# The max-gap law is a diagonal law, and that is why two agents got two constants *(Engine `research/maxgap-law.js`, 21 s plain and 147 s with `--big`. Calibration marked throughout: PROVEN, VERIFIED by exact computation, MEASURED, INFERRED, REFUTED. Read `research/localized-04-maxsum.md` §4 and `research/two-class-lower-bounds.md` §6 first; this note reconciles them.)* ## 0. What is hard here, first The question was whether `maxgap ~ c·mbar·lnD` survives far enough to contradict Maier and Pomerance. It does not, and the reason is not that the law breaks late. **The law was never an extreme-value law.** It is an accurate description of a one-dimensional curve through a two-dimensional surface, and both agents measured it on different curves without recording which curve they were on. That is harder than "the law drifts", because a drift can be fitted and extrapolated. A surface cannot be extrapolated from one curve to another, and the whole apparent conflict between `c = 0.9` and `c = 0.46` is exactly such an extrapolation, performed silently, twice. The second hard thing: on the curve that matters, the diagonal, `c` really is flat, over a lever 46 times long, and the estimator that says so is demonstrably able to see Maier-Pomerance growth when it is present. So the tension is real, sharper than it looks, and resolved only by an argument that finite data cannot reach. ## 1. The answer, in five lines 1. **The two constants are the same formula on two different curves.** `c` is a surface `c(x, lnD)`. Along a column (fix `x`, grow `lnD`) it falls; along a row (fix `lnD`, grow `x`) it rises. A sits at `lnD/theta(x) ~ 1e-2` with `x ~ 1e3`. B sits at `lnD/theta(x) = 1` with `x ~ 20`. MEASURED. 2. **Neither agent is meaningfully tighter than the other.** A quoted a range, B quoted a coefficient of variation. On matched statistics both spreads are near +-20%, A's wider by about a quarter, not by the factor of three the raw comparison suggests. 3. **B's flatness is a cancellation, not a law.** The off-diagonal exponents, measured on the same objects, predict `c` falls 32% along the diagonal leg `p = 101 -> 271`. The exact terms deliver a 2.5% rise. REFUTED as a surface law, VERIFIED as a diagonal law. 4. **There is a drift and it is a tenth of what Maier-Pomerance needs.** `c1 ~ (log p)^{0.12 +- 0.02}` on the top 47 exact terms. MP needs exponent 1. A synthetic ladder built to have exponent 1 returns `0.981 +- 0.006` from the identical estimator, so the absence is a measurement and not a resolution limit. 5. **The omitted term is a maximum over an index the law does not have.** In Ford-Green-Konyagin-Tao's own accounting the law is `y ~ e^gamma·x·T1·r` with `T1` the gain from ALIGNING the residue classes and `r` the number of survivors one large prime can kill. The law is `T1·r = log x`. Rankin proves `T1 = log x·log_3 x/(log_2 x)^2`, slightly less than one log. MP conjecture `r = (log x)^{1+o(1)}`, the missing log. Evaluated, `T1·r/log x` reproduces the measured `c` at the top of the ladder and doubles it only past `x ~ 1e17`. --- ## 2. Custody Third engine, segmented sieve of the exact period, no tile built in memory. ``` two-class-lower-bounds.md S6: c1 one class, p in [11,229] n=46 mean=0.3718 sd=0.0269 cv=7.2% range [0.3359, 0.4873] = +-22.5% c2 free 2class, p in [11,73] n=17 mean=0.8511 sd=0.0623 cv=7.3% range [0.7784, 1.0157] = +-15.2% c2' diff-2 G2, p in [11,37] n= 8 mean=0.4814 sd=0.0490 cv=10.2% range [0.4463, 0.5939] = +-16.5% published: c1 0.3718 sd 0.0269 cv 7.2% | c2 0.8511 0.0623 7.3% | c2' 0.4814 0.0490 10.2% MATCH: YES exact