# T4: is the SIEVE Gaussian maximal law TPC-implying? **YES — in the only form of it that would deliver anything.** Stated for the operative remainder R_H instead of for the H-free potential rho, the sieve maximal law gives an all-positions window of `need/z^2 = 0.49 to 0.61`, flat and below the zone budget at every z from 13 to 43. That is the Gap Reformulation, hence the Zone Postulate, hence TPC. So it cannot be proved by any soft argument, for exactly the reason attack 8 gave for the elementary count. The published above-2 theta column is **not** the maximal law's verdict: it is the maximal law plus two lossy steps taken afterwards, and those steps cost a factor 1.48 rising to 8.40 across z = 13 to 43. **The certificate route retires as a road to TPC.** It survives only in the weaker form the corpus actually wrote down, which is legal precisely because it finishes nothing. **And the reason it has looked close is now measurable.** At the operative window the sharp maximal law is not a generous hypothesis but an approximately exact description of the sawtooth: the ratio `sup|R_H| / (rms * sqrt(2 lnW))` reads 0.56, 0.73, 0.92, 0.79, 0.85 at z = 13, 17, 19, 23, 29, and is transiently above 1 at z = 19. A hypothesis that is true to a few percent, whose truth would give TPC, is not a hypothesis with a soft proof waiting to be found. It is the wall wearing a different hat. **One separate defect, found on the way and worth its own line:** `research/theta-ladder-sup.js`:18 hardcodes u = 3.2, so every exact-supremum number in `theta-ladder.md` sec.5b measures the supremum at a window 39x to 62x longer than the one it draws a conclusion about. Corrected self-consistently, the unconditional requirement at z = 29 is `0.4637 z^2`, not the published `1.0060 z^2`. **The crossing that reading 8 calls "the headline of the run", and that TODO 0 cites as the reason this is not a road to TPC, does not happen.** The route is not refuted by the unconditional data. It is refuted by sec.2. Scratch scripts (not committed): `t4-elem.js`, `t4-sieve.js`, `t4-sup.js`, `t4-probe.js` in the session scratch directory. No repository file was edited. --- ## 0. CUSTODY Three independent reproductions before anything new was computed. ``` ELEMENTARY, attack 8 sec.4, all nine rows, digit for digit: p_n L_need theta L/p_n^2 attack 8 says 13 120 1.8665 0.7101 120 1.8665 0.7101 OK 19 191 1.7838 0.5291 191 1.7838 0.5291 OK 23 312 1.8316 0.5898 312 1.8316 0.5898 OK 31 462 1.7867 0.4807 462 1.7867 0.4807 OK 47 985 1.7902 0.4459 985 1.7902 0.4459 OK 71 1800 1.7584 0.3571 1800 1.7584 0.3571 OK 97 2935 1.7453 0.3119 2935 1.7453 0.3119 OK 199 9744 1.7351 0.2461 9744 1.7351 0.2461 OK 401 26378 1.6984 0.1640 26378 1.6984 0.1640 OK SIEVE crude column against theta-ladder.md sec.2: z=19 need 576.3 theta 2.1588 | ladder 576.28 2.158846 OK z=23 need 938.0 theta 2.1827 | ladder 938.05 2.182686 OK z=29 need 1436.0 theta 2.1589 | ladder 1435.99 2.158887 OK z=31 need 2285.8 theta 2.2523 | ladder 2285.76 2.252324 OK z=37 need 4760.3 theta 2.3451 | ladder 4760.25 2.34513 OK z=41 need 7040.0 theta 2.3857 | ladder 7040.03 2.38567 OK z=43 need 9485.4 theta 2.4347 | ladder 9485.41 2.43473 OK MEAN SQUARE, my bucketed evaluator against the repo's closed form (research/sift-limit-lemmaV.js meanSquare), worst relative error over {H = 7, 50, 200, 651, 3806} plus H = z^3.2, and the plateau: z=13 4.0e-13 z=17 2.2e-12 z=19 3.0e-12 z=23 1.1e-11 z=29 2.0e-11 z=31 1.3e-11 FULL-PERIOD WALK, my prefix-sum walker against the repo's fullPeriodArray: z=13 H=200 sup 2.168831 minT 9 | repo identical z=17 H=340 sup 5.975358 minT 10 | repo identical z=19 H=600 sup 9.758594 minT 14 | repo identical z=23 H=1000 sup 11.169236 minT 23 | repo identical z=29 H=700 sup 13.289889 minT 9 | repo identical ``` The bucketed evaluator is the only new machinery. It is the repo's own closed form regrouped: a divisor pair (i,j) enters `_H` only through `h = H mod g`, `g = gcd(q_i,q_j)`, so one O(N^2) pass bucketed by `(g, x = (c_j-c_i) mod g)` serves every H. The buckets are tiny (3,068 nonzero entries at z = 19; 29,834 at z = 31), which is what makes a full H-scan affordable where the repo's per-H O(N^2) call is not. **No S6 column was used.