# Attack foldL-05: the direct 0c attack — bounding maxsum_m without counting kills *2026-08-19. Attack angle 5 of 5 on the fold-multiplier complex, `TODO.md` item 0c verbatim: a proven upper bound on `maxsum_m(T_x)` (plain and residue-deleted) that is not `G₂` itself, by an argument that never counts kills. Producer: `research/attack-foldL-05-maxsum-direct.js`, formally embedded (`node research/qc/embed.js --streams both --timeout 900`), 270.0 s, all self-tests pass. Legend as in `research/sift-limit-attack.md`: **[PROVEN]** derived or published theorem; **[VERIFIED]** checked computationally here; **[MEASURED]** empirical, finite range; **[INFERRED]** deduction from sourced facts.* --- ## HEADLINE **The brief's premise contradicts a result this repository already proved, and the correction has to come first.** The brief opens with *"today's session raised its payoff: the bridge `L ≤ 1 + max{k : maxsum_k(T_x) ≥ k(2p′−2)}` converts any maxsum bound directly into a fold-multiplier bound"*. It does convert — and it cannot convert far enough, for a reason that has nothing to do with how good the maxsum bound is, and that reason is **already in the corpus**: > **The Bridge Floor.** `maxsum_k ≥ maxsum_1 = G₂(T_x)` for every `k`, so every > `k` below `G₂(T_x)/(3p)` satisfies the bridge's inequality and the bridge's > output is `≥ 1 + ⌊G₂(T_x)/(3p)⌋ ≈ 0.183x` **whatever upper bound on maxsum is > supplied.** — `research/a3-05-bound-L.md` §7 (*"Theorem B can never prove > L ≤ 1 + m for any m below G2(T_x)/(3p), no matter how the gap word behaves"*), > and `research/kappa-not-L.md` §"The wall, located precisely", which adds that > **Theorem C inherits the ceiling, so redirecting `L` to `κ(m)` does not move > the wall.** [PROVEN, already in the corpus; re-derived independently here and > tabulated at the eight exact levels] So **solving 0c perfectly does not deliver the u-frame through the L-bridge, and it does not deliver it through κ(m) either.** 0c's value has to be collected through the copy theorem, which is an identity with index cost zero (`U-FRAME.md` §5a step 2), and not through `L`, which is where the brief and the branch keep routing it. **This is the finding this attack ranks first, and the credit for the argument belongs to A3/A5, not to this run.** What this run adds is the exact-level table (§3) and the demonstration that a *kill-count-free* argument lands on exactly the same floor (§4, R3). **What is new here, and it is one thing.** The threshold-`m` sieve certificate is **unconditionally available, all-positions, and it amortises**: its overhead over the truth is *additive*, not multiplicative, so the overshoot falls from 1.94 at `m = 1` to 1.20 at `m = 8` (`z = 29`), and is **exactly 1.0000 at `z = 13`, `m = 8`**. That is a proven upper bound on `maxsum_m` which is not `G₂` and which costs almost nothing beyond the `m = 1` bound the corpus already ran — and its asymptotic crossover sits at `z^{β₂}·V`, a growing power above where the branch needs it. **And two mean-square routes are closed, one of them tautologically.** | route | verdict | |---|---| | R1 certificate at threshold `m` | **NEW, PARTIAL SUCCESS**, unconditional, exact at five levels; **CLOSED as a route to `L`** — crossover in the wrong place by `x^{β₂−1}·ln x` | | R2 Chebyshev over **alignments** | **NEW, CLOSED, PROVEN, tautologically** — Samuelson forces the exceptional fraction `≥ 1/(p−1) > 1/p` | | R2 Chebyshev over **positions** | **CLOSED, quantitatively** — `maxsum_m ≤ H` is emptiness, shortfall ~20× per fold; Lemma V's version vacuous by 810× to 3470× | | R3 residue-deleted / Merge Lemma | **CLOSES into a well-posed recursion** (a genuine fixed point, kill-count free) **and then busts the Overshoot Budget** in ~20 folds; its arithmetic ingredients are A3's, re-derived | | R4 prior art | see §6 | **Pre-registration held on three of five.