# The Fourier sup bound, extended: does `u_sup` plateau? Brief: extend attack 1's `u_sup` past its four points and decide plateau against slow divergence. Script: `research/lemmaV-sup-extension.js` (new). Parents: `research/history/staging/attack-beta2-01-lemmaV-meansquare.md` §5, `research/lemmaV-parseval.js` S5, `research/sift-limit-attack.md` §7b/§7c. No repository file was edited, nothing was committed, nothing was pushed. --- ## THE VERDICT **`u_sup` rises at every added level. A plateau BELOW `β₂` is rejected; the fast divergence attack 1 feared is rejected too; and the question is not "plateau or divergence" but "plateau ABOVE `β₂`, or divergence" — which for this route is the same verdict.** ``` z 13 17 19 23 29 31 37 41 43 47 u_sup 2.0617 2.3036 2.5518 2.6666 2.7464 2.8924 3.0125 3.1103 3.2026 -- u_sat 2.2850 2.4623 2.7228 2.8281 2.8827 3.0259 3.1644 3.2543 3.3448 3.4306 beta_2 = 4.26645 ``` Nine points for `u_sup` against attack 1's four, ten for `u_sat`. Four statements, in decreasing order of how firmly the data support them. **1. `u_sup` is rising, and a strict plateau is rejected.** It increases at all eight steps, 2.0617 to 3.2026, with no step negative and no step near zero. The constant model's RSS is **1.14** against **2.34e-2** for the best-fitting model — 49× worse, with a maximum residual of 0.666. Over the measured range a flat `u_sup` is not a live hypothesis. **2. Attack 1's `θ(z)/ln z` shape is rejected too, and in the conservative direction.** Residuals against its own endpoint fit `u_sup = 0.2865·θ(z)/ln z + 1.197` are −0.0004, +0.064, +0.076, −0.0003 at the four levels it was fitted to, then **−0.086, −0.189, −0.249, −0.373, −0.535** at the five new ones: monotonically negative and growing. **The rise is real but slower than attack 1 feared**, which is the one piece of good news here and the reason nothing crosses at `z = 45`. **3. Every model that fits crosses `β₂`, including the bounded one, and the location spans 1.5 orders of magnitude.** `a ln z + b` crosses at `z = 147` [141, 158]; `a lnln z + b` at `267` [253, 297]; `a θ(z)/ln z + c` at `73` [71, 79]; and `A − B/ln z`, which is BOUNDED, has asymptote **A = 5.4635 — above `β₂`** — so it crosses at `z = 1544` [1189, 2438]. **The direction is unanimous. The location is undetermined, and no arithmetic here pins it down.** **4. And the mechanism can be read with no model at all, which outranks every extrapolation above.** `u_sat` converges to the log–log slope of `Ssat`, and that slope is **4.99 regressed over all ten levels and 6.20 over the top five — rising, and above `β₂ = 4.26645` in every window.** The profile over the asymptote agrees from the other side: the best-fitting limit is `A = 5.46` for `u_sup` and `5.56` for `u_sat`, and forcing the limit down to `β₂` costs a factor **8.70** in RSS. **So even the bounded reading of this instrument lands above `β₂`.