# The `u_sup` ladder in the `(h,m)` basis: the divergence is the object, and the natural basis is the worse of the two *(2026-08-19, attack E of five. Producer: `research/attack-hm-basis.js`, one run, OUTPUT block written by `research/qc/embed.js` with `--streams both`. Legend: **[PROVEN]** published theorem with source; **[VERIFIED]** computed in this repository; **[MEASURED]** empirical, finite range; **[ABSENT]** searched, with the channel and a same-session calibration named; **[INFERRED]** our deduction. **No literature channel was opened in this session and no `[ABSENT]` claim is made anywhere in this document.** No repository file was edited: every correction below is a draft for a human to apply. Nothing committed, nothing pushed.)* ## HEADLINE **The computation the ledger recorded as never run is now run. The two bases are not rivals — they are nested, the natural one is the outer and weaker member of the pair, and the gap between them is a flat factor of 7.26 across nine levels. So the divergence that closed `u_sup` survives the change of basis with the same growth law, and reading 21's "basis-dependent" turns out to be a statement about term counts that does not transfer to the total price.** 1. **The weights are `K_h(m)`, and attaching them is not an extension of the identity — it is the identity with one summation not yet performed.** [VERIFIED] `K_h(m) = Σ_{i : q_i = m} (w_i/q_i) e(−h c_i/m)`. Its `h = 0` value rebuilds the main term (`Σ_m W(m) = M` to **7.7e-15**), and its aggregation over the moduli divisible by `e` rebuilds attack 1's `Θ_e(a)` at every frequency, worst discrepancy **3.12e-16 over 2309** frequencies at `z = 13` and **5.62e-16 over 30029** at `z = 17`, against a `Θ` rebuilt independently from the raw term list. 2. **(T1), and it decides the shape of the answer before any number is read.** [VERIFIED] The one absolute-value step is taken at the same place in both bases and they differ only in whether the sum over `m` is inside or outside it, so `Ssup_hm(H) ≥ Ssup_e(H)` at every frequency, every `H` and every `z`. Checked directly rather than inherited: **0 violations out of 2309, 30029 and 510509 frequencies** at `z = 13, 17, 19`. **The `(h,m)` basis could never have rescued `u_sup`. What was genuinely open was the size of the gap.** 3. **The size is a constant.** [MEASURED] `Ssat_hm/Ssat_e` = 6.6170, 7.8932, 7.3607, 6.9840, 7.5211, 7.5241, 7.4093, 7.0958, **7.0136** over `z = 13..43`. Geometric mean **7.2599**, total spread 1.19×, and the log-log slope of the ratio against `z` is **0.0133**. Nine levels, no trend. 4. **So the divergence survives, with the same law.** [MEASURED] Per added prime the saturation grows **2.0889** in the `e` basis and **2.1042** in the `(h,m)` basis on the common range, a ratio of **1.0073**. Log-log slopes `d ln Ssat/d ln z` are **4.9872 against 4.8678** over all levels and **6.1967 against 6.1204** over the top five. Both above `β₂ = 4.26645` in every window. **The 2.05-per-added-prime law that closed `u_sup` is not a property of the `e` basis.** 5. **And it survives on the ladder itself, not only on the saturation.** [VERIFIED] `u_sup` in the `(h,m)` basis reads **2.8201, 3.0313, 3.2463, 3.3030, 3.3723, 3.5062** over `z = 13..31`, rising at **5 of 5** steps, beside the `e` basis's 2.0617, 2.3036, 2.5518, 2.6666, 2.7464, 2.8924 rising at 5 of 5. `H_hm/H_e` = 6.995, 7.861, 7.729, 7.353, 8.228, 8.230 — the same flat factor of about 7, now in the window. 