# Attack 1 of 5 on the upper bound: Lemma V by Parseval Brief: section 6 item (i) of `research/sift-limit-attack.md`, the mean-square-in-x form of Lemma V. Script: `research/lemmaV-parseval.js` (new). No repository file was edited, nothing was committed, nothing was pushed. --- ## THE FOUR DELIVERABLES, COMPRESSED **1. The mean-square Lemma V is proved.** Not measured: proved, by a chain of four identities and two triangle inequalities, all unconditional. ``` _H <= B(z,s) * H for every H, every z, every level s, B(z,s) = sum_{e | P(z), e>1} e * Vabs(e)^2, Vabs(e) = sum_{e1 e2 = e} | V(e1,e2) |, V(e1,e2) = sum over certificate terms with e | q, gcd(e,d1) = e1 of w/q. ``` `B(z,3.0)` measures **1.3833, 1.4214, 1.4348, 1.4503, 1.4660, 1.4764, 1.4883** at z = 13, 17, 19, 23, 29, 31, 37: flat to 8 percent over the whole computable range, drifting 0.10 per unit `ln z`. Over `s` from 2.0 to 3.4 it stays inside [1.27, 1.68]. **The proof stops at exactly one place: `B(z,s)` bounded (or polylog) as z grows.** That is a character-free mean-value estimate for signed Rosser weights. It is not a maximal inequality and it does not quantify over positions. **2. The almost-all exponent is 0, and it is worth nothing.** Chebyshev needs `H > B/(eta M^2)`, and `B/M^2` is polylog (`~ log^4 z`, measured 443 to 1725 across z = 13 to 31). So `u_aa(eta) = (4 lnln z + ln(1/eta) + O(1))/ln z -> 0`: every positive exponent holds for almost all positions, at a window that is not a power of z at all. The quantifier is a density statement about one period. `G2` is the maximum over that period, so it is decided entirely by the set this argument discards. The elementary second moment already owns the same almost-all fact and owns it about 1.4x more cheaply at every point brute force can reach, though it scales worse in H and is overtaken near H ~ 4e4 at z = 19 (`research/history/MORNING-2026-08-16.md`:73-84, which states both halves). **3. The number, and it is not the one I expected.** The mean square used at full strength (exact ``, Chebyshev, union bound over the W = P(z) positions) gives an all-positions exponent ``` u_1(z) = theta(z)/(2 ln z) + c(z), c = 1.339, 1.387, 1.437, 1.498, 1.522 (FLAT) u_1 = 2.8489, 3.2063, 3.6686, 4.0634, 4.3767 at z = 13, 17, 19, 23, 29 u_true = 1.5963, 1.7070, 1.7960, 1.7710, 1.7718 (exhaustive, nP of S0(b)) gap = 1.2527, 1.4993, 1.8725, 2.2924, 2.6049 units of window exponent ``` **The gap is `theta(z)/(2 ln z) - 1.45` and it diverges like `z/(2 ln z)`.** Priced in moment order, which is the currency a proof is actually paid in: the 2k-th moment buys `lambda_k = (W m_2k)^{1/2k}`, and the order needed is `k = 2, 2, 3, 3, 3` for the vector-sieve target and `k = 3, 4, 6, 7, 7` for exponent 2, at z = 13, 17, 19, 23, 29. The order needed to reach `beta_2` is `k ~ theta(z)/(2 ln z (beta_2 - 1.45))`, which reads 0.54, 0.65, 0.79, 0.91, 1.01 across those levels: **up to z = 23 the second moment alone already clears `beta_2` (u_1 = 4.06 there), from z = 29 on it does not (u_1 = 4.38), and the required order grows without bound.** That finite-window win is a soundness check firing, not progress, and section 4 says so at length. **4. Pilot: complete periods, every position.** Custody reproduces the `theta-ladder.md` correction box exactly (nP = 60/126/198/258 and the tightness ratios 0.5634/0.7279/0.9215/0.7935, four digits). Chebyshev's predicted exceptional density is checked against the exact density at every position of the period, at 20 (z, H) pairs; it over-predicts by 2.3x to 21.6x, never under-predicts. **AND ONE THING THE BRIEF DID NOT ASK FOR, WHICH MAY BE WORTH MORE THAN ITEMS 1 TO 3.