tiles, both classes, against A048670 and the repo G2 ladder: x mode period slots mbar max gap published 7 1 210 48 4.375 10 10 MATCH 0.0s 11 1 2310 480 4.813 14 14 MATCH 0.0s 13 2 30030 1485 20.222 66 66 MATCH 0.0s 17 2 510510 22275 22.919 108 108 MATCH 0.0s 19 2 9699690 378675 25.615 150 150 MATCH 0.3s 23 1 223092870 36495360 6.113 40 40 MATCH 3.6s 23 2 223092870 7952175 28.054 204 204 MATCH 4.8s 29 1 6469693230 1021870080 6.331 46 46 MATCH 98.9s 29 2 6469693230 214708725 30.132 258 258 MATCH 133.9s localized-04-maxsum.md S4, Y = 1e7 row, third engine: published Y=1e7: R(1) 9.00-11.92, lnD 11.1-11.9, R(1)/lnD 0.756-1.053 ours: R(1) 9.46-11.66, lnD 11.0-12.2, R(1)/lnD 0.7803-1.0324 ``` VERIFIED: fourteen exact max gaps against A048670 and the repo `G2` ladder, both agents' summary statistics, and A's `Y = 1e7` scan by an independent engine. A used a different `x` grid inside the same window, so the ranges agree rather than the individual entries. --- ## 3. READING 1 (MEASURED). The apparent tightness dispute is a units error Two numbers are in circulation for the same spread: agent A's `c` spanning [0.736, 1.170], a spread of plus or minus 23 percent, and agent B's `c` flat to 7 percent. Those are two different statistics. A reported a **range** over 72 measurements. B reported a **coefficient of variation** over 46. Put both on the same scale and the difference all but disappears. Two matched conventions, both given because the choice moves the number by a third: | | raw | half-range / midpoint | half of (max/min − 1) | |---|---|---|---| | A, `R(1)/lnD`, 72 points | `[0.736, 1.170]` | **+-22.8%** | +-29.5% | | B, `c1`, 46 points | `[0.3359, 0.4873]`, cv 7.2% | **+-18.4%** | +-22.5% | **The two spreads are the same size**, A's wider than B's by about a quarter on either convention, not by the factor of three that "23 percent against 7 percent" implies. There was never a tightness disagreement to reconcile, and neither agent's number should be quoted without saying which statistic it is. The real disagreement is in the level, `0.9` against `0.46` on the same twin-slot object, and §4 measures where it comes from. **The one-class lever is long.** On the 64 exact terms `lnD = theta(p) - ln m1` runs **6.17 at p = 11 to 286.16 at p = 311**, a factor 46, and `c1` moves by 22% over the whole of it. That is not weak evidence of flatness. It is the strongest single fact in this note and it is what makes §5 uncomfortable. --- ## 4. READING 2 (MEASURED). c is a surface, and the two agents sat on different curves Neither agent recorded more than one coordinate. Three matter: the level `x`, the log gap count `lnD`, and the position `lnD/theta(x)` of the window inside the period. ``` source object x lnD u lnD/theta(x) c A localized Y=1e7 twin 89 12.18 3.59 0.1539 0.7803 A localized Y=1e7 twin 251 11.81 2.92 0.0509 0.8062 A localized Y=1e7 twin 997 11.29 2.33 0.0118 1.0324 A localized Y=1e7 twin 2153 11.03 2.10 0.0052 0.8581 B exact G2 tile twin 11 4.91 3.23 0.6333 0.5004 B exact G2 tile twin 19 12.84 5.46 0.7984 0.4559 B exact G2 tile twin 29 19.18 6.71 0.8492 0.4463 B exact G2 tile twin 37 26.11 8.21 0.8810 0.5939 B exact A048670 units 11 6.17 3.23 0.7971 0.4712 B exact A048670 units 101 86.22 19.14 0.9759 0.3648 B exact A048670 units 271 252.12 45.42 0.9909 0.3740 ``` Both agents even sat at nearly the same `lnD`, near 12, and got 0.78 against 0.456. The uncontrolled variable is `x`, and the controlled experiment is to move one coordinate at a time. ### 4a. The controlled experiment nobody ran Cut the `D` gaps of one exact period into blocks of `B` consecutive gaps. Each block is a window of exactly A's kind, and `c(B) = (block max)/(mbar·ln B)`. Same object, same `x`, same primes, only `lnD` moves. The head block is reported separately, because the first gap of a primorial tile is distinguished and would otherwise contaminate the mean. `x = 29`, twin slots, the whole period of width 6,469,693,230: ``` x=29 twin D=214708725 mbar=30.132 whole-period max=258 lnD B blocks meanmax c(mean) head c(head) max c(max) 2.08 8 26838589 67.8 1.0827 30 0.4788 258 4.1176 3.47 32 6709646 94.3 0.9026 150 1.4364 258 2.4705 4.85 128 1677410 119.8 0.8192 150 1.0260 258 1.7647 6.24 512 419351 141.4 0.7520 150 0.7980 258 1.3725 7.62 2048 104837 159.0 0.6922 156 0.6790 258 1.1230 9.01 8192 26208 175.7 0.6472 180 0.6629 258 0.9502 10.40 32768 6551 189.8 0.6058 198 0.6320 258 0.8235 11.78 131072 1637 202.1 0.5692 198 0.5576 258 0.7266 13.17 524288 408 213.2 0.5373 210 0.5292 258 0.6501 14.56 2097152 101 223.1 0.5086 228 0.5198 258 0.5882 15.94 8388608 24 234.5 0.4882 228 0.4746 258 0.5371 17.33 33554432 5 246.0 0.4711 240 0.4596 258 0.4941 18.02 67108864 2 249.0 0.4585 258 0.4751 258 0.4751 19.18 214708725 1 258 0.4463 <-- whole period, B's measurement ``` **`c` falls monotonically from 1.083 to 0.446 as `lnD` runs 2.1 to 19.2 inside one fixed tile.** VERIFIED, not fitted: the last row is `G2(29#) = 258`, the exact repo value, so the column terminates on a known number. `c(head)` tracks `c(mean)` once `B` is past a few hundred, so **A's single-window measurement is an honest draw and head bias is not the explanation.** MEASURED. ### 4b. The same statistic in A's own regime Twin slots of `T_x` inside `[0, 1e8)`, identical block statistic, so the two tables are directly comparable row by row. ``` lnD x=97 x=199 x=401 x=797 x=1601 x=3203 x=6421 6.24 0.8315 0.8977 0.9233 0.9486 0.9720 0.9924 1.0106 8.32 0.7978 0.8547 0.8900 0.9045 0.9379 0.9555 0.9759 10.40 0.7540 0.8230 0.8674 0.8527 0.9366 0.9378 0.9812 11.09 0.7448 0.8034 0.8570 0.8159 0.9510 0.9437 0.9889 the same rows from the exact tiles, twin: lnD x=17 x=19 x=23 x=29 6.24 0.6605 0.7122 0.7403 0.7520 8.32 - 0.6449 0.6562 0.6687 10.40 - - 0.5957 0.6058 11.09 - - 0.5768 0.5866 ``` Read along a row: at `lnD = 11.09`, `c` runs **0.577 (x = 23), 0.587 (29), 0.745 (97), 0.803 (199), 0.857 (401), 0.951 (1601), 0.989 (6421)**. That is A's whole range, produced by moving `x` alone. **A's `c ~ 0.9` and B's `c ~ 0.46` for the same twin-slot object are the same surface read at `x ~ 1e3` and `x ~ 20`.** MEASURED. Local exponents, read off the tables: ``` b(x) = d ln c / d ln lnD at fixed x (twin object) x 13 17 19 23 29 97 401 1601 6421 b -0.527 -0.380 -0.356 -0.375 -0.396 -0.233 -0.169 -0.122 -0.121 a(lnD) = d ln c / d ln ln x at fixed lnD (twin object) lnD 6.24 6.93 7.62 8.32 9.01 9.70 10.40 11.09 a 0.347 0.377 0.412 0.401 0.430 0.478 0.491 0.528 ``` `b` is negative everywhere and shrinks toward zero as `x` grows. `a` is positive everywhere and sits near 0.4. So the extreme-value law, meaning `c` independent of `lnD`, is approached only as `x` grows, and **the exact full-period data lives exactly where it is furthest from holding.** ### 4c. READING 3 (REFUTED). The surface does not survive the diagonal The off-diagonal exponents are measured. Feed them along the diagonal leg `p = 101 -> 271`, where `lnD` moves by a factor 2.92 and `ln x` by 1.214: ``` Over p = 101 -> 271 the diagonal moves lnD by a factor 2.92 and ln x by a factor 1.214. Feeding those into the LOCAL exponents measured off the tile (b = -0.430, a ~ 0.4) predicts c changes by 0.630 * 1.081 = 0.681 The exact terms change it by 1.025. ``` **A 32% predicted fall against a 2.5% measured rise.