6. **The bounded reading agrees in both bases.** [MEASURED] Fitting the one bounded model `u = A − B/ln z`, like for like on the same nine levels, `u_sat` reads **A = 5.4512** in the `e` basis against **5.4823** in the `(h,m)` basis; on `u_sup` over six levels, **5.2163** against **5.3737**. All four are above `β₂`. **The most favourable reading the natural basis admits still lands above the target.** 7. **The withdrawn `τ(m)` re-pricing did not come back in, and the check is decisive rather than rhetorical.** If the basis change had smuggled it back, the `(h,m)` total would have come out cheaper. It comes out **more expensive at every level, by 6.62× at the least and 7.89× at the most.** 8. **Two corrections fall out, both moving their own file's conclusion in the direction it already points.** `attack-tau-repricing.md`'s "at most `3·τ(m)`" holds for odd `m` and is **4.5·τ(m)** at even `m`; and `attack-tail-maximal.md` reading 21's "`z^{2s}` moduli" is a cap on the **size** of `m`, not a count — the realised `(h,m)` modulus count is **fewer** than `2^{π(z)}` at every level, falling to **0.474** of it at `z = 43`. **WHAT THIS IS NOT.** The `(h,m)` basis is **not a second independent instrument**. By (T1) it is the `e` basis with one summation moved outside an absolute value, and it agrees because it must. §6 states this at length, because this project has been burned twice this week by exactly that distinction. --- ## 0. Custody Everything numeric below is quoted from `research/attack-hm-basis.js`, run once in this session, with its OUTPUT block written by `research/qc/embed.js`. The divisor-pair term list, the exact mean square and the full-period walk are `research/sift-limit-lemmaV.js`'s; the `e`-basis spectral records are `research/lemmaV-parseval.js`'s `spectralRecords()`; the `(e,a)` sweep kernel and the `H_sup` protocol are `research/lemmaV-sup-extension.js`'s `sweep()` and `findHsup()`. All are imported, never recopied. **Both columns below are produced by one kernel, and differ only in whether the `V`-array is aggregated across moduli or kept per modulus.** A basis comparison run on two implementations would be measuring the implementations. `research/qc.js` and `research/qc/units.js` were read before any file was written. **The published anchors, reproduced before anything new** (§A of the output; the brief named the `u_sup` ladder and the 2.0516 explicitly): | anchor | published | here | source of the published figure | |---|---|---|---| | `Ssat` at ten levels | 1.9602e+1 … 1.2623e+4 | same to 5 figures, worst rel. diff **2.7e-5** | `lemmaV-sup-extension.md` §2 | | `H_sup` at `z = 13..31` | 198, 683, 1833, 4278, 10384, 20586 | **all six exact** | same | | `u_sup` at `z = 13..23` | 2.0617, 2.3036, 2.5518, 2.6666 | **identical** | `lemmaV-parseval.js` S5 | | `S_sat` per added prime, `z = 13..47` | 2.0516 | **2.0516**, rel. diff 5.4e-6 | `attack-tau-repricing.md` §2 (a) | | bounded-family asymptote for `u_sat` | 5.5649 | **5.5649** | same | Two further published figures fall out without being aimed at. §F(c): `N/Ssat_e` = **43.47** at `z = 13` and **25.76** at `z = 43`, the two ends of `attack-tau-repricing.md`'s "23–44× and falling". §F(a): `n/τ` at `m = 2310` reads **3.0000** at `z = 13` (its `96 = 3·τ(2310)`) and **3.5625** at `z = 17` (its `114`). **Convention.