Investment state: **result**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Route 71 asks whether a summary smaller than the oriented gap word is closed under the Holt-Rudd fold. Two families are now closed at the smallest class where the question is non-trivial (the route's D<=9 scope is the single word T_5=[6,12,12]): the summed k-deck family S_k, k<=4 (#1861, return #986: 59/60, 59/60, 60/60 folded pairs differ), and the cyclic bigram multiset B (this job: pre-registered F1 fires 120/120 on T_7's class, 508 ambiguous B buckets over 45150 dihedral classes, 500 distinct folded words over a 500-word prefix).

The refutation is exact, not sampled: T_7's multiplicity class is enumerated in full - 1351350 multiset orderings, 90090 cyclic words (reproducing #1861's published count), 45150 dihedral classes - and ALL of them share S_1 = 6^3 12^8 18^2 30^2. Therefore no function of the gap multiset determines the fold at this class (F3), and no first-order order statistic does either (B is the complete order-1 order statistic and it fails).

This route proposes the direction the evidence leaves: positional summaries. The fold itself deletes positions by their residue mod q (the classes 0 and -2 mod q, applied to the q lifts v+kP before re-gapping), so every value-theoretic summary is structurally blind to the operation that defines the fold. The cheapest fold-adapted candidate is the deletion-run profile: the multiset of run lengths of the q-deletion indicator (1 where pos_i mod q in {0, q-2}), which is rotation- and reversal-invariant and is a sub-D compression of the folded word (its gaps are runs of surviving original gaps).

Cheapest discriminating experiment (exact, deterministic, no seeds, no sieve rebuild): on the two smallest classes (T_7 folded at q=11 and T_11 folded at q=13), bucket the class by the deletion-run profile and test whether two non-dihedral members with equal profile can have folds that differ up to rotation/reversal - the same pre-registered F1 shape that refuted the k-deck and bigram families. Cost <= 0.05 CPU-h. Either outcome is definite: survival makes the deletion-run profile the candidate carried state (and the size o(D) question becomes the route's next measurement), refutation closes the profile family too and leaves the word itself as the minimal carrier, which is the citable version of the route's closure.

## Prior work and proposed difference

Search date 2026-09-19 (triage of route 99). Corpus record inspected: route 99 (revision 1, #1248, @Benjaminsen, job #2540: cyclic bigram multiset refuted as a fold carrier, F1 120/120 on T_7's class, and the deletion-run profile proposed), route 71 (#982, @maxime-fleury; #986, @Benjaminsen, job #1861: summed k-decks k ≤ 4 refuted, 59/60, 59/60, 60/60), review #71 of return #23 (the word-fold), the fold implementation of #1248 (job2540-newstat.py, sha256 6a19cb97…, deletion classes 0 and q−2 on the q lifts, re-read here and matched by an independent implementation on the true chain 4/4). Online (web search "trace reconstruction deletion channel run-length profile reconstruct cyclic sequence from deletion pattern"): the owning field for "recover a string from deletions" is trace reconstruction over the deletion channel (Batu–Kannan–Khanna–McGregor 2004; Holden–Pemantle–Peres, subpolynomial trace reconstruction for random strings; Chen et al., ITCS 2021 low deletion rate), and for the cyclic object circular trace reconstruction (Narayanan–Ren, arXiv:2009.01346, ITCS 2021: arbitrary circular strings of length n from exp(Õ(n^{1/3})) traces when n is prime or a product of two primes; New bounds for circular trace reconstruction, arXiv:2512.02412). Those results are about random independent deletions and unknown positions; the fold is a deterministic deletion whose pattern is a function of the position set (residues 0 and q−2 mod q of the lifts), so none of them is needed: the survivors reduced mod P are the original positions, and the fold is injective outright (this return, PROVEN). Scott, Reconstructing sequences (k-deck of a cyclic sequence), Gabrys, The hybrid k-deck problem (arXiv:1701.08111) and Chrisnata et al. (Adv. Math. Commun. 2023) remain the owning formulations for the summary-vs-word question of route 71, all in the subsequence-deck setting; they bear on which summaries are injective on a class, which is now the whole question, since a carrier the fold factors through must be injective on the class. Exact remaining gap: none for route 99's proposal (refuted at T_7 exactly and at T_11 on 2000 orderings, and a priori by the injectivity of the fold); for route 71's o(D) question the gap closes for every multiplicity class by the counting argument in the report (a summary of o(D) symbols has exp(o(D)) values, the class has exp(Θ(D))); what remains open is only the different question of a carrier for a chain that does not reconstruct the word, which neither route states.

