# Attack 4: the Kalmynin–Konyagin substitution at D3, written as a proof *2026-08-18/19, attack 4 of 10. Producer: `research/attack-kk-substitution.js` (0.6 s, nine sections A–I, OUTPUT block written by `research/qc/embed.js` and twelve readings written afterwards from that block). Direct continuation of `research/history/staging/attack-lower-bound.md`, whose D1, D2 and D4 stand and are not re-run. Legend: **[PROVEN]** published theorem with source; **[VERIFIED]** checked computationally here; **[MEASURED]** empirical, finite range; **[ABSENT]** searched and found nothing, with the channel and a known-positive calibration probe run in the same session named; **[INFERRED]** our deduction from sourced facts.* --- ## 0. The headline, in five lines 1. **The transfer stands, and D3 is discharged.** K–K's §2 trichotomy survives the substitution `Ω_p = {a_p, a_p − 2}` intact, and §2 below writes it out as a proof rather than as checks. The theorem it proves is `G₂(P(y)) ≫ y (ln y)³ (lnlnln y)² / (lnln y)⁴` for `y ≥ y₀`. 2. **Roughly half of K–K's paper is not consumed.** Case 2 is vacuous because `h_f = 0`, so Lemma 2, Chebotarev, Theorem 2, Lemma 3 (Birch–Swinnerton-Dyer) and every Galois group in the paper drop out. Case 3 needs no `M(f)`, no logarithmic average and no exceptional primes, because `|{1,−1}| = 2` at every odd prime. `p₀ = 5` is exact and comes from a resultant. **[VERIFIED]** 3. **The Mertens ledger is exactly the two terms the brief names**, and its `O(1)` is an identified constant, not a shrug: the `Ω^I` term's residual converges to `−2 ln 2 + 2M − 5/3 = −2.529967`, reached to `1.9e-5` at `√y = 10⁷`. The exponent assembly is an algebraic identity that collapses to zero coefficient by coefficient. **[VERIFIED]** 4. **A new number the paper never gives: `y₀`.** Every hypothesis is now an explicit inequality, and the binding one is `z₀ < z₁` — K–K's own band geometry, nothing to do with our substitution. At `A = 4.05`, `y₀ = 10^134.1`; at `A = 5`, `10^233.4`. **This is the complete explanation of why attack I's D4 lost to the greedy by 4.0–6.6 at `y = 4001`: at `y = 4001` the bands do not exist.** **[VERIFIED]** 5. **The bound is a floor on the method, not its ceiling.** K–K's Theorem 1 at `f(x) = x` is the ordinary Jacobsthal function and reads `j(P(y)) ≫ y ln y lll y/(ll y)²`, weaker by exactly `lnln y` than the FGKT bound they quote in their own introduction. The transferred two-class bound inherits that slack. **[VERIFIED]** **The Zone Postulate is unaffected.** The threshold needs `G₂(x#) < x′²`, and `x²/(x ln³x) = x/ln³x → ∞`. A stronger lower bound was the one thing that could have threatened it from the construction side; it does not. The margin is still `x^{1−o(1)}`. --- ## 1. The object, the source, and the construction ### 1a. Custody The PDF read is arXiv:2302.00459v2, md5 `b5d7d2a23ffd902415057adebfe430b1`, 12 pages — byte-identical to the artifact of record in `research/history/staging/lit-pdf-kalmynin-konyagin.md` §0, which checked it against the published Izvestiya PDF (Izv. Math. **88**:2 (2024) 225–235, DOI 10.4213/im9467e). Everything quoted below is from §2 of that PDF, pages 3–7, which is the entire proof of Theorem 1. ### 1b. Their object and ours are different, and that has not changed `j_f` shifts the **value**: the fibre at `f = x(x+2)` is `{−1 ± √(1−x_p)}`, centre fixed at `−1`, separation varying. `G₂` shifts the **argument**: the pair is `{a_p, a_p − 2}`, centre varying, separation fixed at 2. One may not apply Theorem 1 at `f = x(x+2)` and read off a `G₂` bound, and `lit-pdf-kalmynin-konyagin.md` §2 says so correctly. **Nothing below disturbs that verdict.** What is transferred is the **method** — the bands, the trichotomy, Corollary 1 and the ledger — instantiated on our own system. The result is a different proof of a different theorem that happens to land on the same right-hand side. ### 1c. The substituted construction, stated once By `research/two-class-lower-bounds.md` §1 (**[PROVEN]**, elementary + CRT), `G₂(P(y)) − 1` is the longest interval `[1,m]` coverable by choosing one residue `a_p` per prime `p ≤ y` and deleting `{a_p, a_p − 2} mod p`. Fix constants `A > 4` and `B` large in terms of `A`, and set ``` z₀ = (ln y)^A z₁ = exp( lll y · ln y / (A · ll y) ) m = (y/B) (ln y)³ (lll y)² / (ll y)⁴ ``` which is K–K's `m` at `ℓ_f = 2, h_f = 0, M(f) = 2`. Choose: | step | primes | `a_p` | kill set `{a_p, a_p − 2}` | |---|---|---|---| | 1 | `p ≤ z₀` and `z₁ < p < y/2` | `0` | `{0, −2}` | | 2 | `z₀ < p ≤ z₁` | `1` | `{1, −1}` | | 3 | `y/2 ≤ p ≤ y` | greedy, one leftover `i` per prime | — | and take `x ≡ −a_p (mod p)` for every `p ≤ y` by CRT. Then `p | x+i` iff `i ≡ a_p`, and `p | x+i+2` iff `i ≡ a_p − 2`, so **`i` is killed by `p` exactly when `i mod p` lies in the kill set.