# VERIFY: the Kalmynin–Konyagin substitution (attack 4), adversarially *2026-08-18/19, overnight wave. Target: `research/history/staging/attack-kk-substitution.md`, producer `research/attack-kk-substitution.js`. Verifier's producer: `research/verify-kk-substitution.js` (seven sections V1–V7, 3.5 s, OUTPUT block written by `research/qc/embed.js`; `--check` matches on both hashes). Written headline-first and appended to as the check ran. Brief: break it, and default to "cannot verify" over "verified".* --- ## VERDICT **STANDS WITH CORRECTIONS.** The theorem > `G₂(P(y)) ≫ y (ln y)³ (lnlnln y)² / (lnln y)⁴` for `y ≥ y₀` survives every attack made on it here, including an **exhaustive brute-force of the Proposition itself** over four million values of `i`. Five corrections follow, none of which touches the exponent; one of them is a number with no run behind it, one is a hypothesis checked in the wrong place, two are claims that are true in the body and over-stated in the headline and the draft CHANGELOG, and one is a limit on what a numerical collapse can test. I could not break it. That is not the same as it being refereed, and §4 below lists exactly what I imported rather than reconstructed. --- ## 1. Custody **Same document.** `curl -sL https://arxiv.org/pdf/2302.00459` → 148,566 bytes, md5 `b5d7d2a23ffd902415057adebfe430b1`, 12 pages, left-margin stamp `arXiv:2302.00459v2 [math.NT] 3 Dec 2023`. **Identical to the report's stated PDF of record and to artifact A of `research/history/staging/lit-pdf-kalmynin-konyagin.md` §0.** §2 is pages 3–7, as stated. **Read twice, and the second reading mattered.** Everything below was first read from `pdftotext -layout`, then every display this verification leans on — `m`, `z₀`, `z₁`, the `S(m,Ω)` line, and `π(y) − π(y/2)` — was re-read from a 200 dpi `pdftoppm` page image. One of those five reads reversed a finding (§3.5). --- ## 2. What I independently reconstructed, and it holds ### 2a. The bridge to `G₂` is sound `two-class-lower-bounds.md` §1: `G₂(x#) − 1` is the longest `[1,m]` coverable by choosing `a_p` per prime and deleting `{a_p, a_p − 2}`. Re-derived: `x + i` is a twin start iff `(x+i, N) = (x+i+2, N) = 1`, so `i` is killed iff `i ≡ −x` or `i ≡ −x−2 (mod p)`, and CRT makes `a_p = −x mod p` free. The report uses it correctly, and correctly says in §1b that `j_f` at `f = x(x+2)` is a **different object** (fixed centre, varying separation) and is not being invoked. ### 2b. Corollary 1 sees `Ω_p` only through `|Ω_p|` — verified at the source K–K v2 p. 4, verbatim: *"Suppose that for any `p ≤ z` the set `Ω_p ⊂ Z/pZ` contains `g(p)` elements … Then `S(X, Ω) ≪ XV(z)`."* **This is the single fact that makes the substitution legal**, because it is indifferent to whether the classes are fibres of a polynomial or a free translate. The report's §4 table asserts it; it is true as printed. ### 2c. Case 2 really is vacuous, checked at the source K–K v2 p. 6 define `Ω^II_p = {t : ∃ non-linear irreducible factor q(x) of f(x) with q(t) ≡ 0}`. `x(x+2)` is a product of two linear factors over `ℤ` and has no non-linear irreducible factor, so `h_f = 0`, `Ω^II_p = ∅` at every prime, and Case 2's hypothesis quantifies over the empty set. Lemma 2 and Chebotarev enter the ledger **only** through the `Σ_h r_p(q_h)/p` term, which is an empty