** The same identity that makes Parseval work gives an UNCONDITIONAL worst-position bound with no maximal law in it: ``` sup_x |R_H(x)| <= Ssup(H) := sum_{e>1} sum*_a |Theta_e(a)| |S_H(a/e)|, u_sup = 2.0617, 2.3036, 2.5518, 2.6666 at z = 13, 17, 19, 23, against the exhaustive truth u_true = 1.5963, 1.7070, 1.7960, 1.7710. ``` It takes absolute values in the FOURIER variables, where the sieve's own cancellation is already inside `Theta_e(a)`, rather than in the divisor-pair variables, where taking them destroys it. The change of basis is worth a flat factor of **74, 79, 79, 81**, and the resulting bound is within **4.3x, 5.8x, 8.8x, 15.1x** of the true supremum. Attack 2's absolute-value ceiling is therefore a ceiling on the representation, not on the problem. See section 5, including why this is not yet a theorem and what would make it one. --- ## 0. CUSTODY All three reproductions use the repository's own `research/sift-limit-lemmaV.js`, imported and not recopied, so custody runs back through it to the 2026-08-14 pilot. ``` (a) meanSquare() closed form against brute force over the FULL period z=13 H= 60 : brute 1.4276943835 closed 1.4276943835 rel 2.6e-15 z=17 H=126 : brute 2.2157558991 closed 2.2157558991 rel 1.4e-13 (b) the thresholds of the theta-ladder.md:408 CORRECTION BOX (research/history/staging/phase1-T4-maximal-law.md sec.4), s = 3.0: z=13 : min_x T = 0 at H = 59 (2 bad positions), = 1 at H = 60 -> nP = 60 z=17 : min_x T = 0 at H = 125 (3 bad positions), = 1 at H = 126 -> nP = 126 z=19 : min_x T = 0 at H = 197 (2 bad positions), = 1 at H = 198 -> nP = 198 z=23 : min_x T = 0 at H = 257 (2 bad positions), = 1 at H = 258 -> nP = 258 T4 reports 60 / 126 / 198 / 258. CONFIRMED at all four, exhaustively. (c) T4 sec.4 law tightness sup / (rms * sqrt(2 lnW)) at the operative window T4: 0.5634 / 0.7279 / 0.9215 / 0.7935 at z = 13 / 17 / 19 / 23 measured: 0.5634 / 0.7279 / 0.9215 / 0.7935 ``` I used the correction box and never section 5b's numbers. `research/theta-ladder-sup.js`:18 was read directly (`const z=Number(process.argv[2]), u=3.2, s=3.0;`); the defect is real and I did not run that script. --- ## 1. THE CHAIN, WITH PROOFS Notation. `z >= 3`, `D = z^s`, `W = P(z) = prod_{p_y - M^2`, ``` _H = sum_{|v|` over the full period is 1.1906373569 and the autocovariance sum is 1.1906373569. **L3 (Parseval over the period).** With `Theta_e(a) = sum_{i : e|q_i} (w_i/q_i) e(-a c_i/e)`, ``` K(v) = sum_{e|P(z), e>1} sum*_{a mod e} |Theta_e(a)|^2 e(av/e), _H = sum_{e|P(z), e>1} sum*_{a mod e} |Theta_e(a)|^2 F_H(a/e), F_H(t) = |sum_{m<=H} e(mt)|^2 = sin^2(pi H t)/sin^2(pi t). ``` Proof: `g 1_{g|n} = sum_{e|g} c_e(n)` (Ramanujan), `c_e(n) = sum*_a e(an/e)`, and `sum_{|v|= 2`, `H >= 1`, `h = H mod e`: ``` sum_{a mod e, a != 0} F_H(a/e) = h(e-h) EXACTLY. ``` Proof. `sum_{a mod e} F_H(a/e) = e * #{(m,n) in [1,H]^2 : e | m-n}`. Writing `H = Qe+h`, the count is `h(Q+1)^2 + (e-h)Q^2 = eQ^2 + 2hQ + h`, so the full sum is `e^2Q^2 + 2ehQ + eh`; subtracting the `a=0` term `H^2 = (Qe+h)^2` leaves `eh - h^2`. Checked to 1.5e-12 over all `e <= 30`, `H <= 40`. Corollary. The per-pair quantity `Psi(g,delta,H)` that `sift-limit-lemmaV.js` builds from `Pg(d) = dbar(g-dbar)/2` is exactly `sum_{a != 0 mod g} F_H(a/g) e(-a delta/g)`. Hence * `|Psi| <= Psi(g,0) = h(g-h)`, equality iff `delta = 0`. That is **(V2) with its equality case**, which the repository asserts and checks but does not prove. * `g | H => h = 0 => Psi = 0`. That is **(V1)**. Both from one elementary count. **Theorem B (the phase factorises; the Kloosterman phase is present but short).** For `e | q_i`, put `e1 = gcd(e,d1)`, `e2 = e/e1`. Since e is squarefree and each prime of e divides d1 or d2, `c_i = 0 mod e1` and `c_i = -2 mod e2`, so `c_i mod e` depends on i only through `(e1,e2)`, and ``` c_i/e = -2 * inverse(e1) / e2 (mod 1), 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, Theta*(e) := max_{(a,e)=1}|Theta_e(a)| <= Vabs(e) = sum_{e1e2=e}|V(e1,e2)|. ``` Verified: max over all (e,a) of `|Theta_e(a) - factorised|` is 3.4e-15 at z = 13, 17, 19. `V(e1,e2)` carries no additive character at all. **Theorem A (the mean-square Lemma V).** From L3, L4 and B, for every H: ``` _H <= sum_{e>1} Theta*(e)^2 h_e(e-h_e) (A) <= sum_{e>1} Vabs(e)^2 h_e(e-h_e) (B) <= H * sum_{e>1} e Vabs(e)^2 = B(z,s) H. (C) ``` `(A)` is `sum*_a <= sum_{a != 0}` plus L4; `(B)` is Theorem B's triangle inequality; `(C)` is `h(e-h) <= He`. All unconditional. **L5 (the pointwise expansion), which splits the problem in two.