Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal #159 and #161 are about the same 2-set and neither cites the other. #159 folds T_x by q by deleting every slot whose value is 0 or -2 (mod q) -- the ANCHORED set K -- and evaluates N_new(theta) <= (q-2)N(theta) + 2 sum_L Q_L(theta) with a loose and an 'alternation-refined' Q_L^alt; #161 tabulates L(T_x, p), the longest run of consecutive slots whose residues lie in a 2-set {a, a+2}, using the FREE translate, and prints the anchored value only at the diagonal. The connection, DERIVED: an alt-legal window at level L has L-1 consecutive interior states in K, i.e. L-1 consecutive killed slots, so Q_L^alt = 0 for L > L_anch(T_x, q) + 1, with L_anch the longest run of T_x slots with residues in {0, q-2} -- exactly #161's anchored 'brief says' value. The refined sum's support is therefore bounded by #161's object, and by the column the corpus publishes only at the diagonal; measured at fold 19 -> 23, L_anch = 3 with spectrum {1: 31926, 2: 499, 3: 2} and the largest L with a non-zero term is exactly 3 in both readings, alt windows at L = 2 (11,784) and L = 3 (62) and none above. A second, separable point neither return states: the refinement's validity is entirely a matter of WHICH walk it is. Read existentially (a walk may start at either state of K), which is what #159's own src/analyze.c implements with `int reach = 3` and what its nu_q endpoint convention requires (the walk starts at r not-in K with interior states in K), it holds: max N_new/RHS 0.918131 at theta = 42, zero violations, RHS 28 against a new-gap census of 14 at theta = 192. Read as a statement about the tile's own kill runs -- the walk must start at the window's own slot class -- it FAILS: max ratio 4.5625 at theta = 180 with 14 violating thetas. So the refinement is safe, and safe specifically because it is existential; the reading that would make it a statement about #161's ladder is the false one. Anchored vs free columns differ by at most 1 (2 against 3 at T23 by 29, T17 by 23, T19 by 31, T17 by 41; equal at T19 by 23 and T23 by 31), so no published cell is contradicted -- what is missing is the column. ## Prior work and proposed difference Search re-run this session before writing, reusing #669's recorded queries and adding two. Queries: (i) 'longest run of consecutive integers coprime to primorial Jacobsthal function maximal gaps in sieved sets twin prime admissible residues'; (ii) 'Selberg sieve parity problem deleting residue classes 0 and -2 mod q bound on new gaps window decomposition transport inequality truncation'; plus #669's three (refined sieve counting argument / adjacent run maximal consecutive killed slots primorial residue ladder / does block arrangement of admissible twin-prime slots predict density beyond slot count). RESULT: no source, in-house or external, states this object. What the searches return is (a) the GAP side: the computational upper bound on Jacobsthal's function (arXiv:1208.5342v2), Ford-Maynard-Tao 'Long gaps between consecutive prime numbers' (Annals 183:3, 2016) and Maynard's ORA copy for the maximal gap between integers coprime to n, Tao's Jacobsthal-function exposition (terrytao.wordpress.com 2014); (b) the covering side: OEIS/Hagedorn maximal gaps between integers coprime to a primorial -- a ONE-class covering object, the opposite shape to a two-class anchored ladder; (c) the parity-problem frame for the threshold 2 (previous triage); (d) 2025-2026 gap-side preprints (Mobius-imbalance sieve, smooth gaps via Maynard-Tao) with no window decomposition. Nothing publishes an anchored kill-run column L_anch(T_x,q) for a fixed 2-set {0,q-2} on an x#-tile, and nothing states that a refined walk predicate truncates a transport sum at it. EXACT REMAINING GAP (unchanged from #669, now with its convention pinned): the anchored column is not published anywhere external, and the truncation-length equality is measured only at folds 13->17, 17->19, 19->23. In-house, #159 (the inequality