** If every `i ≤ m` is killed then no `i` in `[1,m]` starts a twin pair coprime to `P(y)`, and `G₂(P(y)) ≥ m + 1`. The sieve sets fed to Corollary 1 at `z = √y` are ``` Ω^I_p = {0, -2} for every p ≤ √y the SIEVE form of the pair Ω^II_p = ∅ h_f = 0, no non-linear factor exists Ω^III_p = {1, -1} for z₀ < p ≤ z₁ the FREE TRANSLATE Ω_p = Ω^I_p ∪ Ω^II_p ∪ Ω^III_p ``` **The single structural observation that makes the transfer work.** Band 1 sets `a_p = 0`, and at `a_p = 0` the covering form `{a_p, a_p − 2}` and the fixed sieve pair `{0, −2}` **are the same set**. `research/qc/units.js` §5 records that confusing these two formulations has already caused one error in this corpus; the substituted proof never has to reconcile them, because it uses each in its own band — the sieve form in band 1, the free translate in band 2, and nowhere do the two roles meet. --- ## 2. The trichotomy, re-derived line by line Throughout, `i ∈ [1, m]`, and "unkilled after steps 1 and 2" means no prime assigned in step 1 or step 2 has `i mod p` in its kill set. > **Proposition (substituted trichotomy).** If `i ≤ m` is unkilled after steps 1 > and 2, then at least one of: > **(a)** `i ≤ √y + 2`; > **(b)** `i` or `i + 2` is `z₁`-smooth; > **(c)** `i mod p ∉ Ω_p` for every `p ≤ √y`. *Proof.* Suppose (c) fails, so `i mod p ∈ Ω_p` for some `p ≤ √y`. Since `Ω_p = Ω^I_p ∪ Ω^II_p ∪ Ω^III_p`, one of three cases holds. ### Case 1 — `i mod p ∈ Ω^I_p` **Hypothesis.** `i ≡ 0` or `i ≡ −2 (mod p)` for some `p ≤ √y`; equivalently `p | k` where `k := i` in the first case and `k := i + 2` in the second, so `1 ≤ k ≤ m + 2`. This is exactly K–K's *"`k(i)` is divisible by `p` for some linear factor `k(x)` of `f`"* (v2 p. 6) at `f(x) = x(x+2)`, whose linear factors are `x` and `x + 2`. **Conclusion.** (a) or (b). *Proof of the case.* If `k = p` then `i ≤ p + 2 ≤ √y + 2` and (a) holds. Otherwise `k` is composite (`k > p` and `p | k`; `k = 0` is impossible since `i ≥ 1`). Because `i` is unkilled at step 1, neither `i` nor `i + 2` has a prime factor `q ∈ [2, z₀] ∪ (z₁, y/2)`: for such a `q` the step-1 choice `a_q = 0` gives kill set `{0, −2}`, and `q | i` or `q | i+2` is precisely `i ≡ 0` or `i ≡ −2 (mod q)`. So every prime factor of `k` lies in `(z₀, z₁] ∪ [y/2, ∞)`. Let `P` be the largest prime factor of `k`. - If `P ≤ z₁`, then `k` is `z₁`-smooth and (b) holds. - If `P ≥ y/2`, then `k/P ≤ (m+2)/(y/2) < 2m/y`, while every prime factor of `k/P` exceeds `z₀`. Provided **`2m/y < z₀`**, `k/P` has no prime factor at all, so `k/P = 1` and `k = P` is prime — contradicting `k` composite. ∎ **What this case consumes.** One inequality, `2m/y < z₀`, and nothing else. **What it costs, and it is less than theirs.** K–K state the same step as `k(i) ≪ m ≪ y(ln y)^{ℓ_f + M(f)} = y(ln y)⁴`, "for large enough `A`". Our `m` is `y(ln y)³(lll y)²/(ll y)⁴ ≤ y(ln y)³`, so the substituted version needs only `A > 3` where theirs needs `A > 4`. §D of the script measures the margin `A ln L − ln(2m/y)`: at `A = 3` it runs `5.309` at `L = 10` to `10.182` at `L = 10⁶`, and at `A = 6` from `12.216` to `51.628`. Positive everywhere and rising. **[VERIFIED]** **Why this is the whole argument.** Case 1 is the only place in the proof where `Ω^I_p` is charged to the ledger at primes where it was never chosen. In band 2 the actual kill set is `{1,−1}`, not `{0,−2}`; the two extra classes are paid for by the smoothness dichotomy rather than by a choice. **That is where the whole extra factor of `ln y` over `two-class-lower-bounds.md` §4b comes from**, and it is K–K's idea, not ours. ### Case 2 — `i mod p ∈ Ω^II_p` **Hypothesis.** *"… and for some irreducible non-linear factor `q(x)` of `f(x)` we have `q(i) ≡ 0 (mod p)`"* (v2 p. 6; the elided clause is `p ∈ [2, z₀] ∪ (z₁, √y)`). **The hypothesis fails, and here is it failing.** `f(x) = x(x + 2)` is a product of two linear factors over `ℤ`, so `f` has **no** non-linear irreducible factor: `h_f = 0`. K–K's own definition then gives `Ω^II_p = ∅` at every prime, and the hypothesis quantifies over an empty set. **The case is unsatisfiable and contributes nothing.** **What drops out with it.** Their conclusion in Case 2 is *"then `f(i)` is also divisible by `p`, hence `p` must be sifted after the first step"* — an argument that needs `Ω^II_p` to be non-empty to be about anything. Downstream, the ledger's `Lemma 2` entry and the Theorem-1 factor `((ll y)²/lll y)^{h_f}` both vanish identically. §A3 of the script evaluates them: the factor reads `1.000000` at every `L` tested, and the ledger entry reads `0.000000` against `5.849812` to `8.172412` at `h_f = 1`. **[VERIFIED]** So **Lemma 2, Chebotarev's density theorem in its effective Lagarias–Odlyzko form, Theorem 2, Lemma 3 (Birch–Swinnerton-Dyer), the Galois groups `G_f` and `G_f⁺`, and the whole of K–K §3 are consumed by nothing.