** The crude `need` is recomputed from `row()`'s `plateau`, `M` and `lnW`, which `sift-limit-lemmaV.js`:424 lists as UNAFFECTED by the `elementaryVariance` bug, and the elementary side is computed here from `paper/variance-note.md` Theorem 2 directly, not from the flagged function. --- ## 1. THE TWO STATEMENTS, SIDE BY SIDE This was the first deliverable asked for, and stating it is most of the answer. **E — the elementary version** (attack 8 sec.4). Let `N_H(x) = #{r in (x, x+H] : r and r+2 both p_n-rough}`, `E = delta H`, `Var` from `paper/variance-note.md` Theorem 2. The hypothesis is a sub-Gaussian lower tail plus a union bound over the W = p_n# positions of the period: ``` P(N_H(x) = 0) <= exp(-E^2/(2 Var)) for every x, => min_x N_H(x) >= 1 as soon as E^2/(2 Var) > ln W, equivalently E > sqrt(2 lnW * Var). ``` **S — the sieve version.** The certificate splits exactly as `T(x) = H*M + R(x)` (`sift-limit-lemmaV.js` header), `R` a signed sawtooth over the divisor-pair lattice. The corpus states the hypothesis in exactly one place, `sift-limit-lemmaV.js`:466, echoed by `theta-ladder.md` sec.1: ``` S-crude: sup_y |rho(y)| <= sqrt() * sqrt(2 ln W), then |R_H| <= 2 sup|rho| => H*M > 2 sqrt(2 lnW ) (right side FREE of H) ``` The exact analogue of E is not written down anywhere in the corpus. It is: ``` S-sharp: sup_x |R_H(x)| <= sqrt(2 lnW) * sqrt(_H) => H*M > sqrt(2 lnW * _H) (right side DEPENDS on H) ``` **The difference is not a difference of hypothesis class.** Both say: the supremum of a lattice sawtooth sum over W positions is at most `sqrt(2 ln W)` times its own rms. rho and R_H are members of the same family over the same divisor-pair lattice; R_H is a sawtooth sum of 2N terms. What separates them is where the template is applied and what is done afterwards: | | E | S-crude | S-sharp | |---|---|---|---| | field | the count N_H | the potential rho | the remainder R_H | | fluctuation scale | Var(N_H), grows like H | ``, H-free | `_H`, grows like H | | extra lossy step | none | triangle inequality, factor 2 | none | | criterion | `E > sqrt(2 lnW Var)` | `HM > 2 sqrt(2 lnW )` | `HM > sqrt(2 lnW _H)` | S-crude pays twice: the factor 2 from `|R| <= 2 sup|rho|`, and the replacement of the operative `_H` by the H-free saturation plateau `2` that `_H` only reaches at H far beyond the window in question. At z = 19, H = 191: `2 sqrt() = 4.4508` against `sqrt(_191) = 1.4659`, a factor of 3.04. --- ## 2. THE ANSWER: THE SHARP COLUMN IS BELOW THE BUDGET AND FLAT Fixed family s = 3.0 throughout, matching the ladder. `need/z^2` is the column the Gap Reformulation must get below 1: the sift is over p < z, so a survivor pair inside the zone (w, z^2), w the largest prime below z, is a twin pair. ``` z crude need th_cr cr/z^2 | sharp need th_sh sh/z^2 | cr/sh C_crit 13 147.5 1.9470 0.8728 | 100 1.7954 0.5917 | 1.48 2.225 17 305.4 2.0195 1.0568 | 142 1.7492 0.4913 | 2.15 1.720 19 576.3 2.1588 1.5963 | 191 1.7838 0.5291 | 3.02 1.358 23 938.0 2.1827 1.7732 | 325 1.8446 0.6144 | 2.89 1.418 29 1436.0 2.1589 1.7075 | 436 1.8049 0.5184 | 3.29 1.660 31 2285.8 2.2523 2.3785 | 570 1.8479 0.5931 | 4.01 1.445 37 4760.3 2.3451 3.4772 | 709 1.8178 0.5179 | 6.71 1.564 41 7040.0 2.3857 4.1880 | 913 1.8356 0.5431 | 7.71 1.509 43 9485.4 2.4347 5.1300 | 1129 1.8688 0.6106 | 8.40 1.386 ``` Three readings, in order of what they change. **(a) The sharp column is below the zone budget at every level and is flat.