** §4 says why: the absolute-value step that removes the position quantifier is the step that loses a factor geometric in `π(z)`, measured at 2.05 per added prime across nine levels. --- ## 0. CUSTODY, AND THE INSTRUMENT AGAINST KNOWN POSITIVES The whole file is a re-implementation, so nothing may be believed until it reproduces what already exists. Six reproductions run before a single new level. ``` (a) the fast |sin(pi t)| against Math.sin over 2.4e6 arguments m/e worst absolute error 6.63e-10, worst RELATIVE error 6.63e-10 (relative is the one that matters: G = |Theta|/|sin(pi a/e)| divides by it) (b) the PEELED Theta_e(a) against lemmaV-parseval.js's own factorisation sum, evaluated term by term, over 2309 / 3000 / 3000 / 3000 pairs (e,a): worst |peeled - direct| = 1.96e-17, 1.64e-17, 1.25e-17, 9.22e-18 at z = 13, 17, 19, 23, with 0 CRT-weight mismatches at every one. (c) Ssup(nP), Ssat, SCS and Var(c) against A1.supBoundTable(), which evaluates the same sums at cost phi(e) 2^omega(e): Ssup rel 4.81e-11 / 4.29e-11 / 4.92e-11 / 4.80e-11 Ssat rel 1.14e-11 / 8.99e-12 / 7.05e-12 / 5.99e-12 SCS and Var(c) at machine epsilon (they never touch the fast sine). (d) the spectral mean square from the SAME sweep against the repository's INDEPENDENT O(N^2) divisor-pair meanSquare(), sharing no code path: z=13 H=169 1.161265531 vs 1.161265531 rel 1.46e-10 z=17 H=289 3.013047790 vs 3.013047790 rel 5.64e-11 z=19 H=361 4.217260880 vs 4.217260880 rel 5.07e-11 z=23 H=529 5.049384489 vs 5.049384488 rel 1.79e-10 z=29 H=841 6.769446539 vs 6.769446539 rel 8.72e-11 (e) attack 1 sec.5's published H_sup and u_sup, by its own protocol: z=13 H_sup = 198 u_sup = 2.0617 CONFIRMED z=17 H_sup = 683 u_sup = 2.3036 CONFIRMED z=19 H_sup = 1833 u_sup = 2.5518 CONFIRMED z=23 H_sup = 4278 u_sup = 2.6666 CONFIRMED (f) is closure an up-set in H? (every integer H from 8 to 2 H_sup) z H_min H_sup(protocol) overshoot d u later failures 13 198 198 0.00% 0.0000 0 UP-SET 17 659 683 3.64% 0.0126 21 NOT AN UP-SET 19 1833 1833 0.00% 0.0000 0 UP-SET ``` Item (d) is the strongest of the six: it validates the entire `Theta` pipeline against a computation carried out in the other basis entirely. **The floors, which are the "known positives" a sup bound must never dip under.** ``` z H=nP Ssup(nP) rms(R_nP) true sup|R| Ssup/sup Ssup/rms sup/rms u_true u_max nP/W 13 60 11.4467 1.1949 2.6494 4.32 9.58 2.22 1.5963 3.0196 2.6e-2 17 126 28.4118 1.4885 4.9203 5.77 19.09 3.31 1.7070 3.6390 4.2e-3 19 198 60.0382 1.4478 6.8403 8.78 41.47 4.72 1.7960 4.4637 3.9e-4 23 258 118.3280 1.7365 7.8157 15.14 68.14 4.50 1.7710 5.1308 2.7e-5 ``` `Ssup/sup` reproduces attack 1's 4.32 / 5.77 / 8.78 / 15.14 to four digits, and is never below 1: the bound never dips under the truth. `u_sup` clears `u_true = ln(nP)/ln z` by 0.47 to 0.90 at all four levels. `nP/W` runs from 2.6e-2 down to 2.7e-5 and `u_max = ln W/ln z` stays above `u_sup` everywhere, so no row is near the degeneracy `R_W ≡ 0`. **What attack 1's `u₁` flag means, and why it is the whole point here.** Attack 1 §4(ii) found its mean-square exponent `u₁` below `β₂` at `z = 13, 17, 19, 23` and crossing at `z = 29`, and refused to call that a result: `u₁ = θ(z)/(2 ln z) + 1.45` **diverges**, so "below `β₂`" at toy `z` says only that `θ(z)/(2 ln z)` has not got there yet — and the crossing landed exactly where the law puts it, which is what proved it was a law. `u_sup` sits in precisely that position. Four points below `β₂` prove nothing until the shape is known, and attack 1's own fit for it has the same divergent shape with a smaller constant. §3 is the test of the shape. --- ## 1. THE COST, AND WHAT MADE THE EXTENSION AFFORDABLE Two changes remove the exponential factor from the evaluation of `Θ_e(a)` and almost all of its trigonometry. Neither is a new inequality; both are the same sum, evaluated differently. **(P1) The phase is a product over the primes of `e`.