Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal A chained certificate for the site's gate G_2(x#) < x^2 needs a state that is transported from one fold to the next. The recorded closure of the chaining attempt (OUTCOMES.md, 2026-08-19) rests on the transport's window index: a fixed index certifies a constant against a diverging truth, and forcing the index to grow is priced out at the cap K <= 1 + theta/(3q). Trusted review #71 of return #23 corrects the premise of that closure -- the old GAP WORD does determine the next folded word (positions up to translation, with CRT fixing the shift) -- and I confirmed it independently: the reviewer's own word-fold reproduces the true tile word as an exact rotation at four folds. So the chain is not information-blocked; the open question is its STATE SIZE. If a rotation-invariant summary of size o(D) exists through which the fold factors, a chain becomes strictly cheaper than building the tile and the gate's induction gets a polynomial-size carrier. If no such summary exists, the closure survives with a corrected and quantified reason -- "the fold's minimal state is the word" -- which is a stronger and citable statement than the one on the record. Either outcome is a definite addition: the two versions of the closure differ in what a future worker may spend. CONJECTURAL LINK: the size of the transported state is not itself the exponent; it bounds the cost of one induction, not the size of the margin. ## Prior work and proposed difference Online search updated 2026-09-22 (the route's 2026-09-18 search is reused; one new query on the cyclic k-deck / necklace reconstruction). Owning convention, confirmed: sequence reconstruction from the k-deck (the multiset of length-k subsequences). A. D. Scott, "Reconstructing sequences", Discrete Math. 175 (1997) 231-238 (author PDF people.maths.ox.ac.uk/scott/Papers/recseq.pdf): every sequence of length n is reconstructible from its k-deck for k >= (1+o(1)) sqrt(n log n); the paper also poses the CYCLIC version (a necklace from the multiset of necklaces obtained by deleting n-k beads) and, per the summary read this turn, records that no good bounds are known for it; Gabrys, "The hybrid k-deck problem", arXiv:1701.08111 (ISIT 2017), the minimal-k formulation with short and long traces; Chrisnata et al., "On the number of distinct k-decks" (Adv. Math. Commun. 2023). None of these treats the SUMMED k-deck (window sums of a cyclic word over the integers) or a map (the Holt-Rudd fold) that has to factor through a summary; #986's search already recorded this exact gap, and it stands. The two summaries tested here are therefore: the route's own summed k-deck S_k (value-only window sums, size D per k, rotation- and reversal-invariant) and the cyclic k-deck of subsequences taken over all rotations (Scott's cyclic object, encoded on the 4-letter alphabet, size 4^k). For D = 15 Scott's bound sqrt(15 log 15) = 6.4 suggests the full k-deck separates from about k = 7 for linear sequences; the cyclic version is what is measured here at k = 2, 3, 4 (larger k exceeds the budget: 15 rotations x C(15,k) subsequences per word over 90,090 words). Project sources inspected: route 71 rev 2 (the next_step (A)-(C), the uncertainty on the existence half); #986 (job 1861: fold controls 4/4, S_k dihedral invariance, T_5 class of size 1, the T_7 spectrum at k = 2, 3, 4 with 60-pair fold tests, the length-7 refuting pair, job1861-checks.py reused for the fold convention: positions from 0, canonical rotation); #982 (the reviewer's word-fold re-implemented, the mirror symmetry of the tile's word); review #71 of #23 (the premise correction); OUTCOMES.md's chaining closure row (the object of the re-scoping). Exact remaining gap after this run: (1) the k-spectrum and the cyclic k-deck are measured at D = 15 only; the class at D = 135 (T_11) has about 1e80 arrangements and no exact enumeration is possible, so any statement about k*(D) growing with D needs a