** `sh/z^2` has mean 0.5566, sd 0.0461, range [0.4913, 0.6144], and an OLS slope against ln z of **+0.0104 +/- 0.0414, t = +0.25**: no trend, and the fluctuation band is stated as brief Trap 3 requires. Structurally `need_sharp ~ 0.55 z^2`, so `th_sh = 2 + ln(0.55)/ln z` rises toward 2 **from below** — exactly the finite-size mechanism `theta-ladder.md` sec.0 warns about, operating here in the direction the ladder did not test. The rise of th_sh is not the requirement getting worse; it is a constant below 1 read through a logarithm. The measured `d th_sh/d ln z = 0.0640 +/- 0.0233` (t = 2.75) is exactly what a constant ratio predicts: `d/d ln z [2 + ln c / ln z] = -ln(c)/(ln z)^2 = 0.062` at c = 0.55, ln z = 3.1. The slope is the signature of a constant, not of a drift. **So the decisive column is `need/z^2`, not theta.** Below 1 means TPC. **(b) The crude column's rise above 2 is the lossy steps, not the requirement.** `d th_cr/d ln z = 0.3746 +/- 0.0404 (t = 9.26)` against `d th_sh/d ln z = 0.0640 +/- 0.0233 (t = 2.75)`. **83% of the published upward drift in theta is the growth of the triangle-inequality and plateau losses, and the residual 17% is the finite-size term of a constant ratio, not a real drift** (the predicted finite-size slope is 0.062, the measured 0.064). The loss factor itself grows as a clean power: `d ln(crude/sharp)/d ln z = 1.367 +/- 0.142, t = 9.66`, so `cr/sh ~ z^{1.4}`. `theta-ladder.md` sec.3 attributes the drift to `` tracking the divisor-pair count N at a flat cancellation factor. That diagnosis is correct about `` and it is therefore a diagnosis of the lossy step, because `_H` at the operative window does **not** grow that way: `_H / H` reads 1.99e-2, 1.52e-2, 1.13e-2, 1.18e-2, 1.15e-2, 1.08e-2, 1.07e-2, 1.06e-2, 1.01e-2 at z = 13..43, essentially constant. The sawtooth remainder at the operative lag behaves like a random walk in H; the saturated potential does not, and the ladder prices the saturated potential. **(c) The implication is robust, not delicate.** `C_crit` is the largest constant C for which a maximal law `sup|R_H| <= C sqrt(2 lnW _H)` still yields `need < z^2`. It reads 2.23, 1.72, 1.36, 1.42, 1.66, 1.45, 1.56, 1.51, 1.39 — flat near 1.5 with no downward trend. A maximal law that is loose by 50% is still TPC-implying. The implication does not depend on the constant `sqrt(2 ln W)` being exactly right. ### Why this settles the binary `S-sharp` at level z gives `T(x) > 0` at every x for `H = need_sharp < z^2`. The Brüdern-Fouvry pointwise inequality makes `T(x)` a lower bound for the count of twin-admissible r in the window, so every window of length H holds one; the window anchored at x = w is the zone (w, z^2); a z-rough pair below z^2 is a twin prime pair. That is the strong Zone Postulate at every z, hence TPC. Attack 8 sec.1.3 already showed no quantifier over z buys legality. Checked against the more conservative budget `w^2` (attack 8 sec.1.1's form, w the top sifted prime, rather than the ladder's `z^2`): `need_sharp/w^2` = 0.826, 0.840, 0.661, 0.900, 0.824, 0.678, 0.738, 0.667, 0.672 at z = 13..43. Below 1 at every level, with less margin. The conclusion survives the stricter convention. --- ## 3. THE ELEMENTARY SIDE, REPRODUCED AND EXTENDED Attack 8's nine rows reproduce exactly (sec.0). Extended to twenty-three levels the drift is locally noisy and unambiguous in the aggregate: `d theta/d ln p = -0.0362 +/- 0.0029 (t = -12.3)` and `d ln(L/p^2)/d ln p = -0.447 +/- 0.015 (t = -29.4)`. ``` p_n 13 17 19 23 29 31 37 41 43 47 theta 1.8665 1.7755 1.7838 1.8316 1.7594 1.7867 1.7674 1.7717 1.7877 1.7902 L/p^2 0.7101 0.5294 0.5291 0.5898 0.4447 0.4807 0.4317 0.4283 0.4500 0.4459 p_n 53 59 61 67 71 97 127 199 251 theta 1.7722 1.7500 1.7662 1.7566 1.7584 1.7453 1.7396 1.7351 1.7165 L/p^2 0.4048 0.3608 0.3824 0.3593 0.3571 0.3119 0.2833 0.2461 0.2087 p_n 401 601 1009 2003 theta 1.6984 1.6815 1.6592 1.6340 L/p^2 0.1640 0.1303 0.0947 0.0619 ``` Below 2 at every level; `L/p^2` falls by a factor of 11.5 across the range, at a clean rate `L_need ~ p^{1.55}`. Confirmed: **the elementary maximal law is TPC-implying with a margin that grows.