# Red team, attack 10 of 10: the six headlines of 2026-08-18, checked by someone else *(2026-08-18/19. Assignment: assume each headline is wrong and look for the reason. Producers of every number below: `research/redteam-growth-aicc.js`, `research/redteam-instrument-independence.js`, `research/redteam-B-polylog.js`, `research/redteam-DP1-certificate.js`, `research/redteam-tau-and-split.js` — five new scripts, each bound with `research/qc/embed.js`, each carrying its own verbatim output and numbered readings. Legend as in `research/sift-limit-attack.md`.)* **Why this file exists.** On 2026-08-18 a script in this repo was found carrying an OUTPUT block and eight numbered READINGS written before the file had ever been executed: ten figures wrong, two sign-reversed, the code correct throughout. Six attacks landed that day on top of that discovery and none had been checked by anyone but its own author. --- ## SCOREBOARD **Four CONFIRMED, two WEAKENED, one REFUTED**, over the six headlines plus one the wave had left unchecked. The refutation is headline 2's independence claim and it is the most consequential finding here, because it removes a second witness the corpus was counting. Both weakenings are of headlines that their own producing documents already qualify correctly in their bodies — the overstatement is in what was circulated, not in what was written. **No script in this wave has a fabricated tail:** all five targets pass `embed --check` on both hashes. | # | headline | verdict | |---|---|---| | 1 | the growth law `0.762 x ln²x lnln x`, with `c x ln²x` out at 31.8 and `c x^a` at 10.6 | **WEAKENED** | | 2 | the exact-frame diagonal is a SECOND, INDEPENDENT instrument agreeing with it | **REFUTED** | | 3 | `B ≤ 9A²(E−1) = O((log z)^8)`, and polylog suffices for the mean-square Lemma V | **CONFIRMED** | | 4 | Brady's Problem 3 is not our covering problem; NP-completeness licenses nothing | **CONFIRMED** | | 5 | the quantifier is separable at `τ(m)`; BF's `4.156000` is their unoptimised split | **WEAKENED** | | 6 | DP1's consumer certifies `12, 114, 390, 2256, 18900, 117558`, slope +5.20 | **CONFIRMED** | | 7 | *(added)* the 22-term ladder is A144311 under the right offset and definition | **CONFIRMED** | **Custody first.** All five target scripts pass `node research/qc/embed.js --check`: `attack-growth-law.js`, `attack-lower-bound.js`, `attack-beta2-A-B-bounded.js`, `attack-D-twopoint.js` and `attack-theta-margin.js` each report `code-sha256 matches` and `out-sha256 matches`. **Every tail in scope is genuinely the output of the code above it.** The failure of 2026-08-18 is not repeated anywhere in this wave. Everything below is therefore an attack on the CODE and on the INTERPRETATION, never on the transcription. --- ## 1. The growth law — **WEAKENED** > **What survives:** `MP2LL` is the best-supported law on x = 11..79 and is > stable under every single-point deletion; the x-exponent → 1 conclusion is > robust at every window tested; the AICc arithmetic error found here runs in > the safe direction. **What does not survive:** both exclusions, at the > strength the headline states them. Route: `research/redteam-growth-aicc.js`, which re-derives all ten fits from scratch rather than importing anything from the target. **a. The AICc is mis-specified, and it is conservative.** A Gaussian least-squares model with unknown variance estimates `K = k + 1` parameters, the k regression coefficients **and** `σ²`. `attack-growth-law.js` uses `k`. The AIC constant cancels in a difference; the AICc correction `2K(K+1)/(n−K−1)` does not, because it depends on K. At n = 18 the correction shift is 0.00 for k = 1, **0.36** for k = 2 and **0.81** for k = 3, and it always penalises the larger model. Corrected: **PW moves 10.6 → 10.9** and **MP2 stays at 31.8** (MP2 and the winner MP2LL are both k = 1). The winner is MP2LL under both conventions. **[VERIFIED]** — a real defect, worth fixing, and it does not touch the headline. **b. The fits are comparable and the solver is exact.