** From attack 1's Theorem B, `c_i ≡ 0 (mod e1)` and `c_i ≡ −2 (mod e2)`. Splitting `1/e = Σ_{p|e} b_p/p` mod 1 with `b_p = inverse(e/p) mod p`, every prime of `e1` contributes 0 to `a·c_i/e` and every prime of `e2` contributes `2 a b_p / p`, so ``` e(-a c_i / e) = prod_{p | e2} e( 2 a b_p / p ), Theta_e(a) = sum_{T subset primes(e)} V(e/e_T, e_T) prod_{p in T} e(2 a b_p / p), ``` a multilinear form in `ω(e)` unit roots, each depending on `a` only through `a mod p`, `p ≤ z`. All the trigonometry collapses into tables of size `p`. **(P2) A multilinear form is evaluated on a whole grid by peeling.** Fixing `a mod p₁, …, a mod p_j` collapses the `2^ω` coefficient array to `2^{ω−j}`. Over the mixed-radix enumeration of `(Z/e)*`, peeling the primes in ASCENDING order costs `Σ_j Π_{i≤j}(p_i−1)·2^{ω−j} ≈ φ(e)`, against `φ(e)·2^{ω(e)}` term by term. ``` z N(terms) #recs max e #(e,a) term-by-term saving plan 13 852 31 2.310e+3 2.3090e+3 1.3903e+4 6.0x u_sup + u_sat 17 2236 63 3.003e+4 3.0029e+4 2.8369e+5 9.4x u_sup + u_sat 19 4764 127 5.105e+5 5.1051e+5 3.6742e+6 7.2x u_sup + u_sat 23 9636 243 8.818e+5 2.6494e+6 1.7493e+7 6.6x u_sup + u_sat 29 20700 448 3.432e+6 1.7193e+7 9.1850e+7 5.3x u_sup + u_sat 31 35868 763 9.700e+6 6.0577e+7 3.4427e+8 5.7x u_sup + u_sat 37 76484 1494 3.187e+7 5.2512e+8 3.9789e+9 7.6x u_sup + u_sat 41 125884 2501 7.328e+7 1.7951e+9 1.3648e+10 7.6x u_sup + u_sat 43 183084 3814 8.365e+7 3.7441e+9 2.8070e+10 7.5x u_sup + u_sat 47 293980 6035 2.231e+8 1.0915e+10 8.5411e+10 7.8x u_sat only ``` The `saving` column is the count only. In time the gain is larger, because (P1) turns every one of those `cos`/`sin` pairs into a table lookup; what remains is one `|sin(πa/e)|` per `(e,a)` and one per scanned window length, taken with a degree-13 odd polynomial whose relative error S0(a) measures at 6.63e-10. **The reason the ladder stopped at `z = 23` was a cost estimate, and the estimate was wrong by about 550×.** `Π_{p Ssup(H)`, and a fortiori `H·M > Ssat`, put `T(x) > 0` at EVERY position, with no maximal law and no averaging. `u_sat` is the `H`-free one and is an upper bound for `u_sup` by construction. ``` z #(e,a) Ssat u_sat H_sup u_sup [bracket] u_sat-u_sup 0.51/ln z u_max H_sup/W 13 2.309e+3 1.9602e+1 2.2850 198 2.0617 [2.0598,2.0617] exact 0.2232 0.1988 3.020 8.6e-2 17 3.003e+4 5.0314e+1 2.4623 683 2.3036 [2.3030,2.3036] exact 0.1587 0.1800 3.639 2.3e-2 19 5.105e+5 1.2006e+2 2.7228 1833 2.5518 [2.5516,2.5518] exact 0.1709 0.1732 4.464 3.6e-3 23 2.649e+6 2.4250e+2 2.8281 4278 2.6666 [2.6666,2.6666] exact 0.1614 0.1627 5.131 4.4e-4 29 1.719e+7 5.2311e+2 2.8827 10384 2.7464 [2.7198,2.7464] 0.1362 0.1515 5.709 4.7e-5 31 6.058e+7 9.5263e+2 3.0259 20586 2.8924 [2.8662,2.8924] 0.1336 0.1485 6.578 3.2e-6 37 5.251e+8 2.5727e+3 3.1644 52989 3.0125 [2.9876,3.0125] 0.1519 0.1412 7.207 2.6e-7 41 1.795e+9 4.6495e+3 3.2543 103823 3.1103 [3.0750,3.1103] 0.1440 0.1373 7.980 1.4e-8 43 3.744e+9 7.1072e+3 3.3448 170378 3.2026 [3.1678,3.2026] 0.1421 0.1356 8.867 5.6e-10 47 1.091e+10 1.2623e+4 3.4306 -- -- -- 0.1325 9.639 -- ``` Every row is a legal finite statement. At `z = 43`, for instance: **every window of 170,378 consecutive integers contains an `r` with `r` and `r+2` both 43-rough** — established with no maximal law, no Gaussian hypothesis and no exceptional set. `H_sup/W` falls from 8.6e-2 to 5.6e-10 and `u_max = ln W/ln z` stays above `u_sup` by 1.0 to 5.7 at every level, so no row is anywhere near the degeneracy `R_W ≡ 0`. **The two exponents converge.