** Roughly half the paper is not needed for the substituted proof. ### Case 3 — `i mod p ∈ Ω^III_p` **Hypothesis.** `z₀ < p ≤ z₁` and `i ≡ 1` or `i ≡ −1 (mod p)`. **Conclusion.** `i` is killed at step 2, contradicting the assumption. The step-2 choice `a_p = 1` has kill set `{1, −1}`, which is the hypothesis verbatim. ∎ **Why this is strictly stronger than theirs.** K–K's Case 3 reads *"`z₀ < p ≤ z₁` and `f(i) ≡ y_p`. Then `i` is sifted after the second step"*, and to have a `y_p` at all they must invoke the definition of `M(f)` (their Definition 1), prove it exists (their Theorem 2, via Chebotarev and Birch–Swinnerton-Dyer), and settle for control of `M_p(f)` on **logarithmic average** with exceptional primes absorbed into an `O(1)`. Our `Ω^III_p` has exactly two elements at **every** odd prime, with no average and no exceptions. §A of the script confirms it across all 78 498 primes below `10⁶`: zero odd `p` with `|Ω^III_p| ≠ 2`. **[VERIFIED]** ### The three structural facts, quantified | K–K need | we get | source | |---|---|---| | `|Ω^I_p| = ℓ_f` for `p > p₀`, `p₀` unquantified | `|Ω^I_p| = 2` for every `p > 2` | `Res(x, x+2) = 2` | | `Ω^I, Ω^II, Ω^III` pairwise disjoint for `p > p₀`, `p₀` unquantified | disjoint for every `p ≥ 5`, and `p = 3` is the last overlap | `Res(x(x+2), (x−1)(x+1)) = −3` | | `κ = 3 deg f = 6` in Corollary 1 | `κ = 4` | `|Ω^I| + |Ω^III| = 4`, `Ω^II = ∅` | All three **[VERIFIED]** at §A, both by the resultants (which make them proofs, not sweeps) and by the sweep to `10⁶`, which agrees. **`p₀ = 5` is also the best a two-class choice can do.** §B computes `Res(x(x+2), (x−a)(x−a+2)) = a²(a²−4)` over `a`. The values `a ∈ {0, 2, −2}` are degenerate; among the rest `a = 1` and `a = −1` minimise `|Res|` at `3`, and `3` is the floor because `3 | a²(a²−4)` whenever `3 ∤ a`. So `a_p = 1` is optimal. **[VERIFIED]** --- ## 3. The Mertens ledger, term by term ### 3a. Corollary 1 applies, and at a smaller `κ` `R := #{i ≤ m unkilled after steps 1 and 2}`. The Proposition gives ``` R ≤ S(m, Ω) + O(√y) + 2 Ψ(m + 2, z₁) ``` the factor 2 on `Ψ` because case (b) is a statement about `i` **or** `i + 2`; K–K carry one copy because their `k(i)` ranges over `ℓ_f` linear factors and they absorb the count into the implied constant. It changes nothing. Corollary 1 (v2 p. 4) is applied at `z = √y`, `X = m`, `g(p) = |Ω_p|`. Its hypotheses, checked at §C: - **`g(p) ≤ κ`**, with `κ = 4` against their `κ = 6`. A smaller `κ` is not worse: Lemma 1's implied constant depends on `κ` only. - **`g(p) < p`**. `g(p) = 4 < p` for `p ≥ 5`. The two primes where band 2's count would fail are `p = 2` (`g = 2 = p`) and `p = 3` (`g = 3 = p`), and the **band structure** puts both in band 1, where `Ω^III_p = ∅` and `g(2) = 1`, `g(3) = 2`. Band 2 starts above `z₀ = (ln y)^A`, so this asks `(ln y)^A > 3`, i.e. `y > 3.475` at `A = 5`, which is vacuous against K–K's own `y ≥ 19`. - **`z ≪ X`**: `√y ≪ m`, immediate. - Their escape hatch *"If for some `p ≤ √y` we have `g(p) = p`, then clearly `S(m, Ω) = 0`"* is never triggered: §A finds zero primes with `g(p) ≥ p`. `z = √y` and `X = m` are the same as theirs, so **the substitution introduces no new analytic requirement anywhere.** ### 3b. The two contributions, and they are exactly the two the brief names ``` Σ_{p ≤ √y} g(p)/p = Σ_{5 ≤ p ≤ √y} 2/p + Σ_{z₀ < p ≤ z₁} 2/p + O(1) = 2 lnln y + 2(lnln z₁ − lnln z₀) + O(1) ``` The first sum is `Ω^I` over **all** `p ≤ √y`, with `|Ω^I_p| = 2` replacing K–K's `ℓ_f`. The second is `Ω^III` over **band 2**, with `|Ω^III_p| = 2` replacing K–K's `M_p(f)`. The middle term of their three-way split — `Lemma 2`'s `h_f(lnln y − lnln z₁ + lnln z₀)` — is identically zero by §2 Case 2. **Both are now plain Mertens sums over a constant.** The first was already Mertens for them. The second was Chebotarev for them and is Mertens for us, which is the concrete form of "Case 3 is strictly stronger". **The `O(1)` is an identified constant, not a shrug. [VERIFIED]** With `M = 0.2614972128476428` and `lnln √y = lnln y − ln 2`, ``` Σ_{5 ≤ p ≤ √y} 2/p − 2 lnln y → −2 ln 2 + 2M − 2(1/2 + 1/3) = −2.529967 ``` and §F1 measures the residual-minus-constant at `0.007876, 0.002472, 0.000609, 0.000078, 0.000019` for `√y = 10³ … 10⁷`. §F2 tracks the band sum against `2(lnln z₁ − lnln z₀)` to within `0.002443` at `(z₀, z₁) = (10⁴, 2·10⁷)`. **And the identity is rigorous at the `y` that matter, not only where a computer can see them.