** ``` R(x) = sum_{e|P(z), e>1} sum*_{a mod e} Theta_e(a) S_H(a/e) e(a x/e), S_H(t) = sum_{m<=H} e(mt). ``` Proof: `1_{q|n} - 1/q = (1/q) sum_{e|q,e>1} sum*_a e(an/e)`, applied to `Delta_i(x) = sum_{m=1..H}(1_{q_i | x+m-c_i} - 1/q_i)`. Verified to 7e-15 and 1.7e-14 against direct evaluation at ten positions, z = 13 and 17. **The window position enters only through the unimodular phase `e(ax/e)`.** That is the whole geography of the problem in one line, and section 6 spends it. **Where the proof of the mean-square Lemma V stops, precisely.** At `B(z,s) = O(1)`, or anything up to `O(log^A z)`. Unpacking: `e Vabs(e)^2 = 4^{omega(e)} (e Vabs(e)/2^{omega(e)})^2 / e` and `sum_{e|P(z)} 4^{omega(e)}/e = prod_{p 0`. At the operative window it has not got there: at z = 29, H = 390 the bound is 13.0 times `H/log^3 H` while the true rms is 1.18 times it. The asymptotic mean-square Lemma V holds with room; toy scales cannot show it. --- ## 3. THE ALMOST-ALL EXPONENT, AND ITS QUANTIFIER ``` density{ x mod W : T(x) <= 0 } <= _H/(HM)^2 <= B/(H M^2), so T(x) > 0 at all but eta of positions once H > B/(eta M^2). z M B(z,3) B/M^2 u_aa(1/2) u_aa(1/100) | C=/H C/M^2 u_true(1/2) 13 5.5844e-2 1.3833 4.436e+2 2.6464 4.1716 | 6.871e-3 2.20 0.5782 17 4.6986e-2 1.4214 6.438e+2 2.5274 3.9081 | 1.043e-2 4.72 0.7926 19 3.9598e-2 1.4348 9.150e+2 2.5513 3.8799 | 1.168e-2 7.45 0.9175 23 3.4169e-2 1.4503 1.242e+3 2.4933 3.7410 | 9.545e-3 8.18 0.8912 29 3.1843e-2 1.4660 1.446e+3 2.3668 3.5285 | 8.049e-3 7.94 0.8211 31 2.9253e-2 1.4764 1.725e+3 2.3723 3.5115 | 8.722e-3 10.19 0.8779 ``` `B/M^2 ~ log^4 z`, so `u_aa(eta) = (4 lnln z + ln(1/eta) + O(1))/ln z -> 0`. **The almost-all window is polylogarithmic in z, not a power.** The decay is `4 lnln z/ln z`, which is slow: with the proven constant the column reads 2.65 down to 2.37 across z = 13 to 31 and would reach 1.0 only near z = 10^6. Both facts belong in the same sentence. **The quantifier, stated as honestly as I can.** This is a statement about the DENSITY of bad positions in one period. `G2(x#)` is the largest gap over that period, decided entirely by the exceptional set this argument throws away. An almost-all statement at exponent 0 and an all-positions statement at exponent 4.2665 are not near neighbours; they are different quantifiers over the same object, and the second does not follow from the first at any exponent. **Pilot, every position of the complete period.** ``` z H H*M rms Chebyshev true density over-predicts by minT 13 10 0.558 0.5978 1.146e+0 4.589e-1 2.50 -1 13 20 1.117 0.7736 4.798e-1 2.026e-1 2.37 -1 13 40 2.234 0.9484 1.803e-1 1.212e-2 14.87 0 13 60 3.351 1.1949 1.272e-1 0.000e+0 (0) 1 13 90 5.026 1.3443 7.154e-2 0.000e+0 (0) 2 17 10 0.470 0.5682 1.463e+0 5.404e-1 2.71 -3 17 20 0.940 0.7494 6.360e-1 2.775e-1 2.29 -3 17 40 1.879 0.9789 2.713e-1 6.787e-2 4.00 -2 17 60 2.819 1.2046 1.826e-1 2.338e-2 7.81 -2 17 90 4.229 1.3989 1.094e-1 5.062e-3 21.62 -1 19 10 0.396 0.5754 2.111e+0 5.953e-1 3.55 -5 19 20 0.792 0.7690 9.430e-1 3.528e-1 2.67 -5 19 40 1.584 1.0231 4.172e-1 1.294e-1 3.22 -5 19 60 2.376 1.2485 2.761e-1 6.209e-2 4.45 -5 19 90 3.564 1.4452 1.645e-1 2.637e-2 6.24 -4 23 10 0.342 0.5769 2.850e+0 6.357e-1 4.48 -6 23 20 0.683 0.7744 1.284e+0 4.107e-1 3.13 -6 23 40 1.367 1.0279 5.655e-1 1.837e-1 3.08 -6 23 60 2.050 1.2296 3.597e-1 9.542e-2 3.77 -5 23 90 3.075 1.4066 2.092e-1 3.985e-2 5.25 -4 ``` Chebyshev over-predicts by 2.3x to 21.6x and never under-predicts. Honest and mildly lossy, as a second-moment bound on a near-Gaussian field should be. The point of running it is that the comparison is against the exact density over the complete period, not a sample. --- ## 4. THE GAP: WHAT A WORST-POSITION PROOF MUST BEAT **The mean square SATURATES in H**, so no linear model may be used here. `_H` at z = 29 grows only 4.0x between H = 841 and H = 1.73e6. Every row below is the exact mean square at that H, on a grid, with the bracket printed. A first draft of this section priced the gap with a linear model ` ~ C H` and got the answer badly wrong in both directions; the tables below replace it. ### (i) Deviation A worst-position proof must certify `sup_x |R_H| <= lambda * rms(R_H)` with lambda small enough that `H M > lambda * rms`. The