** `u_sat − u_sup = ln(Ssat/Ssup(H_sup))/ln z`, and `Ssup(H_sup)/Ssat` sits near 0.6 at every level, so the gap tracks `0.51/ln z` and shrinks. That is why `u_sat` is a faithful and far cheaper stand-in at large `z`: it needs one sweep with no window scan, which is what lets it reach `z = 47` where `u_sup` does not. Measured `Ssup(H_sup)/Ssat` is 0.531, 0.637, 0.605, 0.603, 0.592, 0.591, 0.567, 0.569, 0.576 — flat near 0.58, close to the `2/π = 0.6366` an equidistributed `|sin(πHa/e)|` would give. **The increments, which is the form the plateau question actually takes.** ``` step d u_sup d u_sat d ln z d(th/ln z) du_sup/dlnz du_sup/d(th/lnz) 13 -> 17 0.2418 0.1773 0.2683 0.6194 0.901 0.390 17 -> 19 0.2483 0.2605 0.1112 0.8248 2.232 0.301 19 -> 23 0.1148 0.1053 0.1911 0.6671 0.601 0.172 23 -> 29 0.0798 0.0546 0.2318 0.5780 0.344 0.138 29 -> 31 0.1459 0.1433 0.0667 0.8697 2.188 0.168 31 -> 37 0.1201 0.1384 0.1769 0.6287 0.679 0.191 37 -> 41 0.0978 0.0899 0.1027 0.7731 0.953 0.127 41 -> 43 0.0923 0.0905 0.0476 0.8863 1.938 0.104 43 -> 47 -- 0.0858 0.0889 0.7721 -- -- ``` Attack 1 read three increments (+0.242, +0.248, +0.115) and could not tell deceleration from noise. With eight, the raw `d u_sup` column does decline — 0.242 down to 0.092 — **but the `z` steps shorten as well**, and `du_sup/d ln z` is 0.90, 2.23, 0.60, 0.34, 2.19, 0.68, 0.95, 1.94, which is not a decaying sequence at all. The two rows where it explodes (17→19 and 41→43) are the ones where `Δ ln z` is smallest. **Per-level increments are the wrong statistic here and this table is the reason; the fits in §3 are the right one.** The last column, `du_sup/d(θ/ln z)`, IS decaying (0.390 → 0.104), which is the same fact as attack 1's shape being too steep, seen locally. **Tightness: how much the bound loses, at EVERY level.** ``` z H_sup Ssup(H_sup) rms(R_Hsup) Ssup/rms sqrt(2 lnW) ratio 13 198 10.4063 0.8006 13.00 3.936 3.30 17 683 32.0666 2.2635 14.17 4.541 3.12 19 1833 72.5791 2.7877 26.04 5.127 5.08 23 4278 146.1371 3.3467 43.67 5.672 7.70 29 10384 309.6710 4.2195 73.39 6.200 11.84 31 20586 563.0990 5.4216 103.86 6.722 15.45 37 52989 1459.4064 8.4046 173.64 7.214 24.07 41 103823 2644.7690 10.1577 260.37 7.699 33.82 43 170378 4096.3370 13.2210 309.84 8.167 37.94 ``` `sup|R| ≥ rms` always, so `Ssup/rms` is an upper bound on the loss and is measurable at every level, where `Ssup/sup` stops at `z = 23`. It runs **13.0 → 309.8**, growing by about 1.35× per added prime. `sqrt(2 lnW)`, the maximal-law size of `sup/rms`, only runs 3.94 → 8.17, so the final column — **3.30, 3.12, 5.08, 7.70, 11.84, 15.45, 24.07, 33.82, 37.94** — is the part of the loss that is NOT explained by the field's own extreme-value behaviour. **That column is the whole story: the bound is loosening geometrically, and the truth is not moving.