# The theta ladder within one family: does the all-positions exponent turn over? Feeds TODO item 0. Companion to `research/sift-limit-lemmaV.js` (S4b, S5, S6), `research/sift-limit-attack.md` §4.5, and `research/ZONE-POSTULATE.md` §5, whose question about the window/G₂ ratio this file settles on the theta side. ## 0. The question, and the doubt brought to it The all-positions exponent theta answers: at sieve level D = z^s, how long a window H = z^theta does the vector-sieve certificate need before it is positive at EVERY position, conditional on an **unproven Gaussian maximal law for the sawtooth remainder**? Every theta in the conditional column below inherits that hypothesis and must not be quoted without it; the exact column of §5b is the one that owes nothing to it. The column on record read theta = 2.159, 2.183, 2.159, 2.252, 2.308, 2.345 at z = 19, 23, 29, 31, 41, 47 with the last two points from a different family (s = 2.6). The Gap Reformulation needs theta < 2. The column sits above 2 and appears to drift upward. **The doubt.** An upward drift over a range this short is not by itself evidence against theta -> 2. Write need = c(z) * z * ln^2 z. Then theta = ln(need)/ln z = (1+e) + (ln A + 2 ln ln z)/ln z when c(z) ~ A * z^e, so theta -> 1+e, and c ~ z^1 is exactly theta -> 2. The bracket is a finite-size term worth 0.15 to 0.35 across our whole range, and when A < 1 it can rise with z before it falls. A rising theta is therefore compatible with theta -> 2, and reading the trend off theta is the wrong measurement. The right one is the growth exponent e of c(z) inside one family. That is what this file does first. It then does three more things, because e turns out to be the noisiest quantity on the table: it measures how much of the theta column is set by the arbitrary offset between z and the largest prime below it (§4), it checks whether s is a lever on theta at all (§5a), and it throws away the Gaussian bound entirely and measures the EXACT supremum, over complete periods to z = 31 and one-sidedly by prefix to z = 71 (§5b). That last one is the measurement that matters. **The answer to the title question, up front.** The conditional exponent, which rests on the unproven Gaussian maximal law, runs 2.159 to 2.482 across z = 19 to 47 and is still rising at the top: on that column the route loses ground. The unconditional one, which owes nothing to any maximal law and is measured self-consistently, sits INSIDE the zone budget at every exactly-measured level — need/z² = 0.3550, 0.4360, 0.5485, 0.4877, 0.4637, 0.6119 at z = 13..31, all exact, worst margin 39% — and the crossing is real but sits in (31, 47]: at z = 47 a 4·10⁹-position prefix finds min T = 0 at H = 2300 = 1.041 z², and a prefix minimum upper-bounds the period minimum, so the requirement leaves the budget there (`history/staging/theta-selfconsistent.md`). What retires the route as a road to TPC is not the crossing but the maximal law's own price: the sharp form is TPC-implying (`history/staging/phase1-T4-maximal-law.md`). The rest of the file is how firmly each column is established and what it costs the programme. --- ## 1. Method and custody The quantity is the one computed in `sift-limit-lemmaV.js` S6, verbatim: > **⚠ Custody warning on S6, added 2026-08-18, because a reader following this > pointer lands in a block that is half void.** `sift-limit-lemmaV.js`:395 now > carries `!!! WRONG. DO NOT TRUST THIS FUNCTION OR ANY S6 COLUMN DERIVED FROM > IT. !!!` over `elementaryVariance`, whose `Var` is too large by about three > orders. Void: S6's `elementary bound` and `ratio sieve/elem` columns, every > "elementary needs H > …" line, and the two closing constants 0.027·ln⁴z and > 0.741·ln²z. The settled verdict is `history/MORNING-2026-08-16.md`:80-83's > "the two methods are genuinely comparable rather than one dominating", **not** > `history/WAVE7-RESULTS-2026-08-15.md`:50-54's "beats the elementary bound by > roughly two orders", which is the reverse and which quotes the two voided > constants inside the sentence telling you not to quote them. > **What survives is exactly the θ column this document owns** — the script's > own banner lists "the conditional all-positions exponent" under UNAFFECTED, > and `MORNING`:86 says "S6's last block does not touch the broken function and > stands". `research/wave7-logs/lemmaV-meansquare.log` carries **no** flag of > its own, and `history/CHANGELOG.md` has no entry for any of this, so this > paragraph is currently the warning's only home outside the script. need = 2 * sqrt( 2 * lnW * ) / M theta = ln(need) / ln z with = plateau/2 the H-free sawtooth potential, M the position-free main-term density, lnW = theta(z) the Chebyshev function. The bound behind it: R(x) = rho(x) - rho(x+H), so |R| <= 2 sup|rho| <= 2 sqrt() sqrt(2 lnW) under the Gaussian maximal law, and positivity of T = H*M + R at every x needs H*M above that. **VERIFIED: `need` contains no H.** The plateau is H-free by construction, M is H-free, lnW is H-free. So `row(z, u, s)` returns the same theta for every u. The window index u is a formality; the family is set by s alone. ``` $ node -e "const L=require('./research/sift-limit-lemmaV.js'); for(const u of [1.8,2.2,2.8,3.2,4.0]){const r=L.row(19,u,3.0); const need=2*Math.sqrt(2*r.lnW*r.plateau/2)/r.M; console.log('u='+u+' H='+r.H+' ='+r.ms.toFixed(4)+' plateau='+r.plateau.toFixed(6) +' need='+need.toFixed(6)+' theta='+(Math.log(need)/Math.log(19)).toFixed(6));}" u=1.8 H=200 =2.1215 plateau=9.904749 need=576.278257 theta=2.158846 u=2.2 H=651 =6.1809 plateau=9.904749 need=576.278257 theta=2.158846 u=2.8 H=3806 =10.0694 plateau=9.904749 need=576.278257 theta=2.158846 u=3.2 H=12360 =11.1585 plateau=9.904749 need=576.278257 theta=2.158846 u=4 H=130321 =12.1839 plateau=9.904749 need=576.278257 theta=2.158846 ``` What u buys is the other columns (, rms, rms/H*M), not theta. **The invocation.