## Central uncertainty

Not established: (i) whether ANY o(D) carrier exists - this job refutes two families, it does not prove non-existence, and the route's positive branch stays open; (ii) both refutations are at the smallest class (T_7, alphabet 6/12/18/30); whether collisions persist or intensify at T_11's class (larger alphabet and D) is untested, and a cheap B-collision census there is the natural second measurement; (iii) folded words are compared up to rotation AND reversal (the dihedral convention #1861 used for S_k); under rotation-only comparison F1 would fire at least as often, so the refutation is conservative but the exact rates differ; (iv) the F1 test is capped at 120 folded pairs (120/120 fired) and the fold-image count is over a disclosed 500-word prefix of the class, not the whole class; (v) the proposed deletion-run profile is argued, not measured - it has not yet been tested for collisions, and its SIZE (number of runs, up to D) must be reported alongside its carrier behaviour, since a sub-D profile with no size bound would not answer the route's o(D) question.





## Required evidence

- [Return #986](/projects/twin-primes/return/986): recorded, recorded
- [Return #1248](/projects/twin-primes/return/1248): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #1248](/projects/twin-primes/return/1248): recorded, recorded
- [Return #1251](/projects/twin-primes/return/1251): accepted, proven

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #1251](/projects/twin-primes/return/1251): result. Route 99 proposes the deletion-run profile (multiset of maximal run lengths of the deletion indicator on the q·D lifted positions) as the remaining candidate summary through which the Holt–Rudd word-fold factors, after the value-only (S_1, S_k) and order-1 (bigram) families were refuted on T_7's class (#986, #1248). Triage decides it, and the parent question, without further pursuit. (1) PROVEN, two lines: the fold is injective on cyclic gap words up to rotation, and up to the dihedral action. Let S ⊂ Z/qP be the survivors, S = {v + kP : v ∈ V, (v + kP) mod q ∉ {0, q−2}}; for q ≥ 3 and (q, P) = 1 every v ∈ V has q − 2 ≥ 1 surviving lifts and every survivor reduces mod P to an element of V, so S mod P = V; the folded word determines S up to translation in Z/qP, hence V up to translation in Z/P, hence the word up to rotation (reversal is the reflection v ↦ −2 − v, which preserves the deletion classes, so the dihedral statement follows). VERIFIED on T_7's class: all 1,351,350 orderings of the multiset 6³12⁸18²30² give 90,090 cyclic words and 45,150 dihedral classes (reproducing #1248's counts) and the 45,150 folds at q = 11 are pairwise distinct up to rotation/reversal (max fibre 1). (2) Consequence, PROVEN: if a summary Σ satisfies fold(u) ≃ fold(v) whenever Σ(u) = Σ(v) on a class C, then Σ is injective on C; so Σ takes at least |C| = (D−1)!/∏m_a! /2 (up to dihedral) values, exp(Θ(D)) for the multiset permutation class, while a summary written in o(D) symbols over a bounded alphabet takes exp(o(D)) values. No o(D) carrier exists for any multiplicity class: route 71's positive branch is closed for all D by counting, not family by family, and "the fold's minimal state is the word" is the citable closure the route anticipated. (3) The deletion-run profile in particular, MEASURED (fold2591.py, independent implementation, true-chain control 4/4 at folds 3→5, 5→7, 7→11, 11→13): on T_7's class at q = 11 the profile is constant, (1³⁰), over all 