** `s = 3.0` throughout, the level the published ladder uses. `π(z)` is the sieve prime count `#{p < z}`, matching `rosserSupport()`. `u = ln H / ln z`, `u_sat = ln(Ssat/M)/ln z`. The `H`-scan uses the published grid schedule `K = 20` at `z ≥ 29` and `24` below, and **both bases are scanned on the same grid at the same `z`**, because `H_sup` off a geometric grid depends on the grid. --- ## 1. THE WEIGHTS — what they are and where they come from The brief's step 1. `attack-tail-maximal.js` reading 13 says a total price in the `(h,m)` basis "needs the weights attached to each `m`, which are not computed here". Those weights are one object. Each divisor-pair term contributes a sawtooth at its **own** modulus `q_i`, and that modulus's Fourier expansion has frequencies `h/m`, `h = 1..m−1`. Collecting the terms that share a modulus: ``` K_h(m) := sum_{i : q_i = m} (w_i / q_i) e(-h c_i / m) [THE WEIGHTS] R(x) = sum_m sum_{h=1}^{m-1} K_h(m) S_H(h/m) e(h x / m), S_H(t) = sum_{j<=H} e(jt) ``` So the weight attached to `m` is a **signed** sum of `w_i/q_i` over the terms whose lcm is exactly `m`, twisted by the phase of that term's CRT class. It is not a new quantity: `K_0(m) = W(m)` is that modulus's share of the main term, and `Σ_m W(m) = M`. Computing it costs one extra index. For a modulus `m` and a divisor `e | m`, the class `c_i mod e` depends on `i` only through `(e₁,e₂) = (gcd(e,d₁), e/gcd(e,d₁))` — `c_i ≡ 0 mod e₁` and `≡ −2 mod e₂`, so `c_i/e ≡ −2·inv(e₁)/e₂ mod 1` — which is exactly `lemmaV-parseval.js`'s `V(e₁,e₂)` structure with the `m`-restriction not yet dropped. **`hmRecords()` is `spectralRecords()` with one key added.** That is the whole implementation, and it is why the comparison is clean. **Two identities, and both must hold if `K_h(m)` is the right object** — §A(d), [VERIFIED]: | check | `z = 13` | `z = 17` | |---|---|---| | `Σ_m W(m) = M`, relative | 7.7e-15 | 1.7e-14 | | worst `\|Σ_{m : e\|m} K_{a m/e}(m) − Θ_e(a)\|` | **3.12e-16** | **5.62e-16** | | frequencies checked | 2309 | 30029 | The `Θ_e(a)` on the right is rebuilt **from the raw term list**, not from the same factorisation arrays, so this is a check on the construction and not a tautology. §B prints the weights themselves at `z = 13`: all 32 moduli, their term counts, `W(m)`, `max_h |K_h(m)|` and what each pays. The largest single modulus contributes **8.2%** of `Ssat_hm`, so no one modulus owns the answer. --- ## 2. THE TWO BASES, AND WHY THEY ARE NESTED — (T1) Putting a frequency in lowest terms, `h = g·a` with `g = gcd(h,m)` and `e = m/g`, turns the `(h,m)` identity into attack 1's L5: `Θ_e(a) = Σ_{m : e | m} K_{a m/e}(m)`. **One line is the whole difference**, and it is where the sum over `m` sits relative to the absolute value: ``` e basis Ssup_e (H) = sum_e sum*_a | sum_{m : e|m} K_{a m/e}(m) | |S_H(a/e)| (h,m) basis Ssup_hm(H) = sum_e sum*_a sum_{m : e|m} |K_{a m/e}(m)| |S_H(a/e)| ``` **(T1)** [PROVEN, one line of triangle inequality] `Ssup_hm(H) ≥ Ssup_e(H)` for every `H`, `z` and `s`, frequency by frequency. Hence `Ssat_hm ≥ Ssat_e`, `H_hm ≥ H_sup`, and `u_hm ≥ u_sup`. Checked directly in §C rather than rested on: **0 violations** out of 2309, 30029 and 510509 frequencies at `z = 13, 17, 19`. **This is the most important structural fact in the file.