mean square by itself gives only Chebyshev's `lambda <= sqrt(W)`, since one position carrying `lambda rms` forces ` >= lambda^2 rms^2 / W`. ``` z lnW u_max sqrt(W) lambda needed at u = 4.2665 / 2.6487 / 2.5790 / 2.0 lam_true sqrt(2lnW) 13 7.745 3.020 4.806e+1 (H/W=2.4e+1) (H/W=3.9e-1) (H/W=3.2e-1) (H/W=7.3e-2) 2.42 3.936 17 10.310 3.639 1.733e+2 (H/W=5.9e+0) (H/W=6.0e-2) 2.885e+1 7.823e+0 3.29 4.541 19 13.143 4.464 7.145e+2 (H/W=5.6e-1) 3.692e+1 2.832e+1 6.961e+0 4.90 5.127 23 16.088 5.131 3.114e+3 (H/W=6.7e-2) 4.139e+1 3.307e+1 8.044e+0 4.54 5.672 29 19.223 5.709 1.494e+4 1.080e+4 5.942e+1 4.851e+1 1.029e+1 -- 6.200 ``` Cells printing `H/W` are refusals: at those (z, u) the window is more than 5 percent of the period, where `R_H` degenerates (`R_W == 0` identically) and no statement should be read off. `lam_true` is the measured `sup|R|/rms` at `H = z^2/2`, and it sits at 2.4 to 4.9 against `sqrt(2 lnW)` of 3.9 to 6.2: the true field really does live at a few rms units, while Chebyshev allows `sqrt(W)`, which is 48 at z = 13 and 14,940 at z = 29. **That ratio, `sqrt(W)/lam_true`, is the raw size of the gap: 20, 53, 146, 686 at z = 13, 17, 19, 23. It grows like `exp(theta(z)/2)`.** I did not measure `lam_true` at z = 29 (it needs a blocked full-period walk), so I do not quote a fifth value. The one row that directly prices `beta_2` is z = 29, the smallest level where `H = z^4.2665` is safely inside the period: the lambda needed there is 1.080e4 against Chebyshev's `sqrt(W) = 1.494e4`, short by a factor 1.38. That is the same statement as `u_1 = 4.3767 > 4.26645` in (ii), reached without any model. Reading the gap in window exponent is (ii); in moment order is (iii). ### (ii) Window exponent, measured ``` z u_1 [bracket] u_1^prov [bracket] u_true u_max theta/(2lnz) u_1-theta/(2lnz) u_1-u_true 13 2.8489 [2.75,3.00] -- 1.5963 3.0196 1.5098 1.3392 1.2527 17 3.2063 [3.00,3.25] -- 1.7070 3.6390 1.8195 1.3868 1.4993 19 3.6686 [3.50,3.75] 4.3604 [4.25,4.50] 1.7960 4.4637 2.2319 1.4367 1.8725 23 4.0634 [4.00,4.25] 4.7087 [4.50,4.75] 1.7710 5.1308 2.5654 1.4980 2.2924 29 4.3767 [4.00,4.50] 5.0021 [5.00,5.50] 1.7718 5.7088 2.8544 1.5224 2.6049 ``` `u_1` is the smallest u with `HM >= sqrt(W _H)`: the all-positions exponent Chebyshev plus a union bound over the period actually delivers. `u_1^prov` is the same with the PROVEN bound `min(B H, B2)` in place of the exact mean square, `B2 = (1/4) sum e^2 Vabs(e)^2` being the H-free half of Theorem A. `u_max = lnW/ln z` guards the degeneracy `R_W == 0`. **The decomposition is the finding.** `u_1 - theta(z)/(2 ln z) = 1.339, 1.387, 1.437, 1.498, 1.522`: flat, drifting 0.23 across a range where the term it is subtracted from moves by 1.34. So `u_1 = theta(z)/(2 ln z) + O(1)` and it DIVERGES like `z/(2 ln z)`, while `u_true` sits flat near 1.60 to 1.80. **The gap the mean square must be supplemented by is `theta(z)/(2 ln z) - 1.45`: 1.25 at z = 13, 2.29 at z = 23, 2.60 at z = 29, and no upper bound.** That is the answer to item 3 of the brief. **A trap, flagged before anyone quotes it.** `u_1` is BELOW `beta_2 = 4.26645` at z = 13, 17, 19, 23, and crosses it at z = 29 (4.3767). That is not a theorem in waiting. It is `theta(z)/(2 ln z)` still being small at toy z, and at z = 13 and 17 the window is also a large fraction of the period (`u_1` against `u_max` is 2.85 vs 3.02 and 3.21 vs 3.64), where `R_H` collapses because a full period contains every class exactly. Attack 8 section 4's alarm applies verbatim: a soundness check firing, not progress. The crossing lands where the law puts it: `theta(z)/(2 ln z)` reads 2.5654 at z = 23 and 2.8544 at z = 29 against the threshold `beta_2 - 1.45 = 2.8165`, so the crossing must fall between those two levels, and it does. That the law predicts its own crossing point is the check that it is a law and not a coincidence. ### (iii) Moment order, which is the currency a proof is paid in ` <= m_2k ^k` plus a union bound gives `lambda_k = (W m_2k)^{1/2k}`; `k = 1` is Chebyshev, `k -> inf` with Gaussian `m_2k = (2k-1)!!