** One process per z, run in parallel, each calling the repo's exported `row(z, 3.2, 3.0)` with nothing reimplemented locally: ```js const L = require('research/sift-limit-lemmaV.js'); const r = L.row(z, 3.2, 3.0); const need = 2*Math.sqrt(2*r.lnW*r.plateau/2)/r.M; const theta = Math.log(need)/Math.log(z); const c = need/(z*Math.log(z)*Math.log(z)); ``` One change was made to `research/sift-limit-lemmaV.js`: `fullPeriodArray` was added to `module.exports`, for §5b. The function itself is untouched. **CUSTODY.** The four existing s = 3.0 rows reproduce digit for digit against the four rows on record, 2.159, 2.183, 2.159, 2.252: ``` z=19 s=3.0 N=4764 M=3.9598e-2 lnW=13.1432 sqrt()=2.225393 -> H > 576.2783 = 3.498433 * z ln^2 z ; theta = 2.158846 [0.6s] z=23 s=3.0 N=9636 M=3.4169e-2 lnW=16.0876 sqrt()=2.825328 -> H > 938.0464 = 4.148437 * z ln^2 z ; theta = 2.182686 [2.5s] z=29 s=3.0 N=20700 M=3.1843e-2 lnW=19.2231 sqrt()=3.687272 -> H > 1435.9919 = 4.367083 * z ln^2 z ; theta = 2.158887 [12.1s] z=31 s=3.0 N=35868 M=2.9253e-2 lnW=22.5904 sqrt()=4.973780 -> H > 2285.7557 = 6.252746 * z ln^2 z ; theta = 2.252324 [36.7s] ``` Cost: rows are O(N^2) in the divisor-pair count, with N = 4764, 9636, 20700, 35868, 76484, 125884, 183084, 293980 at z = 19..47, giving 2.85e-8 s per pair on this machine, at which z = 31 costs 36.7 s. --- ## 2. The rows Fixed family s = 3.0 throughout. The four new rows are z = 37, 41, 43, 47. `c` is the coefficient in need = c * z * ln^2 z; `c/B` is need/z^2 exactly, the ratio the Gap Reformulation must get below 1. No s = 2.6 row appears anywhere in this file. **Every theta in this table is CONDITIONAL on the unproven Gaussian maximal law for the sawtooth** (§1). The unconditional column, which assumes nothing, is §5b, and at the self-consistent window it is a factor 2.46 to 3.68 smaller, the factor rising with z. ``` z pmax N M lnW sqrt need c(z) theta c/B secs 19 17 4764 3.960e-2 13.143 2.22539 576.28 3.498 2.15885 1.596 1 23 19 9636 3.417e-2 16.088 2.82533 938.05 4.148 2.18269 1.773 2 29 23 20700 3.184e-2 19.223 3.68727 1435.99 4.367 2.15889 1.707 12 31 29 35868 2.925e-2 22.590 4.97378 2285.76 6.253 2.25232 2.379 37 37 31 76484 2.806e-2 26.024 9.25620 4760.25 9.867 2.34513 3.477 174 41 37 125884 2.624e-2 29.635 11.99697 7040.03 12.451 2.38567 4.188 515 43 41 183084 2.444e-2 33.349 14.19345 9485.41 15.593 2.43473 5.130 1120 47 43 293980 2.317e-2 37.110 18.99742 14129.48 20.280 2.48199 6.396 2834 ``` Three things to read off before any fitting. **theta rises monotonically from z = 29 onward** and reaches 2.482 at z = 47. Within the family the drift is real and it has not slowed: 2.159, 2.183, 2.159, 2.252, 2.345, 2.386, 2.435, 2.482. **c(z) rises from 3.50 to 20.28**, a factor of 5.8 while z rises by a factor of 2.5. If c grew like z^1, the factor would have been 2.5. So c is growing distinctly faster than z^1 over this range, which by §0's identity is theta going somewhere above 2, not to 2. **need/z^2, the column that actually decides the Gap Reformulation, runs 1.60, 1.77, 1.71, 2.38, 3.48, 4.19, 5.13, 6.40.** The required window is already too long for the zone at the bottom of the ladder and is four times more too long at the top. Whatever the asymptotics turn out to be, nothing in the measured range is close to closing. --- ## 3. The within-family fit The question is the growth exponent e in c(z) ~ A z^e, because theta -> 1+e and e = 1 is exactly theta -> 2. Fitted by OLS on logs over all eight rows: ``` -- FIT c(z) = A z^e over all 8 points -- e = 1.9933 +/- 0.2562 A = 0.0077 R^2 = 0.9098 95% CI for e: [1.3662, 2.6203] => theta_infty = 1+e in [2.3662, 3.6203] H0: e = 1 (theta -> 2) t = 3.876 REJECTED at 95% H0: e = 0.9 (theta -> 1.9) t = 4.266 REJECTED at 95% drop-one jackknife range for e: [1.8400, 2.3826] last 4 points only: e = 3.0685 +/- 0.2751 first 4 points only: e = 0.9447 +/- 0.3842 ``` **c grows faster than z^1, by a margin this data can see.** But the last two lines are the important ones. **A second parametrisation, because ln z is a bad abscissa here.** Consecutive rows differ by which prime joins the sieve, while ln z moves by whatever the prime gap happens to be; §4 shows what that does. lnW = theta(z) moves only when a prime actually enters, so abscissa and ordinate jump together, and by PNT lnW ~ z so the exponent transfers. Same model, need = A' lnW^(1+eW) ln^2(lnW), same null eW = 1: ``` c_W(lnW) = A' lnW^eW : eW = 1.5256 +/- 0.1413 A' = 0.1101 R^2 = 0.9510 => theta_infty = 1 + eW = 2.5256 95% CI [2.1798, 2.8714] H0: eW = 1 (theta -> 2) t = 3.719 REJECTED at 95% drop-one jackknife range for eW: [1.4737, 1.7251] ``` (A trap worth recording, since it caught this analysis first time round: a raw regression of ln(need) on ln(lnW) must be compared to the null 2 + <2/ln lnW> = 2.649, NOT to 2. That 2.649 is an empirical null for a regression slope and is numerically a near-coincidence with the unrelated vector-sieve threshold 1+sqrt(e) = 2.6487 of §7; the two have nothing to do with each other. The ln^2 factor carries about 0.65 of slope over this range. Testing the raw slope against 2 manufactures a rejection out of nothing.) **Both parametrisations reject e = 1 at 95%, and the drop-one jackknife never comes near it** (worst case 1.84 in z, 1.47 in lnW). Two cautions on that, stated because eight structured points do not deserve a clean p-value. First, the residuals are not independent noise; §4 shows they carry the bookkeeping pattern, so the nominal level is optimistic. Second, and more seriously: **c(z) is not a power law over this range.