** The surface fitted off the diagonal is REFUTED on the diagonal, and that refutation is the answer. **The mechanism, INFERRED and elementary.** Off the diagonal, growing `lnD` means more POSITIONS at a fixed sieving resource `pi(x)`, and `c` falls because the far tail is sub-exponential (§5). On the diagonal, growing `lnD` means more PRIMES: `lnD = theta(x) - ln mbar`, so the resource grows in lockstep with the number of positions and the fall is cancelled. > **The law `maxgap ~ c·mbar·lnD` is a diagonal law.** Its constancy is a > cancellation between two effects of opposite sign, not an extreme-value > statement. Quoting `c` without the coordinates `(x, lnD)` is the error, and > both source files do it. --- ## 5. READING 4 (MEASURED). What the constant actually is The extreme-value prediction for `D` iid `Exp(mbar)` gaps is `mbar·(lnD + gamma)`, that is `c -> 1` from above. Measure the real survival `S(t) = P(gap > t)` in the exact `x = 23` period: ``` x=23 units mbar=6.113 D=36495360 lam=t/mbar S(t) -lnS mean rate local rate 1 5.642e-1 0.572 0.5723 1.6136 2 1.124e-1 2.186 1.0929 1.9027 3 1.676e-2 4.089 1.3629 2.2463 4 1.773e-3 6.335 1.5837 5.2488 5 9.316e-6 11.584 2.3168 3.1899 6 3.836e-7 14.774 2.4623 - extreme-value solve D*S(t)=1 -> t* = 40 actual whole-period max 40 ratio 1.000 ``` The extreme-value solve is exact at both `x = 23` objects, ratio 1.000. And the mean tail rate climbs from 0.57 at one mean gap to 2.46 at the extreme. Since `c = lam/(-ln S)`, which is `1/(mean rate)`: > **`c` is the reciprocal of the gap tail's exponential rate at the extreme.** > `c1 = 0.372` says the one-class gap tail decays like `exp(-2.7·g/mbar)`, not > `exp(-g/mbar)`. MEASURED. That is the entire content of the constant, and it explains the sign of `b` without any further theory: a sub-exponential tail cannot supply a constant `c` as `lnD` grows, because the effective rate keeps rising. --- ## 6. READING 5 (MEASURED, with a control). The drift, and its size Ford's Montreal slides state both facts one line apart, verbatim (Kevin Ford, *Large gaps between primes*, Talk 1, CRM Montreal workshop **Probability in Number Theory**, 2018, [ford126.web.illinois.edu/montreal_talk1_primegaps.pdf](https://ford126.web.illinois.edu/montreal_talk1_primegaps.pdf), slide "Proving large gaps: Jacobsthal's function"): > **Conjecture (Maier-Pomerance, 1990).** `J(T) = T(log T)^{2+o(1)}`. > Random dart model prediction: `J(T) ~ T·Q_T/phi(Q_T) ~ e^gamma T log T`. **The law with `c = 1` IS Ford's random-dart prediction**, so MP requires `c ~ log x/e^gamma`. Over the 64 exact terms `log p` runs 2.40 to 5.74, so a genuine MP `c` must grow by a factor 2.39 across the ladder. ``` dataset n beta(vs log p) beta(vs p) rms c1 one class [11,311] 60 -0.1091 +- 0.0335 -0.0202 +- 0.0084 0.0531 c1 one class [61,311] 47 0.1221 +- 0.0232 0.0245 +- 0.0047 0.0145 c1 one class [97,311] 40 0.1185 +- 0.0236 0.0233 +- 0.0046 0.0097 c1 one class [181,311] 23 0.1994 +- 0.0473 0.0363 +- 0.0087 0.0062 c2 free 2class [11,73] 17 0.0858 +- 0.0988 0.0308 +- 0.0294 0.0673 c2' diff-2 G2 [11,37] 8 0.2236 +- 0.2515 0.0844 +- 0.0833 0.0833 sliding windows of 20 consecutive one-class terms: [11,89]=-0.423 [23,107]=0.077 [41,131]=0.267 [59,151]=0.235 [73,173]=0.008 [97,193]=0.039 [109,223]=0.153 [137,239]=0.200 [157,263]=0.203 [179,281]=0.219 [197,311]=0.166 ``` **There is a drift, it is positive, and it is real.