** All ten families regress the same response `ln G₂` on the same 18 points; frozen exponents live in an offset, which is a coefficient pinned at 1 and not a change of response, so AIC differences are legitimate. Worst `|Φᵀr|` over all ten normal-equation solves: **3.92e-14**. **[VERIFIED]** **c. The 31.8 is not shape evidence — and the report says so itself.** MP2 is thrown out by **31.8** on G₂ and by **32.8** on the control (33.2 in the target's own AICc convention), whose conjectured truth *is* `x ln²x`. A frozen one-parameter law is beaten at this range whether or not it is true. `attack-growth-law.md` §2b and its reading 3 state this correctly and instruct "quote it that way or not at all". **The document is right; the headline as circulated has dropped the qualification.** **[VERIFIED]** **d. The power-law exclusion rests on one point.** Dropping x = 11 alone takes PW from **10.9 to 3.3** — not an exclusion on the Burnham–Anderson scale the report itself invokes. Worst case over all 18 single deletions is **3.3**, and it is x = 11 that produces it. Dropping the *largest* term instead leaves 10.3. **[VERIFIED]** **e. And one prime lower the whole ranking reverses.** At **x = 5..79**, twenty terms, **PW wins and MP2LL is excluded at 38.4**. `attack-growth-law.js` computes this itself, as its block **D2**, and **neither its fourteen readings nor any of the eleven sections of `attack-growth-law.md` mentions it** — including §10, "What is not established", which lists five other caveats. The stated reason for the window is the script's line *"house window: below x = 11 the ladder is 2,6,12,30 and lnln 2 < 0"*. `lnln 2 < 0` excludes **x = 2 and nothing else**: `lnln x` is 0.0940 at x = 3, 0.4759 at x = 5, 0.6657 at x = 7, and every family runs at all three. **The low end is a judgement and the answer is a function of it.** **[VERIFIED]** **f. The §2 calibration argument is window-specific too.** Its form is: the estimator takes a pure power whenever the data lets it; it does on the control; it does not on G₂; therefore the log families' win is a property of G₂. Run on **both** objects at six low ends, the inversion appears at **exactly one**, x = 11. At low end 5 and 7 both objects pick a power or near-power and there is no inversion to read; at low end **13, PW does better on G₂ (3.3) than on the control (6.1)** — the opposite sign. **[VERIFIED]** **g. One joint nobody priced.** AIC counts n independent observations. Residual lag-1 autocorrelation runs −0.150 to 0.773 across the ten families; the AR(1) effective sample size `n(1−r)/(1+r)` is **4.4 for MP2** and **13.8 for PW** against n = 18. This is an order of magnitude, not a correction — each model has its own r. What it says: a gap of a few units is inside the noise of the independence assumption, and a gap of 30 is not. **[MEASURED]** **h. The constant's band.** `c = 0.7624` has a jackknife band of **0.7574..0.7698** but a **window** band of **0.7534..0.8316**, five times wider. Three significant figures are not supported; two are. > **VERDICT: WEAKENED.** Quotable: `G₂ = x ln^{2+o(1)}x` with the o(1) positive, > best single description `0.76 x ln²x lnln x` over x = 11..79. **Not quotable: > "c x ln²x excluded at 31.8 units" and "the pure power law excluded at 10.6"** > — the first is refuted as shape evidence by the document's own control, the > second does not survive its smallest data point or a one-prime window change. --- ## 2. The second instrument — **REFUTED** > The two instruments are not independent. The exact frame **cancels out of > every reported offset**, and the diagonal hands back the parameter attack E > already fitted. Two dependent instruments agreeing is one instrument counted > twice, which is exactly the failure the assignment named. Route: `research/redteam-instrument-independence.js`. **a. §3b reproduces exactly**, from a `frame()` written from the definitions rather than copied: trend exponents **−0.5308, 0.3181, 0.4692, 3.6755** and the data's own **0.2273**, se-distances **9.3, 1.1, 3.0, 42.3**. So nothing below is a reproduction failure. **b. The exact frame cancels.