** R1 was rated most likely to yield an unconditional partial result and did. R2-positions and R3 were predicted to close with mechanism and did. Two predictions were WRONG and are left visible. **(P1)** predicted the Merge index cost at `m = 1` to be `≈ 0.55p`, i.e. 6, 7, 9, 10, 13 at the five folds; the self-consistent solution gives **4, 5, 5, 6, 6**, because the fixed point is reached well below the naive estimate — the recursion is better than the prediction and still busts. **(P6)** predicted the brief's per-gap `2p′−2` floor would prove too strong; it is **exactly right** as a single-gap floor (`2p−2` for `p ≡ 1 mod 6`, `2p+2` for `p ≡ 5 mod 6`), merely weaker than the `3p` pair rate — and both are A3's qualifying-gap law and Alternation Lemma, which the disk already held and which the pre-registration should have checked for before predicting anything. ## 1. THE OBJECT, FIXED `T_x` is the set of twin slots mod `W_x = x#`, `D_x = ∏_{3≤q≤x}(q−2)`, `m̄ = W/D`; `maxsum_m(T)` is the largest sum of `m` consecutive cyclic gaps. Sieve form throughout (`qc/units.js` §5). Every twin slot is `≡ 5 (mod 6)` for `x ≥ 3` [VERIFIED], so **every gap is a multiple of 6** — a fact §2 turns into a theorem. > **The window-count equivalence [PROVEN, VERIFIED].** For half-open windows, > `min_t #{ slots in [t, t+H) } ≥ m ⟺ H ≥ maxsum_m(T)`. *Proof.* (⇐) given `t`, take `s_i` the largest slot `< t`; then `s_{i+m} ≤ s_i + maxsum_m ≤ t−1+H < t+H`, and `s_{i+1}..s_{i+m}` all lie in the window. (⇒) if `H ≤ maxsum_m − 1`, the window `[s_i+1, s_i+1+H)` at the extremal `i` holds only `s_{i+1}..s_{i+m−1}`. ∎ Verified **both directions**, at `H = maxsum_m − 1` and `H = maxsum_m`, over all `W` positions of `T_3, T_5, T_7, T_11` for every `m ≤ 6`. So a bound `maxsum_m ≤ H` **is** an all-positions window-count lower bound at threshold `m`, with no slack in the translation. That is what makes R1 the right first move: the sieve certificate speaks exactly this language. --- ## 2. THE KILL-GAP FLOOR — RE-DERIVED, AND IT WAS ALREADY PROVED HERE This section was run as a fresh derivation before the disk was searched. **It reproduces `research/kappa-not-L.md` in full and adds nothing to it.** Recorded because an independent second route to a proven result is custody, and because the campaign rule (`primeoire-campaign-lessons`) says an absence must check the disk first — this one did not, and the check is here rather than hidden. Fold `T_x` by `p`. The kills are exactly the slots in classes `{a, a−2} mod p` (`U-FRAME.md` §5a step 1). So a kill-gap satisfies `d ≡ 0` or `±2 (mod p)` **and** `6 | d`, because every twin slot is `≡ 5 (mod 6)` [VERIFIED here]. CRT gives the cost of each transition exactly [VERIFIED here at `p = 5..43`]: | transition | `p ≡ 1 (mod 6)` | `p ≡ 5 (mod 6)` | |---|---|---| | `a → a` and `a−2 → a−2` | `6p` | `6p` | | `a → a−2` | `2p−2` | `4p−2` | | `a−2 → a` | `4p+2` | `2p+2` | That table **is** `kappa-not-L.md`'s qualifying-gap law *"{2p−2, 4p+2, 6p} when p ≡ 1 (mod 6), and {2p+2, 4p−2, 6p} when p ≡ 5 (mod 6)"*, **checked there against brute force at all 302 primes from 5 to 1999** — a range 25× wider than this run's twelve. Its Theorem A (Run Cost) is the pair floor `6p`; the Alternation Lemma is the two-state walk that forces it; Theorem B is the bridge; Theorem C carries all of it to `κ(m)`. The one increment this run produces is the additive constant: maximising the discount over `k` transitions with `|n₁ − n₂| ≤ 1` gives > span of `k` consecutive kill-gaps `≥ 3pk − p − 2`, against `kappa-not-L.md`'s `c_min(k) ≈ 3p(k−1)`. [VERIFIED here at the seven folds `5→7 … 23→29` for every `k ≤ 24`: never violated; observed minimum single kill-gap 12, 24, 24, 36, 36, 48, 60 against the floor 12, 20, 24, 32, 36, 44, 56.] It is an additive constant on a bound whose branch is already settled, so it changes nothing, and it is recorded at that size. ## 3. THE BRIDGE FLOOR, TABULATED AT THE EIGHT EXACT LEVELS `L(x,p) ≤ 1 + max{ k ≥ 1 : maxsum_k(T_x) ≥ 3pk − p − 2 }` — A5 Theorem B with §2's constant [VERIFIED against the true `L` at all seven folds]. And then: | `x` | `G₂(T_x)` | bridge floor | u-frame need `0.31p/ln p` | floor / need | source | |---|---|---|---|---|---| | 11 | 42 | 2 | 1.42 | **1.406** | exact, A144311 | | 13 | 66 | 3 | 1.57 | **1.909** | exact | | 17 | 108 | 3 | 1.86 | **1.613** | exact | | 19 | 150 | 4 | 2.00 | **2.000** | exact | | 23 | 204 | 4 | 2.27 | **1.759** | exact | | 29 | 258 | 4 | 2.67 | **1.498** | exact | | 37 | 528 | 6 | 3.18 | **1.889** | exact | | 79 | 1710 | 8 | 5.60 | **1.427** | exact | | 101 | 4292.5 | 15 | 6.78 | 2.211 | model `0.55·θ(x)²` | | 1009 | 510224.5 | 169 | 45.22 | 3.737 | model | | 10007 | 5.396e7 | 1798 | 336.79 | 5.339 | model | | 10⁶ | 5.483e11 | 182779 | 22438.55 | 8.146 | model | **Read the two halves differently, and the note says so rather than blurring them.** The eight exact rows are the finding: the bridge's floor is above the u-frame's need at **every level anyone can compute**, and the ratio has **no trend** across them (1.406 to 2.000, non-monotone). The model rows add only the direction, and they inherit `maxgap-law.md`'s caveat that `0.55` is a whole-period constant with a fitted exponent bracket of 1.3 to 1.9 (`exponent-control.md` §5). **Nothing here proves the divergence; what is proven is failure at every computable level, plus the algebra `0.183x / (0.31x/ln x) = 0.59 ln x` that says which way it goes.** **What is new in this section is the table, not the argument.** `a3-05-bound-L.md` §7 states the ceiling and its exponent caveat; this run puts it beside the u-frame's own requirement level by level, which had not been done. **Consequence for `TODO.md` 0c, stated plainly.** 0c is worth doing, and it must not be collected through `L`. The bridge is a lossy channel with a floor; the copy theorem `maxsum_m(T_{x′}) = max_a Δ_m(x,x′,a)` is a lossless channel with index cost zero (`U-FRAME.md` §5a step 2). Any future 0c work should target `max_a Δ_m` directly and should not quote a bridge-derived `L` as its payoff. --- ## 4. THE FOUR ROUTES ### R1 — the certificate at threshold `m`. PARTIAL SUCCESS, then closed as a route. `research/theta-ladder-sup.js` (rewritten today, window from the caller) reports `minT_H`, the certified all-positions minimum of the vector-sieve count over the **full period**. By §1, `minT_H ≥ m` **is** `maxsum_m(T_{z−}) ≤ H`, where `z−` is the largest prime below `z` (the script sieves by primes `< z`). Driven here on a stride-6 grid (12 at `z = 23`, 48 at `z = 29`), 203 full-period walks: | `z` | tile | `H_cert/maxsum_m` at `m = 1` | at `m = 4` | at `m = 8` | overhead `H_cert − maxsum_m`, `m = 1..8` | |---|---|---|---|---|---| | 13 | `T_11` | 1.4286 | 1.1667 | **1.0000** | 18, 6, 12, 18, 12, 6, 6, 0 | | 17 | `T_13` | 1.9091 | 1.1538 | 1.1842 | 60, 42, 18, 24, 30, 24, 30, 42 | | 19 | `T_17` | 1.8333 | 1.5152 | 1.4375 | 90, 78, 102, 102, 150, 132, 132, 126 | | 23 | `T_19` | 1.7200 | 1.6053 | 1.2857 | 108, 96, 96, 138, 108, 114, 90, 108 | | 29 | `T_23` | 1.9412 | 1.5517 | 1.2045 | 192, 210, 192, 192, 198, 174, 138, 108 | > **Statement R1 [VERIFIED, unconditional, all positions].** At `z = 13, 17, 19, > 23, 29` the two-class vector-sieve certificate proves `maxsum_m(T_{z−}) ≤ > H_cert(z,m)` at every one of the `W = P(z)` positions of the period, for > `m ≤ 8`, with the values tabulated in the producer's §4. **The mechanism, and it is the one useful thing this attack found.