** The exponent itself drifts, from e = 0.94 +/- 0.38 on the first four points to e = 3.07 +/- 0.28 on the last four. So the fitted e is not estimating an asymptotic exponent; it is averaging a moving one. Anything read off it as "the limit" is over-reading. What the fit legitimately supports is the weaker and still decisive statement that **over the measured range c grows faster than z^1 and the gap is widening.** **The mechanism, which is steadier than the fit.** Decomposing need = 2 * sqrt(2 lnW) * sqrt() / M, two of the three factors are tame and known: ``` d ln[sqrt(2 lnW)]/d ln z = 0.5657 +/- 0.0263 (R^2 0.987) [lnW ~ z, so 1/2] d ln[1/M]/d ln z = 0.5551 +/- 0.0345 (R^2 0.977) [1/M ~ ln^2 z, so 2/ln z] d ln[sqrt()]/d ln z = 2.4625 +/- 0.2184 (R^2 0.955) [everything else] d ln[N]/d ln z = 4.5629 +/- 0.1952 (R^2 0.989) cancellation (N/12)/: 80.2 100.6 126.9 120.8 74.4 72.9 75.7 67.9 ``` So theta < 2 requires sqrt() to grow slower than z^{3/2} (after the ln^2 z is accounted for). It is measured at z^{2.46}, and at z^{3.02} on the last four points. The whole problem is in , and is tracking the divisor-pair count N with a cancellation factor that is flat near 75 to 125 rather than improving. That is a cleaner statement of the obstruction than any number in the theta column: **the mean square is not cancelling any harder as the sieve grows, so it inherits N's growth, and N grows like z^{4.6} while the budget allows z^3.** --- ## 4. How much of the theta column is z-versus-p_max bookkeeping z does double duty. It sets D = z^s and ln z continuously, but it enters the arithmetic only through `primesBelow(z)`, which jumps at primes. So a block of z between two consecutive primes shares an identical prime set, an identical W, an identical lnW. Anything theta does across such a block is bookkeeping. **This is not a defect in the ladder.** z = prime is the physically correct convention: the zone is (p, p'^2), the sifted set is the z-rough numbers with z = p', and the Gap Reformulation asks for the gap below p'^2 = z^2. Every z in the ladder is a prime, so every point is a genuine zone. The block sweep below is a CONTROL on how much of theta's numerical value is set by the offset, not a correction to the ladder. Three complete blocks. Every row inside a block has the SAME prime set, the same W, the same lnW; only D = z^3 and ln z move. ``` z pmax D N lnW M sqrt need theta 19 17 6859 4764 13.1432 3.9598e-2 2.225393 576.278 2.158846 <- zone (prime) 20 19 8000 8868 16.0876 3.3424e-2 3.079392 1045.180 2.320616 21 19 9261 9052 16.0876 3.3437e-2 2.963209 1005.380 2.270675 22 19 10648 9372 16.0876 3.4160e-2 2.896904 962.083 2.222260 23 19 12167 9636 16.0876 3.4169e-2 2.825328 938.046 2.182686 <- zone (prime) 24 23 13824 16668 19.2231 2.9934e-2 3.960675 1640.844 2.329402 25 23 15625 17052 19.2231 2.9948e-2 4.013201 1661.815 2.303806 26 23 17576 18308 19.2231 3.0622e-2 3.861137 1563.664 2.257388 27 23 19683 18940 19.2231 3.1346e-2 3.512913 1389.768 2.195768 28 23 21952 19436 19.2231 3.1665e-2 3.300789 1292.677 2.150069 29 23 24389 20700 19.2231 3.1843e-2 3.687272 1435.992 2.158887 <- zone (prime) 32 31 32768 57020 26.0244 2.6266e-2 6.893421 3786.841 2.377356 33 31 35937 57380 26.0244 2.6601e-2 6.736350 3653.914 2.346214 34 31 39304 62116 26.0244 2.6879e-2 8.055576 4324.263 2.374118 35 31 42875 69124 26.0244 2.7415e-2 8.827980 4646.281 2.374964 36 31 46656 69476 26.0244 2.7681e-2 8.405287 4381.314 2.339908 37 31 50653 76484 26.0244 2.8057e-2 9.256201 4760.250 2.345125 <- zone (prime) ``` **What the control says, and it is not what a first look suggests.** In the p_max = 19 block theta falls monotonically by 0.138 across z = 20..23, and in the p_max = 23 block it falls by 0.179 across z = 24..29 (2.329 down to 2.150, with a small rebound at the last step). Either number is as large as the whole drift being read off the ladder. That first look suggests a correctable offset, roughly 0.95 per unit of ln z. The third block REFUTES that. In p_max = 31, z = 32..37, theta moves by only 0.037 and does not fall monotonically; it wanders (2.377, 2.346, 2.374, 2.375, 2.340, 2.345). So there is no offset rule to correct by, and the 0.95 figure does not survive its second test. What does survive is weaker and still worth carrying: **the theta column carries irregular bookkeeping noise of order 0.04 to 0.18, of no consistent sign, and the noise shrinks as z grows.** At the bottom of the ladder that noise is the size of the whole signal; by z = 37 it is a fifth of it. The practical consequence is that no LOCAL slope on this ladder means anything, and only whole-range fits with their error bars should be quoted. §3 does that. **An independent control ladder.** Sampling at z = p+1 instead of z = p puts every point at its block START rather than its block end. This is not a zone, so it is not physical, but it is a second systematic sample of the same arithmetic: ``` z pmax N M lnW sqrt need c(z) theta c/B secs 18 17 4644 3.959e-2 13.143 2.30839 597.83 3.976 2.21193 1.845 1 20 19 8868 3.342e-2 16.088 3.07939 1045.18 5.823 2.32062 2.613 3 24 23 16668 2.993e-2 19.223 3.96067 1640.84 6.769 2.32940 2.849 12 30 29 34820 2.857e-2 22.590 5.17646 2435.58 7.018 2.29270 2.706 49 32 31 57020 2.627e-2 26.024 6.89342 3786.84 9.852 2.37736 3.698 129 38 37 110692 2.558e-2 29.635 11.29829 6799.61 13.523 2.42596 4.709 413 42 41 177972 2.430e-2 33.349 14.20768 9549.59 16.275 2.45186 5.414 1097 44 43 253716 2.258e-2 37.110 17.46291 13326.33 21.150 2.50978 6.883 2102 ``` Same picture, shifted up by about 0.05 in theta, which is the block-position offset and nothing else. The