sum. Confirmed at the source, not at the paraphrase. ### 2d. `κ = 4` is valid, and `κ = 6` was never a requirement `g(p) = |Ω^I_p ∪ Ω^III_p| ≤ 4` everywhere, with 4 attained only on band 2. V2 enumerates every prime to 1000: `p = 2` and `p = 3` are the **only** primes where `{0,−2}` and `{1,−1}` fail to be disjoint pairs of size 2, so `p₀ = 3` (first good prime 5) and `g(p) < p` for `p ≥ 5`. Band 2 starts above `z₀ = (ln y)^A`, so 2 and 3 are in band 1 where `g(2) = 1`, `g(3) = 2`. K–K's `κ = 3 deg f = 6` is a *uniform crude bound over all `f`*, and Lemma 1's implied constant is the only thing `κ` touches. **Invoking it at `κ = 4` is a tightening, not a weakening.** Brief's suspicion 2: not sustained. ### 2e. Case 1, re-derived — and then brute-forced, which is the strongest result here Re-derivation: `k ∈ {i, i+2}`, `p | k`, `p ≤ √y`. If `k = p` then `i ≤ √y + 2`. Else `k` is composite; unkilled at step 1 forces every prime factor of `k` into `(z₀, z₁] ∪ [y/2, ∞)`. If the largest `P ≥ y/2` then `k/P < 2m/y`, and every prime factor of `k/P` exceeds `z₀`, so **`2m/y < z₀` forces `k/P = 1`**, making `k = P` prime and contradicting composite. Hence `P ≤ z₁` and `k` is `z₁`-smooth. **V1 then tests the Proposition exhaustively**, which is possible because the Proposition is a finite statement for any `(y, z₀, z₁, m)`. At `y = 200000`, `z₁ = 300`, `m = 4·10⁶` (so the threshold is `2m/y = 40`): | `z₀` | `2m/y < z₀` | unkilled `i ≤ m` | by (a) | by (b) | by (c) | **counterexamples** | |---|---|---|---|---|---|---| | 100 | yes | 16985 | 10 | 240 | 16735 | **0** | | 60 | yes | 14381 | 9 | 798 | 13574 | **0** | | 45 | yes | 13164 | 8 | 1064 | 12092 | **0** | | 30 | NO | 11228 | 8 | 1355 | 9630 | 235 | | 20 | NO | 10372 | 7 | 1446 | 8120 | 799 | | 10 | NO | 9086 | 6 | 2200 | 3959 | 2921 | The Proposition holds with zero exceptions exactly when the report says it holds, and breaks the moment its one stated hypothesis breaks. **The report's claim that Case 1 consumes "one inequality, `2m/y < z₀`, and nothing else" is not just re-derivable, it is necessary and sufficient at finite scale.** Brief's suspicion 3: not sustained (but see correction 4 for the `A > 3` framing). ### 2f. The covering-form / sieve-form trap is not sprung Brief's suspicion 4 is the one I expected to land. It does not. `Ω^I_p = {0,−2}` is charged at **every** `p ≤ √y`, including the band-2 primes where the construction actually kills `{1,−1}` instead. That is not double-use of the freedom: `S(m,Ω)` bounds **only** branch (c), and any `i` that lands in `Ω^I_p` at a band-2 prime is removed by branch (a) or (b) via Case 1. The trichotomy is a genuine disjunction over all unkilled `i`, so nothing is counted twice and band 1's primes are never re-used as free. V1's exhaustive run is a direct test of exactly this: a leak would appear as a counterexample, and at `2m/y < z₀` there are none. `research/qc/units.js` §5's error 7 does not recur. ### 2g. The Mertens ledger and both `O(1)` constants Independently: `Σ_{5≤p≤√y} 2/p − 2 lnln y → −2 ln 2 + 2M − 2(1/2+1/3) = −2.529967`, measured residual-minus-constant `0.00787636, 0.00247186, 0.00060922, 0.00007794, 0.00001915` at `√y = 10³…10⁷` — the report's §F1 digits. The second constant, `−2 ln 2 + 2M − 1/2 = −1.363300`, is the residual of the **full band-aware sum from `p = 