** The certificate's cost over the truth is `≈ sup|R_H|/M`, which is **additive** and grows only slowly with `H`, while `maxsum_m` grows linearly in `m`. So the threshold-`m` certificate costs **almost nothing beyond the `m = 1` one**: the `m = 1` certificate is the expensive one, and it is the only one the corpus had ever run. Where the truth is `528` (`T_23`, `m = 8`) the certificate says `636`; where the truth is `204` (`m = 1`) it says `396`. **No quantifier price is paid for the `m` threshold** — the certificate is all-positions by construction, so the `C^{π(z)}` wall that closed `u_sup` (`sift-limit-attack.md` §7e) is not re-opened and is not touched: that wall is the price of turning a *mean-value* estimate into a *sup*, and the certificate never takes that step. **And it is closed as a route to `L`.** Asymptotically the unconditional `κ = 2` sieve certifies `N(t) ≫ H·V` for every `t` once `H ≥ z^{β₂+ε}` (`β₂ = 4.26645028414864191641`, DHR; the two-class assembly is `covering-dive.md` §"point 3", **[INFERRED from PROVEN ingredients]**). The threshold-`m` certificate therefore sits at `H ~ z^{β₂}` for every `m ≤ m* = z^{β₂}·V`, and only above `m*` becomes the linear `maxsum_m ≪ m·m̄`. The L-bridge needs `maxsum_k < 3pk` from `k ≈ 0.19x/ln x` upward. Measured: | `x` | 23 | 101 | 1009 | 10007 | 10⁶ | |---|---|---|---|---|---| | `m* = x^{β₂}·V` | 2.300e4 | 6.680e6 | 5.655e10 | 5.711e14 | 8.655e22 | | needed `0.19x/ln x` | 1.394 | 4.158 | 27.717 | 206.419 | 13752.659 | | `m*/needed` | 1.65e4 | 1.61e6 | 2.04e9 | 2.77e12 | 6.29e18 | The ratio grows like `x^{β₂−1}·ln x`. **The certificate is in the right language and at the wrong exponent — the 4.2665-against-2 wall in threshold coordinates.** The amortisation is genuine and it does not reach. **Where the certificate's own soundness is checked.** `H_cert ≥ maxsum_m` at all 40 `(z,m)` cells, with equality at exactly one. A single cell below 1 would falsify the pointwise minorant `Λ₁⁻Λ₂⁺ + Λ₁⁺Λ₂⁻ − Λ₁⁺Λ₂⁺ ≤ θ₁θ₂` that the whole certificate rests on; none is. (The sandwich itself is machine-verified in `research/sift-limit-attack.js`, not re-verified here.) ### R2, alignment form — CLOSED, PROVEN, and the closure is tautological. The copy theorem's quantifier is `max` over the `p` 2-sets. The brief's ask was exact: *"an almost-all bound survives a max over `p` items if the exceptional fraction is `< 1/p` — price exactly that."* Priced: > **[PROVEN, one line].** Samuelson's inequality gives `max − mean ≤ sd·√(p−1)` > for **any** list of `p` numbers. So the Chebyshev exceptional fraction > `sd²/(max−mean)²` is `≥ 1/(p−1) > 1/p` **always**. A Chebyshev bound over the > alignments can therefore **never** be upgraded to the max by a union bound over > the `p` 2-sets, at any level, for any tile. [VERIFIED] at all 18 cells tested: `sd²/(max−mean)²` reads 0.1207 to 12.0000 against `1/p` = 0.0345 to 0.0909. The column is `NO` everywhere, by necessity rather than by accident. **This is not a measurement that came out badly; it is an identity, and it closes the alignment-averaging idea permanently.** **What is left of the alignment route, and it is one instrument.