** The whole-ladder fit is negative only because `p = 11` is a large outlier (`c1 = 0.4712` against a plateau of 0.366). From `p = 61` upward `beta = 0.12 +- 0.02`, from `p = 181` upward `0.20 +- 0.05`, and ten of the eleven sliding windows are positive. **The control, which is the part that makes this a measurement.** Following `exponent-control.md`'s method, synthesise two ladders on the same `p`, `mbar`, `lnD` as the real one, rounded to even integers exactly as a real Jacobsthal value is. LAW has `c` constant, MP has `c` proportional to `log p`. Run the identical estimator: ``` synthetic LAW ladder (true beta = 0): estimator returns -0.0152 +- 0.0083 synthetic MP ladder (true beta = 1): estimator returns 0.9808 +- 0.0063 REAL A048670 ladder: estimator returns -0.1091 +- 0.0335 ``` **The estimator recovers 0.981 when the truth is 1.** So MP's growth is not merely unresolved on this ladder. It is absent by a factor of 5 to 8 in the exponent, and the estimator is demonstrably able to see it. VERIFIED as an estimator property, MEASURED as a fact about the ladder. Term by term against the MP shape: ``` p g(p#) mbar lnD c1 MP c c1/MP 11 14 4.813 6.2 0.4712 1.346 0.3500 59 118 7.475 47.0 0.3359 2.289 0.1467 109 312 8.636 100.2 0.3606 2.634 0.1369 179 550 9.377 160.0 0.3665 2.913 0.1258 241 834 9.925 224.3 0.3746 3.079 0.1217 277 978 10.153 257.7 0.3737 3.158 0.1184 ``` `c1/MP` falls monotonically from 0.35 to 0.12. The MP `o(1)` is not merely negative on this range, it is diverging. --- ## 7. READING 6 (INFERRED, from a PROVEN source). The term the law omits Ford, Green, Konyagin and Tao, arXiv:1408.4505 §1, verbatim. This is the whole answer and it was sitting in the source the repo already cites. **Read from the arXiv:1408.4505v2 PDF (9 Nov 2015, 32 pages), foot of printed page 3 running onto page 4**, with the sub- and superscripts resolved from glyph bounding boxes rather than from flattened text (verified 2026-08-18, `research/history/staging/lit-pdf-fgkt-maier.md` item 1). > a key to all of them being to take a common value of `a_p` for "large" `p`, > say `a_p = 0` for `z < p < delta x` ... The numbers in `[y]` surviving this > first sieving either have all of their prime factors `<= z` (i.e., they are > "z-smooth") or are of the form `pm` with `p` prime and `m <= y/delta x`. One > then appeals to bounds for smooth numbers ... to see that there are very few > numbers of the first kind, say `O(x/log^2 x)`. By the prime number theorem > there are `~ y log_2 x/log x` unsieved numbers of the second kind. **By > contrast, if one were to take a random choice for `a_p` for `z < p < delta x`, > then with high probability, the number of unsifted integers in `[y]` would be > considerably larger, about `y log z/log x`.** > Assuming that `V` is a "random" subset of `[y]`, for every prime > `p in (delta x, x]` there should in fact be a residue class `a (mod p)` > containing `>> log x/(log_2 x)^{O(1)}` elements of `V`. (Roughly, the > heuristic predicts that the sizes of the sets `V n (a mod p)` are Poisson > distributed with parameter `~ |V|/p`.) Run their own accounting. Survivors after stages 1 and 2 are `|V| ~ e^{-gamma}·y·log_2 x/(log x·log z)`, and stage 3 kills `r` of them per prime with `(1-delta)x/log x` primes available, so > `y ~ e^gamma · x · T1 · r`, with `T1 = log z/log_2 x`. The law is `y = e^gamma·x·log x`. So **in the law's own units** > **`c = T1 · r / log x`.** - **`T1` is the Rankin term. PROVEN.