** It means the answer to "does the divergence survive?" was never going to be "no". The `(h,m)` basis is pointwise weaker, so if `u_sup` diverges then `u_hm` diverges, unconditionally and with no measurement at all. **What was genuinely open, and is what §3 and §4 measure, is the size and the shape of the gap** — a bounded factor leaves the growth law untouched, a growing factor would mean the `e` basis flatters the bound by an unbounded amount and the two ladders would be telling different stories about the rate. --- ## 3. THE LADDER, BOTH COLUMNS — the brief's step 2 Every row is a legal unconditional worst-position statement in **both** bases: `H·M > Ssup(H)` puts `T(x) > 0` at every position, with no maximal law and no averaging. §D: ``` z Ssat_e Ssat_hm ratio u_sat_e u_sat_hm diff H_sup_e H_sup_hm u_sup_e u_sup_hm diff 13 1.9602e+1 1.2970e+2 6.6170 2.2850 3.0217 0.7367 198 1385 2.0617 2.8201 0.7584 exact 17 5.0314e+1 3.9714e+2 7.8932 2.4623 3.1915 0.7292 683 5369 2.3036 3.0313 0.7278 exact 19 1.2006e+2 8.8374e+2 7.3607 2.7228 3.4007 0.6779 1833 14167 2.5518 3.2463 0.6945 exact 23 2.4250e+2 1.6936e+3 6.9840 2.8281 3.4480 0.6199 4278 31458 2.6666 3.3030 0.6363 exact 29 5.2311e+2 3.9344e+3 7.5211 2.8827 3.4819 0.5992 10384 85435 2.7464 3.3723 0.6259 31 9.5263e+2 7.1676e+3 7.5241 3.0259 3.6136 0.5877 20586 169427 2.8924 3.5062 0.6138 37 2.5727e+3 1.9062e+4 7.4093 3.1644 3.7190 0.5546 41 4.6495e+3 3.2992e+4 7.0958 3.2543 3.7819 0.5277 43 7.1072e+3 4.9847e+4 7.0136 3.3448 3.8627 0.5179 47 1.2623e+4 -- -- 3.4306 -- -- ``` The `u_sup` scan stops at `z = 31` in **both** bases because it costs the ladder times the grid, and the `(h,m)` basis pays **1.106e+10** frequency points at `z = 43` on its own. `u_sat` carries the comparison to 43, which `lemmaV-sup-extension.md` §2 licenses: it measures `u_sat − u_sup` tracking `0.51/ln z` and shrinking. At `z = 31`, for instance, the `(h,m)` statement is: **every window of 169,427 consecutive integers contains an `r` with `r` and `r+2` both 31-rough**, established with no maximal law and no exceptional set. It is true, it is legally derived, and it is 8.23× weaker than the same statement in the `e` basis. --- ## 4. DOES THE DIVERGENCE SURVIVE? — the brief's step 3 **Yes, and with the same law. Here is the honest size of that.** **(a) The gap between the bases is flat.** `Ssat_hm/Ssat_e` over nine levels: 6.6170, 7.8932, 7.3607, 6.9840, 7.5211, 7.5241, 7.4093, 7.0958, 7.0136. Minimum 6.6170, maximum 7.8932, geometric mean **7.2599**, log-log slope against `z` **0.0133**. In window exponent, `u_sat_hm − u_sat_e` falls 0.7367 → 0.5179 and `(u_sat_hm − u_sat_e)·ln z` stays between **1.8896 and 2.0660** at all nine levels — the difference *is* `ln(7.2599)/ln z`, and it goes to zero. **(b) The growth law is the same in both.** Per added prime, on the common range `z = 13..43`: **2.0889** (`e`) against **2.1042** (`(h,m)`), ratio **1.0073**. Log-log slopes `d ln Ssat/d ln z`: **4.9872 / 4.8678** over all levels, **6.1967 / 6.1204** over the top five, **7.2037 / 6.2285** over the top three. Every window in both bases sits above `β₂ = 4.26645`. **(c) The ladder itself rises identically.** `u_sup` rises at **5 of 5** steps in both. The step sizes track: 0.2418/0.2112, 0.2483/0.2150, 0.1148/0.0566, 0.0798/0.0693, 0.1459/0.1339. **(d) Even the bounded reading lands above `β₂` in the natural basis.