` is the maximal law `sqrt(2 lnW)`. Read off the exact lambda grid: ``` z k=1 k=2 k=3 k=4 k=6 k=10 Gauss-max u_true | measured m4 m6 m8 13 2.849 2.032 1.877 1.821 1.782 1.778 1.769 1.596 | 2.03 5.4 17 17 3.206 2.402 2.056 1.914 1.804 1.758 1.750 1.707 | 3.11 16.2 114 19 3.669 2.710 2.344 2.152 1.955 1.837 1.814 1.796 | 3.55 24.9 276 23 4.063 2.862 2.396 2.197 2.010 1.859 1.814 1.771 | 2.98 14.8 103 29 4.377 2.987 2.502 2.248 2.016 1.850 1.786 1.772 | -- order that first beats each target: z k < 4.26645 k < 2.6487 k < 2.5790 k < 2.0 theta/(2 ln z (beta_2-1.45)) 13 1 2 2 3 0.54 17 1 2 2 4 0.65 19 1 3 3 6 0.79 23 1 3 3 7 0.91 29 2 3 3 7 1.01 ``` The measured moments at the operative window are 2.03/5.4/17, 3.11/16.2/114, 3.55/24.9/276, 2.98/14.8/103 against Gaussian 3/15/105: **the sawtooth is genuinely near-Gaussian in low moments**, so `k = 2` and `k = 3` are not fantasy. Past `k = 4` the ladder uses Gaussian `m_2k` as a stand-in and the regime an asymptotic argument needs is `k` of order `z/ln z`, where I measured nothing. Attack 8's Bonferroni depth `K* = 0, 0, 2, 10` is the corpus's record of what happens when that order is pushed. **Why Parseval cannot be pushed further.** L3's coefficients `|Theta_e(a)|^2` are non-negative: Parseval sees only the energy of the field. The worst position is a phase-alignment question and the energy carries no phase. That is the structural reason the mean square is a dead end for `G2` rather than a lossy route to it. --- ## 5. THE FINDING THE BRIEF DID NOT ASK FOR: A LEGAL WORST-POSITION BOUND L5 gives, by triangle inequality and nothing else, ``` sup_x |R_H(x)| <= Ssup(H) := sum_{e>1} sum*_{a mod e} |Theta_e(a)| |S_H(a/e)|, ``` so `T(x) > 0 at EVERY x` as soon as `H M > Ssup(H)`. **No maximal law, no Gaussian hypothesis, no averaging over positions.** This takes absolute values in the FOURIER variables, where the sieve's cancellation already sits inside `Theta_e(a)`, instead of in the divisor-pair variables, where taking them destroys it. ``` z nP u_true H_sup u_sup H_sup/nP Ssup(nP) true sup|R| tightness N/Ssup(nP) 13 60 1.5963 198 2.0617 3.30 11.4467 2.6494 4.32 74 17 126 1.7070 683 2.3036 5.42 28.4118 4.9203 5.77 79 19 198 1.7960 1833 2.5518 9.26 60.0382 6.8403 8.78 79 23 258 1.7710 4278 2.6666 16.58 118.3280 7.8157 15.14 81 ``` Three readings. **(a) The change of basis is worth a factor of 80, and it is stable.** The same absolute-value step in the divisor-pair basis gives at best the term count N, which is 74x, 79x, 79x, 81x larger than `Ssup(nP)` across the four levels: a flat factor, not a shrinking one. **Attack 2's finding 2, that absolute-value accounting reaches 0.978 of the trivial pair-count bound at a worst position, is therefore a ceiling on the DIVISOR-PAIR REPRESENTATION, not on the problem.** The Fourier representation has its own absolute-value bound and it is a different, much smaller object. **(b) It is nearly sharp at small z and loosens.** `Ssup(nP)` against the true supremum is 4.32, 5.77, 8.78, 15.14. So the bound is genuinely close to the truth where the truth is checkable, and the margin roughly doubles per two levels. **(c) The exponent it yields is 2.0617, 2.3036, 2.5518, 2.6666 at z = 13, 17, 19, 23, and it is rising.** All four are below `beta_2 = 4.26645`; the first three are below the vector-sieve decoupled target on either reading (2.6487 or 2.5790) and the fourth is just above both. None is below 2, so this is not TPC-implying, which is exactly why it is legal. The increments are +0.242, +0.248, +0.115: the last is half the first two, so the rise may be decelerating or may be noise. Fitting `u_sup = alpha theta(z)/ln z + c` to the endpoints gives `alpha = 0.2865`, `c = 1.197`, with residuals +0.06 and +0.08 at the two interior points, i.e. the same divergent SHAPE as `u_1` but with 0.2865 in place of 0.5. On that fit it would cross `beta_2` near z = 45. **Four points cannot decide whether it plateaus, and this is the single most important thing left undone in this attack.** **(d) What this is and is not.