2`**, not of the same sum; the two are different objects and both are right. No inconsistency. ### 2h. The exponent assembly, hand-derived ``` lnln z₀ = lnA + lll lnln z₁ = llll + ll − lll − lnA SUM = 2ll + 2(lnln z₁ − lnln z₀) + O(1) = 4ll − 4lll + 2llll − 4lnA + O(1) exp(−SUM) = (ln y)^−4 (ll)^4 (lll)^−2 A^4 m·exp(−SUM) = (y/B)(L³ lll²/ll⁴)·L^−4 ll⁴ lll^−2 A⁴ = A⁴ y/(B ln y) ``` which is K–K's own displayed `A^{2M(f)−2h_f} y/(B ln y)`, **confirmed on the rendered page 7**, at `M(f) = 2, h_f = 0`. Evaluated at 21 `(L,A,B)` triples out to `L = 10³⁰⁰`: ratio 1 to `4.547e-13`. ### 2i. `y₀` and the D4 retirement Bisecting `A² ln²L < L lnln L` independently gives `L₀ = 3.0867e+2, 4.0795e+2, 5.3747e+2, 1.2733e+3, 3.1008e+3` at `A = 4.05, 4.5, 5, 7, 10` — every figure of the report's table, with `z₀ < z₁` binding. At `y = 4001`, `z₁ = 2.07` and `z₀ = 5.26e+3` at `A = 4.05`, so band 2 is empty at every `A > 4` and the construction does not exist there. **The consequential claim is sound**: attack I's D4 comparison at `y = 4001` was not a test of this construction. Brief's suspicion 6: not sustained on substance (see correction 1 on the number). ### 2j. The calibration in §6 is right, and it limits the claim K–K's own p. 2 quotes `j(P(y)) ≫ y ln y lnlnln y / lnln y` (FGKT). Their Theorem 1 at `f(x) = x` (`ℓ_f = 1, h_f = 0, M(f) = 1`) gives `y ln y lnlnln y / (lnln y)²`. **Weaker by exactly one `lnln y` than the bound they cite in their own introduction**, so the transferred two-class bound inherits the same slack. Confirmed at the source. The report is right to call its result a floor on the method rather than a ceiling. --- ## 3. CORRECTIONS **1. `§5`'s "1.5e6" is not a computed number.** The report says *"at `y = 4001` … `z₀ = (ln y)^A` with `A > 4` is already `1.5e6`, past `y`."* At `y = 4001`, `ln y = 8.29430`, so `z₀ = 4.7328e+3` at `A = 4` and `5.2609e+3` at `A = 4.05`. `1.5e6` requires `A = 6.722`. It appears in no section of the producer's OUTPUT block. The conclusion is unaffected (`z₀` already exceeds `y = 4001`, and `z₁ = 2.07 < z₀`), but the figure is 285× off and is quoted as if measured. **2. Corollary 1's load-bearing hypothesis is checked in the wrong place.** K–K's Lemma 1 (v2 p. 3) lists `g(p) ≤ κ`, `g(p) < p`, `|r_d| ≤ g(d)` and `z ≪ X`, and concludes `S ≪ XV(z)` with **no remainder term**. An upper-bound sieve with support `ξ` carries `Σ_{d≤ξ², d|P(z)} 3^{ω(d)}|r_d| ≪ ξ²(ln ξ)^{11}`, and `z ≪ X` constrains `ξ` not at all. §3a checks the hypotheses **as written** (*"`z ≪ X`: `√y ≪ m`, immediate"*) and inherits the omission. **The missing hypothesis does hold**, so the theorem is untouched: at the naive support `ξ = z = √y` the remainder is `y(ln y)^{11}` against a main term `y/(B ln y)`, too big by `(ln y)^{12}`; at `ξ = √y/(ln y)⁷` it is `y(ln y)^{−3}`, i.e. `(ln y)^{−2}` of the main term, while the main term loses only `(1 − 2K lnln y/ln y)^κ → 1`. **What is missing is the check, not the fact**, and the report should say which support it is using. **3. The headline mis-routes half of K–K §3.** Headline 2 reads *"Case 2 is vacuous because `h_f = 0`, so Lemma 2, Chebotarev, **Theorem 2, Lemma 3** (Birch–Swinnerton-Dyer) and every Galois group … drop out."