** The sieve version is TPC-implying with a margin that is constant (sec.2a, slope t = -0.25). Both are on the same side of the line, and the difference between them is a rate, not a verdict. **Cross-check against a settled result, not a re-derivation.** `research/history/MORNING-2026-08-16.md`:74 settled that the sieve mean square loses to the elementary second moment by about 1.4x. Comparing the two criteria at *matched prime sets* (sieve at z uses p < z, so it pairs with the elementary at p_n = the largest prime below z): ``` sift primes <=11 <=13 <=17 <=19 <=23 <=29 elementary 88 120 153 191 312 374 sieve sharp 100 142 191 325 436 570 ratio 1.14 1.18 1.25 1.70 1.40 1.52 mean 1.37 ``` The 1.4x reappears at a completely different observable. That is a consistency check on both computations, and it is the settled number, not a new one. --- ## 4. A SEPARATE DEFECT FOUND ON THE WAY: THE EXACT-SUPREMUM COLUMN IS NOT SELF-CONSISTENT IN H This one matters independently of the binary, because the exact-supremum walk is the instrument TODO 0 says to keep. `research/theta-ladder-sup.js`:18 hardcodes `u = 3.2`. Every `need_true` in `theta-ladder.md` sec.5b is therefore `sup|R| / M` with the supremum measured at `H = z^3.2` and the conclusion asserted about a window 39x to 62x shorter. Since `sup|R_H|` grows with H toward saturation, the published column systematically **overstates** the unconditional requirement. Measured self-consistently — the smallest H from which `min_x T(x) >= 1` holds onward, by exhaustive full-period walk at every integer H in the scanned range: ``` z published need_true /z^2 | self-consistent nP /z^2 th_true | overstated by 13 -- -- | 60 0.3550 1.5963 | -- 17 -- -- | 126 0.4360 1.7070 | -- 19 313.83 0.8693 | 198 0.5485 1.7960 | 1.58x 23 448.97 0.8487 | 258 0.4877 1.7710 | 1.74x 29 846.06 1.0060 | 390 0.4637 1.7718 | 2.17x ``` At z = 19 the published number uses `sup|R| = 12.4270` measured at H = 12360; the supremum at the window the conclusion is about, H = 198, is 6.8403. The overstatement grows with z (1.58, 1.74, 2.17) because the mismatch between `z^3.2` and the operative `~0.5 z^2` widens like `z^{1.2}`. This is brief Trap 2 in its purest form: two quantities compared at different levels. **The consequence is that the crossing does not happen.** `theta-ladder.md` reading 8 — "the headline of the run" — states `need_true/z^2 = 0.869, 0.849, 1.006, 1.178` and concludes "the route's actual requirement fitted inside the zone budget only at z = 19 and z = 23 ... and stopped fitting at z = 29". Measured self-consistently at z = 29 by exhaustive walk over all 223,092,870 positions of the period at every integer H from 250 to 700: `min_x T(x) >= 1` from H = 390 onward, and 390/29^2 = **0.4637**. The certificate fits inside the zone budget at z = 29 with 54% to spare. It fits at every exactly computable z, not at two of them. TODO 0's "the unconditional supremum, measured exactly, crossed the zone budget between z = 23 and 29" is refuted at the point where it says the crossing happens. **The prefix rows at z = 37 to 71 do not survive either, and not because they are prefixes.** They are lower bounds on `sup|R_{z^3.2}|`, hence lower bounds on a quantity that is itself 1.6x to 2.2x larger than the self-consistent one. A lower bound on an overstatement bounds nothing. Their one-sidedness is real for the question they answer and useless for the question TODO 0 asks of them. **What the corrected column actually says**, unconditional and exhaustive over complete periods: the vector-sieve certificate is positive at **every one of the W positions** of the period at a window of only 0.36 to 0.55 of the zone budget, at z = 13, 17, 19, 23, 29. That is a stronger unconditional statement than the corpus currently records. It is a finite verification, not a theorem. **Law tightness at the operative window.