fit agrees with the main ladder within its error bar and reproduces the same non-stationarity: ``` e = 1.6242 +/- 0.1829 R^2 0.9293 H0 e=1: t = 3.413 REJECTED at 95% drop-one jackknife [1.4843, 1.7534]; first 4: 0.9877 +/- 0.4305; last 4: 2.2237 +/- 0.3522 eW = 1.3653 +/- 0.1134 R^2 0.9603 H0 eW=1: t = 3.222 REJECTED at 95% drop-one jackknife [1.3035, 1.4940] need/z^2: 1.85 2.61 2.85 2.71 3.70 4.71 5.41 6.88 ``` So the conclusion does not depend on which systematic position in the block is sampled. Both ladders reject e = 1, both put the point estimate well above it, and both show the same drift of the exponent from about 1 on the first four points to 2.2 or 3.1 on the last four. (A ninth control row at z = 48, the block start matching the main ladder's z = 47, was launched and had not finished at 50 minutes when this was written. It is not needed for anything above: the control ladder already spans p_max = 17 to 43, the same span as the main ladder, and the two fits already agree inside their error bars.) --- ## 5. Two probes that change how the column should be read ### 5a. theta does not move with s: raising the level is not a lever The s = 2.6 and s = 3.0 families are not comparable, and the reason is stronger than incomparability: at s = 2.6 the main term is below the sifting threshold 1+sqrt(e) = 2.6487 and is positive only by finite-size effects, so those rows do not describe an asymptotically valid certificate at all. But empirically theta barely notices s: ``` z=19: s D N M sqrt need theta z=19: 2.8 3806 2460 3.6949e-2 2.138971 593.599 2.168904 z=19: 3.0 6859 4764 3.9598e-2 2.225393 576.278 2.158846 z=19: 3.4 22272 5908 4.2085e-2 1.865714 454.579 2.078281 z=19: 4.0 130321 8004 4.3511e-2 2.562481 603.883 2.174737 z=19: 4.6 762547 8628 4.3621e-2 2.497732 587.144 2.165190 z=23: 2.8 6499 5412 3.2270e-2 2.647928 930.895 2.180245 z=23: 3.0 12167 9636 3.4169e-2 2.825328 938.046 2.182686 z=23: 3.4 42646 15876 3.8156e-2 3.756713 1116.951 2.238358 z=23: 4.0 279841 21020 3.8908e-2 3.427819 999.466 2.202913 z=23: 4.6 1836319 25340 3.9025e-2 4.579069 1331.135 2.294307 z=29: 2.8 12437 16668 2.9934e-2 3.960675 1640.844 2.198490 z=29: 3.0 24389 20700 3.1843e-2 3.687272 1435.992 2.158887 z=29: 3.4 93789 38140 3.4690e-2 6.297258 2251.147 2.292402 ``` Over a factor of 200 in D, theta stays inside a band of about 0.1, the same size as the bookkeeping noise, with no monotone trend. **Raising s is not a lever on theta**: the extra divisor pairs a higher level buys inflate at almost exactly the rate they inflate M. Two consequences. First, there is no cheap escape here; the exponent will not be bought by pushing the level. Second, the two s = 2.6 points at z = 41, 47 are not materially biased in value: mixing families was a validity error, not the source of the drift, and separating them does not by itself change the reading. ### 5b. The exact supremum, which is not conditional on anything `need` assumes the Gaussian maximal law as an equality and then applies |R| <= 2 sup|rho|. Both are loose. Where the full period can be walked, the true sup|R| is available and the whole conditional apparatus can be bypassed. `fullPeriodArray` handles z <= 29 (W = 223092870); past that W + H exceeds the 2^32-1 typed-array limit, so a blocked walk was written for z = 31 and custody-checked against z = 29 first: ``` $ node research/theta-ladder-sup.js 29 z=29 s=3 W=223092870 H=47827: sup|R|=26.9407 minT=1496 H*M=1522.9407 ... th_cond=2.1589 th_true=2.0018 true/z^2=1.0060 [18.6s] (fullPeriodArray gives sup|R| = 26.9407, identical) ``` Every z for which the full period is walkable, s = 3.0 throughout. `th_cond` is the conditional column of §2; `th_true` uses the actual supremum in place of the Gaussian bound. z = 20 is included as a non-prime control (it is not a zone, so it is not a physical point): ``` z W sup|R| rms sup/rms sqrt(2lnW) ratio slack need_cond need_true th_cond th_true true/z^2 19 510510 12.4270 3.3404 3.720 5.127 0.726 1.8363 576.28 313.83 2.1588 1.9524 0.8693 20 9699690 18.8274 4.3393 4.339 5.672 0.765 1.8555 1045.18 563.28 2.3206 2.1143 1.4082 23 9699690 15.3410 3.5291 4.347 5.672 0.766 2.0893 938.05 448.97 2.1827 1.9477 0.8487 29 223092870 26.9407 5.2735 5.109 6.200 0.824 1.6973 1435.99 846.06 2.1589 2.0018 1.0060 31 6469693230 33.1030 6.2204 5.322 6.722 0.792 2.0199 2285.76 1131.63 2.2523 2.0476 1.1776 ``` Three readings, and this is the part of the file that is not conditional on an unproven law. 1. **The conditional column overstates the true requirement by a factor 2.46 to 3.68 at the self-consistent window, and the factor rises monotonically from z = 17** (`history/staging/theta-selfconsistent.md`). The theta column is pessimistic at every measured z, which keeps it safe as a ceiling and useless as a floor. 2. **That slack is not a constant**, so the old argument that the conditional and the true theta share a limit (a constant slack vanishes in the exponent) does not go through on measurement; whether the two columns share a limit is OPEN. > **⚠ CORRECTION 2026-08-18, amended 2026-08-19: THE z = 29 CROSSING DOES NOT HAPPEN, AND EVERY `need_true` > IN THIS SECTION IS MEASURED AT THE WRONG WINDOW.** > `research/theta-ladder-sup.js`:18 reads `const z=Number(process.argv[2]), u=3.2, > s=3.0;` — **z comes from the caller and u never does.