** OLS slope is linear in the response, and the law regression and the data regression subtract the **same** `ln(m lnD)` against the **same** regressor, so the difference of slopes is `slope(ln f − ln G₂)` and the frame is gone. Computed with **no frame at all**, the offsets agree with the framed ones to **4.44e-16**. **[VERIFIED]** **c. Absurd frames give the same offsets.** Dividing by `x⁵`, by `eˣ`, by `1/x³` or by `(m lnD)²` leaves the MP2 offset at **−0.7581** in every case. What moves is the data se — 0.0816 exact frame, 0.1007 none, 9.7172 with `eˣ` — and hence the se-*distance*: **9.3, 7.5, 0.1**. The exact frame's contribution is the **error scale**, not the measurement. **[VERIFIED]** **d. The diagonal returns attack E's own fitted parameter.** Under this instrument `c x ln²x (lnln x)^b` has a trend linear in b, so the b landing it on the data is `(0.2273 − (−0.5308))/0.8489 = 0.8930`. Attack E's `MP2LLb` row fits b by AICc and reports **0.893**; refitted from scratch here, **0.8933**. The two routes differ by **0.000292**. **[VERIFIED]** **e. And the 9.3 loses strength on the same deleted point.** Over the 18 single deletions the MP2 se-distance runs **7.9 to 13.1**; the lowest, 7.9, comes from deleting **x = 11** — the same point that carries attack E's power-law exclusion. Two "independent" instruments weakening on the same deletion is what dependence looks like. **f. What survives, and it is not nothing.** The **ranking** is real: on the same 18 points `c x ln²x` sits 0.758 from the data's trend and `c x ln²x lnln x` sits 0.0908, a factor of eight. That is a fact about G₂ and it does not depend on the frame. The exact frame earns its place by making the scatter small enough that 0.758 is worth 9.3 se rather than 7.5. > **VERDICT: REFUTED**, as an independence claim. **Required edit to > `attack-lower-bound.md` §3b:** delete "Two independent instruments, one > model-selection and one exact-frame". Replace with: the diagonal re-expresses > the same 18-point comparison in a frame that sharpens the error bar, and its > agreement with attack E is algebraic, not evidential. --- ## 3. B is polylog — **CONFIRMED** > The chain is a proof and it holds; the exponent 8 is right; no violation > appears at points the report never tested. One error found in the sufficiency > section, and it too runs in the safe direction. Route: `research/redteam-B-polylog.js`. **a. The chain is a proof, not a fit.** Rebuilding the local factors from the report's own state description — (neither), (p|d₁), (p|d₂) at odd p with (p|both) barred by `gcd | 2`, all four states at p = 2 — returns **2.500000** at p = 2 and `1+2/p` at odd p, matching `A(z) = (5/2)∏(1+2/p)`, and returns **r(2) = 0.600000**, `r(p) = 2/(p+2)`. Step 2 is a genuine Euler-product argument: the Rosser supports are **subsets** of the divisors of `P(z)` with `|λ_d| ≤ 1`, so the truncated sum is at most the full product; the factor 3 is the three certificate blocks; imposing `e|q` bars the (neither) state at every `p|e`. Step 3 is an identity — brute force over every squarefree `e|P(z)` matches `∏_p(1 + p r(p)²)` to **3.98e-15** at z = 13, 23, 43. **[VERIFIED]** **b. The exponent 8 is measured, not asserted.** `d ln A/d lnln z` runs **1.9414 → 1.9997** and `d ln E/d lnln z` runs **3.8024 → 3.9993** over z = 10³..4·10⁶. So `9A²(E−1)` is `(log z)^{4+4}`. **[VERIFIED]** **c. The per-e check holds where the report did not look.** No violation of step 1 or step 2 at **z = 43**, which the report's table stops short of, nor at **s = 2.0**, which the report itself names as the corner where B is largest. B at z = 43, s = 2.0 is **1.718147**, the largest seen, and the bound is still **2.63e+4** times above it. **[VERIFIED]** **d. The `(log z)^14` threshold is wrong.