# The exponent of G2, measured against a control that knows the answer *(2026-08-17. Script: `research/exponent-control.js`. Every number below is in that file's pasted output and reproduces in about a second.)* ## What is hard here, first The question is what exponent governs G2(x#), and the honest obstacle is not that our ladder is short. It is that **the estimator does not work at these sizes, and we can prove it does not, because the same estimator run on fifty-eight terms of an object whose answer is known reports 1.282 when the truth is 1.** That is the whole content of this note. Everything else follows. ## The control The ordinary Jacobsthal h(p#) is our object one dimension down. Iwaniec 1978 proves exponent at most 2; Maier and Pomerance conjecture p·(log p)^{2+o(1)}, so exponent 1 + o(1); Erdos 1962 is the ancestor and Holt's 1402.1970 tabulates it as the maximum gap in the cycle. OEIS A048670 carries 64 terms, out to p = 311 (58 on the entry face; the b-file tail a(59)..a(64) is Bozek's, single-witness, adopted 2026-08-20 after an exact 58/58 overlap check — the calibration fits of §1 to §4 are index-pinned to the first 58, and §5's 22-term refit reads its control off the full 64). The two-class objects are Ziller and Morack's adversarial paired Jacobsthal h2 (A288815, 21 terms, out to p = 73) and our own G2 — 12 exact custody terms out to p = 37, extended to 22 trusted terms out to p = 79 by the adopted A144311 ladder (`research/a144311-full-ladder.js`, series rule of 2026-08-20) — with G2 ≤ h2 at every shared term. **None of the objects is ours except G2.** The reduction "h2(n) < p_n² − p_n implies TPC" is Ziller and Morack's, 2017. Custody: the h/G2 ratio row of PRIOR-ART.md, the nineteen U-FRAME §6a margins, the four THE-DIALS §2 margins, and A288815 = 6·A072753 + 6 all reproduce digit for digit, and so do the three pilot numbers on record, 1.191, 1.282, 1.801. ## 1. The bias does not shrink. It grows, then sticks at +0.28. Sliding windows of every width, over all 58 control terms, true exponent 1: | width | windows | mean | sd | min | max | bias | |---|---|---|---|---|---|---| | 10 | 49 | 1.262 | 0.094 | 1.126 | 1.549 | +0.262 | | 12 | 47 | 1.267 | 0.086 | 1.152 | 1.470 | +0.267 | | 19 | 40 | 1.280 | 0.058 | 1.197 | 1.415 | +0.280 | | 21 | 38 | 1.282 | 0.053 | 1.214 | 1.400 | +0.282 | | 30 | 29 | 1.283 | 0.036 | 1.236 | 1.349 | +0.283 | The scatter collapses by a factor of three. The centre does not move. **The estimator converges, and it converges to 1.28.** Nested ranges say the same thing: 1.191 on ten terms, 1.238 on nineteen, 1.282 on fifty-six, and 1.259 on the disjoint tail [179, 271]. There is no drift toward 1 anywhere in the data. Two immediate consequences. The +0.19 on record is an unlucky-low draw from a distribution centred on +0.26; the window [5, 37] sits near the bottom of it. And the correct correction to use is the distribution mean at matched width, which moves G2's corrected reading from 1.610 down to 1.539. ## 2. Model discrimination, and the result is mostly negative Residual structure, not R². Sign runs against the (n+1)/2 expected under white noise, and lag-1 autocorrelation. **Control, all 56 terms, p in [5, 271]:** | model | k | rms | max | runs/exp | ac1 | AIC | param | |---|---|---|---|---|---|---|---| | c·p^a | 2 | 0.0552 | 0.159 | 27/28.5 | 0.118 | −320.3 | a = 1.282 | | c·p·log^a p | 2 | 0.0840 | 0.239 | 17/28.5 | 0.495 | −273.4 | a = 1.000 | | c·p·log²p | 1 | 0.2812 | 1.239 | 6/28.5 | 0.751 | −140.1 | c = 0.134 | | c·p² | 1 | 0.7051 | 2.141 | 2/28.5 | 0.891 | −37.1 | c = 0.028 | | c·θ^a | 2 | 0.0442 | 0.139 | 15/28.5 | 0.584 | −345.4 | a = 1.204 | | c·θ² | 1 | 0.8303 | 2.650 | 2/28.5 | 0.890 | −18.8 | c = 0.037 | Read that table with the answer in hand. **The pure power law wins by 47 AIC units, its residuals are white (27 runs against 28.5 expected, ac1 = 0.118), and its exponent is wrong by 0.28.