** Samuelson itself, used forwards: `max_a Δ_m ≤ mean_a Δ_m + sd_a·√(p−1)`, exact and unconditional. NEW here (`sd_a` is not in `research/attack-0c0e-01-deleted-family.js`, whose min/mean/max this run reproduces exactly as custody). Priced over 48 `(fold, m)` cells: **Samuelson/max = 1.0391 to 1.3497**, best at `(17→19, m = 6)`, worst at `(7→11, m = 2)`. It is never near-tight — `(max−mean)/sd` runs 0.289 to 2.878 against `√(p−1)` = 3.162 to 5.292, so the extremal one-outlier configuration never occurs. **Verdict: it converts one unproven object (`max_a Δ_m`) into two (`mean_a` and `sd_a`) at a cost of 4 to 35 percent. Worth recording, not worth pursuing** — the two new objects are maxima over positions themselves, so nothing has been quantifier-reduced. ### R2, position form — CLOSED, quantitatively. Chebyshev at threshold `m` over positions is unconditional and exact here (the tile's second moment is computed exactly from the pair-distance distribution). The tile is **under-dispersed**: Fano `Var/mean` = 0.267, 0.350, 0.403, 0.454, 0.501 at `T_7 … T_19` for `m = 1`, and 0.185 to 0.250 at `m = 4`. Chebyshev over-states the true bad fraction by about 5× (0.25140 against 0.05199 at `T_19`, `m = 1`). **None of that is the obstruction.** By §1, `maxsum_m ≤ H` is the statement that the bad set is **EMPTY**, so the fraction must be pushed below `1/W`, not below any constant. Measured, `Cheb/(1/W)` = **2.80e1, 4.07e2, 6.12e3, 1.15e5, 2.44e6** at `x = 7, 11, 13, 17, 19` — about 20× lost per fold, forever. An almost-all statement over positions is not a partial result on the way to `maxsum_m`; it is a different quantity, and the gap between them is `W`. **Lemma V's mean square at threshold `m`, with explicit constants.** The proved form `⟨R²⟩_H ≤ B(z,s)·H` (`B ≤ 9A²(E−1) = O((log z)^8)`, unconditional, `history/staging/attack-AB-bounded.md`) gives > `frac{ x : T_H(x) < m } ≤ B·H/(H·M − m)²`, and with `H = A·m/M` this is > `A·B/(M·m·(A−1)²)`. All positions needs that below `1/W`, i.e. `m > A·B·W/(M(A−1)²)`. At `A = 2` and the measured `B(z,3.0)`: | `z` | 13 | 19 | 29 | 37 | 41 | |---|---|---|---|---|---| | `m` needed | 1.094e5 | 3.357e7 | 1.835e10 | 1.923e13 | 7.562e14 | | `m/D` | **810** | **1510** | **2310** | **3090** | **3470** | A threshold above `D` is not a statement about the tile at all. **The almost-all/all gap is `W`, and Lemma V's mean square does not begin to pay it — by three orders and widening.** This reproduces, in threshold coordinates, `attack-beta2-01`'s own verdict that *"the almost-all exponent is 0"* and `sift-limit-attack.md` §7e's *"almost-all is not the currency this problem trades in"*. **And the u-frame's "average" is over FOLDS, not positions.** `gate-multiplies.md` §8's threshold `L ≤ 0.19–0.31 p/ln p` is stated *"on average over the ladder"* — the average runs over levels `p ≤ x`, and **each level's term still needs the worst position at that level**. So there is no averaging slot for an almost-all-positions statement to fill. That is the precise gap the brief asked for, and it does not close: the chain is `almost-all positions → (needs emptiness, gap W) → worst position at level p → (averaged over p) → the u-frame`, and the failure is at the first arrow. ### R3 — the residue-deleted structure. CLOSES, then busts the budget. §2 inverted is the kill-count-free inequality the brief asked for: a window of span `S` holds at most `1 + ⌊(S+p+2)/(3p)⌋` kills, so > `S := maxsum_m(T_p) ≤ maxsum_{ m + 1 + ⌊(S+p+2)/(3p)⌋ }(T_x)`, an implicit inequality in `S` alone, with no kill count anywhere. **It closes.** [VERIFIED] the least fixed point above `maxsum_m(T_x)` exists at every cell tested, reached in a few iterations, at index 4 to 13 — no runaway. **And then it busts.** Cost against the truth: 1.3077 to 2.0909, i.e. **0.2683 to 0.7376 nats per fold**, against a per-fold replenishment `2 ln(p/x)` of 0.2225 to 0.9040 nats. At `m = 1` the burn exceeds the replenishment at **three of the five folds** (0.7376 vs 0.3341; 0.4700 vs 0.2225; 0.3857 vs 0.3821). The **LIFETIME** Overshoot Budget is 0.88 to 1.19 nats in total (`gate-multiplies.md` §5), so the chain is exhausted inside about twenty folds. *(Note the definitional difference: "burn" here is `ln(bound/truth)` per fold; `gate-multiplies.md` §8's burn is `ρ·m̄·L/G₂`. The two are not the same number and are not compared.)