** It is the gain of an ALIGNED `a_p` over a random one. It is capped by the smooth-number step, which needs the `z`-smooth survivors to be `O(x/log^2 x)`, forcing `z = x^{c log_3 x/log_2 x}` and `T1 = log x·log_3 x/(log_2 x)^2`. That is LESS than one log, which is exactly why the best proven bound sits BELOW the random-dart level `e^gamma x log x` and does not contradict the law. The best proven bound is FGKMT's eq. (1.2), `Y(x) >> x·log x·log_3 x/log_2 x`, a single power of `log_2 x`; against `e^gamma x log x` its ratio is `log_3 x/(e^gamma log_2 x) -> 0`, so it sits below by a diverging margin. **Confirmed against the source.** - **`r` is the Maier-Pomerance term. CONJ.** `r = 2` for a positive proportion is proved, and it buys their constant `1.31256 e^gamma` and nothing in the exponent. `r = (log x)^{1+o(1)}` is the conjecture and is the whole of the missing log. > **Both omitted factors are maxima over an index the law does not have.** The > law maximises over POSITION. `T1` maximises over the ALIGNMENT of the residue > classes. `r` maximises over the CLASS chosen for each large prime, and is > itself an extreme-value statement, over `p` Poisson counts rather than over > positions. A random-dart model has neither index. That is why it can be off by > a whole log and still fit everything anyone can compute. **The Maier-Pomerance statement, pinned to the source.** Checked against `arXiv:1412.5029` (FGKMT, JAMS 31 (2018) 65-105) on 2026-08-17, so it does not need re-checking; if it is ever re-checked, check it against that version. - FGKMT p. 4, verbatim: *"It is conjectured by Maier and Pomerance that in fact `Y(x) << x(log x)^{2+o(1)}`."* - `Y(x) = j(P(x)) - 1` exactly, their **eq. (1.3)**; `Y` itself is their **Definition 1**. Their **eq. (1.2)** is a different statement, the proven lower bound `Y(x) >> x log x log_3 x/log_2 x`. - The squared form `x log x log_3 x/(log_2 x)^2` is **Rankin's**, and FGKMT print it only to say they improve on it: *"This improves on the bound `Y(x) >> x log x log_3 x/(log_2 x)^2` obtained by Rankin."* Their own Theorem 1 is `G(X) >> log X log_2 X log_4 X/log_3 X` with an effective implied constant; the arbitrary-constant `R` statements belong to the earlier Rankin-shape literature and not to FGKMT. - The equality form `J(T) = T(log T)^{2+o(1)}` is Ford's Montreal slides, §6, and the slides are located: Kevin Ford, *Large gaps between primes*, Talk 1, CRM Montreal workshop **Probability in Number Theory**, 2018, [ford126.web.illinois.edu/montreal_talk1_primegaps.pdf](https://ford126.web.illinois.edu/montreal_talk1_primegaps.pdf). The same slide corroborates FGKMT's single `log_2 T` independently: *"Lower bound (FGKMT, 2018). `J(T) >> T log T log_3 T/log_2 T`."* The form is also in print in A048670's OEIS comment (*"Maier & Pomerance conjecture that Max_{n <= x} A048669(n) = log(x)(log log x)^{2+o(1)}"*). ### 7a. The ledger, evaluated `r` solves `max` over `p` classes of `Poisson(1/log x)`, that is `r·log(1/lambda) + log r! = log p`. ``` x log x T1 r c_pred=T1*r/logx c_pred/c_pred(229) 1e1.6 3.61 0.547 2 0.3032 0.825 1e2.4 5.43 0.998 2 0.3674 1.000 1e3.0 6.91 1.219 2 0.3528 0.960 1e6.0 13.82 1.934 4 0.5601 1.524 1e10.0 23.03 2.675 5 0.5810 1.581 1e20.0 46.05 4.216 8 0.7324 1.994 1e50.0 115.13 7.960 17 1.1753 3.199 1e100.0 230.26 13.181 29 1.6601 4.519 1e1000.0 2302.59 78.627 191 6.5221 17.753 ``` Three readings. 1. **`c_pred(229) = 0.3674` against the MEASURED `c1 = 0.3722`.** The ledger's own value at the top of the exact ladder, with `r` frozen at MP's PROVEN `r = 2` and `T1`'s implicit constant set to 1, lands on the measured constant. Do not over-read it: `T1` carries an unfitted `O(1)`. What is not a coincidence is the shape. 2. **Why `c` looks flat.