** The one model in `lemmaV-sup-extension.md` §1 item 3 that is bounded, `u = A − B/ln z`: | fit | levels | `A` | `B` | RSS | |---|---|---|---|---| | `u_sat`, `e` basis | 10 (`z = 13..47`) | **5.5649** | 8.5739 | 4.772e-2 | | `u_sat`, `e` basis, restricted | 9 (`z = 13..43`) | **5.4512** | 8.2419 | 3.660e-2 | | `u_sat`, `(h,m)` basis | 9 (`z = 13..43`) | **5.4823** | 6.3603 | 2.665e-2 | | `u_sup`, `e` basis | 6 | **5.2163** | 8.0799 | 1.660e-2 | | `u_sup`, `(h,m)` basis | 6 | **5.3737** | 6.5156 | 1.467e-2 | The restricted row matters and is why it is printed. Fitted over ten levels the `e` basis reads 5.5649, **above** the `(h,m)` basis's 5.4823, which would look like a violation of (T1). It is not: like for like on the same nine levels the `e` basis reads 5.4512 and the `(h,m)` basis is the higher of the two, as (T1) requires. **The apparent inversion is the tenth level and nothing else.** This is the sort of comparison the units file's window-convention entry exists to catch, and it fired here. **(e) The mechanism, and it is worth recording.** The mean number of moduli sharing one frequency **grows**: 6.81, 10.56, 14.30 at `z = 13, 17, 19` with maxima 16, 32, 62. The mean cross-modulus collapse `Σ_m|K| / |Σ_m K|` grows with it: 9.2472, 10.5341, 10.0385. **More moduli pile onto each frequency and the cancellation among them grows in step**, which is why the total ratio in (a) does not move. The `e` basis's advantage is real and it is not accumulating. ### What this licenses, and what it does not **It licenses this:** the `e`-basis divergence recorded in `sift-limit-attack.md` §7e is not an artefact of where the summation over moduli sits. That was named as an open escape route by `attack-tail-maximal.md` §7 and by `TODO.md` item 3, and it is now closed by computation. **It does not license calling this an independent confirmation.** By (T1) the `(h,m)` basis is the `e` basis with one summation moved, so it agrees because it must. The corpus's own two recent injuries were exactly this shape — a "second independent instrument" that turned out to be the first in other coordinates, and a variable that should have been `θ(x)` rather than `x`. **The correct statement is: one specific alternative representation was named, it has now been computed, and it is strictly worse.** Two bases are not all bases, and nothing here bears on a representation nobody has written down. --- ## 5. THE PRICE CHECK — the brief's step 4 The test is simple and it is decisive rather than rhetorical: if the basis change had smuggled the withdrawn `τ(m)` re-pricing back in, the `(h,m)` total would have come out **cheaper**. It comes out **more expensive at every level, by 6.6170× at the least (`z = 13`) and 7.8932× at the most (`z = 17`)**. **(a) The bookkeeping identity reproduces from the modulus side.** `Σ_m n_m = N` exactly at all six levels tested, 852 through 35868 — the identity `attack-tau-repricing.md` §2 (a) rests on, recomputed here from a different direction. **(b) The three-way nesting, which is the cleanest way to read the whole file.** §F(c), all three at `s = 3.0`: | `z` | `N` | `Ssat_hm` | `Ssat_e` | `N/Ssat_hm` | `Ssat_hm/Ssat_e` | `N/Ssat_e` | |---|---|---|---|---|---|---| | 13 | 852 | 1.2970e+2 | 1.9602e+1 | 6.57 | 6.6170 | **43.47** | | 23 | 9636 | 1.6936e+3 | 2.4250e+2 | 5.69 | 6.9840 | 39.74 | | 31 | 35868 | 7.1676e+3 | 9.5263e+2 | 5.00 | 7.5241 | 37.65 | | 43 | 183084 | 4.9847e+4 | 7.1072e+3 | 3.67 | 7.0136 | **25.76** | **The Fourier gain the corpus records as "23–44× and falling" is now split.** At `z = 43` the 25.76× is a factor **3.67 from summing coherently inside a modulus** and a factor **7.0136 from summing coherently across moduli**. The first is falling; the second is flat. **All of the corpus's shrinking Fourier advantage lives in the within-modulus half.