# The fold-factor field M_p, derived down to its comb: a formula that hit blind at a second fresh anchor *(2026-08-20. Staging note; nothing here is integrated into a live document. Producers, both formally embedded: `research/attack-mp-derive-01.js` (0.2 s, no sieving — every number parsed from committed embeds) and `research/attack-mp-derive-02-anchor.js` (15.8 s, one fresh 2·10⁹ sieve). The blind test's pre-registration `mp-window-prereg.md`, same folder, was committed ALONE at `303711b` before the producer existed. Calibration marked on every claim: PROVEN, VERIFIED by exact computation, MEASURED, PREDICTED, REFUTED.)* ## 0. The answer `perfold-error-model.md` left M_p "MEASURED, not derived" with the comb-weight correlation 0.654 as its recorded lead. The lead was right, and it sharpens into a formula: > **M_p ≈ k · W1(θ_p) · exp(−δ · θ_p/m̄_p)**, where > **W1(v) = ∏_{q|v} (q−2)/(q−4) · ∏_{q|v−2 or q|v+2} (q−3)/(q−4)** over the > folded primes 5 ≤ q < p — the **exact endpoint comb** of the two kill > classes, zero parameters — m̄_p = 6·∏_{5≤q (k, δ) = (1.9468, 0.2771) the only fitted scalars. Equivalently, as a law: > **E[X_p] = kills · (A·k) · W1(θ_p) · exp(−(c+δ)·θ_p/m̄)** with c+δ = 1.359 — > the law's own two constants re-estimated with the comb in place, and no > degree of freedom the law does not already carry. Fitted on the 37 pooled folds p ≤ 293 only, the formula drops the field's roughness from sd(ln M) = 0.560 to a train residual of **0.177**, beats every rival on both held-out sets (10–40 nats), and then **hit a pre-registered blind test at a second fresh anchor**: at `[1.32·10¹¹, 1.32·10¹¹ + 2·10⁹)` — anchor never used, bands sealed from arithmetic plus the two train scalars, NO per-fold measured M anywhere in them — it scored **34 of 37 inside the 90% bands (needed ≥ 28), 0 of 37 outside the 99.73% bands (allowed ≤ 1), and a +25.4-nat likelihood margin over the constant-bias rival (needed > 0; predicted +23.6)**. VERDICT under the sealed rule: **HIT**. **Grade, against the brief's own bar: PARTIAL DERIVATION.** The comb half is derived with zero parameters and carries the within-depth spread; the two scalars are refits of constants the law already had; and a real residual sub-field (sd ≈ 0.18, a far-tail flattening, fold 631, four flagged folds) is deterministic-looking and NOT derived — §5 names it exactly. ## 1. Where W1 comes from, and why it is not the recorded comb A slot n of the twin comb dies at fold q iff n ≡ 0 or −2 (mod q). For a PAIR of slots at distance v the forbidden residue set is {0, −2, −v, −v−2}, whose size is 4 minus the coincidences: two if q | v, one if q | v−2, one if q | v+2 (at most one of the three can hold for q ≥ 5). **[PROVEN, elementary residue counting]** the pair-survival weight of distance v relative to a generic distance is therefore exactly > q | v → (q−2)/(q−4); q | v±2 → (q−3)/(q−4); else 1, per fold, which is W1. Since θ_p = 2(p−η), the q | θ class is the divisor structure of p−η and the q | θ∓2 classes are the divisor structures of p−η∓1 — the fold's exact arithmetic, in the corpus's anchored sense. The recorded comb W0 = ∏_{q|θ}(q−1)/(q−2) is the twin-prime singular-series form: right divisor classes on the θ side, wrong magnitudes, and no θ±2 classes at all. Using W1 as the gap-count weight is a modeling step (endpoints exact, interior smooth), and everything below is its MEASURED scorecard. No prior-art search was run on W1 itself; the object is singular-series-shaped and nothing here is claimed as new. ## 2. The target table, custody first Producer 01 re-sieves nothing. It assembles the whole measured field from the embedded OUTPUT blocks (perfold-01's pooled table and Stage-4 pools, perfold-02's blind X, window-5's B2/C1, the four scaling windows + offset, import-stein's B2, import-thinning-03's K column) and must reproduce, before anything is fitted: the 48 printed pooled rows, the recorded 0.654 probe (got 0.6535), the printed band means 0.959/0.858/0.520, the blind 33/37, the Slam = 1.1211·λ_{2·10¹¹} identity, the analytic m̄ chain against the embedded one (worst 0.44%), and the deep pooled M̂ against the printed rows (worst 0.0005). **[VERIFIED, all of it]** The import-stein unification extends from the record's 8 of 8 to **14 of 14** printed folds. Field = 102 folds in [101, 709]: TRAIN the 37 with p ≤ 293, TEST the 65 with p ≥ 307 plus the 37 blind-window folds — split declared before any fit. ## 3. The candidate battery, calibrated, and the scores The identical pipeline first reads two synthetic controls at the real exposure grid (sfc32 seeded): a planted comb+depth field is recovered (δ̂ = 0.247 vs 0.25 planted; free exponent β̂ = 1.09 vs 1) and a pure lognormal field yields neither depth (δ̂ = 0.091) nor a comb advantage (D4 − D0 = −6.9 nats). All five gates PASS; the run aborts otherwise. The real field, candidates fitted on TRAIN only, scored on held-out Poisson-lognormal log-likelihood (each candidate with its own train roughness): | candidate | train sd(ln resid) | deep PLN LL (65 folds) | blind PLN LL (37) | |---|---|---|---| | M ≡ 1 (the raw law) | 0.634 | −128.2 | −132.4 | | D0 constant bias b = 0.750 | 0.563 | −118.3 | −133.5 | | D1 k·W0 (the recorded lead) | 0.494 | −116.2 | −130.1 | | D2 k·W1 (comb only) | 0.353 | −122.7 | −123.0 | | D3 k·e^{−δz} (depth only) | 0.535 | −107.8 | −132.1 | | **D4 k·W1·e^{−δz}** | **0.177** | **−88.6** | **−112.7** | | D4q curvature variant | 0.180 | −101.6 | −112.8 | Two mechanisms, both needed, and the free-exponent diagnostic reads **β = 1.031 on ln W1 where the derivation says exactly 1** [MEASURED]. The candidate list the brief posed closes as: (i) the comb lead becomes D4; (ii) the Stein route M_p ≡ λ_derived/λ_model is exact by construction where λ_derived exists (14/14) but extending λ_derived to all folds without sieving IS the histogram-derivation problem — it collapses into (i); (iii) **REFUTED as a field predictor**: K = Ψ/Φ² spans a factor 1.009 across folds 101/211/421 while M̂ spans 2.28 — the Fold Moment Identity's deviation does not carry the field; (iv) the divisor structures of p±1, p±2 are exactly W1's classes and are subsumed. **The field is a function of θ, not of p** [MEASURED]: the 13 twin-θ pairs (twin primes share θ) agree at rms |Δln M̂| = 0.155 against 0.764 for adjacent non-twin pairs — though at 2.25× the Poisson-pull scale, so a ~10–15% non-θ (or anchor-jitter) residual exists. ## 4. The blind test at the second fresh anchor Sealed before the producer existed (`mp-window-prereg.md`, committed alone at `303711b`; producer-01 pinned there by its embed fingerprint): window `[1.32·10¹¹ +2·10⁹)`, the 37-fold population, X_p ~ PLN(λ_model·M_D4, s0 = 0.177), all 74 band integers, and three criteria. Producer 02's Stage A reproduces the embedded [0, 2·10⁹) calibration 11 of 11 digit for digit (the engine is the record engine); Stage B re-derives all 37 sealed band pairs integer-exactly from the committed chain; Stage C sieves; Stage D scores: > **(a) 34 of 37 in-90 (needed ≥ 28; incumbent-truth expectation 34.6); > (b) 0 of 37 outside 99.73% (allowed ≤ 1); (c) margin over D0 = +25.4 nats > (predicted +23.6). HIT.** Misses: 101 (279 vs [146, 274], by 5), 137, 199. > For the record, unscored: N2 = 39, ΣX = 20,317,819, last L ≥ 2 fold = 397. This is the evidence class the existing test sets could not supply: every prior scoring window was measured before the formula existed. It also adds a third anchor at which the field's per-fold structure is reproduced by one fixed object — consistent with anchor-constancy, though bands this wide (s0 = 0.177) do not tighten redteam T1.c's ~6–10% one-replicate jitter bound. ## 5. What fails, exactly (the underived residual, named) - **The far tail decays slower than the train extrapolation.