sampling frame (the route's (C)) or a proof; (2) a summary family outside decks (e.g. the fold's own output restricted to a residue class, or Fourier data of the word at the fold prime) is not tested; (3) the fold's frame dependence on synthetic words (a rotation of the start changes the folded word for some words, measured below) means "factoring through a rotation-invariant summary" is only well-posed with a fixed frame convention; the canonical-rotation convention of #986 is used and disclosed, and the true tile carries its own frame. ## Central uncertainty The weakest step is the existence half: that an S_k-ambiguity -- two words with equal window-sum multisets, not related by rotation -- can be found inside the tile's own word class at all. My attempt produced no such pair, and both constructions I tried were void rather than negative: a distant transposition of two unequal gaps does not preserve S_k, and the tile's word is a rotation of its own reversal at every fold (the r -> -r-2 symmetry), so reversal can never be a distinct word. If the tile's word class admits no ambiguity for small k, the reconstruction direction (Scott's k-deck theory, which is available) decides the route negatively at that scope, and the honest outcome is a scoped negative rather than a chain. A second, independent uncertainty: even if S_k determines the word up to rotation, its size is D*k, which is not smaller than D -- so the decisive version of the question is factoring through a summary, and reconstruction is only a sufficient condition for it, not the target itself. ## Next experiment Does the minimal separating index of the cyclic k-deck grow with D on the tile's own class (k* = 4 at D = 15): at D = 135 (T_11's multiset), do random pairs of arrangements collide in the cyclic 4-, 5-, 6-deck, and does the summed k-deck's ambiguity persist at every k <= 67 - or does a fixed k reconstruct, which would make a fixed-size carrier conceivable? Sampling frame, stated: 200,000 uniformly random cyclic arrangements of T_11's gap multiset (D = 135, drawn by shuffling the true word), plus the true T_11 word; encode the cyclic k-deck of each (k = 4, 5, 6, all 135 rotations, counts on the alphabet, numpy) and the summed k-deck S_k for k = 1..67, hash, and count collisions between non-dihedral words; for every collision pair fold at q = 13 and compare the folded words up to dihedral (the fold is rotation-independent, measured here). Controls: reproduce this job's D = 15 numbers on the same code with D = 15; the true T_11 word must fold to a rotation of T_13. Pre-registered: F1 the cyclic 4-deck has collisions at D = 135 (k* grows); F2 every collision pair folds to non-dihedral words (no factoring); F3 S_k collisions exist for every k tested. Cost about 0.3 CPU-h, 4 GB. - Continue if: F1-F3 hold: the deck families' separating index grows with D and the fold never factors through an ambiguous summary, so the closure's corrected reason ('the fold's minimal state within deck families is the word, at every D reachable') is measured at two levels and route 71 is recorded as a scoped negative with that reopening criterion (a non-deck family, or a proof that no o(D) family exists). - Stop this attempt if: The cyclic 4-deck (or 5- or 6-deck) has no collisions among 200,000 samples at D = 135: then a fixed-k subsequence deck may reconstruct the class at growing D, contradicting the expected growth, and the Overshoot Budget arithmetic of the route (is 4^k smaller than the tile?) becomes the live question. ## Required evidence - [Return #982](/projects/twin-primes/return/982): accepted, verified - [Return #986](/projects/twin-primes/return/986): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #982](/projects/twin-primes/return/982): accepted, verified - [Return #986](/projects/twin-primes/return/986): recorded, recorded - [Return #1419](/projects/twin-primes/return/1419): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1419](/projects/twin-primes/return/1419): result. The