** `c_pred/r = T1/log x = log_3 x/(log_2 x)^2`, and that function has a stationary point where `log_3 x = 1/2`, that is `log x = 5.200` and **`x = 181`**. The exact one-class ladder, `p = 11` to `271`, brackets that point, and `r` is frozen at 2 the whole way. **The flatness of `c` over the only data anyone has is a property of where the data sits on `log_3 x/(log_2 x)^2`.** INFERRED, and it is the most useful line in the note. 3. **Where it breaks.** `c` reaches twice its measured value at `x ~ 1e17` and ten times at `x ~ 1e406`. Under MP itself, which is the fast reading, the factor 2 arrives at `x ~ 1e5` and the factor 10 at `x ~ 1e24`. ### 7b. The four live extrapolations ``` x law measured drift ledger MP 1e2.4 0.3722 0.3722 0.3722 0.3722 1e3.0 0.3722 0.3831 0.3574 0.4732 1e6.0 0.3722 0.4163 0.5674 0.9463 1e10.0 0.3722 0.4426 0.5886 1.5772 1e20.0 0.3722 0.4810 0.7420 3.1545 1e100.0 0.3722 0.5835 1.6819 15.7723 1e1000.0 0.3722 0.7692 6.6076 157.7229 ``` All four agree to within 20% over the entire exact ladder, which is exactly why the ladder cannot decide between them. **The law is not wrong in our range and it is not right asymptotically. Its failure is not observable, only derivable.** --- ## 8. READING 7 (REFUTED). One repo projection is wrong, and it is not the one anyone worried about `FOLD-PROFILE.md` §12 projects > localized: `M(x, x'^2) ~ 1.2·x·ln x` against `x^2`, **margin `~ x/(1.2 ln x)`** The law says `M(x, x^2) = c·mbar·ln(x^2/mbar) ~ 4.8·c·ln^3 x`, which is polylog, not linear in `x`. Measured directly: ``` x M(x,x^2) mbar lnD c 4.8 ln^3x 1.2 x ln x ratio 211 498 71.1 6.44 1.087 736 1355 2.7 401 630 89.9 7.49 0.936 1034 2884 4.6 797 924 114.2 8.62 0.938 1431 6390 6.9 1601 1452 140.4 9.81 1.054 1928 14175 9.8 3203 1722 170.3 11.01 0.919 2524 31025 18.0 6421 2832 204.2 12.22 1.135 3235 67554 23.9 9973 2868 227.1 12.99 0.972 3747 110193 38.4 ``` `M(x,x^2)/ln^3 x` sits between 2.9 and 4.2 over a 47-fold range in `x` (the "flat at 3.2 to 3.7" that stood here was narrower than either engine that measured it; `history/staging/audit-cross-document-constants.md` A6), and `c` is flat at 0.92 to 1.14 with no trend. **`FOLD-PROFILE`'s projection is a factor 38 high at `x = 9973` and the error grows like `x/ln^2 x`.** REFUTED. This is an internal contradiction the repo already contained. `LOCALIZED-GAP.md` §5 records `M/(k ln^3 x)` flat in 1.2 to 1.6, which at `k = 2` is `M ~ 2.4` to `3.2 ln^3 x`, and the measurement above settles it in `LOCALIZED-GAP`'s favour. It errs in the safe direction for the Zone Postulate: **the true localized margin is `x^2/(4.2 ln^3 x)` at this engine's worst case, `x^2/(4.4 ln^3 x)` across both engines that measured it (3.5 is the midpoint, not a bound), not `x/(1.2 ln x)`.** --- ## 9. Which repo extrapolations inherit the error The rule from §4: an ON-DIAGONAL use of the law (window = the whole tile `x#`) is safe, for the cancellation reason. An OFF-DIAGONAL use (window `x^k` at fixed `k`) needs the off-diagonal `c`, which is 0.85 to 1.05 and itself moves with `x`. | use | curve | verdict | |---|---|---| | `two-class-lower-bounds.md` §6, `G2(41#)` | diagonal, one step | **SAFE**. Band 476 to 633; 513 at the full-sample mean `c2' = 0.4814`, 488 if the `x = 37` term is an outlier and 633 if it is a level shift. Ledger drift over `37 -> 41` is 4%. | | `two-class-lower-bounds.md` §8, measured law `≈ 1.2 x ln^2 x` | diagonal | **SAFE as a description of `x <= 41`.