** δ refit on the deep folds alone (diagnostic) reads 0.252 vs the train 0.277; band [300,500): predicted 0.423 vs measured 0.520; [500,710): 0.233 vs 0.357. The linear-in-z residual decay flattens beyond z ≈ 8, and that flattening is the structure a third parameter would chase. (Shallow bands land: 0.946 vs 0.959 and 0.872 vs 0.859.) - **Four flagged folds** (|pull| > 3 against the formula): 311 (+3.10), 331 (+3.21), 409 (+3.28), 631 (+3.36) — all HIGH. 331 and 631 are the two comb-loudest deep folds (W1 = 5.26, 5.55), so part of the residual grows with W1; 311 and 409 (W1 = 1.22, 1.36) say not all of it does. - **Fold 631, the twice-seen extinction fold.** D4 makes it comb-loud but not the loudest (top expected deep events sit at 541, 547); its two events (2·10¹¹ once, the 6.6·10¹⁰ blind window once) are a 2.8·10⁻³ Poisson event at that fold under D4 (P = 0.17 that some p ≥ 521 fold shows two). The far-tail TOTAL is D4's cleanest deep win: 2 events observed vs 2.30 expected, against 9.60 for the constant-bias rival (P(N ≤ 2) = 3.8·10⁻³). The perfold record's rider — the extinction extreme's anchor spread is partly deterministic — survives in sharpened form: W1(1260) = 5.55 explains a preference for 631, and does not explain two sightings. - M̂(631) itself: 4.482 on the six-window pool (the record's 4.5), 8.9 with the blind event; the formula says 0.338. Tiny exposure, honest tension. ## 6. What this does not show Nothing here proves H″ or touches the exponent, the aggregate law, or its WEAKENED grade. W1's use as a gap-value weight is a modeled leading term — the interior of the gap is treated as smooth — and (k, δ) remain fitted, so this is not a zero-parameter derivation of M_p; it is a zero-parameter derivation of the field's comb structure plus a recalibration of constants the law already owned. The PLN roughness s0 = 0.177 prices the formula's own residual, so its bands are wider than the measured-field NB bands, which remain the sharper predictive (the prereg's consequence fixes this: formula bands are a RIVAL line in future preregs, never the replacement). The deep TEST folds pool six windows dominated by 2·10¹¹ exposure; per-fold deep M̂ below Sx ≈ 10 is Poisson-limited. And the depth term e^{−δz} is phenomenological: no mechanism here forces the residual exponent shift δ = 0.277, and §5's flattening says linear-in-z is itself only a local description. ## 7. Proposed consequences, for adjudication rather than applied - **`perfold-error-model.md` §4's** "M_p is MEASURED, not derived; the comb correlation is a lead, not a mechanism" can now carry: *the lead closed — M_p = k·W1(θ_p)·e^{−δθ_p/m̄} with W1 the exact endpoint comb (zero parameters, β = 1.03 measured), (k, δ) the law's own two constants re-estimated, blind-validated 34/37 at a second fresh anchor (`mp-derivation.md`); the residual sub-field (sd ≈ 0.18, far-tail flattening, fold 631) stays measured-only.* - **`prop-thinning-null.md` §5**: the same sentence-level update where the 2026-08-20 block says "the comb-weight correlation 0.654 is a lead". - The law's statement gains the comb clause: the ~20% hot systematic and the deep-decade overprediction are the exposure-weighted image of k·W1·e^{−δz} − 1, and a comb-aware refit (A·k, c+δ) removes most of the per-fold first-moment roughness the smooth law misses. ## 8. Reproduction ``` node research/attack-mp-derive-01.js # 0.2 s, no sieving node research/attack-mp-derive-02-anchor.js # 15.8 s, one fresh 2e9 sieve STAGE=predict node research/attack-mp-derive-02-anchor.js # stops before the fresh anchor node research/qc/embed.js --check research/attack-mp-derive-01.js node research/qc/embed.js --check --timeout 200 research/attack-mp-derive-02-anchor.js ``` --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [../CHANGELOG.md](../CHANGELOG.md), indexed by document.*