** Rosser–Schoenfeld 1962 Thm 20 gives `|Σ_{p≤x} 1/p − lnln x − M| < 1/(10 ln²x) + 4/(15 ln³x)` for `x ≥ 286`; `M` cancels in a band difference. §F4 evaluates the resulting bound at the **real** band endpoints: `2.475e-4` at `L = 10³` falling to `1.911e-5` at `L = 10⁹`. §F3 assembles the full ledger at the largest `y` where the bands fit inside a computable sieve range, and lands `0.12`–`0.21` from the identified constant `−2 ln 2 + 2M − 1/2 = −1.363300`. That gap is `z₀` sitting at 14 to 32, an order of magnitude below the `x ≥ 286` where Rosser–Schoenfeld applies. It is the small-band artifact, not a defect in the decomposition. ### 3c. The assembly, which is an identity ``` S(m, Ω) ≪ m ∏_{p≤√y}(1 − g(p)/p) ≪ m exp(−Σ g(p)/p) ≍ m (ln y)^{−2} (ln z₀ / ln z₁)² ``` Expanding against K–K's own displayed `A^{2M(f) − 2h_f} y/(B ln y) = A⁴y/(B ln y)`, every coefficient cancels: ``` (ln m − L) − 2 lnL + 2(ln ln z₀ − ln ln z₁) − 4 lnA + lnB + lnL = 3 lnL + 2 ln lll − 4 ln ll − lnL + 2(2 lnA + 2 ln ll − ln lll − lnL) − 4 lnA = 0 lnL: 3−1−2, lll: 2−2, ll: −4+4, lnA: 4−4 ``` §G evaluates that difference at 30 `(L, A, B)` triples from `L = 10²` to `10³⁰⁰` and gets between `−5.684e-13` and `8.527e-14`, `max |ratio − 1| = 5.684e-13`. **[VERIFIED]** Since the implied constant depends only on `κ`, and not on `A` or `B` — K–K's own remark, v2 p. 7 — taking `B` large against `A` gives `S(m,Ω) ≤ y/(4 ln y)`. ### 3d. The smooth-number step, and the greedy `2Ψ(m+2, z₁) = o(y/ln y)` is the one imported estimate K–K also assert without proof. §E evaluates it. With `u = ln m/ln z₁ ~ A·ll y/lll y`, the requirement holds at every `A ≥ 4` tested and fails at `A = 3` from `L = 10⁶` upward, which is the crude `A > 4` showing up. The bisected least admissible `A` rises `2.3196, 2.7134, 2.9068, 3.1093, 3.2709, 3.3666, 3.5879, 3.7460, 3.8185` across `L = 10²` to `10³⁰⁰`, approaching 4 from below and never reaching it. **The substituted proof needs `A > 4`, which is exactly the `ℓ_f + M(f) = 4` K–K's own version of this step asks for.** The estimate is used well inside its range: the Hildebrand slack `ln z₁/(lnln m)^{5/3}` runs `6.364e-1` at `L = 10²` to `2.339e5` at `L = 10⁹`. **[VERIFIED]** Hence `R ≤ y/(4 ln y) + o(y/ln y) ≤ y/(3 ln y)`. Step 3 assigns each remaining `i` its own prime `p ∈ (y/2, y]` with `a_p ≡ i`. §G2 makes the counting explicit rather than asymptotic: Rosser–Schoenfeld give `π(y) − π(y/2) > y/L − 0.62753 y/(L − ln 2)`, which exceeds `y/(3L)` exactly when `0.62753 L/(L − ln 2) < 2/3`, i.e. `L > 11.807294`, i.e. **`y > 1.3423e5`**. Comfortably below every `y₀` in §5. **[VERIFIED]** The two-class system has slack here that K–K do not: each band-3 prime could kill two leftovers if they happened to differ by 2. Nothing uses that. --- ## 4. The theorem, with every hypothesis > **Theorem.** There is an absolute `y₀` such that for all `y ≥ y₀` > > ``` > G₂(P(y)) ≫ y (ln y)³ (ln ln ln y)² / (ln ln y)⁴ > ``` > > with an absolute implied constant. **Inputs consumed, all [PROVEN] with source:** | input | source | |---|---| | Lemma 1, the fundamental lemma of sieve theory, at `κ = 4` | K–K v2 p. 3, citing Halberstam–Richert Thm 2.2 | | Corollary 1, which sees `Ω_p` only through `\|Ω_p\|` | K–K v2 p. 4 | | Mertens' theorem, with Rosser–Schoenfeld Thm 20 for the explicit error | Rosser–Schoenfeld, *Illinois J. Math.* 6 (1962) | | `Ψ(x,z) ≪ x ρ(u)` in Hildebrand's range | standard; asserted without proof by K–K too | | `π(y) − π(y/2)` lower bound | Rosser–Schoenfeld, both directions | | `G₂(P(y)) − 1` = longest coverable interval, one free pair per prime | `two-class-lower-bounds.md` §1, elementary + CRT | **Inputs NOT consumed:** Lemma 2 and Chebotarev; Theorem 2 and the Galois groups `G_f`, `G_f⁺`; Lemma 3 and Birch–Swinnerton-Dyer; the definition of `M(f)`; and Theorem 3. All of K–K §3. **What is conditional.** Nothing beyond the imported inputs. The deduction in §§2–3 is written out here in full. The place a sceptic should push is the smooth-number estimate of §3d, which K–K also assert; §E shows it is used with five orders of magnitude of slack against its range condition at the parameters that matter. **Calibration, stated plainly.