** The family that contains the truth loses. So a clean power-law fit with white residuals carries no evidence about the asymptotic exponent at these sizes, and better diagnostics do not repair it. The Maier-Pomerance shape is worse than imprecise here, it is refuted on this range: frozen at log², rms 0.281 and a maximum log residual of 1.239, a factor of 3.5. As a noiseless truth it would report 1.791 over [5, 37] and 1.540 over [5, 271]; the data reports 1.191 and 1.282. Over the whole available ladder h(p#) tracks p·log p (which would report 1.270), not p·log²p. The o(1) is strongly negative below p = 271. **That refutation is of the FROZEN log² shape, and the o(1) is not static.** `research/maxgap-law.md` §6 measures the drift directly in the extreme-value frame: the constant `c1` in `g(p#) = c1·m1·lnD` grows like `(log p)^{0.12 ± 0.02}` on the top 47 exact terms, which is in Maier-Pomerance's direction at a tenth of the rate their conjecture needs. A synthetic ladder built on the same p, m1 and lnD to have exponent exactly 1 returns 0.981 ± 0.006 from the identical estimator, so the shortfall is a measurement and not a resolution limit. Read together: the shape is wrong on this range and the residual drift is real, positive, and an order of magnitude too small to be the missing log. **Two classes, best data, h2 on 19 terms, p in [5, 73]:** | model | k | rms | runs/exp | ac1 | AIC | param | |---|---|---|---|---|---|---| | c·p^a | 2 | 0.1133 | 7/10.0 | 0.066 | −78.8 | a = 1.847 | | c·p·log^a p | 2 | 0.1129 | 9/10.0 | −0.174 | **−78.9** | a = 2.448 | | c·p·log²p | 1 | 0.1655 | 6/10.0 | 0.498 | −66.4 | c = 1.579 | | c·p² | 1 | 0.1648 | 8/10.0 | 0.445 | −66.5 | c = 0.577 | | c·θ^a | 2 | 0.0875 | 8/10.0 | 0.122 | −88.6 | a = 1.712 | **c·p^1.847 and c·p·log^2.448 p are separated by 0.1 AIC units.** Those two hypotheses differ by a factor of x^0.85 in the limit. Nineteen terms cannot tell them apart, and ten terms of G2 spread all seven models across 15 AIC units with no model showing clean residuals. The honest answer to "what is the exponent" is that this data does not determine it. One thing the comparison does say. At matched n and matched range the frozen quadratic loses 74.7 AIC units on the control and only **12.2** on h2. Two classes genuinely look far more quadratic than one, on the residuals. That is a statement about model fit at these sizes and not a measurement of the exponent: the same comparison run on the control prefers a model whose exponent is wrong by 0.28. ## 3. Why exponent 2 is still disfavoured, and it is not the standard error The nominal fit gives 1.847 ± 0.035, which puts 2 at 4.4 sigma. That number is worthless: the control's fits have nominal errors of ±0.008 around a value that is wrong by 0.28. The real argument is one-sided-ness. The control's bias is positive in **all 40** windows, minimum 1.197. It arises because the truth carries a positive power of log inside it, and any positive log power biases a finite-range power fit upward. Nothing about the two-class object suggests a negative log power. So the bias transfers in sign even if not in size, and the true exponent lies **below** the raw 1.847, not above it. Two hard checks pin the readings from the other side. - **h2 ≥ h pointwise**, verified at all 21 terms, and elementary: the adversary choosing two classes per prime may take the first to be the one-class optimum. So exponent(h2) ≥ exponent(h) = 1 + o(1). This **refutes** the proportional-bias correction (scale the control's bias by the ratio of fitted effective log powers, 0.48 → 1.98), which returns 0.67 to 0.83. It overshoots. - **G2 ≤ h2 pointwise**, verified at all 12 shared terms, so any upper bound on h2's exponent is one on G2's. ## 4. The transfer done structurally, which needs no bias Divide each maximum gap by its **exact** mean gap: M1 = p#/φ(p#) ~ e^γ·log p for one class, M2 = p#/∏(q−2) ~ c·log²p for two. That removes the entire sieve-dimension difference. What remains, Q = (h2/M2)/(h/M1), is the residual cost of the second class. If Q is bounded, h2 and h share a power exponent. | range | n | slope of Q | |---|---|---| | [5, 73] | 19 | 0.329 ± 0.028 | | [17, 73] | 15 | 0.234 ± 0.025 | | [23, 73] | 13 | 0.200 ± 0.033 | | [41, 73] | 9 | 0.100 ± 0.069 | and Q turns over at the end: 2.487, 2.511, 2.473, 2.392, 2.302. **Do not bank this.