** So every `need_true` > here is sup|R| at a window 61x to 123x longer than the operative window (an > earlier 39x-62x phrase mixed two denominators). Verified by reading the line. Measured self-consistently by > exhaustive full-period walk, `need_true/z²` is **0.5485 / 0.4877 / 0.4637** at > z = 19 / 23 / 29 against the 0.869 / 0.849 / 1.006 printed below — overstated > by 1.58x, 1.74x and **2.17x**. The certificate is POSITIVE at every one of the > 223,092,870 positions at H = 0.46 z². **Nothing crosses at any level this box > measured — z ≤ 31, all exact, all inside the budget — and the real crossing > sits in (31, 47]** (`history/staging/theta-selfconsistent.md`). The prefix > rows past z = 31 below are lower bounds on an overstated quantity and > therefore bound nothing for this question. The numbers below are left in place as the custody record of > what was run; do not quote them. Source: > `research/history/staging/phase1-T4-maximal-law.md`. 3. **The true exponent crosses 2 inside the computable range, upward.** th_true reads 1.9524, 1.9477, 2.0018, 2.0476 at the four physical points z = 19, 23, 29, 31, and need_true/z^2 reads 0.869, 0.849, 1.006, 1.178. The route's actual requirement FITTED inside the zone budget at z = 19 and z = 23, with 13 to 15 percent to spare, and stopped fitting at z = 29. This is the single most important measurement in the file, because it is exact arithmetic over complete periods and it is conditional on nothing at all. **Past z = 31 the sup is still reachable one-sidedly.** The supremum over a PREFIX of the period is a lower bound on the supremum over the period, hence a lower bound on need_true and on th_true, and the prefix walk costs O(prefix), not O(W). Four billion positions each, s = 3.0, about 210 seconds at every z tried: ``` z W H sup|R| need_true th_true need_true/z^2 secs 37 2.0056e11 104291 45.9279 1636.96 2.0495 1.1957 202 41 7.4207e12 144847 63.3621 2414.81 2.0975 1.4365 202 43 3.0425e14 168694 70.9670 2903.62 2.1200 1.5704 205 47 1.3083e16 224240 83.1568 3589.54 2.1261 1.6250 211 53 6.1489e17 329369 94.2254 4259.08 2.1048 1.5162 211 59 3.2589e19 464223 113.5770 5297.18 2.1030 1.5217 215 61 1.9228e21 516482 128.2555 6269.47 2.1269 1.6849 218 67 1.1729e23 697332 142.1060 7152.35 2.1108 1.5933 218 71 7.8583e24 839512 153.5875 8013.87 2.1088 1.5897 218 ``` **The flattening past z = 47 is an artifact of the fixed prefix and must not be read as the true value levelling off.** The prefix is 4e9 positions at every z while W runs from 2.0e11 to 7.9e24, so the fraction covered collapses (3e-7 of the period at z = 47, 5e-16 at z = 71) and the bound weakens monotonically. For a field with Gaussian tails the sup over a window of length P scales like sqrt(2 ln P) rather than sqrt(2 lnW), so the bound understates by roughly sqrt(lnW / ln(4e9)), which is 1.09 at z = 37, 1.30 at z = 47 and 1.61 at z = 71. Correcting by that heuristic factor would put th_true near 2.22 at z = 71, still climbing. The correction is a heuristic and is NOT claimed; only the printed inequalities are. Putting the exact and the one-sided rows in one column, s = 3.0, every point a zone: ``` z 19 23 29 31 37 41 43 47 59 71 th_true 1.9524 1.9477 2.0018 2.0476 >2.0495 >2.0975 >2.1200 >2.1261 >2.1030 >2.1088 need/z^2 0.8693 0.8487 1.0060 1.1776 >1.1957 >1.4365 >1.5704 >1.6250 >1.5217 >1.5897 status exact exact exact exact prefix prefix prefix prefix prefix prefix ``` **The statement this licenses, with nothing conditional in it:** the vector-sieve certificate's actual all-positions window exceeds the zone budget z^2 at every z from 29 to 71 that was tested, by a factor of at least 1.5 from z = 43 onward. It fitted inside the budget only at z = 19 and z = 23, the two smallest points available. That is the run's firmest result and it is the one that does not depend on the Gaussian maximal law at all. --- ## 6. The three-way verdict The brief asks which of three the data supports: theta turns over below 2; theta converges to exactly 2; theta settles above 2. Taking them in the order the evidence separates them. ### "theta turns over below 2": the range does speak to this, and it argues against The exponent fit sharpens the trend and does not settle where the LIMIT sits. Branch A is settled anyway, because §5b drops the conditional bound entirely and measures the exact supremum: ``` z 19 23 29 31 37 41 43 47 53 59 61 67 71 th_true 1.9524 1.9477 2.0018 2.0476 >2.05 >2.10 >2.12 >2.13 >2.10 >2.10 >2.13 >2.11 >2.11 need/z^2 0.8693 0.8487 1.0060 1.1776 >1.20 >1.44 >1.57 >1.63 >1.52 >1.52 >1.68 >1.59 >1.59 status exact exact exact exact prefix prefix prefix prefix prefix prefix prefix prefix prefix ``` Measured self-consistently, the actual all-positions requirement stays inside the zone budget at every exactly-measured level — need/z^2 = 0.3550..0.6119 at z = 13..31, all exact — and leaves it in (31, 47] (§5b, `history/staging/theta-selfconsistent.md`). The table above is the custody record of the mis-windowed run — its suprema were taken at windows 39x to 62x too long, and its prefix rows are lower bounds on an overstated quantity; do not quote either. The conditional column, which rests on the unproven Gaussian maximal law, is where the route loses ground, with need/z^2 running ``` z 19 23 29 31 37 41 43 47 need/z^2 1.596 1.773 1.707 2.379 3.477 4.188 5.130 6.396 ``` What argues against branch A is the conditional column and, past z = 31, the self-consistent crossing; and the route is retired regardless: the sharp maximal law it needs is TPC-implying (`history/staging/phase1-T4-maximal-law.md`). ### "exactly 2" against "above 2": the fits reject "exactly 2", with one real caveat The honest answer here is nuanced, and it took the whole apparatus of §3 and §4 to see why. **Nominally, e = 1 is rejected.