** This is a proof, not a citation. Every input is published and sourced; the deduction has not been refereed. The corpus's own practice is to write **[PROVEN]** for a complete elementary derivation from published inputs (`two-class-lower-bounds.md` §1 does exactly this), and that is the tag used here — but the honest reading is "derived here, complete, not refereed", and it should be quoted that way outside the repo. **Where it sits. [VERIFIED]**, §I2: | statement | form | status | |---|---|---| | `two-class-lower-bounds.md` §4b unconditional base | `x ln x` | superseded | | free, `G₂ ≥ g` via FGKMT | `x ln x lll x / ll x` | superseded, still **[PROVEN]** | | `two-class-lower-bounds.md` §4b INFERRED analogue | `x ln²x lll x / ll x` | superseded, and it was **conditional** | | **this theorem** | `x ln³x (lll x)² / (ll x)⁴` | **unconditional** | | Maier–Pomerance, `two-class-lower-bounds.md` §4b ledger | `x ln^{3+o(1)} x` | **CONJ**, above | Measured ratios: transferred/free runs `1.5637e2` at `L = 10²` to `1.5733e20` at `L = 10¹²`; transferred/§4b runs `1.5637e0` to `1.5733e8`, which is the one factor of `ln x` the brief predicted and which is worth only **1.56** at `L = 10²`; transferred/MP runs `5.1856e-3` down to `1.8898e-5`. **A lower bound sitting below a conjectured truth is the only consistent arrangement**, and the ratio falls like `(lll x)²/(ll x)⁴`, so it never crosses. Against the standing upper bound `G₂ ≪_ε x^{4.26645+ε}` there is no contact at all, and nothing here moves that exponent. **Note that the §4b analogue this supersedes was itself conditional** — §4b tags it *"contingent on the Rankin/FGKMT refinements transferring, which is not free: they need survivor counts in arithmetic progressions, that is, a level of distribution for the dimension-2 sieve"*. The transferred bound needs no such thing: Corollary 1 is an upper-bound sieve on `[1,m]` and asks for no level of distribution. **So the new bound is one log higher AND on firmer ground.** --- ## 5. What "for large enough `y`" costs, and what it explains K–K's proof is asymptotic and the paper never quantifies `A`, `B` or `y₀`. Every hypothesis of the substituted proof is now an explicit inequality, so §H bisects for `y₀`. **[VERIFIED]** | condition | inequality in `L = ln y` | |---|---| | H1 `z₀ > 3` | `A ln L > ln 3` | | H2 `z₀ < z₁` | `A² ln²L < L lnln L` | | H3 `z₁ < √y` | `2 lnln L < A ln L` | | H4 Case 1 | `ln2 − lnB + 3 lnL + 2 ln lll − 4 ln ll < A ln L` | | H5 smooth count | `ln2 + ln(m/y) + ln ρ(u) + ln3 + lnL < 0` | | H6 greedy | `L > 11.8073` | | `A` | `L₀ = ln y₀` | `y₀` | binding | |---|---|---|---| | 4.05 | `3.0867e2` | `10^134.1` | H2 | | 4.5 | `4.0795e2` | `10^177.2` | H2 | | 5 | `5.3747e2` | `10^233.4` | H2 | | 7 | `1.2733e3` | `10^553.0` | H2 | | 10 | `3.1008e3` | `10^1346.7` | H2 | `B` does not move `y₀` at all — `B = 10` and `B = 1000` give the same `L₀` to five figures — because `B` enters only H4, which §2 Case 1 shows is slack. **The binding condition is always H2, `z₀ < z₁`.** That is a property of K–K's own band geometry and has nothing to do with the substitution: their construction has the same `y₀` at the same `A`. Read it as the price of an asymptotic Erdős–Rankin argument, not as a defect in one. **It is also the complete explanation of attack I's D4.** `attack-lower-bound.md` §4 records that the three-band construction certifies at five accessible `y` but "loses to the §5d greedy by 4.0 to 6.6". At `y = 4001` the bands do not exist — `z₀ = (ln y)^A` with `A > 4` is already `1.5e6`, past `y`. **No finite computation can exhibit this construction working as designed, so D4 was never evidence about the exponent in either direction.** That was stated as a suspicion in attack I; it is now a computation. --- ## 6. What the bound is worth: K–K's Theorem 1 does not recover the classical case This is the calibration that matters most for anyone tempted to treat the transferred exponent as final. **[VERIFIED]**, §I1. Take `f(x) = x`. Then `j_f` **is** the ordinary Jacobsthal function, `ℓ_f = 1`, `h_f = 0`, and `M(f) = 1` (every fibre of the identity is a point; consistent with their own Theorem 3(i), `M(x^d) = τ(d)`, at `d = 1`). Theorem 1 reads ``` j(P(y)) ≫ y (ln y)⁰ · ( ln y · lll y / (ll y)² )¹ = y ln y lll y / (ll y)² ``` against the bound K–K themselves quote on their own page 2, from Ford–Green–Konyagin–Tao: `j(P(y)) ≫ y ln y lll y / ll y`. **Their general theorem is weaker than the known special case by exactly one factor of `lnln y`.** The measured ratio is `4.6052, 6.9078, 9.2103, 13.8155, 20.7233` at `L = 10² … 10⁹`, matching `lnln y` to every digit printed. That is not an error in their paper — a general theorem is entitled to be weak at a special case, and theirs generalises in a different direction. It comes from using a **fixed `A`** in `z₀ = (ln y)^A` rather than the optimised Erdős–Rankin band choice. **But the transferred two-class bound inherits the same slack**, so > `x ln³x (lll x)²/(ll x)⁴` is a **floor on what this method gives**, not the > method's ceiling. **[INFERRED]** — an optimised band choice, or importing Pintz's or FGKMT's refinements into the substituted frame, should recover one or two factors of `lnln x`. That is not done here and is not claimed. It is the natural next attack, and unlike everything else on the lower side it is a *paper* attack of known size. --- ## 7. Novelty, re-verified this session with calibration Every probe below was run in this session with `curl`, with a known-positive calibration on the same channel and the same call shape. Standing negatives already settled in `research/SEARCH-CONVENTIONS.md` §"do not redo" are **not** re-run, per that file's own instruction. ### 7a. A144311 carries no formula and no reference. [ABSENT] Channel: `oeis.org/search?q=id:A144311&fmt=text`, full record retrieved, 1193 bytes. **Zero `%F` lines and zero `%D` lines.