** The turnover coincides with three large upward jumps in the denominator h at p = 67 and 71 (local exponents 2.33 and 3.17), so it is probably numerator-denominator noise. It is a hypothesis with nine points of weak support, and its consequence if true is large: margin growing like x/log³x. ## 5. The answer, with error bars | reading | value | what it rests on | |---|---|---| | raw fit, h2, 19 terms, x-frame | 1.847 ± 0.035 (nominal) | nothing, once §1 is read | | raw fit, G2, 22 trusted terms, x-frame | 1.777 ± 0.029 (nominal) | same | | control-corrected, h2 | **1.57 ± 0.06** stat, systematic unquantified | equal-bias transfer | | control-corrected, G2, 22 terms | **1.50 ± 0.05** stat, systematic unquantified | equal-bias transfer at width 20 | | θ-frame corrected, h2 | 1.49 | equal-bias transfer, θ frame | | θ-frame corrected, G2, 22 terms | 1.43 | equal-bias transfer, θ frame | | hard floor | ≥ 1 | h2 ≥ h, elementary | | proportional-bias | 0.67 to 0.83 | **refuted** by the floor | | structural, if Q bounded | 1 + o(1) | nine weak points | The G2 rows are the 2026-08-21 refit on the 22-term trusted A144311 ladder (script §S11), run with the same discipline as everything above: raw log-log slope over p in [5, 79] (n = 20, p = 2 and 3 excluded as in every fit here), the control's bias read off its sliding windows at the matched width 20 (+0.279 over the 64-term h, sd 0.052, 45 windows; the pinned 58-term windows give +0.281 and move the corrected reading by 0.002), and the bracket as [corrected − 2·window-sd, raw] rounded outward. Both moves of the refit push the central DOWN: the raw fit fell 1.801 → 1.777 where the control's own raw fit RISES with prefix length (1.191 at 10 terms, 1.238 at 19, 1.282 at 56), and the matched-width bias grew (+0.262 at width 10 to +0.279 at width 20). **Central estimate 1.50 on G2's own 22 trusted terms (h2's corrected figure stays 1.57), practical bracket 1.3 to 1.8, hard floor 1, and exponent 2 disfavoured by the one-sided bias argument rather than excluded by the data.** Chris's ~1.6 sits inside the bracket; the bias-transfer route to it does not. ### 5a. The certificate ladder reads lower, and most of the difference is priced `research/two-class-lower-bounds.md` §5 runs the same control-corrected estimator on a certified greedy covering ladder to x = 4001, 108 times further out than the exact terms, and reads **1.11 to 1.25, falling with range**. That is the same object: by the CRT identity of that file's §1, G2(x#) − 1 is exactly the longest interval coverable by choosing a residue pair per prime, and the greedy emits a verified cover. So the two numbers are one exponent read through two proxies, not two exponents. Two measured effects account for most of the gap between them, and they point in opposite directions. | effect | size | source | |---|---|---| | a certified lower-bound ladder biases the exponent **down**, at a 28% terminal shortfall | −0.09 | `h2-scoping.md` §5b, priced on the control | | the exact ladders stop at x = 73 and x = 79; the control's own reading rises with prefix length, 1.245 at 22 terms to 1.282 at 56 | up to +0.04 on the short ladders | §1 | Adding the first back gives about 1.29 from the certificates against 1.50 for G2 on trusted terms. **Quote 1.57 for h2 and 1.50 for G2, note that the long certificate ladder reads 1.2 and that about a third of the gap is the greedy's own downward bias, and treat the residual 0.2 as unexplained.** It is small against the 4.2665-to-2 gap it would be used on, and it does not change any verdict in this note: every reading, corrected or raw, sits below 2 and above 1. ## 6. Consequence for the Zone Postulate margin Two things about how the margin must be read, then the number. **The index convention changes the sign of the trend.** With p_{n+1}² the h2 margin trend is −0.084 ± 0.045 over [5, 73]; with p_n² it is +0.154 ± 0.035. The two agree asymptotically and disagree at n = 19, because the OLS slope of log p_{n+1} on log p_n is 0.88 rather than 1 on this range: consecutive log-primes crowd together as n grows. Any margin trend quoted from a short ladder must say which convention it uses. **The frame mixing overstates the comfort.