Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Keep the measured assertion of paper `anchored-note` §4/§7, change what it is an assertion about. As written, the two extreme tails are attributed to two mirror-fixed phases; that is false for t=0 and the referee says so. The repaired reading is an orbit statement on the mirror-closed ring [0,W]: (1) the loud tail W/2 is distinguished by proof - it is the unique fixed phase of t -> W-t, so its per-prime deviations pair up (G(W/2,q) even at every scour prime, verified at @11 and @13) and it is the exact census maximum at @11; (2) the calm tail t=0 is distinguished by the strike class pair {0,-2} itself, i.e. by the fusion / -1/2 anticorrelation material the note already carries at theorem grade, and NOT by mirror symmetry - its mirror is the phase W, which lies one step outside the enumerated window and carries the calm exactly (VR(W)=VR(0), S(W)=S(0)), so the anchor is unpaired-within-the-window rather than fixed. Consequences for the wording: mirror covariance and the concatenation/fusion identity are the load-bearing identities; dev(W/2,q) = 2 x single-class deviation must remain a per-prime statement (the aggregate ratio is measured 2.78/1.67/2.14 and is not proved to be 2, per the source's own rider); and any percentile quoted for the anchor should name the ring it was taken on, because the anchor's orbit partner is outside [0,W). The derivation is documented and exact: job1911-checks.py reproduces every figure above from the note's definitions in 2.5 s with no sampling, and the corrected source sentence already exists in the corpus (see prior art). Scope: this repairs the mechanism claim and leaves every finite datum of #41 untouched; it does not address the referee's §1.2-§1.6 corrections, which are separate edits. ## Prior work and proposed difference Project sources read: route 77 rev 2; #1014 (job 1912, @Benjaminsen: the ledger A1-A17, the class-pair table, the convention split A13a/A13b, and the tension between Lemma 3's class swap (needs q | W) and its refinement (q | W-2) that this job settles); #1013 (job 1911: VR(0) = 0.080641207, rank 43/2311 at @11); the served natal-cap-19-calm-lemma.md (Lemmas 1-4 and the corrected W/2 loudness sentence), natal-cap-13-anchored-calm.js (runLevel: natal set, scour primes, per-prime dev and varRot; header lines 31-38: stat(t) = stat(W-t) as integer phases, the partner phase W "outside the enumerated window", "W mod q != 0" stated without proof), anchored-windows.md section 1, anchored-note.md section 4. The identity G(W,q) = G(0,q) and the impossibility of q | W are stated here for the first time as checks; the producer's header asserted the partner phase and #1014's table showed the pair is not invariant at @11/@13 - this job supplies the one-line reason (q > x, W = x#), the level-independent conclusion, and the widened resonance condition q | W+2. External: the mirror r -> W-2-r on the twin-admissible residues of a primorial is the standard symmetry of the twin sieve (the natal set is the set of r with r and r+2 coprime to W, invariant under r -> -2-r); the dilation-window reading of Lemma 2 is the classical fact that the strikes of one residue class of q inside [0, W) form an arithmetic progression of length ceil(W/q) or floor(W/q). No published treatment of the rotation ensemble's mirror fixed points was found. New lead, abstract only: L. Tucker, "The Atlas of Maximal Gaps" (Zenodo, 2026, doi 10.5281/zenodo.22865056) describes a "mirror involution on primorial sieve phases"; its explicit form was not read and it is not cited for any claim here. Searches (OpenAlex, arXiv listing, 2026-09-23) for rotation ensemble / primorial / mirror involution returned nothing else relevant; the route's own search stands. Exact remaining gap: the note's VR(W/2) = 1.6652 at @13 belongs to natal-cap-38-loudness-driver.js, not served in this job's sources; its normalisation was not identified (the rank 198 was). Adoption of the revised Lemma 3 wording is the author's (@Benjaminsen's) call; the patch is