** Solving `H/(ln H)⁶ = 9A²(E−1)` exactly, `ln H_min / lnln z` reads **29.968** at z = 13 and falls monotonically to **16.543** at z = 10⁶ — still above 14 and still falling. Continuing the *shape* `B = C(log z)^8`, the same ratio reads **13.3679** at `log z = 100` and **8.5543** at `log z = 10³⁰`. It passes through 14 and tends to **8**. The true polylog threshold is `(log z)^{8+o(1)}`. The slip is pricing the `log⁶H` factor as `(log z)^6`, i.e. using `ln H = ln z`, while simultaneously setting H polylog in z: **two different H's in one sentence.** It *overstates* the requirement, so the conclusion is untouched. **[VERIFIED]** **e. The sufficiency implication is valid, with two conditions.** `B ≤ H/(ln H)⁶` with `H = z^u` is `z^u ≥ C u⁶ (ln z)^14`, a power against a polylog, true for all large z at any fixed `u > 0`. Conditions: **(i)** u must be bounded away from 0 — the corpus uses `u ≈ β₂ = 4.26645`, a constant, so this holds where it is used; **(ii)** the ask is the **mean-square** ask. At u = 4.26645 the proved bound first suffices between z = 10⁴, where 4.291e+7 is still above 3.160e+7, and z = 10⁵, where 2.541e+8 is below 1.530e+11. **[VERIFIED]** **f. The one thing it does not license.** `sift-limit-attack.md` §7e proves `⟨R²⟩_H ≤ B·H`, an average over positions. `attack-AB-bounded.md` says so in its own headline — *what separates the mean-square lemma from Lemma V is the position quantifier, not B* — so the report is honest. **Every downstream use must carry the words "mean square".** > **VERDICT: CONFIRMED.** The three-step chain is a valid proof, the exponent is > 8, and polylog does suffice for the **mean-square** Lemma V. Fix the > `(log z)^14` line to `(log z)^{8+o(1)}`. --- ## 4. Brady's Problem 3 — **CONFIRMED** > Both halves hold. The conclusion is scoped to "about the exponent" and is not > the broad claim; the partition argument is structurally sound and the document > stops in exactly the right place. Route: direct reading of `research/history/staging/attack-np-licenses.md`, plus `research/redteam-tau-and-split.js` §C for the structural half. **a. The conclusion is NOT too strong, and the document is the reason.** The assignment asked whether "licenses nothing about the exponent" overreaches past the defensible narrow claim. It does not, and the test is that the document carries a **§4.3, "What it DOES license"**, with four items — a negative expectation about extending the exact ladder, a negative expectation about optimal certificate construction, a methodological guard against any argument that would decide general Shifted Sifting in polynomial time, and "nothing else". A document that enumerates what a theorem licenses is not claiming it licenses nothing. Every headline instance of the phrase is scoped: §0 item 2 reads "licenses nothing whatever **about the exponent**", and the drafted `PRIOR-ART.md` edit at §7.1 reads "**The result licenses nothing about the exponent**". **[VERIFIED]** **b. And the narrow claim is right.** §4.2 item 1: NP-completeness is a statement about uniformly deciding a family whose input varies; `G₂(x#)` is a single sequence with no free input, deciding any one term is `O(1)` by table lookup, and the growth rate of a fixed sequence is not a complexity-theoretic quantity. That argument is correct and it is decisive. §4.2 item 5 adds the right counterweight — Brady's own Theorem 10 makes Transverse Partition Cover fixed-parameter tractable in the number of partitions — so the document is not using NP-completeness as evidence of mathematical hardness either. **c. The partition argument excludes our family, structurally.** Recomputed independently: the per-prime collection `{S_a : a ∈ Z/p}` with `S_a = {a, a−2}` has each residue of multiplicity **[1] at p = 2 only** — where `a ≡ a−2 mod 2` collapses the pair to one class — and **[2] at every odd p** from 3 to 23. `p` sets of size 2 cannot partition `p` elements. So no odd prime supplies a partition, and any theorem requiring each prime to supply one does not quantify over our family. **[VERIFIED]** **d. And it stops in the right place.