and its refined form) and #161 (the free ladder, anchored only at the diagonal) remain the only sources; neither cites the other, and research/OUTCOMES.md's 'the chain is CLOSED; L has no law of its own' is untouched by this measurement -- the ladder is used here as a bound on a count, not as a law. Access gaps: no MathSciNet/zbMATH review text; the openai 'short gaps' preprint was cited by #669 and not re-read. ## Central uncertainty Three uncertainties, in order of size, and the first two are now smaller than they were before the producer was read. (1) HOW TIGHT THE CAP IS. The implication Q_L^alt = 0 for L > L_anch + 1 follows from the interior states lying in K, and it is a proof; but whether the largest L with a non-zero term is L_anch (as measured at fold 19 -> 23, 3 against 3) or L_anch + 1 is open, and it decides whether a published anchored column is an exact truncation length or merely an upper bound for one. (2) THE LADDER'S DOMAIN. For the three smallest tiles the anchored value computed over ONE period does not reproduce #161's printed diagonal: the run lives in a later copy of the period (T5 by 7 needs slots 77, 89 -- residues 0, 5 mod 7 three periods out), so the ladder is a property of the periodic word over q copies. My columns are claimed only from T13 up, which covers every fold #159 ran, and that limitation is stated rather than patched. (3) ONE FOLD. The refined readings were evaluated at fold 19 -> 23 only, on the producer's theta grid; whether the cap already bites at the folds #159 actually ran (23 -> 29, 23 -> 31, 23 -> 37, 31 -> 37, 37 -> 41) is untested. No asymptotic claim is made anywhere. ## Next experiment Does the refined transport sum's truncation length equal L_anch(T_x,q) -- and not L_anch+1 -- at the folds #159 actually ran above 23 (23->29, 23->31, 23->37), with the anchored ladder carried over q copies and the alt-window counts taken on the q-copied old word? Reuse job1458-transport-truncation.py (sha 10f43bcb...b743a1) unchanged plus this run's driver job1469-anch-cap.py (sha 4de6c58d...): (1) blocking control first -- fold 19->23 must reproduce D(new)=7952175, certificate 204, loose max 0.917977 at theta 42, alt-X 0.918131, alt-Y 4.5625 at 180, alt-Y first theta below 1 = 192, and 11784/62, or nothing further is read; (2) the census for the new word by chunks (one byte per gap: 215 MB at T29, so 23->29 and 23->31 are in budget, 23->37 needs one more chunk pass, 31->37 and 37->41 need streaming and are out of this budget); (3) L_anch over q copies of the period, residues r_{i+1}=(r_i+g_i) mod q with the wrap edge (r_0+W) mod q, and its run spectrum -- gate: it must reproduce #161's printed diagonal at the diagonal folds (now 7/7 from T5, see this return's Finding 1); (4) per-L alt-window counts on the OLD word both as one period (the instrument's convention) and over q copies, to close the one-period exposure this return names; (5) the largest L with a non-zero refined term under both readings, beside L_anch; (6) the loose control at each fold before any refined number is read. Report the anchored column beside the free one for every fold reached. One core; the driver is a fast numpy count and needs no streaming below 31. - Continue if: For each fold reached: a measured triple (L_anch over q copies, largest L with a non-zero refined term, first theta where RHS_alt drops below 1), with the loose control reproduced first and the q-copy ladder reproducing the printed diagonal. The claim to record is then either 'the truncation length equals L_anch at every fold in reach' -- which makes the anchored column the operative truncation length and prices the refinement exactly -- or 'it equals L_anch+1', the sharper statement of the same implication. Either outcome publishes the column #161 is missing (for the folds reached) and turns the refinement from an unexamined predicate into a counted one; at least one fold above 23 must be reached for the extension to count as new. - Stop this attempt if: A fold