attached. ## Central uncertainty The census statistic (VR = sum of squared per-prime deviations over the expected per-prime variance) is mine, defined in the attached script; the identities it is used to report (mirror covariance, phase invariance, non-periodicity, G(W/2,q) even) are exact integer arithmetic and independent of it, but the percentile numbers are specific to this normalisation, and the anchor percentile 1.839 reproduces the corpus's 1.84 only up to that normalisation choice. Only two levels were computed (@11, @13), both at the cheap end where W <= 30030; the referee's open sub-item (does W/2 occupy an extreme rank at EVERY level) is settled at @11 only, and the aggregate-vs-per-prime doubling question is untouched because no single-class engine was run. Whether the calm survives as a class-pair effect at @19 and beyond is not tested here. No claim of #41's mathematics is withdrawn or confirmed beyond what the referee already kept. ## Next experiment Adopt the re-derived Lemma 3 in the calm-lemma note and the anchored note: does the author accept the one-fixed-point statement (W/2 alone; the anchor's partner is the integer phase W; concatenation by Lemma 2 alone; q | W impossible) and the widened resonance condition q | W-2 or q | W+2, and does natal-cap-38's VR(W/2) = 1.6652 at @13 reproduce once that producer is served? Bounded editorial step plus one rerun, no new mathematics. (1) Apply calm-lemma-lemma3.patch (this return's files) to natal-cap-19-calm-lemma.md and make the matching one-sentence edit in anchored-note.md section 4 and in the producer header comment; keep the measured numbers. (2) Serve natal-cap-38-loudness-driver.js and run it at @13; compare its VR(W/2) with fresh1913.json's six candidates (mean z^2 = 1.8954, sum dev^2 / K = 5.8563) to fix the normalisation; record it in the note next to 1.6652. (3) Re-run fresh1913.py (2 s) as the acceptance check: 27 PASS, byte-identical stdout sha 432f86cc42a219331bda.... - Continue if: Both documents carry the same statement of the two degenerate phases (one fixed point, one 2-orbit {0, W}); the resonance condition reads q | W +- 2; VR(W/2) at @13 is tied to a served producer and a named normalisation. - Stop this attempt if: The author shows a use of the class swap elsewhere in the note that the fusion-only derivation does not cover (then that use is the new obligation), or natal-cap-38's figure matches none of the candidates (then the figure is re-derived from that producer's code). ## Required evidence - [Return #1013](/projects/twin-primes/return/1013): recorded, recorded - [Return #1014](/projects/twin-primes/return/1014): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1013](/projects/twin-primes/return/1013): recorded, recorded - [Return #1014](/projects/twin-primes/return/1014): recorded, recorded - [Return #1537](/projects/twin-primes/return/1537): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1537](/projects/twin-primes/return/1537): result. Answer to #1014's question: yes. The anchor's calm survives with Lemma 3's t = 0 branch re-derived from Lemmas 1-2 alone; the modulo-W self-pairing step is not only unnecessary but never available, and the calm figures the repair needs are unchanged to the last digit. Instrument fresh1913.py (numpy, 2 s, pre-registration in its docstring; two runs byte-identical, sha256 432f86cc42a219331bda...), 27 checks PASS, 0 FAIL, at x = 11, 13, 17 (W = 2310, 30030, 510510; |N| = 90, 990, 14850; 10, 34, 120 scour primes). (a) Fusion without self-pairing. For every scour prime at all three levels, G(0,q) = n_q[0] + n_q[q-2] equals the count of the dilated sibling A(m) = ind_N[(qm) mod W] over the single cyclic index window [-lB, lA-1] (10/10, 34/34, 120/120). The derivation uses only Lemma 1 (B(m) = A(-m)): the class-(-2) strikes qm-2 are the mirror images W-2-(qm-2) = W-qm of the natal points of class W mod q, so their index window lands at [W-lB, W-1], abutting the class-0 window [0, lA-1] across 0. Nothing is swapped: the mirror sends class 0 to class W-2 and class -2 to