** `u_sup` rises because `Ssup` is losing ground, not because `G₂` is. --- ## 3. THE TREND, FITTED Five models, least squares, on all points and on the top half, each with a leave-one-out band on the extrapolated crossing of `β₂ = 4.26645`. The models are chosen to bracket the plausible shapes: one bounded and constant, two unbounded and slow, one unbounded and fast (attack 1's), one bounded and rising to an asymptote. ``` u_sup (9 points, z = 13, 17, 19, 23, 29, 31, 37, 41, 43) model a b RSS max|res| crosses beta_2 at z constant u = c 0.00000 2.72756 1.14e+0 0.6658 never (constant) u = a ln z + b 0.89093 -0.17935 2.57e-2 0.1079 146.9 u = a lnln z + b 2.81952 -0.58524 2.34e-2 0.0922 267.4 u = a th(z)/ln z + c 0.18687 1.63550 4.31e-2 0.1380 73.0 u = A - B/ln z -8.78899 5.46353 2.72e-2 0.1070 bounded, A = 5.4635; 1543.9 leave-one-out bands on the crossing (refit dropping each level in turn): u = a ln z + b [141.1, 157.9] u = a lnln z + b [252.9, 296.5] u = a th(z)/ln z + c [ 71.0, 79.0] u = A - B/ln z [1189.4, 2437.9] the same models on the TOP 5 levels only (z = 29, 31, 37, 41, 43): constant u = c a= 0.00000 b= 2.99285 RSS=1.29e-1 never u = a ln z + b a= 1.02743 b= -0.68267 RSS=4.29e-3 123.6 u = a lnln z + b a= 3.65749 b= -1.66571 RSS=4.27e-3 158.0 u = a th(z)/ln z + c a= 0.14588 b= 1.93254 RSS=1.90e-3 89.0 u = A - B/ln z a=-13.00565 b= 6.63515 RSS=4.27e-3 A = 6.6351; 242.4 u_sat (10 points, z = 13 ... 47) u = a ln z + b 0.85489 0.10053 2.97e-2 0.1051 130.7 [126.2, 135.5] u = a lnln z + b 2.72962 -0.31437 3.50e-2 0.1170 211.8 [198.8, 233.5] u = a th(z)/ln z + c 0.16730 1.89894 4.06e-2 0.1191 73.0 [ 73.0, 79.0] u = A - B/ln z -8.57390 5.56487 4.77e-2 0.1360 A = 5.5649; 737.6 [499.7, 1049.9] ``` **(1) A strict plateau is rejected.** RSS 1.14 against 2.34e-2 for the best model, max residual 0.666 against 0.092. That is not a close call. **(2) Attack 1's `θ(z)/ln z` shape is rejected too.** On nine points its RSS is 4.31e-2, the worst of the four rising models, and its residual against the endpoint fit grows monotonically negative to −0.535 at `z = 43`. Its crossing at `z = 73` is the earliest any model gives and the data are already below it. **(3) The four rising models are statistically indistinguishable and disagree about everything that matters.** RSS 2.34e-2, 2.57e-2, 2.72e-2, 4.31e-2 across `lnln z`, `ln z`, `A − B/ln z`, `θ/ln z` — a spread of 1.8×, on nine points. They cross `β₂` at 267, 147, 1544, 73. **Every one crosses. None of them can be preferred.** On the top five levels the ordering even flips (the `θ/ln z` model becomes the best fit, RSS 1.90e-3), which is what "indistinguishable" looks like. **(4) The profile over the asymptote is the sharpest way to price it, and it is the one test that answers the question directly.** Fix a candidate limit `A` in the bounded family `u = A − B/ln z`, minimise over `B` only, and read the cost: ``` A best B RSS RSS/RSS_min verdict at beta_2 = 4.26645 3.00000 1.0036 9.136e-1 33.62 the bound would stay under beta_2 3.50000 2.5837 5.903e-1 21.72 the bound would stay under beta_2 4.00000 4.1639 3.400e-1 12.51 the bound would stay under beta_2 4.26645 5.0059 2.365e-1 8.70 the bound would stay under beta_2 4.50000 5.7440 1.628e-1 5.99 the bound would cross beta_2 5.00000 7.3241 5.856e-2 2.15 the bound would cross beta_2 6.00000 10.4844 6.922e-2 2.55 the bound would cross beta_2 8.00000 16.8049 9.669e-1 35.58 the bound would cross beta_2 5.46364 8.7894 2.718e-2 1.00 best-fitting asymptote ``` **Every limit compatible with the data to within a factor 3 in RSS lies between about 4.8 and 6.2, and all of it is above `β₂`.