** That is a fact about the instrument that nothing in the corpus had isolated, and it is what a per-modulus price like `τ(m)` is blind to by construction. --- ## 6. DRAFTED CORRECTIONS — I have edited nothing **(C1) `research/qc/ledgers.js`, `ABSENCE_VERIFIED`, key `'TODO.md|not run'`.** The entry's own closing clause says it "goes stale the moment anyone attaches the weights". That has happened. The entry should be retired with a pointer to `research/attack-hm-basis.js` and this report, not merely re-dated. **(C2) `TODO.md` item 3, "Compute the `u_sup` ladder in the `(h,m)` basis".** Done. Its premise, "`attack-tail-maximal.md` reading 18 shows the `e`-basis divergence that closed `u_sup` is basis-dependent", is the part that needs correcting rather than deleting: reading 18 compares **per-modulus term counts**, and the total price in the `(h,m)` basis is **7.26× the `e` basis's, flat over nine levels**, with the same per-added-prime factor to 0.7%. **(C3) `research/history/staging/attack-tail-maximal.md` reading 21 and `research/attack-tail-maximal.js` reading 12.** Both set "`2^{π(z)}` moduli" in the `e` basis against "about `z^{2s}` moduli" in the `τ` basis and conclude the `τ` basis is the more expensive of the two on modulus count below `z ≈ 200`. `z^{2s}` is the cap on the **size** of `m`, not a count: every `m = [d₁,d₂]` here is squarefree and `z`-smooth, so `m | P(z)` and the `(h,m)` modulus count is capped by `2^{π(z)}` as well. Measured: 32, 64, 128, 244, 468, 828, 1528, 2536, **3880** against `2^{π(z)}` = 32, 64, 128, 256, 512, 1024, 2048, 4096, **8192**, a ratio falling 1.000 → **0.474**, while `z^{2s}` at `z = 43` is 6.32e+9. The `(h,m)` basis has **fewer** moduli at every level, and the frequency count it actually pays is 2.587 to 3.220× the `e` basis's, flat. **(C4) The same two files' "no claim is made about the total here".** Superseded: the total is `Ssat_hm` = 1.2970e+2 … 4.9847e+4 over `z = 13..43`, and the `(h,m)` basis is the weaker of the pair by (T1). **(C5) `research/history/staging/attack-tau-repricing.md` §1 table and §2 (a), and `research/attack-tau-repricing.js` check (iv): "the per-modulus term count is at most `3·τ(m)`".** True for **odd** `m` (measured maximum 3.0000 over odd `m` at all six levels) and false for even `m`: it is **4.5000·τ(m)** at `m = 2` at every level, and at `m = 2310` it reads 3.0000 at `z = 13`, 3.5625 at `z = 17` and 19, then 4.5000 from `z = 23` on as the level truncation stops biting. The reason is structural — an ordered pair `(d₁,d₂)` with `lcm = m` and `gcd | 2` sends each odd prime of `m` to exactly one of `d₁, d₂`, but may send the prime 2 to `d₁`, to `d₂` **or to both** — so the ceiling is `4.5·τ(m)` at even `m`. Zero moduli exceed it anywhere. What was checked in that file was the ceiling at `m = P(z)` specifically; the general statement is what needs narrowing. **This makes the `τ` price looser than that file assumed, which is the direction its own verdict already runs.** **(C6) `research/sift-limit-attack.md` §7e** could gain one sentence: the `C^{π(z)}` law is basis-independent across the pair `(e, (h,m))` — the natural basis pays a flat 7.26× more and grows at 2.1042 per added prime against 2.0889. It should **not** gain the word "independently confirmed", for the reason in §4. **(C7) `research/G2-STATE.md`** `u_sup` row and **`research/ZONE-POSTULATE.md`** could record that the closure survives the change to the natural basis, with the same caveat. --- ## 7. WHAT THIS FILE DOES NOT ESTABLISH - **The `(h,m)` basis is not a second independent instrument.