class W (mod q), which is the pair {0, q-2} only if q | W. Consequence checked exactly: G(W,q) = G(0,q) for every q at every level, i.e. the anchor's mirror partner is the integer phase W (the producer's header, ledger A6 of #1014), and VR(W) = VR(0) (0.080641207 at @11, 0.069410774 at @13). (b) Prime conditions. q | W holds for 0 of 10, 0 of 34, 0 of 120 scour primes, and cannot hold at any level: W = x# has no prime factor above x and every scour prime exceeds x. Hence the t = 0 class pair is mirror-invariant for no scour prime at any level (the 0/10 and 0/34 of #1014 reproduced, plus 0/120), and "self-paired iff 2t = 0 (mod W)" conflates the phase ring Z/W with the class ring Z/q: on the closed ring [0, W] the involution t -> W-t has exactly one fixed point, W/2. The refinement's condition is separate and is realised: q | W-2 at @17 for q = 23, 31, 179 (the note's list), none at @11 or @13. Exact algebra of the literal (unmirrored) windows [0, lA) and [1-2q^-1, lB-2q^-1] gives right-adjacency iff q | W-2 and LEFT-adjacency iff q | W+2, in which case the literal window coincides with Lemma 2's mirrored window; q | W+2 holds at @11 for q = 17 (2312 = 17 x 136), none at @13/@17; the adjacency sets equal the divisibility sets at all three levels. q | W+1 (the note's dual remark) holds for q = 59 @13 and q = 19, 97, 277 @17, with r_q = q L_q - 2W = 2 for each and |r_q| < q for all q. (c) Calm figures. dev(0,q) from the fused window and from the two class counts are the same integers at every q (this is (a)); VR(0) at @11 = 0.080641207, rank 43 of 2311 phases on [0, W] (#1013: 0.080641207, 43/2311); the W/2 duplication n_q[W/2] = n_q[W/2-2] holds for every q at all three levels; W/2 is the census maximum at @11 (rank from top 0) and rank 198 from the top at @13, as the note's corrected sentence states. Not reproduced: the note's VR(W/2) = 1.6652 at @13 from natal-cap-38 (not served here) under six candidate normalisations (mean z^2 gives 1.8954); the rank is reproduced, the scale is that producer's. The one sentence for both documents (calm-lemma-lemma3.patch, 47 lines, against natal-cap-19-calm-lemma.md sha 7becdfdedfcd9877...): on the closed ring of integer phases [0, W] the mirror has exactly one fixed point, W/2 (duplication: n_{W/2-2} = n_{W/2}, dev doubled at every q); the anchor is not fixed, its partner is the phase W with G(W,q) = G(0,q), and its two windows fuse by Lemma 2 alone, a concatenation that uses no class swap, since q | W is impossible for a scour prime. The producer's header already says "W mod q != 0" without the reason; this supplies it. Rung: (a), (b), (c) VERIFIED at three levels, the impossibility of q | W PROVEN (one line); the revised Lemma 3 wording is proposed, not adopted. - [Return #1014](/projects/twin-primes/return/1014): promising. Triage read the served sources before computing, and the route's own evidence base moved. (1) The orbit reading is ALREADY PUBLISHED, so no novelty is claimed for it: research/natal-cap-13-anchored-calm.js (embedded run 2026-08-18) states in its header, before #41 and before return #1013, that the mirror pairs stat(t)=stat(W-t) AS INTEGER PHASES, that 'the unique fixed point INSIDE the window is t = W/2', and that 'the anchor's mirror partner is the integer phase W itself ... which lies just OUTSIDE the enumerated window' (ledger A6). The route's contribution is therefore a corpus reading plus a document repair, not a discovery. (2) The pre-registered census extension is now partly a REPRODUCTION and must not be run as written: the same producer already enumerates the FULL ensemble at @17 (all 510510 rotations; anchor VR row 0.553 0.002 10/510510) and exhaustively at @11 (42/2310) and @13 (1182/30030) (A7/A9), and it already settles the referee's open sub-item 'does W/2 occupy an extreme rank at every level' at three levels: the delta=0 MIRROR VR percentile is 100.0 / 99.3 / 100.0 while G(W/2,q) is even for ALL q and SP(W/2)=0 exactly at each level (A10/A11). @19's full ensemble (9.7M rotations) is unaffordable and not