* **Why it cannot be repaired, in one line.** The index cost per fold is `⌊(S+p+2)/(3p)⌋ ≈ G₂/(3p) ≈ 0.183p` — the Bridge Floor again, in its other costume. The `3p` of §2 is the *worst-case* kill spacing; two of every `p` slots are killed, so the *average* kill spacing is `p·m̄/2`, which is `406.8` at `T_23 → 29` (`m̄ = 28.054`) against the worst case `3p = 87`. The over-count factor is `m̄/6`, which is `4.68` at `T_23` and grows like `ln²p`. **A kill-count-free argument pays for being kill-count-free with exactly that factor**, and it is the same `ln p`-shaped deficit the branch has met at every entrance. --- ## 5. THE SINGLE BEST SURVIVING STATEMENT > **Theorem candidate (threshold-`m` certificate).** Let `z ≥ 3`, let `T` be the > set of `r` with `r` and `r+2` coprime to `P(z) = ∏_{p `V = ∏_{p `m ≥ 1`, > > > `maxsum_m(T) ≤ C · max( z^{β₂+ε}, m/V )`, > > with `β₂ = 4.26645028414864191641` the dimension-2 sifting limit and `C, ε` > absolute. > > **Hypotheses, all of them.** (i) The Diamond–Halberstam two-dimensional > lower-bound sieve at `κ = 2` [PROVEN, Cambridge Tracts 177, Table 17.1]. > (ii) Trivial interval remainders at level `H^{1−ε}`, uniform in window > position [PROVEN, this corpus, `beta2-note` §3]. (iii) The assembly of (i)+(ii) > into a two-class Jacobsthal-type bound, which is folklore-available and > **unwritten** [INFERRED from PROVEN ingredients, `covering-dive.md` §"point 3", > absence sourced]. (iv) Nothing conditional: no EH, no GRH, no maximal law, no > quantifier trade — the certificate is all-positions by construction. > > **What is new here.** The `m` in the statement. The corpus had only the `m = 1` > case. The `max` says the threshold-`m` bound equals the `m = 1` bound for every > `m ≤ z^{β₂}·V` — a saving of a full factor `m` over the trivial `m·G₂` — and > becomes linear in `m` above it. > > **Exact support** [VERIFIED, no hypotheses at all]: the full-period certificate > at `z = 13..29` gives `H_cert/maxsum_m` falling from 1.43–1.94 at `m = 1` to > 1.00–1.29 at `m = 8`, with additive overhead. > > **Why it does not close the branch.** Its crossover `m* = z^{β₂}·V` exceeds the > L-bridge's needed `0.19z/ln z` by 1.65e4 at `z = 23`, growing like > `z^{β₂−1}·ln z`. --- ## 6. R4 — PRIOR ART The order-`m` object is the natural generalisation of the Jacobsthal function ("smallest `H` such that every `H` consecutive integers contain at least `m` coprime to `n`"). `SEARCH-CONVENTIONS.md` §1 owns the `m = 1` case in three conventions (A144311 for the two-class primorial object; A048670 for the one-class one; MathOverflow 88323 for the general framing). **The order-`m` question was run in this session as a separate owning-convention search; its result is recorded in §6a below and must be read before any "absent" sentence about `maxsum_m` is written anywhere in the corpus.** Boundary check the brief asked for, and it holds: Holt's machinery is bounded by spans `< 2p_{k+1}` throughout (`LOCALIZED-GAP.md` §9, quoting arXiv:2603.25915 §1) and never bounds a maximum gap. That restriction is about the *span*, not about the *threshold*, so it applies to `maxsum_m` for every `m` exactly as it applies to `maxsum_1`: any `maxsum_m` at full-tile scale is `≫ 2p`, hence outside their range. **The boundary is confirmed to carry over.** ### 6a. The order-`m` object IS in print, and 0c's absence sentence has to change **PRESENT.