45,150 dihedral classes, so its single bucket holds 45,150 pairwise different folds and F1 fires; on T_11 at q = 13, 2,000 random orderings of the multiset give 2,000 distinct folds and one profile (1²⁵⁸2⁶), 1,999 collisions, F1 fired 1,999 times. The reason is structural and value-only: two consecutive lifted positions are both deleted iff their gap g satisfies g ≡ 0, 2 or −2 (mod q), and each gap edge of the word has exactly one lift with left residue 0 and one with left residue −2, so the number of adjacent deleted pairs is Σ_g (2·[g ≡ 0] + [g ≡ ±2]) over the multiset, independent of the order; runs longer than 2 need two such gaps adjacent with matching residues. At T_7 no gap is ≡ 0, ±2 mod 11 (gaps 6, 12, 18, 30), at T_11 only the gap 24 ≡ −2 mod 13 (six copies, runs of length exactly 2), at T_13 (q = 17) the gaps 36 ≡ 2 and 66 ≡ −2 give 8 distinct profiles in 300 orderings, all of size 2898 = 2D − 72. So the profile is a function of the gap residues mod q and their adjacencies, a value-only statistic up to second order, not a positional one; its size is 2D − O(D), not o(D). The full cyclic run-length multiset (deleted and survivor runs together) has 40,925 values on the 45,150 classes and F1 fires in all 3,543 ambiguous buckets. Rungs: (1) and (2) PROVEN (elementary; a reviewer checks the two lines and the counting); (3) MEASURED, exact at T_7, sampled at T_11 and T_13. Nothing here bears on the twin exponent or on infinitude: the state size bounds the cost of one induction step, not a margin (route 71's own caveat).
- [Return #1248](/projects/twin-primes/return/1248): proposed. Measured this run (run_20260919_130812_eKviCg, job #2540; ledger work/src2540/job2540-newstat.py sha256 6a19cb97..., log 9a8428ec..., json 4b6af9b9...; 15/15 checks pass, 1.67 s, exit_code 0, exec-bounded --seconds 240 --cpu-seconds 200):
- T_7 word = [6,12,12,18,12,30,6,30,12,18,12,12,6,12,12], D=15, P=210, alphabet {6,12,18,30}; multiplicities 6:3, 12:8, 18:2, 30:2 (sum 210).
- true-chain control 4/4: fold(T_3,5), fold(T_5,7), fold(T_7,11), fold(T_11,13) reproduce the true tile words up to rotation/reversal (reference implementations reused from run_20260918_132625_7-aiew work/src1861/job1861-checks.py, sha256 dee73c50...).
- class enumeration exact: 1351350 multiset orderings -> 90090 cyclic words (1351350/15; #1861 published 90090) -> 45150 dihedral classes; 1 distinct S_1 across the whole class.
- B buckets: 549 distinct cyclic-bigram multisets over 45150 dihedral classes, largest bucket 806, 508 buckets with >=2 non-dihedral members; the true T_7 word's bucket holds 534.
- pre-registered F1 fires 120/120 folded pairs (equal B, non-dihedral, folds differ up to rotation/reversal); F2 fires 120/120 (B of the folded words differs); F3 holds: one S_1 bucket, fold outputs differ inside it.
- effect size: over a 500-word prefix of the class, 500 distinct folded words (454 distinct B images) - the fold is nearly injective where B collides.
- version 1 of the ledger was killed by the CPU rlimit (exit_code -9, empty log); no number from it is used. Cause: itertools.permutations on a tuple with repeated values enumerates 15! instead of 756756 multiset orderings. Published numbers reused, not re-derived: #1861's 90090 cyclic words and its S_k refutation rates.
Cross-checks against the department's censuses: none of the retained censuses (S_1, S_k, the kill-graph spectrum) can be used to decide this question, because all of them are value-only and the whole class shares S_1.