** Pulling the limit down to `β₂ = 4.26645` costs 8.70×; to 4.00 costs 12.5×; to 3.50 costs 21.7×. The `u_sat` profile says the same with the same shape (best `A = 5.5649`, `β₂` costs 7.21×). **This is the answer to "plateau or divergence": if it plateaus, it plateaus above `β₂`.** **(5) The mechanism agrees, with no model at all.** `u_sat = (ln Ssat + ln 1/M) / ln z`, and `M·log²z` is flat in [0.336, 0.377] across `z = 13..47`, so `u_sat` converges to the log–log slope of `Ssat`. Regressed: ``` d ln Ssat / d ln z = 4.987 over all ten levels (RSS 0.511) = 6.197 over the top five (RSS 0.019) cumulative slope from z = 13/17/19/23/29/31/37/41/43 to z = 47: 5.03, 5.43, 5.14, 5.53, 6.59, 6.21, 6.65, 7.31, 6.46 ``` **Above `β₂` in every window, and rising.** The best-fitting asymptotes of (4), 5.46 and 5.56, sit inside that range, which is the two readings agreeing. **(6) The `s`-panel says the trend is not an artifact of the level.** ``` z s=2.6 s=3.0 s=3.4 fit u_sat = a th(z)/ln z + c: 13 2.4875 2.2850 2.3298 s=2.6: a = 0.13794, c = 2.04159 17 2.5134 2.4623 2.4187 s=3.0: a = 0.20596, c = 1.72140 19 2.6452 2.7228 2.6954 s=3.4: a = 0.25537, c = 1.54454 23 2.7727 2.8281 2.9145 29 2.8012 2.8827 3.0049 31 2.9664 3.0259 3.1924 ``` Rising at all three levels of `s`, with the slope increasing in `s`. `s = 2.6` sits just below `1+√e = 2.6487` and is a stress test only, not a usable level; even there the slope is positive. No choice of `s` in the usable range flattens the trend. **What this prices, and it is the reason to stop measuring.** The two shapes the data cannot separate are `u ~ a ln z` (so `H ~ exp(a ln² z)`) and `u ~ (ln C)·π(z)/ln z` (so `H ~ C^{π(z)}`). They differ only in whether the driver is `ln² z` or `π(z)` — and over the measured range those two are almost the same function: ``` z 13 17 19 23 29 31 37 41 43 101 211 pi(z) 5 6 7 8 9 10 11 12 13 25 46 ln^2 z 6.58 8.03 8.67 9.83 11.34 11.79 13.04 13.79 14.14 21.25 28.60 ``` They agree to within 9% at `z = 43`, cross near `z ≈ 60`, and part company only well after. **Nine levels cannot separate them because at nine levels they are the same function.** Separating them needs `z ≈ 200`, where `π(z) = 46` against `ln²z = 28.6`. The spectral sum is 3.74e9 points at `z = 43` and 3.36e10 at `z = 53` (measured, both), growing about 3× per level, so `z ≈ 200` is of order `10^17` points with the `H`-scan on top. **Brute force cannot settle this.** Only an analytic estimate for `Ssat` can, and §4 says what it would have to be. --- ## 4. WHAT WOULD HAVE TO BE PROVEN The chain is finite and explicit at every `z`; nothing in it is asymptotic. To turn a measured level into a theorem, exactly one statement is needed: ``` Ssat(z,s) <= z^u * M(z,s) for every z >= z0, with u FIXED, Ssat(z,s) = sum_{e | P(z), e>1} sum*_{a mod e} |Theta_e(a)| / |sin(pi a/e)|, Theta_e(a) = sum_{e1 e2 = e} e(2a * inverse(e1) / e2) * V(e1,e2), V(e1,e2) = sum_{i : e | q_i, gcd(e,d1) = e1} w_i/q_i, M = sum_i w_i/q_i, the vector-sieve main term, M ~ c/log^2 z. ``` Then `H = floor(z^u)+1` gives `H*M > Ssat >= Ssup(H) >= sup_x |R_H(x)|`, so `T(x) > 0` at EVERY `x`, so every window of length `z^u` holds a two-class survivor pair, so `G2(P(z)) <= z^u`. No maximal law, no Gaussian hypothesis, no exceptional set, no averaging over positions. **It is a mean-value estimate and not a maximal inequality — and that is exactly why, at the measured rate, it is not strong enough.