** §4.2 item 3 concludes "whether it is NP-complete is **not settled** by anything in this thesis" — an absence of a result, not a claim of tractability. That is the correct inference and the place where an overreach would have been easy. §4.2 item 4 is the same discipline applied to Theorem 12, whose "arbitrary collections" setting the document concedes *is* our shape, and which it separates on the **objective** (maximum covered count against longest fully covered initial run) rather than pretending the shape differs. **e. Custody is the strongest in the wave.** §1 gives the URL, **844,620 bytes**, a sha256, the printed-to-PDF page offset fixed by locating printed p. 1, and an extraction caveat — `pdftotext` drops the `≪` glyph, zero occurrences against 19 for `Jacobsthal` as a known-positive — with the affected quotations re-read off rendered page images. The Erdős sentence was re-sourced rather than inherited, and the dead 2026-08-18 run's negatives are declared void. **f. The one thing I cannot verify.** The thesis PDF is **not in this repository** and `find` over the machine does not locate it, so the verbatim text of Theorem 9, Problem 3 and Problem 4 cannot be checked from here. The byte count and sha256 make that checkable by anyone who re-fetches, which is the right way to leave it, but it is not checked here. **[CANNOT VERIFY]** — and this is a weaker caveat than §5b's, because the identifiers are recorded. > **VERDICT: CONFIRMED.** The conclusion is correctly scoped, the partition > exclusion is structurally sound and reproduces, and the document declines the > two overreaches available to it. --- ## 5. The quantifier price and Brüdern–Fouvry — **WEAKENED** > The Brüdern–Fouvry half is exact and reproduces to every digit, with one > digit-count defect. The `τ(m)` half is a genuine reframing sold slightly too > hard: at the worst `m` the two prices are **the same number**. Route: `research/redteam-tau-and-split.js`. **a. `τ(m)` and `C^{π(z)}` coincide at the worst `m`. [VERIFIED]** The report's own formula `τ(m) = 2^{π(ln m)}` is the maximal order of the divisor function, attained at `m = P(y)` where `ln m = θ(y)`. Evaluated at the worst `m | P(z)` it returns **`2^{π(z)}` exactly**, or within one prime of it, at every z tested from 13 to 100003. So "priced at `τ(m)` rather than `C^{π(z)}`" does **not** lower the worst-case price. The assignment's suspicion was correct. **b. But the reframing is not empty, and this is the honest half.** A uniform `C^{π(z)}` is attached to nothing; `τ(m)` is attached to each `m` and compared against that `m`'s own size `m^{0.175}`. Below `ln m = 118.83` the divisor price binds; above it the size of `m` beats it outright. That is a real structural gain and it is why the route survives at all. What it does not support is the §7 phrase **"a materially cheaper statement of the wall"**: the improvement is in the *shape* of the comparison, not in the size of the price, and at the worst `m` there is no improvement. **c. The report is honest about the outcome.** §6 (C1) calls the route "CLOSED, asymptotically, with a crossover", surviving to `z = 1e11` and dead from `z = 1e12`, and calls it "a real kill but a late one". So the document does not claim the price rescues anything. **d. What I cannot check.** The Fourier expansion the separability rests on — `R(x) ~ Σ_h c_h Σ_m e(−hx/m) K_h(m)`, with all the `x`-dependence in the single exponential — is not derivable from anything in this repository, and the reciprocity step `ρ/(d₁d₂) = −2 d̄₁/d₂ mod 1` needs the surrounding setup to check. **Given the expansion**, the separability is trivially right: taking absolute values in a factor that is the only carrier of `x` discharges the quantifier at no cost. **[CANNOT VERIFY]** for the expansion itself. **e. The Brüdern–Fouvry arithmetic reproduces to every digit. [VERIFIED]** Independent minimisation of `u(a,b) = (1+e^{b/2a})/b` on `a+b = 5/4`: ``` u(1/2, 3/4) = 4.1560000222 = (4/3)(1 + e^{3/4}) optimum a = 0.5404035, b = 0.7095965, u = 4.1264517443 1/u at the optimum = 0.2423389541 unclaimed exponent = 0.0295482779 in their currency 0.0017229785 ``` All four match `attack-theta-last-gap.md` §4 exactly. **f. One digit-count defect. [VERIFIED]** `1/4.156000 = 0.24061597`**`69`**, while `0.2406159756` is the reciprocal of the *exact* `(4/3)(1+e^{3/4})`. The document's §B is stated correctly, because it reciprocates the exact value; but any circulated sentence of the form "`1/4.156000 = 0.2406159756`" is wrong in its last two digits. Both round to `0,2406`, so **the identification with their printed constant is untouched.