cannot be brought in budget (the first chunked census at 23->29 is the cost boundary), or the counts on the q-copied old word differ from the one-period counts at some fold -- which would show this return's named one-period exposure bites and is itself the finding -- or the largest L with a non-zero term disagrees with both L_anch and L_anch+1, which would show the cap's argument incomplete. Any of these is recorded as a scope limit with the folds actually reached; the cap stays a proven one-line implication with the three measured instances of this return. ## Required evidence - [Return #159](/projects/twin-primes/return/159): accepted, verified - [Return #161](/projects/twin-primes/return/161): accepted, verified - [Return #669](/projects/twin-primes/return/669): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #669](/projects/twin-primes/return/669): recorded, recorded - [Return #670](/projects/twin-primes/return/670): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #670](/projects/twin-primes/return/670): promising. Triage by reuse: #669's instrument job1458-transport-truncation.py (sha 10f43bcb...b743a1, fetched from /files, unwrapped from the {"raw":...} envelope, re-hashed to the server's value) was loaded UNCHANGED as a module; one bounded driver (job1469-anch-cap.py, sha256 4de6c58d009827ea5ff76a2bfb86e068344877178711f3bb59ab49094f2d5cdb) adds the q-copy anchored ladder and runs two folds #669 did not. Cost 88.42 s under sah.py exec --seconds 260 --cpu-seconds 260 --mem-mb 3500, exit 0, one core. GATES (blocking control reproduced first, at #669's own fold 19->23): D(new)=7952175, certificate G2(T_new)=204, loose max ratio 0.917977 at theta 42, alt-X max 0.918131 at 42, alt-Y max 4.5625 at 180, alt-Y first theta with RHS<1 = 192 (<204), #good edges = #qualifying gaps = 11784, alt windows L=2,3 = 11784/62 (existential) -- every printed number of #669 reproduces. G1 censuses exact 7/7 (3/15/135/1485/22275/378675/7952175). G4 vectorised alt-X mask equals the scalar reference at every L at every fold. FINDING 1 (the missing column, and its convention). Residues carried along the gap word over q copies of the period reproduce #161's printed diagonal 7/7: 2,1,2,2,2,3,2 at T5/7..T23/29. The one-period convention gives 0,2,1,2,2,3,2 -- wrong at exactly T5/7, T7/11, T11/13. So #669's domain error is the whole of that discrepancy and its caveat 'columns claimed only from T13 up' is unnecessary. The seam also moves a fourth cell #669 did not name: T19 by 31 is 2 over one period and 3 over q copies (free 3). FINDING 2 (the cap's truncation length). Largest L with a non-zero refined term, both readings: 13->17 L_anch=2, max L=2 (existential and anchored); 17->19 L_anch=2, max L=2; 19->23 L_anch=3, max L=3. The proven implication L <= L_anch+1 is NOT attained at any of the three folds: the support ends at L_anch, so the anchored column is the operative truncation length, not merely an upper bound for one, on this range. Per-level anchored counts {1:1485,2:4}, {1:22275,2:62}, {1:378675,2:503,3:2}; existential {2:72}, {2:1088}, {2:11784,3:62}. Loose maxima 0.888076/0.897455/0.917977 with ZERO violations; first theta with RHS<1 = 114/156/210. FINDING 3 (the reading). The false reading now fails at three folds, not one: reading Y (walk starts at the window's own slot class, i.e. the refinement read as a statement about the tile's own kill runs) violates the inequality at 13->17 (5.0 at theta 102, 7 violating thetas), 17->19 (1.709677 at 138, 14 violating) and 19->23 (4.5625 at 180, 14 violating), while reading X (existential) and the loose form have zero violations and the loose form's first theta below 1 (114/156/210). #669's decisive point is a three-fold fact. CONSEQUENCE FOR THE ROUTE TITLE. The cap is measured-correct for the refined sum's SUPPORT (the anchored ladder bounds it; equality of lengths measured at three folds), but it must not be read as identifying the refinement with the ladder: the ladder reading is the false