** Both parametrisations, both ladders, all four tests: ``` main ladder, c vs z : e = 1.9933 +/- 0.2562 t = 3.876 REJECTED main ladder, c_W vs lnW: eW = 1.5256 +/- 0.1413 t = 3.719 REJECTED control ladder, c vs z : e = 1.6242 +/- 0.1829 t = 3.413 REJECTED control ladder, c_W vs lnW: eW = 1.3653 +/- 0.1134 t = 3.222 REJECTED ``` and no drop-one jackknife on any of the four comes within 0.3 of e = 1. Taken at face value that settles it: theta does not go to 2, it goes above. **But it should not be taken quite at face value, for one specific reason.** The fitted exponent is not stable. On the first four points of the main ladder it is 0.94 +/- 0.38, exactly the theta -> 2 value; on the last four it is 3.07 +/- 0.28. The control ladder does the same, 0.99 then 2.22. A single power law does not describe c(z) here, so the OLS e is not estimating a limit, and a test against e = 1 is a test against a model the data has already rejected. The rejection is real as a statement about the measured range and is not licensed as a statement about the limit. So the fits establish this much, robustly: **over z = 19 to 47, c grows faster than z^1, the margin is widening, and the exponent shows no sign of turning down.** They do not establish where it ends up. The reason the separation of the limit fails is §4. Consecutive rows on this ladder differ by which prime enters the sieve, and the abscissa ln z moves by whatever the prime gap happens to be, so local slopes swing from 0.2 to 5.4 and carry no information. Only the whole-range fit is meaningful, and over ln z from 2.94 to 3.85 the leverage is not enough. The ceiling on what this ladder can settle is therefore a mechanism, not an assertion. **Extending the ladder will not fix it.** §9 prices z = 71 at 36 hours and it would move ln z only from 3.85 to 4.26. The error bar on e scales like 1 over the spread in ln z, so a day and a half of compute buys roughly a tenth off the error bar, and it cannot repair a fit whose exponent tripled across the points it already has. The limit cannot be settled by more of this ladder, and that is a statement about the method, not about the budget. ### The three-way verdict, stated plainly 1. **theta turns over below 2: NO — it dips and comes back.** Measured self-consistently the unconditional requirement sits inside the budget at every exact level (need/z^2 = 0.3550..0.6119 at z = 13..31, falling 19 to 29 and rising from 31) and leaves it in (31, 47], where min T = 0 inside a prefix at H = 1.041 z². The conditional budget ratio rose 1.60 to 6.40 over z = 19 to 47. Both columns now point the same way, and the route is retired regardless, because the sharp maximal law it needs is TPC-implying. 2. **theta converges to exactly 2: NOT SUPPORTED.** The null e = 1 is rejected at 95% in all four fits and no drop-one jackknife approaches it. The caveat of §3 applies and is important: the exponent is not stationary, so this rejects a model rather than a limit. But nothing in the data pulls toward 2, and the direction of the instability is away from it, not toward it. 3. **theta settles above 2: SUPPORTED as to "above 2", NOT ESTABLISHED as to "settles".** Every estimator centre on both ladders in both parametrisations puts theta above 2, with the range of point estimates 2.37 to 2.99. But "settles" is not observed: the fitted exponent climbs from about 1 on the first four points to 2.2 or 3.1 on the last four, and the mechanism of §3 says why, namely that tracks a divisor-pair count growing like z^{4.6} with a cancellation factor that is flat near 75 to 125 rather than improving. **The answer is (3), with an amendment.** theta is above 2, it is rising, and there is no evidence of it settling anywhere within reach. A fourth possibility is live beside the three: that theta has no visible asymptote in this range at all. **The decisive instrument is the exact supremum, not the fit.** The fit cannot separate a limit, for the reasons above. The supremum removes the conditionality outright, which no amount of laddering the conditional column could have matched. --- ## 7. What each branch costs the programme The threshold arithmetic of `sift-limit-attack.md` §4.5, restated with our parameters. Component levels D1 = D2 = z^s, window H = z^u, theta_total defined by D1*D2 = H^{theta_total}, so **theta_total = 2s/u**. | accounting | theta_total | positivity needs | against beta_2 = 4.26645 | |---|---|---|---| | coupled, absolute-value remainders | 1 | u > K_BF = 5.1581 (the 2(1+sqrt e) = 5.2974 here before 2026-08-20 was Ford-Halberstam's constant, 2.7% worse) | loses | | break-even | 1.2090 | u > 4.26645 | ties | | full decoupling (Lemma V as stated) | 2 | u > 1+sqrt e = 2.649 | wins by 1.62 | | **this ladder, s = 3.0, u = theta** | **6/theta** | u > theta | see below | At the measured theta = 2.159 to 2.482, theta_total = 6/theta sits at **2.4 to 2.8**, i.e. ABOVE full decoupling: the component level D = z^3 exceeds the window H = z^{2.5}. Two things follow, and both need saying plainly. **(i) The exponent, if the maximal law held, is still inside the interesting range, but only just.** theta = 2.482 at z = 47 is below beta_2 = 4.2665 by 1.78, and below the full-decoupling target 2.649 by 0.17. It sits below 2.649 because s = 3.0 buys margin over the sifting threshold 2.6487, which moves the binding constraint off the sifting limit and onto the remainder. But the column is rising: across z = 37 to 47 theta gains 0.137 while ln z gains 0.240, a local slope of 0.57 per unit ln z. Extrapolating that linear trend, theta reaches 2.649 near z = 63; under the fitted power law of §3, nearer z = 130. Both are extrapolations and §3 says why neither should be trusted as a limit. But either way the margin below the full-decoupling target is 0.17 and closing, so the claim in (i) has a short measured runway, and it would be checkable at z = 59 for about five hours of compute if anyone wanted the deadline pinned. **(ii) But this regime is OUTSIDE the range Lemma V is stated in.