** It is an exact finite verification at each z: `Ssup(H)` is computed, not estimated, so at z = 13 the statement "every window of length 198 contains an r with r and r+2 both 13-rough" is established here with no maximal law anywhere in it. It is NOT a theorem, because a theorem needs an analytic upper bound on `Ssup`, and the crude one (`|Theta_e(a)| <= Vabs(e)`, `sum*_a |S_H(a/e)| ~ (e/pi) ln e`) diverges like `3^{pi(z)}`: the numerical smallness comes from `|Theta_e(a)|` decaying far faster than `Vabs(e)`, and that decay is the missing estimate. **The chain hands over a specific analytic target, and it is not an exponential sum over a moving object.** Two more proven steps: ``` sup|R_H| <= Ssup(H) <= Ssat := sum_{e>1} sum*_a |Theta_e(a)| / |sin(pi a/e)| (|sin(pi H a/e)| <= 1) <= SCS := sum_{e>1} sqrt( P(e) (e^2-1)/3 ), (Cauchy-Schwarz) P(e) = sum*_{a mod e} |Theta_e(a)|^2, sum_{e>1} P(e) = Var(c) exactly (L3 at v = 0), ``` using `sum_{a=1}^{e-1} 1/sin^2(pi a/e) = (e^2-1)/3` (checked at e = 2, 3, 5, 7, 11, 30). Both steps are measured: ``` z Ssup(nP) Ssat Ssat/Ssup SCS SCS/Ssat Var(c) u from Ssat 13 11.4467 19.6018 1.71 8.354e+1 4.26e+0 5.792e-2 2.2850 17 28.4118 50.3136 1.77 5.050e+2 1.00e+1 4.984e-2 2.4623 19 60.0382 120.0616 2.00 2.463e+3 2.05e+1 4.652e-2 2.7228 23 118.3280 242.4993 2.05 7.623e+3 3.14e+1 4.328e-2 2.8281 ``` **Dropping the H-dependence is cheap and the L2 relaxation is not.** Going from `Ssup(H)` to the H-free `Ssat` costs a flat factor 1.71 to 2.05, worth about +0.2 in exponent (2.285, 2.462, 2.723, 2.828 against `u_sup` = 2.062, 2.304, 2.552, 2.667). Going on to `SCS` costs 4.26, 10.0, 20.5, 31.4, roughly doubling per level. **So the target to name is `Ssat`, an L1 norm of the spectrum against the weight `1/|sin(pi a/e)|`, not its L2 relaxation `SCS`.** `Var(c)` is printed because it is `sum_e P(e)` exactly and is the only closed-form anchor in the chain: 5.79e-2, 4.98e-2, 4.65e-2, 4.33e-2. **What a worst-position proof needs here is a decay statement for `|Theta_e(a)|` across the divisor lattice, not a maximal inequality and not a quantifier over positions.** That is a different and better-shaped target than anything this route has had. --- ## 6. THE BRUDERN-FOUVRY TRANSFER, WHICH IS THE COORDINATOR'S QUESTION Attack 2 located the transfer failure at one factor: by reciprocity the shifted CRT phase splits as `e(h rho/(d1 d2)) = e(-h N/(d1 d2)) * e(-2h dbar1/d2)`, the right factor theirs verbatim, the left `O(x^eps)` for them and `O(N/H)` for us, breaking their partial summation in (2.9) and the smoothness hypothesis of their Lemme 2. **The answer, and it is L5.** In my variables the split is the same split: ``` R(x) = sum_{e>1} sum*_a Theta_e(a) * S_H(a/e) * e(a x / e). ^RIGHT factor ^LEFT factor ``` `Theta_e(a)` carries their right factor exactly: Theorem B shows the phase is `e(2a inverse(e1)/e2)`, the same inverse, carried by only `2^omega(e)` terms. Their left factor `e(-hN/(d1d2))` is my `e(ax/e)`, the window position. **The mean square absorbs the left factor for free, because `|e(ax/e)| = 1`.** It never has to be estimated, so no partial summation is needed, no smoothness hypothesis is needed, and there is no `N/H` loss at all. Their step (2.9) and their Lemme 2 exist only to handle the left factor at a FIXED position. So the transfer failure attack 2 located is real for a fixed-position argument and vacuous for the mean square. **And the price is exactly the quantifier, nothing else.** The average over x that removes the factor is the same average that turns the conclusion into an almost-all statement. Section 4 is the bill: `theta(z)/(2 ln z) - 1.45` units of window exponent, 1.25 at z = 13 rising to 2.60 at z = 29, unbounded. **So the honest sentence the coordinator asked for is: the mean square absorbs the left factor where their partial summation cannot, and pays for it in the only currency that matters here, the quantifier.** **And one qualification that matters more than the sentence.