** U-FRAME §6a's α = 1.653 on all 21 terms is the θ frame; the same 21 terms read **1.924** against x, and the Zone Postulate threshold p_n² is an x-frame quantity. The control's bias is +0.282 in x and +0.220 in θ. **The corrected figures below are on a different range and the range must travel with them**: the correction is applied to the [5, 73] fit, 19 terms rather than 21, whose raw x-frame slope is 1.847 and whose raw θ-frame slope is 1.712. Subtracting the bias there gives 1.567 in x and 1.493 in θ, and most of the disagreement cancels. Subtracting the bias from the 21-term figures instead would give 1.642 and 1.433, which is why the two ranges must not be quoted in one breath. The rule that follows: a θ-frame exponent may not be quoted against the critical exponent 2, which is an x-frame threshold. **The margin itself, on 19 terms instead of 4.** Against p_{n+1}², h2 gives 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. Flat at 2.2, 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. Extrapolated margin against x'², anchored at x = 73 where h2's is 2.38: | x | a = 1.847 (raw) | a = 1.567 (corrected) | x/log³x (structural) | |---|---|---|---| | 10³ | 3.55 | 7.39 | 7.8 | | 10⁶ | 10.2 | 147 | 9.8e2 | | 10⁹ | 29.4 | 2930 | 2.9e5 | All three grow. None is proven. **The proven bound is still exponent 4.2665 (`paper/beta2-note.md`), and the job is still to bring an exponent down to 2.** ## 7. Three things this note settles for the rest of the repo These are the readings any other file should be using. `research/THE-DIALS.md` §0 carries all three. 1. **The exponent is not sitting at 2, and the frozen quadratic is the model the residuals reject.** On G2's own ten terms the two frozen quadratics are the worst two of seven models, AIC −32.3 for c·p² and −25.8 for c·θ² against −41.5 for the best, and c·θ² has rms 0.249 with two sign runs against 5.5 expected and ac1 0.648. A fitted constant near 0.6 is right *for that model* (c = 0.656), which is exactly why quoting it as a law is misleading. The exponent is unresolved between 1.3 and 1.8 with a central estimate near 1.50 on G2's own 22 trusted terms (§5). 2. **The margin is flat, not drifting.** A drift read off four rows is an artifact of the p_{n+1} index convention plus the x = 37 outlier, whose local exponent of the 29 to 31 step is 4.49 against a ladder mean near 2, and the 31 to 37 step reads 2.356; x = 37 is an outlier in the c2' column (0.594 against a 0.446 to 0.500 band), not in the local exponent. On 19 terms of the dominating sequence the margin against x′² is flat at 2.2, slope +0.018 ± 0.045, and its minimum of 1.880 at x = 17 is behind us. 3. **Quote the control line.** Any exponent read off a ladder in this repo is reported alongside what the same estimator does to A048670, whose answer is known: *58 terms, true exponent 1, measured 1.282 ± 0.008, no drift.* ## 8. What would change the answer - **More exact terms — now tested in bulk, and the prediction held.** The per-term prediction (measured on 48 control cases: adding one term to a ten-term fit moves the exponent by 0.022 on average, 0.078 at worst) said single terms would not move the question. The ten trusted terms x = 41..79 adopted from A144311 moved the corrected central by −0.04, from 1.54 to 1.50 (§5) — inside one sliding-window sd. The question is bias-limited, not length-limited. - **Extending h2 past 21 terms is not available and would not help.** `research/h2-scoping.md` reproduces eight terms digit for digit, then prices the frontier from Ziller's own cost curve: term 22 costs about 3.4 months on this hardware and term 25 about 85 years. And the diagnostic would not become readable in any case, because precision was never the constraint. Sliding-window sd is 1.05/width, so at 19 terms it is already 0.057 against a 0.5 question. The constraint is bias, and on nested control prefixes the wrong model's AIC lead grows monotonically from a tie at 10 terms to −47.0 at 56. **There is no reachable ladder length at which the exponent becomes readable.** Three cheaper proxies were tested and all three are refuted: a head window does not exist for h2, since by CRT an adversarial cover can be placed anywhere; a certified lower-bound ladder biases the exponent down about 0.09; randomised restarts bias it down 0.40. - **Deciding whether Q is bounded** would settle the exponent outright, and it is capped at the 19 points now available. --- *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.*