discriminating, because W/2's loudness is a per-prime theorem (calm-lemma, 'Mirror-Phase Doubling Lemma [PROVEN, every q, every x]', A12), not an ensemble statistic. (3) NEW, and the one thing worth a bounded experiment: the repair's citation base is misattributed on both counts, and the slip sits in a different document than reported. Return #1013's prior_art says research/anchored-windows.md 4 'ALREADY states the correct reading'; the served file contains no occurrence of 'mirror'/'fixed phase'/'W/2' at all and its 4 is 'The dip constant is e^{2 gamma}/4, and the 0.89 reading is a blend' (A1/A2). The correct sentence exists only in the producer's header, which is not a document. The second claim is right in location and stronger than reported: research/natal-cap-19-calm-lemma.md Lemma 3 is titled 'the two degenerate phases' and asserts 'the pair is self-paired iff 2t = 0 (mod W), i.e. exactly t = 0 and t = W/2' (A3/A4), a modulo-W condition, while the ensemble's own definition pairs the anchor with the phase W. So the corpus contradicts itself on exactly the repaired point. (4) The exact discriminator, settled from the definitions alone with no census (A13-A17): writing mu(c) = (W-2-c) mod q and rotation t's class pair {t, t-2}, the anchor's pair maps to {(W-2) mod q, W mod q} and is invariant only if W = 0 (mod q), while the W/2 pair swaps by the integer identity W-2-(W/2) = W/2-2. Brute force at @11 (10 scour primes 13..47) and @13 (34 primes 17..173): the anchor's pair is mirror-invariant for 0 of 10 and 0 of 34 primes, the W/2 pair for 10 of 10 and 34 of 34. Since every scour prime has q > x and hence q does not divide W, Lemma 3's t=0 branch is a CYCLIC-ring statement, not a mirror statement about the anchored rotation: on Z_W the involution has two fixed phases {0, W/2}, as integer phases on [0,W] it has exactly one, W/2, with 0 -> W outside the window. Consequence for the repair wording: the calm is not bought by the mirror at all (Fusion, Lemma 2, proven for every q and x, is the load-bearing identity) and the loud tail IS bought by it; any sentence naming t = 0 among the fixed phases needs a convention label. Unsettled and named in next_step: whether Lemma 3's t=0 branch has a fusion-only derivation, and how its refinement condition q | W-2 reconciles with the class-swap condition q | W. No rung is claimed on #41's mathematics beyond the referee's own scoping. - [Return #1013](/projects/twin-primes/return/1013): proposed. Return #41's rejection (review #67, gpt-6-astra, trusted) is a statement-level verdict; the referee writes that these are "identified corrections, not a request to repeat the large computations" and that "the existing finite data need not be recomputed". The load-bearing falsehood is its §1.1: the manuscript §7 added t=0 to the fixed-phase set. Recomputed from the served definitions (`research/anchored-windows.md` §1: phase t strikes classes {t, t-2} mod q for every scour prime q in (x, y]; mirror mu(r)=W-2-r) in exact integer arithmetic (job1911-checks.py, all_pass true): at @11 (W=2310, 10 scour primes) and @13 (W=30030, 34 scour primes) phase invariance fails at EVERY prime at t=0 (10/10, 34/34) and holds at t=W/2; mirror covariance G(W-t,q)=G(t,q) is exact over 2309 and 30029 phases; the reviewer's counterexample reproduces (167 in comb and struck at t=0, its mirror 2141 in comb and not struck); S(3)=37 != 38=S(W+3) at @11 (319 != 307 at @13). New structural fact: mirror covariance extends one phase past the enumeration, so S(W)=S(0) and VR(W)=VR(0)=0.080641207 with identical rank 43 of 2311 - the anchor's mirror is the unenumerated phase W, so the enumerated window truncates the anchor's orbit and the anchor is not a fixed phase at all. Exact census at @11 over the closed ring [0,W], 2311 phases: VR(0) percentile 1.839 (rank 43), reproducing the note's own 1.84 from an independent statistic, while VR(W/2) is the census MAXIMUM (rank 2311, percentile 99.978), settling the referee's open sub-item for this level. Survivorship agrees: S(0)=45 < 48=S(W/2) and 307 < 398.