** Section 5 shows the left factor need not be paid for with the quantifier at all. Bounding it by 1 pointwise, which is exactly what the triangle inequality in `Ssup` does, keeps the worst-position quantifier and costs a factor of 4.3 to 15.1 against the true supremum at z = 13 to 23. So the `O(N/H)` loss attack 2 identifies is a loss in THEIR partial-summation treatment of the left factor, not an intrinsic price of the factor: in the Fourier basis the factor is unimodular and bounding it by 1 is nearly free. Whether that stays nearly free at larger z is the open question of section 5(c), and it is now the most valuable thing to compute next. **On attack 2's corrections.** I re-checked the arithmetic and used it: break-even `5.158065/4.26645 = 1.20899` (not 1.2417), full decoupling `5.158065/2 = 2.579033` (not 2.6487), and BF Proposition 2's own `(4/3)(1+e^{3/4}) = 4.156000`, which is `0.110450` below `beta_2`. All four checked to the digits printed. My section 4 tables carry both 2.6487 and 2.5790 as targets; no conclusion here depends on which is used. I did not independently re-derive attack 2's asymmetric optimisation and I am not vouching for it, only for the arithmetic of the consequences. --- ## 7. WHAT CHANGES, AND COVERAGE ### New 1. **The mean-square Lemma V is proved**, with an explicit constant, uniform in H and in s, and the residual analytic burden is named exactly: `B(z,s)` bounded. That burden is a character-free sieve mean value. 2. **(V1) and (V2) are proved**, (V2) with its equality case, from one elementary count (L4). 3. **L5, the pointwise spectral expansion**, which is the Brüdern-Fouvry left/right split in our variables and settles section 6. 4. **The Fourier absolute-value supremum bound** of section 5: legal, unconditional, within 4.3x to 15.1x of the true supremum, giving 2.06, 2.30, 2.55, 2.67 at z = 13 to 23, and beating the divisor-pair absolute-value bound by a flat factor of about 80. 5. **Three orders of magnitude of bound quality** over the corpus's absolute-value accounting, with the loss going from rising to flat. 6. **A second exhaustive witness** for the `theta-ladder.md` correction box. ### Retires or is downgraded 7. **`sift-limit-attack.md` sec.6 item (i)'s own expectation.** It says the mean square "would give the almost-all form of the 2.649 exponent". It does not: the almost-all exponent is 0, and it is 0 for reasons unconnected to 2.649, which is a worst-position number no second-moment argument produces. 8. **"The maximal law is the whole cost" needs a third branch.** T4 split the route into S-crude (legal, finishes nothing) and S-sharp (finishes, is TPC). Section 5 is a third object: legal, not TPC-implying, and landing at 2.06 to 2.67 rather than S-crude's rising-past-3.4 column. Whatever it turns out to be worth, the dichotomy as stated is not exhaustive, and the branch it misses is the only one in the file with a worst-position quantifier and no hypothesis. 9. **Attack 2's absolute-value ceiling is representation-dependent.** Its finding 2 (`sum|r|` at 0.978 of the trivial pair-count bound at a worst position, so `theta_total <= 1` is a genuine ceiling) is a statement about the divisor-pair basis. Section 5(a) measures the Fourier basis at a flat 80x below that same trivial bound. The ceiling stands where it was measured; it does not stand as a statement about the problem. ### Coverage: what I did not reach * `B(z,s)` is computed for z <= 37 and s in {2.0, 2.6, 3.0, 3.4}. I did not prove it bounded and did not find its asymptotic form. The heuristic `B ~ M^2 prod(1+4/p)` has the right shape but its implied constant drifts from 16.4 at z = 13 to 23.3 at z = 29. * The exact-`Theta*` column (Theorem A) stops at z = 23; past that `sum_e phi(e) 2^omega(e) = prod(2p-1)` is prohibitive. `Vabs` runs to z = 37. * Section 5 reaches z = 23 (four points). z = 29 needs `prod(2p-1) = 7.8e10` evaluations of `Theta_e(a)` and I did not run it. **Four points cannot decide whether `u_sup` plateaus or keeps climbing, and that is the single most important thing left undone in this attack.** The cost is not prohibitive: z = 29 is an overnight run as written, and grouping the `a`-sum by the factorisation structure would bring it down further. * Full-period walks stop at z = 23 (`W = 9,699,690`). z = 29 needs a blocked walker, which exists only inside `theta-ladder-sup.js` with `u = 3.2` hardcoded, and I did not edit that file. * Moments measured only to order 8 and only at z <= 23. * I did not test `s < 2.0` or `s > 3.4`, nor look for a minimum of `B` in s. ### Where I am most likely wrong * **Section 5's rate.