** The record is `%I`, `%S/%T` (22 terms, `1,5,11,…,1709`), `%N`, one `%C` (*"For n > 1, a(n) == 5 (mod 6)"*), two `%H`, one `%e`, `%Y` (*"Cf. A048670, A049300, A058989"*), `%K nonn,more,hard`, `%O`, `%A` Andrew Carter Sep 17 2008, `%E` Alekseyev (a(8)–a(16), Nov 18 2009) and Jinyuan Wang (a(17)–a(22), Nov 26 2024). **Calibration, same channel, same call shape, same session:** `oeis.org/search?q=id:A048670&fmt=text` returns 6364 bytes carrying **five `%F` lines and one `%D` line**. So the channel does return formula and reference lines when a record has them. **Calibration 2, numeric:** `q=1,5,11,29,41,65,107,149,203` returns A144311. Channel live on the object itself. ### 7b. A048670 carries the FGKMT bound, as a `%C`. [VERIFIED] Verbatim from the retrieved record: > `%C A048670 Ford, Green, Konyagin, Maynard, & Tao show that j(x#) >> x log x > log log log x / log log x and hence a(n) >> n log^2 n log log log n / log log > n. - _Charles R Greathouse IV_, Mar 29 2018` Pintz's bound is present as both a `%C` and a `%F`. **One refinement to `attack-lower-bound.md` §5b**, which does not say which tag the Maier–Pomerance line carries: it is a **`%F`**, not a `%C` — > `%F A048670 Maier & Pomerance conjecture that Max_{n <= x} A048669(n) = > log(x)*(log log x)^(2+o(1)) which suggests a(n) = n*(log n)^(3+o(1)). - > _Charles R Greathouse IV_, Mar 29 2018` So the free bound is recorded for the **one-class** object and absent for the two-class one, exactly as the corpus claims. ### 7c. Nobody has cited Kalmynin–Konyagin. [ABSENT] | channel | query | result | |---|---|---| | OpenAlex, forward | `works/https://doi.org/10.48550/arxiv.2302.00459` → W4319049890 (2023) | `cited_by_count = 0` | | OpenAlex, forward | `works/https://doi.org/10.4213/im9467` → W4393170300 (2024) | `cited_by_count = 0` | | OpenAlex, forward | `works/https://doi.org/10.4213/im9467e` → W4393954820 (2024) | `cited_by_count = 0` | | **OpenAlex, reverse** | `works?filter=cites:W4319049890` | **0 results** | | **OpenAlex, reverse** | `works?filter=cites:W4393170300` | **0 results** | | **OpenAlex, reverse** | `works?filter=cites:W4393954820` | **0 results** | | Semantic Scholar | `paper/arXiv:2302.00459/citations?fields=title,year&limit=100` | **0 entries**, twice | The OpenAlex reverse probe is **new this session**; attack I ran only the forward `cited_by_count`. It is the stronger negative, because it enumerates the citing set rather than reading a cached counter. **Calibration, same channel, same call shape, same session.** - **OpenAlex, forward:** `works/https://doi.org/10.4007/annals.2016.183.3.4` (Ford–Green–Konyagin–Tao, *Large gaps between consecutive prime numbers*) returns `cited_by_count = 46`. - **OpenAlex, reverse:** `works?filter=cites:W2964097236` returns **48 results**, first *"Long gaps between primes"*. - **Semantic Scholar, identical endpoint and parameters:** `paper/arXiv:1408.4505/citations?fields=title,year&limit=5` returns **5 entries** (2026 Gilbreath, 2026 quantum processors, 2026 bounded gaps, 2026 Goldbach, 2025 random polynomials). The record endpoint `paper/arXiv:1408.4505?fields=title,citationCount` returns `citationCount = 80`. All three probes are live and return non-zero on the neighbouring literature. Semantic Scholar's calibration required a 200-second backoff — its record endpoint returned HTTP 429 repeatedly, and the two negatives quoted above come from calls that returned HTTP 200 with an empty `data` array, not from a rate-limited call. ### 7d. No two-class lower bound turns up on a topical sweep. [ABSENT] Channel: OpenAlex `works?search=Jacobsthal function primorial lower bound`, 5 results. None is a lower bound for the two-class object. The nearest is Ziller–Morack, *On differences between consecutive numbers coprime to primorials*, arXiv:2007.01808 (2020), whose abstract was retrieved in full: it studies **which even numbers fail to occur** as a difference below `j(p_k#)`, not the size of `G₂`. Same channel, same call shape; calibration is 7c's. ### 7e. Channels that failed, with no negative claimed from them - **arXiv API.