# The single-alignment recursion for M(x, x³): measured, decomposed, and where it closes *(2026-08-17. TODO 0d's first move, executed. Calibration marked throughout: PROVEN, VERIFIED, MEASURED, REFUTED, PREDICTED-AND-REFUTED. Script: `research/localized-single-alignment.js` — predictions were written into the script and frozen before the first run; two of the five were refuted and are reported below with the confirmations. Deepest run: x = 1613, window Y = 4.20·10⁹, 240 measured folds, 36 s.)* ## 1. The verdict The route asked whether the head's fold recursion — single-alignment, because copy 0 receives only a = 0 — gives the per-fold multiplier a proven-shaped handle the max-over-alignments recursion lacked. The answer splits, and both halves matter: **The multiplier IS different in shape, decisively (MEASURED).** The fixed-window multiplier is exactly 1 at 225 of the 240 folds from x = 53. Its total spend over the ladder is 1.09 nats against the 7.12 nats a tile-shaped `ln c ≈ 2 ln p / p` ladder would spend; a tile-shaped M(1613) would be 3.7·10⁵, and the measured value is **2220**. The "different shape" premise of TODO 0d is confirmed, not refuted. **And the route to a growth law through the recursion CLOSES anyway.** The object's growth is initiated, at every record of the deep ladder, by the window's own expansion into fresh ground — a term the tile recursion never had, because the tile's new territory is p aligned copies of the old tile while the head's new territory is genuinely new. The boundary term is not a fold statement at all: it is the localized gap problem at the new scale, re-posed. A fold recursion cannot bound the object whose growth happens where no fold reaches. This closure is forced in the following sense: any window that must grow with x carries the boundary term, and the window of M(x, xᵏ) must grow with x by definition. What survives is two countable laws (§6) and a compute observation (§8). ## 2. The object and the statement, precisely Definitions exactly as `localized-01-ladder.js`: rough[r] = 1 iff r has no prime factor ≤ x (all primes folded, 2 and 3 included); a twin slot is r with r and r+2 both rough; M(x, Y) is the largest gap between consecutive twin slots whose left endpoint is < Y, straddle included. This note takes Y = x³, the smallest k at which the head is neither frozen (Impact Lemma: [0, p²) is untouched) nor crystallised (the standing caveat: at k = 2 the slots are twin primes and everything is TPC-hard by construction). **Single-alignment (PROVEN, trivial once said).** For x ≥ 11, x# > x′³, so the whole window [0, x′³) lies inside copy 0 of the fold by p = x′, and the kills in it are exactly the twin slots r ≡ 0 or −2 (mod p): one deterministic 2-set, no maximum over p alignments. The honest per-fold decomposition is > M(x′, x′³) = max( M(x′, x³) , boundary term ), where M(x′, x³)/M(x, x³) is the single-alignment multiplier and the boundary term is the largest gap with left endpoint in the fresh ground [x³, x′³). The window GROWS under the fold — the tile recursion never had this term, and it turns out to be the story. ## 3. Predictions on record, and what the measurement said Registered in the script before any run; the runs did not edit them. Controls first, per the house rule: the incremental engine was verified against a from-scratch full scan at x = 61, 307, 1009, 1289, 1613 (C1), against the independent engine `localized-01-ladder.js` at M(307) = 870 and M(491) = 990 (C2), and against FOLD-PROFILE §12b's tile-computed kill counts 33, 41, 63 at p = 19, 23, 29 (C3). **All controls passed on every run.** | # | prediction | outcome | |---|---|---| | P1 | M/ln³x flat in [3.6, 4.8] to x = 1289 | **REFUTED as stated** — climbs to 6.41 (x = 739), falls back to 5.51 (x = 1613); see §5 | | P2a | tile-shaped multiplier would give M ≈ 2.4·10⁵; refutable by two orders | **REFUTED as predicted**: measured 2220 | | P2b | lifetime budget 3 ln(ln x₁/ln x₀) ≈ 1.8 nats, ~3/p per fold, a factor (2/3)ln p tighter than the tile's | **CONFIRMED**: spend 2.001 nats vs budget 1.862 — 107%, no slack | | P3 | kills in [0, p³) = c_K·p²/ln²p, c_K ∈ [0.6, 1.1] flat | **CONFIRMED**: c_K ∈ [0.778, 0.852] over 240 folds, mean 0.807, thirds 0.812/0.803 | | P4 | boundary carries ≥ 2/3 of record increments by count and nats | **REFUTED as registered** — kills are co-equal: 12 KILL events (0.993 nats, 42%) vs 9 BOUNDARY (1.394 nats, 58%); by count kills lead 12 to 9 | | P5 | a record-moving fusion contains 1–3 kills, never more | **CONFIRMED**: exactly 1 at every event from p = 19 on (2 at p = 7, 17) | ## 4. MEASURED: the ladder x = 5 to 1613, window x³ to 4.20·10⁹, 253 folds, every fold recorded. | quantity | reading | |---|---| | M(1613, 1613³) | 2220 (all controls passed; record at r = 397,901,681) | | folds moving M at all | 21 of 253 | | fixed-window multiplier = 1 exactly | 238 of 253 folds (225 of 240 from x = 53) | | M/x² | falls monotonically, 0.107 (53) → 0.00085 (1613) | | m̄ | 2.39 ln²x at the top, the standing law | | total spend from x = 53 | 2.001 nats = 45% fixed-window + 55% boundary | | kill