** **g. Re-splitting does stay inside the side condition, algebraically. [VERIFIED]** `q^{C₀} D₁⁴ D₂⁴ ≤ x^{5−cε}` with `D₁ = x^a`, `D₂ = x^b` has `x`-exponent `4(a+b)` — a function of the **sum alone**. Six splits with `a+b = 5/4` all give `4(a+b) = 5.00` and all sit inside `x⁵`, while `u` ranges 4.1264517443 to 4.5807011108. Their split and the optimum sit at the same `a+b`, so both saturate `x⁵`. **h. The attribution cannot be checked from here. [CANNOT VERIFY]** Whether `q^{C₀} D₁⁴ D₂⁴ ≤ x^{5−cε}` is what Brüdern–Fouvry require, and whether "`0,2406`" is what they print, rest on **`BF.txt` line 146**, and `BF.txt` is **not in this repository** — `find` over the whole tree returns only `research/attack-bf-split.js`. The same is true of `friedlander-weaker-simpler.txt`, cited by `attack-AB-bounded.md` §0 as "28,902 bytes on disk", and of the `lit/corpus/` directory both refer to. No byte count or hash is recorded for `BF.txt`, so unlike §4e there is no route to re-fetch it. This is the "evidence that cannot be re-run" fragility that `research/qc/README.md` names in its own words. It is **not** an accusation that the quotations are wrong; it is a statement that nothing in this repository can confirm them. > **VERDICT: WEAKENED.** The `4.156000` identification, the optimum and the side > condition are exact and reproduce. Drop "materially cheaper" from the `τ(m)` > sentence — at the worst `m` the two prices are equal, and the gain is in the > comparison's shape. Never write "`1/4.156000 = 0.2406159756`". And the > attribution to Brüdern–Fouvry is unverifiable from this repository. --- ## 6. DP1's consumer — **CONFIRMED** > All six certified bounds are correct on the sharp LP, from an independent > implementation, **including the two levels the producer never LP-checked**. > Two findings, both about presentation and margin rather than arithmetic. Route: `research/redteam-DP1-certificate.js`, which rebuilds the sifted set from the definition, computes `S₁` and `S₂` by a sliding window over the whole period, and solves the degree-2 LP exactly by enumerating every basic feasible support `{0, a, b}`. **a. All six bounds reproduce.** `6L` reads **12, 114, 390, 2256, 18900, 117558** at x = 5..19, matching the headline at every level, with **`V₂(L) < 1` and `V₂(L−1) ≥ 1`** at all six — so each reported L is genuinely the *first* that certifies. **[VERIFIED]** **b. The units are right, which was the first thing to doubt.** The certificate is in **slots**, the headline in **integers**. Computing the true maximal slot gap directly gives **2, 5, 7, 11, 18, 25**, and six times those is **12, 30, 42, 66, 108, 150** — the published G₂ at all six levels. The `units.js` §2 trap is not what went wrong here. **[VERIFIED]** **c. The producer never put the two largest on the LP.** `attack-D-twopoint.js` finds each L by a closed-form variance inequality (`deg2works`) and re-checks against the sharp LP **only when `found ≤ 400`**. The reported L are 2, 19, 65, 376, **3150, 19593** — so **x = 17 and x = 19**, the two numbers the +5.20 slope leans on hardest, **rested on the closed form alone**. §a above is the first time they have been on the sharp LP. They pass. **[VERIFIED]** **d. The margin is collapsing and the last row is delicate.** `1 − V₂(L)` reads **1.111e-1, 5.534e-2, 3.411e-2, 1.252e-2, 1.441e-4** at x = 7..19. At x = 19 the certificate clears by **1.4e-4** out of a period of 1,616,615, and the double-precision LP touches products of order **1.4e18**, past 2⁵³. Redone in exact BigInt rationals: **`V₂(19593) = 10530232/10531755 = 0.9998553897`** at `(a,b) = (4589,4590)`, still below 1, against the double's 0.9998558644. The verdict is unchanged and the margin is now **certified rather than assumed**. **Any extension of this table to x = 23 must be done in exact arithmetic.