** The position `x` has already been discarded, exactly, by `|e(ax/e)| = 1`. What remains is an `l1` norm of a fixed finite arithmetic object against the weight `1/|sin(pi a/e)|`. No supremum over anything appears in it. By the taxonomy this project uses, that is the tractable class, and attack 1 was right to say so. But the measurement prices it, and the price is the whole difficulty. ``` per added prime, C = Ssat(next level) / Ssat(this level): 13->17 17->19 19->23 23->29 29->31 31->37 37->41 41->43 43->47 2.567 2.386 2.020 2.157 1.821 2.701 1.807 1.529 1.776 geometric mean 2.0516 over all nine, 2.2729 over the first four, 1.9079 over the last four. ln Ssat regressed on pi(z): slope 0.7202 (all ten, RSS 0.317) slope 0.6184 (top five, RSS 0.103) ``` With `M·log²z` measured flat in [0.336, 0.377] over `z = 13..47`, a per-prime factor that settles at any fixed `C > 1` gives ``` u_sat = ln(Ssat/M)/ln z ~ (ln C) pi(z)/ln z + 2 lnln z/ln z ~ (ln C) z/ln^2 z, ``` (antecedent REFUTED at fixed `s`: `ln Ssat <= 2s ln z + O(ln ln z)`, so the average per-prime factor tends to 1; `attack-0829n-rml-proof.md` §4.1, `verify-0830-usup-convention.md` §3.2, 2026-08-30) which diverges. **The measured factor is declining — 2.27 over the first four steps, 1.91 over the last four — so `C` fixed is not what the data show either. What they show is the decline being too slow.** For `u_sat ≤ u` at fixed `u`, the per-step increment `Δ ln Ssat` must not exceed `u·Δ ln z`. Measured against `β₂·Δ ln z`: ``` step d ln Ssat beta_2 * d ln z step d ln Ssat beta_2 * d ln z 13->17 0.9427 1.1445 31->37 0.9935 0.7549 17->19 0.8697 0.4745 37->41 0.5918 0.4380 19->23 0.7030 0.8151 41->43 0.4243 0.2032 23->29 0.7688 0.9890 43->47 0.5744 0.3795 29->31 0.5994 0.2845 cumulative 13 -> 47: 6.4676 against 5.4834, a ratio of 1.18 ``` **So the statement a theorem needs is not "bound a sieve mean value"; it is "prove that `d ln Ssat/d ln z` falls below `β₂` and stays there", and over the measured range that derivative is 4.99 rising to 6.20.** It is going the wrong way by about 18% cumulatively and by more at the top. Whether it turns is exactly what nine levels cannot see — but nothing in them suggests it does. **Where the tractable version breaks, and where the wall comes back.** The exponential factor enters at exactly one step, and the step can be pointed at: ``` sup_x |R_H| = sup_x | sum_{e,a} Theta_e(a) S_H(a/e) e(ax/e) | <= sum_{e,a} |Theta_e(a)| |S_H(a/e)| <--- HERE ``` Taking the absolute value inside the `a`-sum discards every cancellation among the `phi(e)` frequencies of each modulus, and the `e`-sum runs over the divisors of some `q_i <= D^2`, at most `min(2^pi(z), z^{2s})` moduli: every divisor of `P(z)` at `z <= 19`, 6035 of 16383 at `z = 47`, polynomially many from `z = 223` on (`verify-0830-usup-convention.md` S3; corrected 2026-08-30). Any bound keeping `u` fixed must therefore retain the phase `e(2a*inverse(e1)/e2)` and exploit cancellation across `a`. But that phase sits beside `e(ax/e)`: keeping one means keeping the other, and the position `x` re-enters. **The moment the estimate is strong enough it is a statement uniform in `x`, which is this project's recurring wall; the moment it is only a mean value it is too weak by a factor measured to grow geometrically over `z = 13..47` and bounded at fixed `s` by `z^{2s+o(1)}` (`attack-0829n-rml-proof.md` §4.1, PROVEN; corrected 2026-08-30).