** `export.arxiv.org/api/query` returned **zero bytes** on the known-positive calibration query `all:"Jacobsthal function"`. Channel dead, the same failure attack I's producing session found. **No negative is quoted from it.** - **Semantic Scholar's record endpoint** `paper/{id}?fields=citationCount` returned HTTP 429 on every attempt for the K–K id, including after two backoffs, so **no `citationCount` figure is quoted for K–K.** Its *citations* endpoint did answer, is calibrated in §7c, and is the only Semantic Scholar channel quoted. ### 7f. What is and is not novel, stated narrowly `SEARCH-CONVENTIONS.md` already settles that the **multi-class Erdős–Rankin construction is Kalmynin–Konyagin's, not ours** (row: *"Is our multi-class Erdős–Rankin claim novel? **No** — Kalmynin–Konyagin"*). Nothing here reopens that, and §1c credits the mechanism to them throughout. What §7a–7d support is narrower and, given 7c, close to mechanical: > **[INFERRED]** K–K's method has not been instantiated at the free-translate > difference-2 system, and the resulting bound for `G₂` is not in print. The > reason is not that anyone tried and failed: **nobody has built on the paper at > all**, on the strongest citation channel available and in both directions. `covering-dive.md` §4.2 named the open step verbatim — *"substituting an arbitrary 2-element set requires re-deriving that trichotomy for `Ω_p = {a_p, a_p − 2}`"*. **That step is discharged by §2 above.** --- ## 8. What this changes in the corpus Four edits. **None is applied here**; this is a staging report and it may not edit the corpus. **(a) `two-class-lower-bounds.md` §3's headline sentence is now false.** It reads *"This is the strongest lower bound available for `G₂` today"* of the free bound `x ln x lll x/ll x`. §4 above is two logs above it. The **[PROVEN]**-with-source status of §3's bound is untouched and it remains the strongest bound with a published left-hand side; the sentence needs the qualifier. **(b) `two-class-lower-bounds.md` §4b's INFERRED analogue is superseded, and by something unconditional.** §4b lands at `x ln²x lll x/ll x` and tags itself contingent on a level of distribution for the dimension-2 sieve. §4 above lands one log higher and needs no such thing. §4b's three-row ledger — base 1, survivor density +1, Maier–Pomerance +1 — should gain a fourth reading: **K–K's band-2 device buys the survivor-density log a second time**, because `Ω^I` is charged at band-2 primes and paid for by smoothness rather than by a choice. That is the mechanism §4b's accounting does not have. **(c) `covering-dive.md` §4.2's named blocker is discharged.** The file records the trichotomy re-derivation as the unchecked step. §2 above is that re-derivation, and §2's Case-2 paragraph is the answer to why it was easier than the file expected. **(d) `ZONE-POSTULATE.md` §3, line 100, survives with one word changed.** The claim *"apparently the first lower bound recorded for the two-class Jacobsthal problem"* was about the free bound; the corpus now has a stronger one, so the sentence should point at §4 above instead. **The Postulate's safety is unaffected and should be restated with the new number**: the threshold needs `G₂(x#) < x′²`, and `x²/(x ln³x) = x/ln³x → ∞`, so the margin is still `x^{1−o(1)}`. A stronger construction-side bound was the one thing that could have threatened it. It does not. ### Draft CHANGELOG entry *(For `research/history/CHANGELOG.md`. Not applied.)* > **2026-08-18/19 — Attack 4: the Kalmynin–Konyagin substitution is finished at > D3, and the two-class lower bound gains a log.** `covering-dive.md` §4.2's > named blocker — re-derive K–K's Cases 1–3 for `Ω_p = {a_p, a_p−2}` — is > discharged in `research/history/staging/attack-kk-substitution.md` §2, written > as a proof rather than as checks. **New theorem:** > `G₂(P(y)) ≫ y (ln y)³ (lnlnln y)²/(lnln y)⁴` for `y ≥ y₀`, derived here from > PROVEN inputs (K–K's Lemma 1 / Corollary 1 at `κ = 4`, Mertens with > Rosser–Schoenfeld, the standard smooth-number estimate, CRT) and **not > refereed**. **Case 2 is vacuous** (`h_f = 0`), so Lemma 2, Chebotarev, > Theorem 2, Lemma 3 and all of K–K §3 are consumed by nothing; **Case 3 is > strictly stronger**, `|Ω^III_p| = 2` at every odd prime rather than on > logarithmic average, verified across all 78 498 primes below 10⁶; > **`p₀ = 5` is exact**, from `Res(x(x+2), (x−1)(x+1)) = −3`, and `a_p = 1` is > the optimal band-2 choice. **Case 1 reduces to the single inequality > `2m/y < z₀`**, needing `A > 3` where K–K need `A > 4`. **The Mertens ledger > decomposes exactly as `2 lnln y + 2(lnln z₁ − lnln z₀) + O(1)`**, with the > `O(1)` identified as `−2 ln 2 + 2M − 5/3 = −2.529967` and reached to `1.9e-5` > at `√y = 10⁷`; the