** Must not be quoted as an asymptotic: with the ledger's drift it is 3x larger by `x = 1e50`. The `≈` is the target's own symbol and carries the distinction: this is a measured approximation, not an asymptotic `~`. | | `two-class-lower-bounds.md` §6 caveat, "`c1` should eventually grow like `ln x` and does not" | diagonal | **CORRECT, now quantified**: it grows like `(ln x)^{0.12}` on the top 47 terms against MP's 1. | | `localized-04-maxsum.md` §4, `M(x,Y) ~ mbar·ln(Y/mbar)` | off diagonal | **SAFE in its own window.** The constant 0.74 to 1.17 is specific to `lnD ~ 11-16` and `x ~ 1e3`. Carrying it to the full tile gives 0.46 for the same object. | | `FOLD-PROFILE.md` §12, localized `M(x,x'^2) ~ 1.2 x ln x` | off diagonal | **REFUTED**, §8. | | `exponent-control.md` §6 margin extrapolations | neither | **UNAFFECTED**. Those are power-law fits, not uses of this law. | | any use of `c` as a universal constant | both | **There is no such constant.** Report `c` with `(x, lnD)` attached or do not report it. | The band 476 to 633 that `two-class-lower-bounds.md` §6 predicts for `G2(41#)` is confirmed here from the diagonal, and the ledger's drift over the single step `37 -> 41` is 4%, well inside the band's own width. The prediction stands as a real test. --- ## 10. What survives, and what does not **Does the law contradict Maier-Pomerance?** No. It is Ford's own random-dart prediction, which he prints one line below the MP conjecture without flagging a conflict, because there is none: the truth is expected to exceed the random-dart level by the correlation factor `T1·r`, and no computable `x` can show it. **Where does it break?** It does not break at an `x`. It breaks off the diagonal, immediately, and that is measurable today: `c` runs from 0.45 to 1.09 across the surface we can reach. Along the diagonal it is stable to 22% over a 41-fold range in `lnD` and drifts upward at `(log p)^{0.12}`, and the ledger says that drift accelerates to double the constant only past `x ~ 1e17`. **What is the omitted term?** `T1·r/log x`, with `T1` the alignment gain that Rankin proves and `r` the multi-kill count that MP conjecture. Both are maxima over indices the law does not carry. **The uncomfortable residue.** `c_pred = T1·r/log x` reproduces the measured `c1` at `x = 229` to 1.3% with nothing fitted, and it predicts flatness across exactly the range where flatness is observed, for a reason (`log_3 x/(log_2 x)^2` is stationary at `x = 181`) that has nothing to do with extreme-value theory. If that is a coincidence it is a good one. If it is not, then the constant this repo has been quoting as a Poisson constant is really a smooth-number constant wearing a disguise, and the right way to extrapolate it is the ledger, not the law. --- ## 11. Reproduction ``` node --max-old-space-size=6144 research/maxgap-law.js # 21 s, tiles to x = 23 node --max-old-space-size=8192 research/maxgap-law.js --big # 147 s, adds x = 29 ``` Section 1 is custody against A048670, the repo `G2` ladder, both agents' summary statistics and A's `Y = 1e7` scan. Sections 3 and 4 are the block statistic, on exact periods and on localized windows respectively. Section 5 is the gap tail. Section 6 is the drift with its synthetic control. Section 7 is the FGKMT ledger. Section 8 is the affected-extrapolation audit. Engine note. The exact periods are swept by a segmented sieve with `O(1)` memory, 16.7M per segment, so `29# = 6,469,693,230` is walked without ever building a tile: 99 s for the units pass and 35 s for the twin pass. The block maxima for all 27 block sizes are accumulated in one pass, and the head block is excluded from every mean, because the first gap of a primorial tile is a distinguished object and dominates the mean at large block size. --- *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.*