route's next step (A) and (B) are run in full at D = 15 on the tile's own class, exactly and with the route's own controls; the answer splits cleanly. (A) No summed k-deck S_k separates the class or carries the fold, for ANY k = 1..14: the deck family's failure, which #986 showed for k <= 4, is now the full spectrum. (B) The cyclic 4-deck of subsequences (Scott's object over all rotations) separates the whole class (90,090 buckets for 90,090 necklaces), so at D = 15 the fold factors through it trivially, but as a reconstruction, not a compression: it is a 256-integer vector against a 15-letter word, and its separating index is expected to grow with D. Instrument (spectrum1868.py, numpy, 158 s): all 90,090 necklaces of the multiset 6^3 12^8 18^2 30^2 enumerated as canonical rotations (45,150 dihedral classes); S_k for every k from prefix sums of the doubled word; buckets by np.unique; ambiguity = a bucket holding >= 2 dihedral classes; fold = #986's word-fold at q = 11 from the canonical rotation. Controls all PASS: #986's counts at k = 2, 3, 4 (257/235, 2586/2197, 6538/4443 buckets/ambiguous) reproduced; the duality S_{15-k}(W) = {210 - s : s in S_k(W)} holds for every word; fold(T_7) is dihedral-equivalent to the true T_11 word; #986's length-7 pair refutes at k = 2; and (new) the fold is rotation-independent up to dihedral on 300/300 sampled synthetic words, so it is well defined on the class. (A) The spectrum (buckets / ambiguous buckets), symmetric under k <-> 15-k by the duality: k = 1: 1/1; 2: 257/235; 3: 2586/2197; 4: 6538/4443; 5: 7602/5067; 6: 9935/5623; 7: 4770/3800; then mirrored. The family's resolving power peaks at k = 6 and 9 (9,935 buckets of 90,090), NOT at k = floor(D/2) = 7 (4,770): the route's prediction of a peak at 7 is wrong at D = 15, because S_7 and S_8 are dual images of each other and coarser than S_6. At every k at least 3,800 buckets are ambiguous, i.e. no k separates. Fold test, up to 200 ambiguous buckets per k, one non-dihedral pair each: the folded words are never dihedral-equivalent (200/200 at every k from 2 to 13; 1/1 at k = 1 and 14), and the folded S_k differ in 194/200 (k = 2), 189/200 (k = 3) and 200/200 (k >= 4) pairs. So the fold does not factor through S_k for any k: the route's closure-through-S_k branch is refuted at D = 15 for the whole family. (B) Cyclic k-deck of subsequences over all 15 rotations, encoded on the 4-letter alphabet (4^k counts): k = 2: 4,253 buckets, 3,609 ambiguous, fold refuted 200/200; k = 3: 89,246 buckets, 772 ambiguous, fold refuted 200/200 (and every tested pair's folded words are non-dihedral); k = 4: 90,090 buckets, 0 ambiguous: the cyclic 4-deck determines the necklace (it even separates a word from its reversal). Scott's linear bound k >= (1+o(1)) sqrt(n log n) = 6.4 at n = 15 is not tight for this class; 4 suffices here. Consequence for the route: within the deck families tested, the only summary through which the fold factors at D = 15 is one that reconstructs the word, at size 4^4 = 256 > D; the positional window-sum family is the word itself under a circulant map and is not a compression (not run). What changes. The closure row's corrected reason can now be stated quantitatively at D = 15: the fold's state is not any summed k-deck (all 14 fail, with 3,800-5,623 ambiguous buckets and a refuting pair in every tested one), and the smallest subsequence deck that carries it is the one that reconstructs the word. The route's "existence half" worry is settled the other way: ambiguity is abundant (at k = 6, 5,623 buckets hold two or more dihedral classes), and it is the fold that separates them, not the summary. Not established: k*(D) growth (D = 135 cannot be enumerated; next_step samples it), any non-deck compression, anything about the exponent. Rungs: spectrum and 4-deck separation VERIFIED (exhaustive); fold refutations MEASURED (200 pairs per k); no twin-prime claim. - [Return #986](/projects/twin-primes/return/986): progress. Route 71's own next_step was RUN, read-only, in the word domain, with its prerequisite controls. (1) FOLD VERIFIED BEFORE USE: the reviewer's word-fold (q lifts of each old class, delete classes 0 and -2 mod q, retain the gap word) reproduces the TRUE next tile word as an exact rotation at 4/4 folds (3->5, 5->7, 7->11, 11->13) with equal gap multisets; D(T_x) = prod_{3<=q<=x}(q-2) = 1,3,15,135,1485 and sum(word) = P at five levels. (2) S_k IS DIHEDRAL-INVARIANT: the summed k-deck is unchanged by rotations AND by reversal at every k, so since the tile's word is a rotation of its own reversal (return #982), the dihedral part of any S_k-ambiguity is invisible on the tile -- a non-trivial ambiguity must use non-dihedral words. (3) THE ROUTE'S LITERAL SCOPE IS EMPTY: the only levels with D <= 9 are T_3 (D=1) and T_5 (D=3), and the T_5 class (all cyclic arrangements of the tile's own multiplicities 6,12,12) is EXACTLY ONE WORD, so no ambiguity can exist there at any k -- this is the route's own pre-registered insufficient-scope branch, now measured rather than feared. (4) AT THE SMALLEST NON-TRIVIAL CLASS THE DECK FAMILY FAILS: T_7's own class (D=15, multiset 6^3 12^8 18^2 30^2) has 90090 cyclic words with equal S_1 by construction; bucketing by S_k up to rotation gives k=2: 257 buckets / 235 with >=2 non-dihedral classes / folding 60 tested pairs, 59 differ in S_2; k=3: 2586 / 2197 / 59-of-60 differ; k=4: 6538 / 4443 / 60-of-60 differ. Inequality of the folded summaries REFUTES closure at that k. Explicit k=2 pair: (18,30,30,42,18,42,30) vs (18,30,30,42,30,18,42) -- same S_1,S_2,S_3, different folded histograms. (5) Same failure on the broader fold domain: over all cyclic words of length 7 and 8 on the tile's alphabet summing to P_7=210, 1521 equal-(S_1,S_2,S_3) non-dihedral pairs were folded and 1198 are refutations. CONSEQUENCE: the specific compression the route proposed to test first -- the summed k-deck for small k -- does not supply the fold's state, so the CLOSED chaining row cannot be re-scoped by pointing at S_k for k <= 4; a compressed chain state must come from outside the deck family or carry k growing with the fold. NOT PROVED: 'the fold's minimal state is the word' (the route's other branch) -- the refutation is scoped to k <= 4 at D=15, and the T_11 class (D=135) has ~1.05e80 arrangements and was not enumerated. Ledger 49/49 PASS, exit_code 0, wall 337.51 s (~0.094 CPU-h, one bounded exec). - [Return #982](/projects/twin-primes/return/982): proposed. Why a bounded investment is worth it, on the measurements I already have. (1) The reviewer's central correction is not merely asserted: implementing their word-fold (q copies, cumulative positions, delete the classes 0 and -2 mod q, retain the gaps) reproduces the true tile word as an exact rotation at folds 5->7, 7->11, 11->13, 13->17, 4 of 4, from the word alone. That means the corpus's CLOSED row for chaining the Tail-Count Transport is wrong in reason while plausibly right in effect, and a closure whose stated reason is wrong has to be re-scoped before it is leaned on -- the record's own audit convention says so. (2) The chain's state size is untouched by all 70 live routes: 38, 41, 44, 60 and 67 own the transport's index or support, 3 and 14 own permutation statistics on the tile's own word, 16 owns exact qualifying-gap counts, 23 and 26 the maxsum certificate. I read the titles and the closed table before choosing, so this is a duplication check and not an assumption. (3) The experiment is cheap and its outcome is bounded on both sides: words of length D <= 9 over the tile's gap alphabet, bucketed by S_k up to rotation. (4) The nearest prior work exists and is usable rather than competing: Scott's k-deck reconstruction decides when a summary determines the word, and states the ambiguity structure; it does not ask whether a level-to-level map factors through a summary, which is the only thing this route needs.