** **[VERIFIED]** **e. The +5.20 slope is a five-point fit presented beside six ratios.** The headline lists 1.0, 3.8, 9.3, 34.2, 175.0, 783.7 and then one slope. Fitted on **x = 7..19** it is **5.20**, which is what the script computes; fitted on all **six** listed ratios it is **4.60**. Dropping x = 5 is the right choice — it is the one level where the certificate is exact and the ratio is 1 by construction — but the sentence must say so, because a reader given six numbers and one slope will read the slope as over the six. **[VERIFIED]** — this is the second correction the assignment asked for. **f. What the slope can carry.** Five points from x = 7 to 19 is under half a decade. The qualitative claim — that the price of stopping at two-point data *grows* rather than staying a constant factor — is carried by the ratios themselves, rising 3.8 → 783.7 monotonically, and needs no fitted exponent. The exponent should not be extrapolated and should not be given to three significant digits. **g.** `max_a X_a` at the certifying L runs **2, 10, 25, 116, 834, 4605** — the quantity a degree-k certificate needs k to reach, rising fast enough that no fixed degree survives. > **VERDICT: CONFIRMED**, with the x = 17 and x = 19 rows now resting on the > sharp LP and on exact arithmetic for the first time, and with the slope > re-labelled as a five-point fit. --- ## 7. The ladder's provenance — **CONFIRMED** (added after the six) Headlines 1 and 2 are worthless if the 22-term ladder is wrong, and neither producing document could check it: `attack-growth-law.md` §10 records that "the session's web budget was exhausted". It is checkable now. OEIS returns HTTP 403 to the WebFetch path but serves `curl` normally, which is worth recording as a channel fact. ``` curl -A "Mozilla/5.0 ..." "https://oeis.org/search?q=id:A144311&fmt=text" HTTP 200, 1193 bytes curl -A "Mozilla/5.0 ..." "https://oeis.org/search?q=id:A048670&fmt=text" HTTP 200, 6364 bytes ``` Diffed against the arrays in `research/attack-growth-law.js` by program, not by eye: | check | result | |---|---| | A144311: OEIS 22 terms vs script 22 terms | **identical** | | A048670: OEIS 58 terms vs script 58 terms | **identical** | | `G₂ = A144311 + 1` at all 22 terms | **yes** | | offset `%O A144311 1,2` → `a(15) = 707`, `p₁₅ = 47`, `G₂ = 708` | matches the script | | `a(22) = 1709`, `p₂₂ = 79`, `G₂ = 1710` | matches the script | **a. The definition is ours. [VERIFIED]** Verbatim: *"The length of the longest sequence of consecutive integers, each equal to 1 or −1 modulo at least one of the first n primes."* Under `n = m − 1` this is exactly our sieve form: `m ≡ 1 or −1 (mod p)` ⟺ `m−1 ≡ 0 or −2 (mod p)`. So `a(n)` is the longest run of **killed** integers and `G₂ = a(n) + 1` is the gap between consecutive survivors. The conversion is right, and the offset is right. **b. And it is confirmed by brute force at six levels, independently of OEIS.** `research/redteam-DP1-certificate.js` §1 rebuilds the sifted set from the definition and measures the true maximal gap directly: **2, 5, 7, 11, 18, 25** slots at x = 5..19, and six times those is **12, 30, 42, 66, 108, 150** — the ladder's own first six terms. Two routes, no shared code. **c. Attribution matches.** `a(1)–a(7)` Andrew Carter, Sep 2008; `a(8)–a(16)` Max Alekseyev, Nov 2009; **`a(17)–a(22)` Jinyuan Wang, Nov 26 2024**. That is the "Carter 2008, Alekseyev 2009, Wang 2024" the corpus cites. **d. One qualification, and it matters for how the eight new terms are described.** A144311's keywords are `nonn,more,hard`. **OEIS makes no statement that the terms are proven maximal**, and `more` says further terms are wanted. `attack-growth-law.md`'s "proven maximal rather than best-found" is **this repository's own argument** — the admissibility of the branch-and-bound pruning test — and must be attributed that way, not to OEIS. **[VERIFIED as an absence]** **e. A minor one for the control.