y = 10⁷, 10⁸ and 10⁹.** Optimal windows overshoot Y by zero; edge windows reach at most 76 to 83 percent of maxsum. ## 9. Attribution The fusion mechanism is Holt and Rudd 2014, Lemma 3.1 of arXiv:1408.6002, restated in arXiv:2603.25915 §1: *"the minimum span between fusions is 2p_{k+1}. So provided that |s| < 2p_{k+1}, the possible fusions in s all occur in separate images of s."* Their minimum span is 2p because the minimum gap between generators is 2. Ours is p − 2 because the two kill classes sit 2 apart, which is why the two-class version needs Fact A. Not found in their work or elsewhere: the two-class form, and the use of the bound as an upper chain on a maximum gap inside a localized window. Holt restricts to spans below 2p throughout and records that Jacobsthal values exceed 2p_k, which is §3's "why it must be localized" from the other side. ## 10. The compute observation, which outlives the lemma Every localized object needs a segmented sieve of [0, x^k) and never the tile of width x#. The ladder is capped by x^k, not by x#. The runs above cover 820 folds to x = 6323 in 27 seconds, against the nine-point ladders the rest of the repo works with. Any question here phrased about a bounded window rather than the whole tile can be asked three decades further out than it currently is. ## 11. Reproduction ``` node research/localized-01-ladder.js 3e7 3 # 0.6 s node research/localized-02-fixed-window.js # 44 s node research/localized-03-merge-lemma.js 4e7 2 # 27 s node research/localized-04-maxsum.js # 18 s ``` Custody: the k = 2 runs reproduce known maximal twin-prime gaps, since below x′² the slots are twin primes. M(x, 10⁷) = 1,722 and M(x, 3·10⁷) = 2,190 at the top of the ladder, both consistent with the tabulated record twin gaps. --- *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.*