* Lemma 2 and Chebotarev do drop for that reason. **Theorem 2, Lemma 3 and the Galois groups compute `M(f)`**, which drops for the *Case 3* reason — `|Ω^III_p| = 2` exactly rather than on logarithmic average. §3b states this correctly; the headline and the draft CHANGELOG entry do not. Both reasons hold, so "all of §3 unconsumed" stands; only the attribution is wrong. **4. The draft CHANGELOG drops a hypothesis the body carries.** It reads *"Case 1 reduces to the single inequality `2m/y < z₀`, needing `A > 3` where K–K need `A > 4`"* with no qualifier. §1c fixes `A > 4`, and §3d says plainly *"the substituted proof needs `A > 4`"* because the smooth count asks for `A > ℓ_f − 1 + M(f) + 1 = 4`. The `A > 3` is true **for that step only**. As drafted the CHANGELOG line is the exact signature `qc.js transfers` exists to catch: a result copied upward keeping its numbers and losing a hypothesis. It should read *"`A > 3` for that step, against their `A > 4`; the proof as a whole still needs `A > 4` from the smooth count."* **5. The numerical collapse tests less than the headline implies.** Headline 3 pairs the identified `O(1)` with *"the exponent assembly … collapses to zero coefficient by coefficient"* and quotes `5.7e-13`. The collapse happens because the `O(1)` is dropped on **both** sides — `2 lnln √y` is replaced by `2 ll`. It tests coefficient bookkeeping and **cannot test the `O(1)` at all**. §F1 and §F4 test the `O(1)` separately and pass. Both halves are sound; the headline reads as one check when it is two. --- ## 4. What I did NOT reconstruct Named, per the brief, rather than waved past. - **Lemma 1 itself.** Imported by K–K from Halberstam–Richert Thm 2.2; I did not read that source. Correction 2 is about its *application*, not its truth. - **`G(z^u) ≫_{κ,u} V(z)^{−1}`**, used in correction 2's repair. Standard for a bounded-dimension sieve; asserted here, not proved. - **`Ψ(x,z) ≪ x ρ(u)` in Hildebrand's range.** Imported, exactly as K–K import it. The report's §E range-slack measurement was not re-run. - **§7's novelty probes.** Not re-run. `research/qc/units.js` and the campaign lessons both record that an absence claim is the one thing no check can test; §7's calibrations are the right shape, and that is all I can say. Note the report's own §7f keeps the claim narrow ([INFERRED], construction credited to K–K), which is the correct calibration. - **`two-class-lower-bounds.md` §4b's INFERRED analogue**, which §8(b) says is superseded. I checked the arithmetic of the ratio (`ln x · lnlnln x/(lnln x)³`, worth 1.56 at `L = 10²`, which the report states) but not §4b's own derivation. --- ## 5. Gate ``` node research/verify-kk-substitution.js node research/qc/embed.js --check research/verify-kk-substitution.js # both hashes match node research/qc.js ``` Script tail: `code-sha256 524c4f9c89a6203a…`, `out-sha256 d84216d0138bb8b6…`, 237 lines of output, 3.5 s. `node research/qc.js` reports **0 findings for this file and for `attack-kk-substitution.js`**; the gate's one remaining finding is `tail-does-not-belong-to-this-code` on `research/attack-bonferroni-degree.js`, another agent's concurrent script, and is not mine to fix. I also re-ran the report's own custody claim: `node research/qc/embed.js --check research/attack-kk-substitution.js` matches on both hashes. --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [../CHANGELOG.md](../CHANGELOG.md), indexed by document.*