** `theta-ladder.md` sec.5b reading 2 reports `sup/rms` running at 0.73 to 0.82 of `sqrt(2 lnW)`, "the maximal law being approached from below". That ratio is also measured at H = z^3.2. At the operative window it is larger, and it does not approach cleanly from below: ``` z 13 17 19 23 29 sup / (rms * sqrt(2 lnW)) 0.5634 0.7279 0.9215 0.7935 0.8507 last H at which S-sharp fails 1 35 227 197 none ``` At z = 19 the sharp law is violated at H = 191 by 0.6%, which is why the sharp bound (191) comes out 3.5% below the truth (198). At z = 29 it holds at every one of the 451 integer H scanned, with 15% to spare at the operative window. **The Gaussian maximal law at the operative window is not a generous hypothesis: it is an approximately exact description of the sawtooth, accurate to a few percent.** That is the cleanest statement of why it cannot be proved softly — it is as hard as the truth, and the truth there is TPC. --- ## 5. WHAT RETIRES, AND WHAT SURVIVES ### RETIRES 1. **TODO item 0's first move is answered: NO.** No weak form of the sawtooth maximal law is both soft-provable and sufficient. The two forms are disjoint: the form that finishes (S-sharp, `need/z^2 ~ 0.56`) is TPC-implying and therefore unprovable by any second-moment, concentration or soft argument; the form that is legal (S-crude, `need/z^2` rising past 3.4) finishes nothing. **The certificate route retires as a road to TPC.** Per TODO 0's own charter that is a win, not a loss. 2. **The theta ladder retires as an instrument for deciding TPC-reachability.** Its column measures S-crude, and 83% of its upward drift is the growth of two lossy steps rather than of the requirement (sec.2b). `theta-ladder.md` sec.6's three-way verdict ("theta settles above 2: SUPPORTED as to above 2") is a verdict on S-crude only; the maximal law's own exponent is 1.75 to 1.85 and nearly flat. `theta-ladder.md` sec.7 Branch A ("theta turns over below 2 => TPC") is not a hope to be tested: it is already true of S-sharp, which is precisely why S-sharp is unprovable. 3. **`sift-limit-attack.md` sec.4.5's "the maximal law is the whole cost" needs splitting.** There are two maximal laws and they have different prices. A maximal law for the H-free potential rho alone is legal and buys an exponent reading 2.43 at z = 43 and 2.48 at z = 47, still rising, against beta_2 = 4.2665. A maximal law for R_H at the operative lag is TPC. The entire difference is the lag-H cancellation between rho(x) and rho(x+H) — which is a genuine mathematical distinction, not a bookkeeping one, and which the repo's own structural fact (V1: `g | H` kills the pair contribution) identifies as where all the cancellation lives. 4. **The Lemma V sub-question is moot, and moves further out of range.** At the corrected working point `u = th_sh = 1.75 to 1.85`, so `s/u = 3.0/1.8 = 1.62 to 1.72` and `theta_total = 2s/u = 3.25 to 3.43`. TODO 0 records `s/u = 1.2` as already outside Lemma V's stated range `s <= u`; the corrected point is further outside, not closer. No weak form of the maximal law brings the working point into Lemma V's range, because the two are not the same statement and never were: `sift-limit-attack.md` sec.4.5 already says so and is right. ### SURVIVES 1. **The exact-supremum prefix walk, but only if run self-consistently in H.** TODO 0 is right that it is unconditional and one-sided in the safe direction, and it remains the best-conditioned instrument here. As currently coded it answers a different question than the one asked (sec.4). Fixing it costs one line: take H from the caller instead of hardcoding `u = 3.2`, and iterate to the fixed point of `H*M = sup|R_H|`. 2. **The unconditional finite verifications, which are stronger than recorded.