** Both halves of that dichotomy are measured here rather than argued, which is new. That is also the precise slot Brüdern-Fouvry fill with Deshouillers-Iwaniec. `Theta_e(a)` carries their RIGHT factor — a modular inverse, by attack 1's Theorem B — so cancellation across `a` in `sum*_a Theta_e(a) S_H(a/e) e(ax/e)` is a Kloosterman-sum estimate uniform in `x`. §7b's location of the transfer failure at one factor, `e(-hN/(d1 d2))`, is the same sentence in their variables. **What survives, and it is not nothing.** The chain gives a legal, unconditional, worst-position exponent at every finite `z` that it reaches, with no hypothesis anywhere in it: every window of 198, 683, 1833, 4278, 10384, 20586, 52989, 103823, 170378 consecutive integers contains an `r` with `r` and `r+2` both `z`-rough, at `z = 13` through `43`. These are exact finite verifications rather than asymptotic theorems, and they are the only worst-position numbers in this corpus carrying no quantifier debt. What is reachable as a theorem at fixed `s` is `Ssat <= CAP(z) <= z^{2s} (2s ln z + 1) 6 prod_{p Ssup(H)}` only if the predicate is an up-set. It is not. Testing every integer `H` from 8 to `2·H_sup`: at `z = 13` and `z = 19` the published value IS the minimum and there are no later failures; at `z = 17` the true minimum is **659**, not the published **683**, and closure fails again at 21 values of `H` above it. The overshoot is 3.64% in `H`, 0.0126 in `u`. Every `u_sup` in this corpus, attack 1's and this file's alike, is therefore an UPPER estimate. The bias is one-signed and works against a plateau reading. **C2. "It would cross `β₂` near `z = 45`" (attack 1 §5(c) and READING 6) is 24% low in `z`.** Their fit `u_sup = 0.2865·θ(z)/ln z + 1.197` reaches `β₂` at `θ(z)/ln z = 10.7136`, first attained at `z = 59`; at `z = 45` the same fit gives `u = 3.9900`. A small slip. It does not touch the shape claim, and the shape claim is what this file tests. **C3. "A flat factor of about 80" (attack 1 §5(a) and READING 6) is not flat.** It was measured as `N/Ssup(nP)` at `H = nP`, a comparison window that changes with `z`. The `H`-free ratio `N/Ssat` falls monotonically from 44.4 at `z = 17` to **23.3 at `z = 47`**, a factor 1.9 across the range, with no sign of levelling. The Fourier advantage is real and large (23–44×) and it is eroding. **C4. Attack 1 over-priced its own next step by about 550×, which is why the ladder stopped at 23.** It states that `z = 29` "needs `prod(2p−1) = 7.8e10` evaluations of `Theta_e(a)`" and is "an overnight run as written". The true term-by-term count is **9.185e7**. `Π_{p **Corrected 2026-08-20 (mismatch adjudication #30).** Both occurrences of > `P(29)` above read `6.5e8` until this date. `P(z)` is `prod_{p primes strictly below `z`, as `sift-limit-lemmaV.js`'s `primesBelow(z)` > builds them — so `P(29) = 2 x 3 x ... x 23 = 223,092,870 = 2.23e8`. The S2 > cost table pins it: `max e` equals `P(z)` exactly at `z = 13, 17, 19` > (2310, 30030, 510510), and the same prime set gives the `5.06e10` quoted > beside it. Nothing produces `6.5e8`. The argument is unchanged: `max e` of > `3.432e6` is 77x below `P(29)` rather than 190x.