** §4, stated there at length. - **Two bases are not all bases.** (T1) is about this pair. - **`u_sup` is neither reopened nor re-closed here.** The closure rests on measurements this file does not repeat. What is removed is one named escape route, by computation. - **The `u_sup` column stops at `z = 31` in both bases**, not at 43. `u_sat` carries it to 43. - **`s = 3.0` throughout, one level.** No claim is made about how the ratio in §4 moves with `s`. - **`H_sup` off a geometric grid depends on the grid.** The published schedule is used so the `e` column reproduces exactly, and both bases share it. A finer grid returns a smaller and equally legal `H_sup` in either basis. - **No literature channel was opened in this session**, and no `[ABSENT]` claim is made anywhere in this document. --- ## 8. DRAFT `CHANGELOG.md` ENTRY (for the adjudicator to apply) > **2026-08-19 — the `u_sup` ladder computed in the `(h,m)` basis.** > `research/attack-hm-basis.js`, report `history/staging/attack-hm-basis.md`. > > - **The absence recorded in `qc/ledgers.js` under `ABSENCE_VERIFIED`, > `'TODO.md|not run'`, is now false and should be retired.** The weights are > `K_h(m) = Σ_{q_i = m}(w_i/q_i) e(−h c_i/m)`; `Σ_m K_0(m) = M` to 7.7e-15 and > `Σ_{m : e|m} K_{a m/e}(m) = Θ_e(a)` to 3.12e-16 over 2309 frequencies. > - **(T1): the two bases are NESTED, not rival.** `Ssup_hm(H) ≥ Ssup_e(H)` > frequency by frequency, one line of triangle inequality; 0 violations out of > 2309, 30029 and 510509 checked frequencies. > - **The gap is a flat factor.** `Ssat_hm/Ssat_e` = 6.6170 … 7.0136 over > `z = 13..43`, geometric mean 7.2599, log-log slope 0.0133. > - **So the divergence is basis-independent across this pair.** Per added prime > 2.0889 (`e`) against 2.1042 (`(h,m)`) on the common range; log-log slopes > 4.9872/4.8678 all levels and 6.1967/6.1204 top five; `u_sup` rises at 5 of 5 > steps in both, reading 2.8201 … 3.5062 in the `(h,m)` basis over `z = 13..31`; > the bounded family's asymptote is 5.4512 (`e`) against 5.4823 (`(h,m)`) on > matched levels, both above `β₂`. > - **It is NOT a second instrument.** By (T1) it is the same instrument with one > summation moved outside an absolute value. > - **`attack-tail-maximal.md` reading 21 corrected**: `z^{2s}` caps the size of > `m`, not the number of moduli. Realised `(h,m)` modulus counts are 32 … 3880 > against `2^{π(z)}` = 32 … 8192, ratio falling to 0.474 at `z = 43`. > - **`attack-tau-repricing.md`'s "at most `3·τ(m)`" narrowed**: true at odd `m`, > `4.5·τ(m)` at even `m` (attained at `m = 2` at every level), because the > prime 2 may go to both `d₁` and `d₂`. Looser, in the direction that file > already argues. > - **The corpus's "23–44× and falling" Fourier gain is split**: at `z = 43` the > 25.76× is 3.67 from coherence inside a modulus and 7.0136 from coherence > across moduli. The falling half is entirely the within-modulus one. > - **Custody**: `Ssat` at ten levels to 5 figures, `H_sup` = 198, 683, 1833, > 4278, 10384, 20586 all exact, `S_sat` per added prime 2.0516, bounded-family > asymptote 5.5649, `N/Ssat_e` 43.47 and 25.76, `n/τ(2310)` 3.0000 at `z = 13` > and 3.5625 at `z = 17` — every one reproduced before anything new was > computed.