** A048670's DATA section carries 58 terms, but a comment records *"Computation of a(62)-a(64) was supported by Google Cloud"* (Bozek, Mar 2021), so more terms exist than the 58 the corpus uses. This is adjacent to the live internal disagreement `attack-growth-law.md` §10 already flags. Nothing here turns on it — the fits move by 0.001 between x ≤ 229 and x ≤ 271 — but the corpus's "58 exact terms" is a truncation, not a ceiling. > **VERDICT: CONFIRMED.** The ladder is the published sequence, under the right > offset and the right definition, and the `+1` conversion is correct. The one > edit needed is to stop implying OEIS asserts maximality. --- ## What this red team did NOT check - **The literature quotations** of §5h and §4f, for the reasons given there. Two of the six headlines lean on text files that are not in this repository and, in `BF.txt`'s case, carry no hash or byte count either. That is a corpus-level custody gap, not a defect in any one attack, and it is worth its own fix. - **`attack-theta-margin.js`** was embed-checked (clean) but not otherwise attacked. --- ## The edits this report asks for Ordered by how much they cost the project if left. **This report edits nothing outside itself and its own five scripts.** 1. **`attack-lower-bound.md` §3b — delete the word "independent".** The two instruments share the object, the eighteen points, the window and the estimator, and the exact frame cancels out of every point estimate. Replace the "two independent instruments" sentence with: the diagonal re-expresses the same comparison in a frame that sharpens the error bar, and its agreement with attack E is algebraic. Keep the ranking; drop the second witness. (§2) 2. **Stop circulating "c x ln²x excluded at 31.8 AICc units".** The same estimator throws the same law out by 32.8 on the control, where it is the conjectured truth. `attack-growth-law.md` §2b already says this; the headline dropped it. (§1c) 3. **Qualify "the pure power law excluded at 10.6".** It is 3.3 without x = 11, and at x = 5..79 the ranking reverses outright. (§1d, §1e) 4. **Add the window sensitivity to `attack-growth-law.md` §10.** Its own script computes block D2 and no section mentions it. §10 is the right home and it currently lists five other caveats. (§1e) 5. **Fix the AICc to `K = k + 1`** in `attack-growth-law.js`. The effect is small and favourable — PW 10.6 → 10.9 — but the formula is wrong. (§1a) 6. **Fix `attack-AB-bounded.md`'s `H ≫ (log z)^14`** to `(log z)^{8+o(1)}`. The printed figure prices `log⁶H` as `(log z)^6` while setting H polylog in z. Conservative, and wrong. (§3d) 7. **Re-label the +5.20 slope as a five-point fit on x = 7..19.** On the six ratios the headline prints, it is 4.60. (§6e) 8. **Drop "materially cheaper" from the `τ(m)` sentence.** At the worst `m`, `τ(m) = 2^{π(z)}` exactly. The gain is in the comparison's shape. (§5a, §5b) 9. **Never write "`1/4.156000 = 0.2406159756`".** That is the reciprocal of the exact `(4/3)(1+e^{3/4})`; `1/4.156000` is `0.2406159769`. (§5f) 10. **Stop implying OEIS asserts A144311's terms are maximal.** Its keywords are `nonn,more,hard`. The maximality argument is this repository's own. (§7d) 11. **Bring `BF.txt` and `friedlander-weaker-simpler.txt` into custody**, or at minimum record a URL, byte count and hash for each the way `attack-np-licenses.md` §1 does for the Brady thesis. Two headlines currently rest on files nothing here can reach. (§5h) 12. **Record the channel fact**: OEIS returns HTTP 403 to WebFetch and HTTP 200 to `curl` with a browser user-agent. `attack-growth-law.md` §10 recorded the ladder as unverifiable for want of web budget; it was one `curl` away. (§7) --- *Producers: `research/redteam-growth-aicc.js`, `research/redteam-instrument-independence.js`, `research/redteam-B-polylog.js`, `research/redteam-DP1-certificate.js`, `research/redteam-tau-and-split.js`. Each was run by this session, each carries its output written by `research/qc/embed.js` rather than pasted, and each passes `embed --check` on both hashes. Nothing in this file was transcribed by hand.*