** The certificate is positive at every position of the complete period at H/z^2 = 0.36, 0.44, 0.55, 0.49, 0.46 for z = 13, 17, 19, 23, 29 — five complete periods, the largest 223,092,870 positions. `sift-limit-attack.md` sec.4.5's pilot records exhaustive positivity at u = 2.2 through z = 20; this is the same fact at the sharper exponent 1.60 to 1.80, and two primes further. 3. **The Brüdern-Fouvry certificate and the exact mean-square machinery.** The closed form in `sift-limit-lemmaV.js` `meanSquare` reproduces to 1e-11 or better against an independently regrouped evaluator at six z and six H, and `fullPeriodArray` reproduces exactly against an independent prefix-sum walker at five z. Neither is implicated in anything above. 4. **The exponent programme, repriced upward in difficulty.** `theta-ladder.md` reading 11 already reprices the route as exponent-improvement rather than TPC. That stands, with one correction: the legal target is now specifically **a maximal law for rho**, not "the maximal law". What it would buy is the published rising column, 2.35 at z = 37 and 2.48 at z = 47, which reading 11 already declines to extrapolate. The sharper-sounding exponent 1.85 that the correctly-stated law delivers is not available at any price: it is TPC. 5. **Attack 8 sec.4's alarm, now generalised.** Its rule — "anyone who computes the conditional and finds theta < 2 should read that as a soundness check firing, not as progress" — applies verbatim to the sieve side. This report is an instance of the rule firing, deliberately. --- ## 6. COVERAGE **What I did not reach.** - z = 47 for the sharp column was launched and had not finished when this was written (the O(N^2) bucket build is 4.3e10 pairs). The column is z = 13..43, nine points, against a published crude column reaching 47. Nothing in the nine points suggests `sh/z^2` is about to leave [0.49, 0.61], but the last point would be worth having and costs about half an hour. - The self-consistent exact supremum reaches z = 29. **z = 31 is not verified**: W = 6.47e9 exceeds the 2^32-1 flat typed-array limit, and `theta-ladder-sup.js`'s blocked walker handles it but hardcodes u = 3.2, so using it would need the one-line change I am not permitted to make here. So the published 1.1776 at z = 31 is uncorrected by measurement; the correction factor grows 1.58, 1.74, 2.17 across z = 19, 23, 29, and 1.1776/2.2 = 0.54 would continue the pattern, but that is an extrapolation, not a measurement. z = 29 is the point where the published column says the crossing happened, and that point IS measured. - The z = 29 scan covered H = 250 to 700. `min_x T` is negative at H = 350 and positive from 390 on, so the threshold is inside the scanned range, but I did not verify H < 250 (irrelevant, since I want the LAST failure) nor H > 700 (where H*M keeps growing against a sup that is flat near 13). - I did not test s other than 3.0. `theta-ladder.md` sec.5a measured theta as nearly insensitive to s over a factor of 200 in D for the crude column; I did not check that the sharp column inherits that insensitivity. - I did not compute the sixth moment or any non-Gaussian tail. The question was posed about the Gaussian maximal law and I answered that. - **A discrepancy in the task, recorded so nobody hunts for it.** The task asked me to reproduce attack 8's elementary result "on the same 18 exact levels". `attack-block-08-secondmoment.md` sec.4's table has **nine** rows (p_n = 13, 19, 23, 31, 47, 71, 97, 199, 401); its sec.6 table has ten, a different set. There is no 18-level table in that report. I reproduced all nine of sec.4 exactly and extended to twenty-three levels. **What I suspect but could not prove.