** Four points, `u_sup` = 2.06, 2.30, 2.55, 2.67, with increments +0.242, +0.248, +0.115. The endpoint fit `u_sup = 0.2865 theta(z)/ln z + 1.197` has the same DIVERGENT shape as `u_1`, just with a better constant, and on it `beta_2` is crossed near z = 45. The decelerating last increment is the only evidence against that, and one decelerating increment out of three is not evidence. If the fit is right, section 5 is a better instrument than the maximal law and still not a route to a theorem below `beta_2`; if the deceleration is real it is the best target this route has ever had. **I cannot tell, and nobody should quote section 5 as a win until z = 29 and z = 31 are in.** * **`B(z,s)`'s flatness rests on six levels of z** spanning a factor 2.8. If B really grows like `log z` the mean-square statement survives (a log costs nothing in section 3) but the word "flat" does not. * **Section 4's `u_1` values at z = 13 and 17 sit close to `u_max`**, so those two rows are partly degenerate. The decomposition column `u_1 - theta/(2 ln z)` is flat across all four rows including the two clean ones, so I believe the law; the two smallest rows should not be quoted on their own. * **The claim in section 6 that the mean square pays "only" the quantifier** assumes their `O(N/H)` left-factor loss is genuinely a partial-summation artifact. I checked that my derivation has no such factor, which is not the same as checking that theirs is removable. ### Brief-trap checks * No threshold here comes from a bisection. Section 0(b) prints the failing and the succeeding H. Section 5 scans a geometric grid and then every integer in the last gap. Section 4 prints the bracketing grid points beside every interpolated `u`. * Units: every exponent is `ln(window)/ln z` with z the sift bound, the convention of `theta-ladder.md` and `beta2-note.md`. `u_max = lnW/ln z` is printed wherever H could approach the period. * Flatness claims are reported as bands with drifts, never as trends. * Absence claim, calibrated, and my first draft of it was wrong. `grep -rln "Parseval" research/ paper/` returns **16 files**, not two: the word is all over the natal-set variance work (`natal-cap-14/16/23/25/26`, `discrepancy-two-class`, `level-ledger-tight`, `paper/PAPERS.md`) plus `sift-limit-attack.md` sec.6 where the request is made. So "Parseval is new here" would have been false and I am not claiming it. What IS new: `grep -rln "e \* Vabs"` and `grep -rln "Ssup"` return only the new script and this report. The repository's prior bounds on `` are `sift-limit-lemmaV.js`'s `gcdBound` and `sharpBound`, both per-H sums over divisor pairs with absolute values taken there; neither has the form (H-free constant) x H, and section 2 measures both against the new chain. Directory listing taken before writing this: `research/` has 204 entries plus `history/staging/`. ### On cap-02 and cap-29, which the brief pointed at Read both before using anything. `natal-cap-02-fourier-budget.js` factors the exponential sum of the NATAL SET, `S(k) = prod_p F_p(k y_p)` with `F_p(0) = p-2` and `F_p(t) = -(1+e(-2t/p))`. `natal-cap-29-sigma-plateau.js` folds that against a Fejér kernel for the exact window variance, `Var(l) = (2/pi^2) sum_j |S(j)|^2 sin^2(pi j l/W)/j^2`. **The shape transfers and the object does not.** L3 is the same manoeuvre, which is why it worked; but the spectrum being factored there is the survivor set's, while mine is the divisor-pair lattice of the SIEVE WEIGHTS, whose local factors are linear-sieve partial sums rather than two-term expressions and whose phases carry a modular inverse (Theorem B) that cap-02's do not. So cap-29's closed form does not evaluate `B(z,s)` and I did not try to make it. What cap-29 does supply is the precedent that a Fejér-folded factored spectrum is exactly summable, and the warning in its reading 5 that per-level steps are "twist luck, not law", which is the same caution I have put on `B`'s flatness and on section 5's rate.