** Fintan Costello and Paul Watts, *An upper bound on Jacobsthal's function*, **Math. Comp. 84, no. 293 (2015) 1389–1399**, MR3315513, DOI S 0025-5718(2014)02896-2, preprint [arXiv:1208.5342](https://arxiv.org/abs/1208.5342). Their §2 defines our object **in the inverse indexing** — exactly the convention flip `SEARCH-CONVENTIONS.md` §2 warns about: > `π(b, m, i)` = the number of integers `b+1, …, b+m` coprime to `P_i`; > `π_min(m, i)` = the lowest value of `π(b, m, i)` across all `b`. and their abstract states it in words: *"π_min(m, k) … represents the smallest number x such that every sequence of m consecutive integers contains at least x integers coprime to P_k."* That is §1's `min_t N([t, t+H))`, and `π_min(m,k) ≥ x ⟺ maxsum_x ≤ m`. **The owning convention for `maxsum_m` is `π_min`, and this repository had no row for it.** Their Theorem 4.4 lower-bounds `π_min(m,k)` for **all** `m` and `k`. Two limits matter for us: (i) it is a **recursion evaluated by PARI**, not a closed form, and **no asymptotic in `m` is stated anywhere**; (ii) the authors only ever use it at the `m = 1` threshold (`0 < π_min(m,k) ⇒ h(k) ≤ m`), giving `h(k) ≤ 0.27749612254 k² log k` for `50 ≤ k ≤ 10000`. **Values of `π_min(m,k)` for `m > 1` appear nowhere in print**, and the paper has only 2 citing works (Mercer, INTEGERS 18 (2018) #A26; Ziller, arXiv:2007.01808), neither taking `π_min` up for `m > 1`. **One class per prime, primorials only.** **The near-hit is withdrawn, and the search run refuted it independently.** Costello–Watts, *Jacobsthal's function and a generalisation of Euler's totient*, [arXiv:1209.3464v1](https://arxiv.org/abs/1209.3464) (2012), Theorem 2.3, is the closed-form order-`m` bound one would want — `m·φ(Q_k)/Q_k ± k(k+1)/2` — which would give `maxsum_m ≤ (m + k(k+1)/2)·n/φ(n)` directly, and at `m = 1` would beat Iwaniec by a log with an explicit constant. **That is the tell, and it is withdrawn**: the arXiv record carries *"This paper has been withdrawn due to an error on page 6"*, zbMATH carries it as `WITHDRAWN`, and the search run located the error (page 6 asserts `⌊φ(Q_{i−1})·m/(Q_i q_j)⌋ ≤ φ(b, F, Q_{i−1})` where the valid chain gives only `φ(Q_{i−1})·⌊m/(Q_i q_j)⌋`) and refuted the conclusion by exhaustive computation: at `k = 11`, `b = 100279391410`, `m = 1707309`, the deviation is **66.146264 > 66 = k(k+1)/2**. Do not resurrect it. **Erdős 1962 does not contain the object.** Read in full from the primary PDF (users.renyi.hu/~p_erdos/1962-12.pdf). He defines `g(n)` (order 1), `C(r) = max g(n) − 1` over `ω(n) = r`, and a two-argument object `φ_n(x, x+B)` which is the **counting function**, not a Jacobsthal-type minimum; he never inverts it and never bounds `min_x φ_n(x, x+B)`. His Theorem III is an **averaged** statement with an exceptional set of `ηn` positions — the wrong quantifier, exactly as §4's R2 says. That averaged line is owned by Hooley (Acta Arith. 8 (1963) 343–347) and Montgomery–Vaughan (*On the distribution of reduced residues*, Ann. of Math. (2) **123** (1986) 311–333), all averaged over `q` or `x`, none worst-case. **Two-class order-`m`: ABSENT**, and it compounds `covering-dive.md` §2.2's existing absence rather than contradicting it. Ziller–Morack (1706.00317, 1706.03668), Ziller (1903.11973, 2007.01808), Kalmynin–Konyagin (2302.00459), Paseman (1311.5944) grepped in full-text PDF for "at least k/m/two integers", "π_min", "order": zero hits in all. **Channels and calibration** (all in one session, all calibrated): arXiv API (https only; `abs:"Jacobsthal function"` → 16 entries), MathSciNet `mrlookup` (`au=Iwaniec&ti=Jacobsthal` → MR499895), zbMATH Open (`ti: Jacobsthal` → 324 records), OpenAlex (`title.search:Jacobsthal` → 62 works), WebSearch. The decisive query was WebSearch on *"Jacobsthal function" generalization "at least k" integers coprime to n consecutive integers upper bound* — **an owning-convention query, not a house-vocabulary one**, which is the whole point. **Residual gaps, stated so they are not read as coverage.