exponent assembly is an algebraic identity that collapses > to zero coefficient by coefficient, checked to `5.684e-13` at 30 parameter > triples out to `ln y = 10³⁰⁰`. **New number the paper never gives:** `y₀`, > `10^134.1` at `A = 4.05` rising to `10^1346.7` at `A = 10`, with `z₀ < z₁` > always binding — a property of K–K's own band geometry. **That retires > `attack-lower-bound.md` D4's factor-5 loss to the greedy as evidence of > anything**: at `y = 4001` the bands do not exist and no finite computation can > exhibit the construction. **Calibration that limits the claim:** K–K's > Theorem 1 at `f(x) = x` is the ordinary Jacobsthal function and lands > `y ln y lll y/(ll y)²`, weaker by exactly `lnln y` than the FGKT bound they > quote in their own introduction, so the transferred bound is a floor on the > method and not its ceiling. **Supersedes:** `two-class-lower-bounds.md` §3's > "strongest lower bound available today" and §4b's INFERRED > `x ln²x lll x/ll x`, the latter also being conditional on a dimension-2 level > of distribution the new bound does not need. **The Zone Postulate is > unaffected**: `x²/(x ln³x) → ∞`, margin still `x^{1−o(1)}`. **Prior art** > re-verified this session on calibrated channels: A144311 carries zero `%F` and > zero `%D` (calibrated against A048670's five `%F` and one `%D`, same call > shape); all three OpenAlex records for K–K report `cited_by_count = 0` and the > **reverse** probe `filter=cites:` returns 0 results for each, calibrated > against FGKT at 46 and 48. Semantic Scholar's > citations endpoint returns 0 entries for K–K against 5 for FGKT on the > identical call. **No negative is quoted from the arXiv API**, which returned > zero bytes on a known-positive query. --- ## 9. What this report does not claim - **It does not move 4.26645.** Nothing on the lower side can. - **It does not prove Lemma 1 or the smooth-number estimate.** Both are imported as **[PROVEN]** with their sources, exactly as K–K import them. §E shows the second is used with five orders of magnitude of range slack; it does not prove it. - **It does not claim the construction is novel.** The multi-class Erdős–Rankin construction is Kalmynin–Konyagin's, as `SEARCH-CONVENTIONS.md` already settles. What is discharged here is the *substitution*, which `covering-dive.md` §4.2 named as the open step. - **It does not claim a refereed theorem.** §4's calibration paragraph says so explicitly, and any statement of it outside this repo should carry that. - **It does not verify the construction at any `y`.** §5 shows no computable `y` exists at which it could be verified, which is itself the finding. - **It does not re-litigate the measured growth law.** `attack-lower-bound.md` §3 settles that the transferred bound and attack E's `0.762 x ln²x lnln x` agree on the x-exponent, do not cross until `x = 10^7327`, and are different kinds of statement. Nothing here disturbs it, and §4's table is consistent with it: the transferred bound sits **below** the Maier–Pomerance conjectural size at every computable `x`. - **It quotes no negative from an uncalibrated channel.** §7e names the two that failed and what is not being claimed from them. --- ## 10. Reproduction and gate ``` # the PDF of record, md5 b5d7d2a23ffd902415057adebfe430b1 curl -sSL -o kk_v2.pdf https://arxiv.org/pdf/2302.00459 pdftotext -layout kk_v2.pdf - # section 2 is pages 3-7 # the script, and the only sanctioned way to put its run into it node research/attack-kk-substitution.js node research/qc/embed.js research/attack-kk-substitution.js node research/qc/embed.js --check research/attack-kk-substitution.js # the probes of section 7 curl -s "https://oeis.org/search?q=id:A144311&fmt=text" curl -s "https://oeis.org/search?q=id:A048670&fmt=text" curl -s "https://api.openalex.org/works/https://doi.org/10.48550/arxiv.2302.00459" curl -s "https://api.openalex.org/works?filter=cites:W4319049890" curl -s "https://api.semanticscholar.org/graph/v1/paper/arXiv:2302.00459/citations?fields=title,year&limit=100" ``` Script tail: `code-sha256 eeeb366a016925a5…`, `out-sha256 0be2b9d1f40d6ba5…`, 444 lines of output, node v22.21.0, 0.6 s; `--check` reports both matching. `node research/qc.js` **contributes 0 findings across all eleven checks for both files added here**, before and after. The gate's non-zero TOTAL during this wave is entirely `uncited-script` and `no-banner-title` on scripts written by the other nine attacks running concurrently; `attack-kk-substitution.js` appears in none of them. `node research/qc/selftest.js`: all 22 known positives fire, all 14 controls silent. --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [../CHANGELOG.md](../CHANGELOG.md), indexed by document.*