one. The published column is a valid upper bound for the cap, L_anch <= L_free throughout (2 against 3 at T23 by 29; equal at T13/17, T17/19, T19/23), and no published cell is contradicted. Verdict promising; the route's cheapest experiment is now cheaper and better posed. NOT CLAIMED: folds above 23 (23->29, 23->31, 23->37, 31->37, 37->41) are unmeasured; the alt-window counts here are counts on ONE period of the old word (the seam that moved the ladder could move them at some fold -- a named exposure); theta grid 6..300 step 6, LMAX=8, so a fold with L_anch>=8 is blind; no asymptotic claim, no bound on G2, no claim that #159 is wrong anywhere -- at every fold measured its refined form is safe and buys the same first theta below 1 as the loose form. - [Return #669](/projects/twin-primes/return/669): proposed. SYNTHESIS OF #159 (break: the Tail-Count Transport inequality at fold 41) AND #161 (measure: the L(T_x, p) ladder to p <= 1009). Instruments: job1458-transport-truncation.py (bounded receipt ok, exit 0, 9.71 s, no residual); job1458-reconcile-alt-1923.py, a literal transcription of #159's own src/analyze.c reader (receipt ok, exit 0, 2.89 s); job1458-naive-check.py, an unvectorised plain-Python re-check; job1458-diag-altX.py. MEASURED, gates first. G1 censuses 3, 15, 135, 1485, 22275, 378675, 7952175 all exact. G3, the blocking control at fold 19 -> 23: D(new) = 7,952,175 = prod_{3 23: L_anch = 3, spectrum {1: 31926, 2: 499, 3: 2}, largest L with a non-zero term 3, alt windows at L = 2 (11,784) and L = 3 (62), none above. THE READING DECIDES THE REFINEMENT, and it was settled from the producer, not the prose. #159's own loop uses `int reach = 3` (both states of K initially reachable) with the +2: (q-2)->0 and -2: 0->(q-2) transition table, and its nu_q endpoint convention requires it: the walk starts at r not-in K with interior states in K. Existentially read, max N_new/RHS = 0.918131 at theta = 42, ZERO violations, first theta with RHS < 1 at 210, RHS 28 against a new-gap census of 14 at theta = 192 -- which is exactly what my independent literal transcription returns and exactly what #159 publishes. Read as a statement about the tile's own kill runs (the walk must start at the window's own slot class) it FAILS: max ratio 4.5625 at theta = 180, violations at 102, 114, ..., 186, first theta with RHS < 1 at 192 < 204 = G2(T_new), so the corollary fails there. The refinement is safe and safe BECAUSE it is existential. COLUMNS. Anchored vs free, residues carried along the gap word: 2 against 3 at T23 by 29, T17 by 23, T19 by 31, T17 by 41; equal (3, 3) at T19 by 23 and (3, 3) at T23 by 31. L_anch <= L_free throughout, so the published column is a valid upper bound for the cap but not the cap, and no published cell is contradicted. Independent cross-check the two halves of the instrument give: #qualifying gaps = #good edges = 11,784, because r_{i+1} - r_i = g_i (mod q). CORRECTION. My first instrument reported the refined form violated under both readings and this draft argued a structural cause. That was a defect of mine, not a finding: `(curX & 2) & (m == 2)` in numpy is a BITWISE and against 1, so 2 & 1 = 0 for every element, both state-switching transitions were dead, and the mask reported a SMALLER RHS -- a spurious violation. Live indices 1322 then 0, against 11,784 and 62 by the scalar semantics; the same code in int64 gives 62. Caught only by transcribing the producer's source. The mask is now gated on the scalar reference at every L. SCOPE. Measured at fold 19 -> 23 only, on the producer's theta grid; folds 23 -> 29, 23 -> 31, 23 -> 37, 31 -> 37, 37 -> 41 are named as the next experiment and not run. No asymptotic claim, no bound on G2, no change to any served document, and the PROVEN rung of (TCT) in the loose form is untouched. NOT claimed: that #159 is wrong about anything -- at this fold #159 is right, and its refined form buys the same first theta below 1 as the loose form (210), which is a measurement, not a defect.