** Iwaniec, Demonstratio Math. 11 (1978) is still not obtainable; Kanold 1967 was assessed from the zbMATH review rather than the GDZ full text (`PPN=GDZPPN002297787`); **Kanold 1975 and Kanold 1977** (*Neuere Untersuchungen über die Jacobsthal-Funktion g(n)*, Monatsh. Math. 84, 109–124) have no zbMATH reviews and **were not read** — the one place a German-language order-`k` variant could still hide. Books remain unsearchable. ### 6b. WHAT THIS OBLIGES THE CORPUS TO CHANGE (not done here — staging only) 1. **`TODO.md` item 0c's sentence** *"nobody has tried bounding the residue-deleted maxsum directly. That 'nobody' is this repository, not the literature"* is still true for the **two-class** object and is now **wrong as a blanket statement about `maxsum_m`**: the one-class order-`m` object has a published upper-bound machine. 2. **`SEARCH-CONVENTIONS.md` §1 needs a row**, and its absence is why five waves could not reach this: | object | our name | **OWNING convention — search THIS** | where it lives | |---|---|---|---| | order-`m` max span of `m` consecutive survivor gaps | `maxsum_m` | **`π_min(m, k)`**, *"the smallest number x such that every sequence of m consecutive integers contains at least x integers coprime to P_k"* | Costello–Watts, Math. Comp. **84** (2015) 1389–1399 | 3. **`covering-dive.md` §2.2** records Costello–Watts 2015 inside the 82 titles of the Iwaniec citation graph and triaged it as *"a one-class `h(k)` bound"*. That triage is correct about the paper's **conclusion** and missed its **machinery**: `π_min` is invisible from the title. Worth a sentence there, because it is the exact shape of failure that file exists to prevent. --- ## 7. WHAT THIS ATTACK DID NOT REACH - **`T_29` alignments beyond `m = 8`, and `T_29 → T_31`.** The fold `23→29` is the deepest alignment family computed here; `29→31` needs the streaming technique of `research/gate-multiplies-03.js` and was not run. - **A stride-1 certificate scan at `z = 29`.** The grid there is stride 48 (14 full-period walks at 11 s each). `H_cert` at `z = 29` is therefore the true threshold rounded up by at most 47. The rounding is relatively *larger* at large `m`, so it understates the amortisation rather than manufacturing it. - **`z = 31` and above.** `W(31) = 6.47e9`; one walk is minutes. The `theta-selfconsistent.md` run reached it for `m = 1` only. - **A proven bound on `mean_a Δ_m` or `sd_a Δ_m`.** Samuelson needs both and neither was attacked. - **The `κ(m)` coordinate is NOT open** — it was listed as unreached in a draft of this note and that was wrong. `kappa-not-L.md`'s Theorem C already transfers the ceiling to `κ(m)`: *"Theorem C inherits the ceiling, so redirecting L to κ does not move the wall."* Nothing here needs to be run. - **The threshold-`m` certificate against `κ(m)` rather than `L`.** R1's amortisation was priced against the L-bridge only. Whether it pays anywhere in the `κ(m)` formulation was not asked. --- ## 8. REPRODUCTION AND CUSTODY ``` node research/attack-foldL-05-maxsum-direct.js # 270 s, 203 full-period walks node research/qc/embed.js --streams both --timeout 900 research/attack-foldL-05-maxsum-direct.js ``` Custody, all inside the run: `D(T_x) = ∏(q−2)` and the `G₂` ladder 6, 12, 30, 42, 66, 108, 150, 204 reproduced from a generator that never sieves; the same eight terms cross-checked against **OEIS A144311 + 1**; the copy theorem `max_a Δ_m = maxsum_m(T_p)` re-verified for `m ≤ 8` at five folds; and `min_a/mean_a/max_a` reproduce `research/attack-0c0e-01-deleted-family.js`'s embedded table figure for figure. All self-tests pass. --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [../CHANGELOG.md](../CHANGELOG.md), indexed by document.*