** Lemma V is written for s from about 0.63u up to u - eps, i.e. s <= u. Here s/u = 1.62–1.72 > 1. So the theta column is NOT the statement "Lemma V suffices". It is the statement "the actual sawtooth, if it obeys a Gaussian maximal law, is smaller than Lemma V's stated range would need". Closing it needs the maximal law, which `sift-limit-attack.md` §3 already names as "the parity wall in concentration clothing". `research/sift-limit-attack.md` §§3 and 4.5 is the home and the authority for this: Lemma V is not the operative assumption at the measured working point. TODO 0's two open items, (a) the drift and (b) the maximal law, are therefore not independent halves. (b) is the whole price; (a) only tells you what you would be buying. Now the three branches. **Branch A: theta turns over below 2.** Then the certificate is positive at every position in a window shorter than z^2, which is the Gap Reformulation, hence the Zone Postulate, hence TPC. But conditional on the maximal law throughout. Even the win branch is not TPC; it converts the programme's open problem from "find an exponent" into "prove one maximal inequality", which is better posed and not obviously easier: it is the same all-positions quantifier that `natal5-variance.js` reading 6 shows no ensemble statement can close. **Branch B: theta -> exactly 2.** This is a MISS, not a tie, and more is owed than it looks. theta -> 2 means need = A z^2 ln^2 z, which exceeds z^2 by a factor A ln^2 z. The Gap Reformulation needs the gap below p'^2 = z^2, so the real requirement on the coefficient is not "c grows like z^1" but c(z) < B(z) = z / ln^2 z , equivalently need / z^2 < 1. That column is in §2 and it is above 1 throughout and rising. So e = 1 exactly loses by A ln^2 z, which with the e := 1 refit coefficient A = 0.2445 is a factor 3.6 at z = 47 and grows without bound. Branch B leaves the route exactly where branch C does. **Branch C: theta settles above 2.** This is what the data says, and the route becomes an exponent-improvement programme that should be repriced as one. Conditional on the maximal law it currently gives a two-class gap exponent of 2.48 at z = 47 against the unconditional 4.2665. That would be a real theorem if the law were ever proved, and it is the first thing in this repo below 2.649. It is not and never becomes TPC. And per (i) the margin below 2.649 is 0.17 and shrinking, so even the "first below full decoupling" claim has a short measured runway. --- ## 8. Readings 1. **VERIFIED.** The four existing s = 3.0 rows reproduce digit for digit with this invocation: theta = 2.158846, 2.182686, 2.158887, 2.252324 at z = 19, 23, 29, 31. The blocked supremum walker reproduces `fullPeriodArray` exactly at z = 19, 20, 23, 29 (sup|R| = 12.4270, 18.8274, 15.3410, 26.9407) and minT = 468 at z = 20 matches the archived S5 log. 2. **VERIFIED.** `need` and theta are H-free. `row(z, u, s)` returns the same theta for u = 1.8, 2.2, 2.8, 3.2, 4.0, so the family is set by s alone and the window index is irrelevant. 3. **MEASURED.** Four new s = 3.0 rows at z = 37, 41, 43, 47, plus a control ladder at z = p+1 and three complete inter-prime blocks. Full tables in §2 and §4. 4. **MEASURED.** The within-family growth of the coefficient. c(z) = 3.498, 4.148, 4.367, 6.253, 9.867, 12.451, 15.593, 20.280 at z = 19..47, fitted as c ~ A z^e with e = 1.9933 +/- 0.2562 (and eW = 1.5256 +/- 0.1413 in the lnW parametrisation, 1.6242 and 1.3653 on the control ladder). All four reject e = 1 at 95%. **c grows faster than z^1.** The strong caveat is reading 9. 5. **MEASURED.** theta does not move with s. Over D from z^2.8 to z^4.6, a factor of 200 in level, theta stays inside a band of about 0.1 with no trend. Raising the sieve level is not a lever on the exponent: the divisor pairs it buys inflate at the rate they inflate M. 6. **REFUTED.** Mixing the s = 2.6 and s = 3.0 families is not the source of the drift. Separating them is correct for VALIDITY, since s = 2.6 is below the sifting threshold 1+sqrt(e) = 2.6487 and its main term is positive only by finite-size effects. But reading 5 shows theta is nearly insensitive to s, so the two s = 2.6 points were not materially biased in value, and cleaning the family up does not change the reading. 7. **MEASURED.** The theta column carries irregular bookkeeping noise of 0.04 to 0.18 from the offset between z and the largest prime below it, of no consistent sign, shrinking with z. The guess that it behaves as a fixed offset of about 0.95 per unit ln z holds in two blocks and is REFUTED by the third. Consequence: no local slope on this ladder carries information, and only whole-range fits should be quoted. 8. **SUPERSEDED 2026-08-18 — this was called "the headline of the run" and it does not survive.** Every `need_true` below is measured at u = 3.2, hard-coded in `theta-ladder-sup.js`:18 and taken from no caller, 61x to 123x longer than the operative window. Self-consistently the values are 0.5485 / 0.4877 / 0.4637 at z = 19 / 23 / 29 (0.6119 exact at z = 31), not 0.869 / 0.849 / 1.006; the certificate is positive at all 223,092,870 positions at H = 0.46 z², and the real crossing sits in (31, 47] (`history/staging/theta-selfconsistent.md`). The original text follows as the custody record of the run; it must not be quoted. See §5b's correction box and `history/staging/phase1-T4-maximal-law.md`. *(original, superseded)* **MEASURED, exact, and conditional on nothing. The headline of the run.