** - That `need_sharp / z^2` converges to a constant near 0.56 rather than drifting. Nine points with a range of [0.49, 0.61] and no trend is consistent with a constant, and also with a very slow drift either way. Brief Trap 3 applies: I am not calling a trend, and I am not calling its absence a proof of one. - That the law-tightness ratio `sup/(rms sqrt(2 lnW))` at the operative window eventually settles above 1. It reads 0.56, 0.73, 0.92, 0.79, 0.85 at z = 13..29: rising then wobbling, with no clean trend over five points, and it already exceeds 1 transiently at z = 19 (the law fails there for every H up to 227, by 0.6% at H = 191). If it settles above 1 the sharp law as literally stated is false, and the honest hypothesis is `sup <= C sqrt(2 lnW _H)` with C somewhat above 1 — which sec.2(c) shows is still TPC-implying for any C below about 1.4. **So the binary does not turn on this**, but the correct statement of the hypothesis does, and I have not pinned C. **Where I think I am most likely wrong.** - **The strongest objection to the whole report is that S-sharp is a strictly stronger hypothesis than S-crude and I have no right to substitute it.** My defence is in sec.1: both are the same template applied to members of the same family of lattice sawtooth sums, the corpus's justification for one contains nothing that fails for the other, and the elementary criterion attack 8 used is the H-dependent one, so applying the H-free one on the sieve side is comparing two different methods. If an adjudicator rejects that, the report degrades to: *the corpus's stated sieve maximal law is not TPC-implying, but a statement of the same type about the same lattice is, and the two differ by a factor 1.5 to 6.7 that is growing.* That weaker form still retires the route, because the gap between the legal form and the finishing form is widening, not closing. - Sec.4's claim against `theta-ladder.md` reading 8 rests on my reading of `theta-ladder-sup.js`:18 (`u = 3.2` hardcoded) plus two self-consistent measurements. I did not re-derive the published `need_true` values from the script's own output; I reproduced `sup|R|` at the published H via `fullPeriodArray` (custody, sec.0) and checked that `sup/M` gives the published need_true at z = 19 (12.4270 / 0.0395977 = 313.83, published 313.83). That is strong but it is one point. - The `C_crit` scan runs H from z^2 to 4x the crude need. If `_H` dips below its scanned minimum somewhere past that, `C_crit` is overstated. The bound `_H <= 4` makes the test automatic past `2C sqrt(2 lnW )/M`, which for C = 4 is exactly the scan top, so the scan covers C up to 4 and the reported values (all under 2.3) are inside it. **Calibration of searches.** `grep -rl "maximal law" research/ paper/ TODO.md` returns 27 files, 8 of them working documents (`paper/PAPERS.md`, `paper/wall-note.md`, `research/G2-STATE.md`, `research/ZONE-POSTULATE.md`, `research/sift-limit-attack.md`, `research/sift-limit-lemmaV.js`, `research/theta-ladder.md`, `TODO.md`), so the pattern is very much live. The known positive: `grep -rn "sup_y |rho(y)|" research/` returns exactly one non-report hit, `sift-limit-lemmaV.js`:466, which is the rho form. The absence claim: `grep -rn "_H|sqrt(2 lnW \* )|sqrt()" research/ paper/` returns hits only inside this report. **The corpus states the maximal law in symbols exactly once, and it is the rho form; the sharp form appears nowhere.** Directory listing taken before writing that: `research/` has 196 entries, `research/history/staging/` was listed as well. **Brief trap checks.** Trap 1: every threshold in this report comes from a full integer scan with the last-failure recorded, never a bisection; `monotone=yes` is printed for the sharp column at every z, and the exact column's scan range is stated. Trap 2: the units are stated in sec.1's table and the budget convention is checked both ways (z^2 and w^2) in sec.2. Trap 3: the flatness of `sh/z^2` is reported as a band, not a trend. Trap 6: the absence claim is calibrated above.