census c_K = kills·ln²p/p² | 0.807 ± 0.04 over the whole ladder, no trend | The per-fold cost question — is the damage countable? — has a clean yes for the CENSUS: the kill count in [0, p³) is deterministic given residues, measured at 0.81·p²/ln²p dead flat, exactly the density prediction, with the elementary cap 2·rough(p²) ≈ 1.1·p²/ln p one log above (every kill is p·t or p·t−2 with t p-rough, t < p²). Converting the census to gap damage is where the old difficulty reappears: a kill's damage is the size of the adjacent gap, which is M-like, and the doubling guess M → 2M is already REFUTED (TODO 0b's ledger). ## 5. The P1 refutation, located M/ln³x does not stay in the registered band. It is 4.63 at x = 307, 6.41 at 739, 6.34 at 1151, 5.51 at 1613. The excursion is one event, not a trend: the patch born at r = 397,901,849 carries the whole overshoot, and the 76 folds after x = 1151 add nothing while ln³x catches up. The frame the repo actually states the law in survives: with M = c·m̄·ln(Y/m̄) (maxgap-law §4), c reads 0.81, 0.86, 1.12, 1.09, 0.98 at x = 61, 307, 1009, 1289, 1613 — inside maxgap-law's measured off-diagonal surface range 0.74 to 1.17. So the crude ln³ constant is the wrong frame; the surface law is intact, and the correct statement stays: **M(x, x³) is polylog in x with an event-lumpy constant; a decade of new ground did not bend the law and did not settle the constant.** ## 6. The mechanism, from the event log Every increase of M over 253 folds, typed by cause (28 events). The deep ladder's records are ASSEMBLED, in two moves: 1. **Birth, by boundary.** p = 739 creates the record patch at r = 397,901,849 — inside the strip [719³, 739³) added by that very fold — jumping 1266 → 1848 (0.379 nats, the largest single event on the ladder). 2. **Finish, by single kills.** p = 997 fattens the same patch to 2052 and p = 1151 to 2220, one kill inside the record gap each time. This is the same assembly mechanism U-FRAME §7 found at the top of the tile ladder ("the last three folds did not extend the old maximum but assembled a fresh one"), now visible in a controlled window with the cause of each move typed. The misalignment principle, spent at last: with the max over alignments gone, the a = 0 fold behaves like a typical alignment — kill census at the density level, record touched a dozen times in 253 folds — and that is the saving. It is real (two orders of magnitude against the tile shape) and it lands on the minority share of the growth. ## 7. Why this closes the route, and what would reopen it The chain the route hoped for was: bound M(x′, x³)/M(x, x³) by something countable (it is: §4), re-base every fold to escape NFP (it does: no accumulating index anywhere above), conclude a growth law for M. The measurement says the conclusion does not follow, because 55% of the nats and the birth of every deep record sit in the boundary term, and the boundary term is the original problem: the largest gap of the CURRENT sieve on fresh ground of width ~x³, which is M's own extreme-value question at the same scale. The recursion reduces M to itself plus a countable correction. That is a relocation with a smaller remainder than any previous chain in this branch — the correction really is countable, which no tile chain achieved — but the remainder is still the whole problem. Reopen conditions, stated so they are checkable: (i) a direct bound on fresh-ground extremes — max twin-slot gap of T_x in [x³ − w, x³) for w ~ x³ — that does not route through G₂; that is not a fold statement and it is where this measurement says the object lives. Or (ii) the Merge Lemma's gate M ≤ (p−2)/4 turning on, which at k = 3 happens near x* ≈ 2.4·10⁴ — see §8. ## 8. Compute, and the reach Cost observed: x = 1613 (Y = 4.20·10⁹, 4.2 GB flat array): 36.2 s. x = 1289 (2.1 GB): 16.0 s. Control x = 307: 0.3 s. Node v22, single thread. The flat Uint8Array engine caps at Y < 2³²; past that the segmented walk of maxgap-law.md is needed. At this cost per decade, the k = 3 regime where the Localized Merge Lemma's own gate turns on — x* ≈ 2.4·10⁴, Y ≈ 1.4·10¹³, called "out of computational reach by four decades" in LOCALIZED-GAP §3 — is roughly single-thread days of a segmented version, not impossible. Whether it is worth the days is a different question: inside the gate the merge chain still dies on the Deficit Lemma, which is alignment-blind, so the gate regime would sharpen §6's forensics, not revive the chain. ## 9. Reproduction ``` node research/localized-single-alignment.js 307 # 0.3 s control, all values known independently node research/localized-single-alignment.js 1289 # 16 s, 2.1 GB node research/localized-single-alignment.js 1613 # 36 s, 4.2 GB (the run quoted here) ``` Custody: M(307) = 870 and M(491) = 990 reproduce `localized-01-ladder.js` digit for digit; kills 33/41/63 at p = 19/23/29 reproduce FOLD-PROFILE §12b's tile-side computation; the incremental record equals a from-scratch full scan at five checkpoints on every run. --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [history/CHANGELOG.md](history/CHANGELOG.md), indexed by document.*