** Replacing the Gaussian maximal bound by the true full-period supremum gives th_true = 1.9524, 1.9477, 2.0018, 2.0476 at z = 19, 23, 29, 31, and a 4e9-position prefix supremum extends the column one-sidedly to th_true > 2.05, 2.10, 2.12, 2.13, 2.10, 2.10, 2.13, 2.11, 2.11 at z = 37, 41, 43, 47, 53, 59, 61, 67, 71, at about 210 seconds each. The corresponding need_true/z^2 is 0.869, 0.849, 1.006, 1.178 exact, then above 1.20, 1.44, 1.57, 1.63, 1.52, 1.52, 1.68, 1.59, 1.59. **The route's actual requirement fitted inside the zone budget only at z = 19 and z = 23, the two smallest points available, and exceeds it by at least half again at every z from 43 to 71.** The apparent flattening past z = 47 is an artifact of the fixed prefix length against a period growing to 7.9e24, not a levelling of the true value. The conditional column is a factor 1.70 to 2.09 pessimistic with no trend in that factor, so both columns share a limit. 9. **MEASURED, and it limits reading 4.** c(z) is not a power law over this range. The fitted exponent drifts from e = 0.94 +/- 0.38 on the first four points to e = 3.07 +/- 0.28 on the last four (control ladder: 0.99 to 2.22). So the OLS e is averaging a moving exponent, not estimating a limit, and no extrapolation off it should be quoted. What survives is the range statement: over z = 19 to 47, c grows faster than z^1 and the margin widens. Extending the ladder does not fix this; by reading 13 it costs 36 hours to move ln z from 3.85 to 4.26, which would tighten the error bar by about a tenth. 10. **REFUTED.** "theta -> 2 exactly would be the boundary case." It is a miss, not a tie. theta -> 2 means need = A z^2 ln^2 z, which exceeds the zone budget z^2 by A ln^2 z: with the e := 1 refit coefficient A = 0.2445 that is a factor 3.6 at z = 47, and it grows without bound. The requirement on the coefficient is c(z) < z/ln^2 z, not c ~ z^1. 11. **INFERRED (the repricing for TODO 0).** The exponent road survives as an exponent-improvement programme and not as a route to TPC. It should be repriced at: a conditional two-class gap exponent of 2.48 at z = 47 and rising, against the unconditional 4.2665, whose entire price is one unproven maximal inequality for the sawtooth. Note the margin below the full-decoupling target 2.649 is now only 0.17 and closing (§7 (i)), so even the modest version of the claim has a measured deadline. TODO 0's items (a) and (b) are not independent halves; (b) is the whole cost and (a) only prices the goods. Item (a), the drift, is now answered as far as this method can answer it and should leave TODO. Item (b), the maximal law, is the live question and should be restated as the whole of item 0. 12. **INFERRED (next move, if the route is kept).** The exact-supremum direction is better conditioned than the exponent ladder and far cheaper per unit of information: it costs O(prefix length) rather than O(N^2) in the divisor-pair count, it needs no hypothesis, and its error is one-sided in the safe direction. Four billion positions cost about 200 seconds at any z in this range, against 47 minutes for one conditional row at z = 47. If the route is kept, this is the instrument to push, not the ladder: it already reaches past where the conditional ladder can afford to go, and the quantity it bounds is the one the Gap Reformulation actually needs. Two refinements worth having: a longer prefix at fixed z, to measure how fast the prefix sup approaches the period sup and hence how much the one-sidedness is costing; and the full period at z = 37 (4.6 hours), which would give a fifth exact point and calibrate the prefix bounds against a known truth at a z where they are already in use. 13. **MEASURED.** The upper ladder is 3x to 10x more expensive than the divisor-pair count suggests at first glance (§9), and by reading 9 it would not separate the hypotheses anyway. --- ## 9. Reproduction ``` node research/theta-ladder-row.js # one row: row(z,3.2,3.0) + the S6 need line node research/theta-ladder-sup.js --fixed-point --band=80 # self-consistent H*, then one band walk node research/theta-ladder-sup.js --Hs=a:b[:st] # a whole H band in ONE period walk (z <= 31 exact) node research/theta-ladder-sup.js 4000000000 --H= # prefix sup at a caller-set window: a LOWER bound node research/theta-ladder-sup.js 1e18 gauss --u= # conditional columns; the window must be named ``` Rows are O(N^2); at s = 3.0 budget about 2.85e-8 s per divisor pair on this machine. The whole ladder above ran in parallel, one process per z, inside 70 minutes of wall clock. **Cost.** Priced against the divisor-pair counts, which are exact and cheap to get: ``` z=47 N=293980 est 41 min = 0.7 h z=53 N=466340 est 103 min = 1.7 h z=59 N=774812 est 285 min = 4.8 h z=61 N=1025812 est 500 min = 8.3 h z=67 N=1574044 est 1177 min = 19.6 h z=71 N=2148388 est 2192 min = 36.5 h ``` So z = 59 is about 5 hours and z = 71 is a day and a half. Given §6, none of it is worth buying: going from z = 47 to z = 71 moves ln z from 3.85 to 4.26 and would move the c-fit's error bar by less than the bookkeeping noise of §4. The exact-supremum walk (`theta-ladder-sup.js`) is the cheaper and better-conditioned direction. Exact means walking the whole period, which is linear in W = P(z): z = 29 is 19 s, z = 31 is 5 minutes, and z = 37 is 4.6 hours. The prefix mode is linear in the prefix instead, so four billion positions cost about 200 seconds at every z tried, from 37 to 71, and it returns a one-sided bound in the safe direction. That is the whole reason the unconditional column of §5b reaches further than the conditional ladder does. Custody for the blocked walker against `fullPeriodArray`: sup|R| = 12.4270, 18.8274, 15.3410, 26.9407 at z = 19, 20, 23, 29, identical both ways, and minT = 468 at z = 20 matches the S5 line of `wave7-logs/lemmaV-meansquare.log`. --- *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.*