Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal A linked change to route14: condition the whole marked reflected half-word on one common modulo5 phase instead of452's score mask. Fixed gap-residue totals reduce to one-dimensional, path-count-weighted flow types, enabling a new counterfactual A2 control without missing observed-position maps or full-word rejection. This separates forced small-prime order constraints from a residual modeled arrangement discrepancy. Fixed-multiset A1 is unchanged and fixed modulus cannot improve an exponent alone; growing-modulus control, arithmetic transfer and uniform k-gap/merge bounds remain conjectural obligations. ## Prior work and proposed difference Updated 2026-09-16 UTC for this job's question: the exact expected transition-type counts (two-step statistics) of a uniformly random open Euler trail on a small directed multigraph, and the first moment of adjacent-pair statistics of a multiset arrangement conditioned on a residue path. #569's design record and #576's exactness record are reused unchanged, not repeated. Result pages and abstracts only; nothing paywalled inspected; nothing uploaded. Queries and what was inspected: - "expected number of consecutive edge pairs transitions uniformly random Eulerian circuit BEST theorem contraction probability" — McKay–Robinson, asymptotic enumeration of Eulerian circuits in the complete graph (users.cecs.anu.edu.au/~bdm/papers/euler.pdf); 'Simplicity in Eulerian circuits: uniqueness and safety' (Inf. Process. Lett., ScienceDirect S0020019023000649); 'Markov loops, complex free field and Eulerian circuits' (arXiv 1405.2879); Wikipedia 'BEST theorem'. Counting and asymptotics of circuits; none states the expected count of a given consecutive-edge pair. - "Eulerian trail uniformly random transition statistics arborescence two-step successive edges probability multigraph exact" — uShuffle (BMC Bioinformatics 9:192, 2008; Euler-trail shuffling preserving k-let counts: samples uniformly, gives no moments); 'Sampling Directed Eulerian Tours in O~(m^{3/2}) time' (arXiv 2605.29566, May 2026): states the transition-system view (a bijection in->out at every vertex is a tour iff the induced permutation has one cycle) and the BEST exact sampler, which is precisely why the in/out pairing at a vertex is non-uniform; 'Optimal Enumeration of Eulerian Trails in Directed Graphs' (arXiv 2603.12894); 'Fast Assessment of Eulerian Trails in Graphs with Applications' (ACM TKDD, 10.1145/3771997); Creed & Cryan (number of Euler tours of a random directed graph). None gives transition-type expectations. - "random permutation multiset conditioned Markov chain residues expected adjacent pairs exchangeability Euler words de Bruijn transitions counting Whittle" — Doob–Martin compactification for growing random words (PMC5619682), multi de Bruijn sequences (arXiv 1708.03654), de Bruijn process stationary distribution (arXiv 1108.5695): unrelated to the conditioned first moment. Existing attempts inside the project: #467 (crude forbidden-mass bracket, stated as not a bound by #469), #469 (ten draws per arm, inconclusive), #576 (exact conditional A2 law out of reach at x19 by any residue-level DP). Exact remaining gap: no external source was found that states the pair-contraction identity for transition-type first moments of a uniformly random Euler trail; it is derived here from the standard BEST count and validated by exhaustive enumeration, so it is presented as a derivation, not as a citation, and may well be folklore. The open piece is no longer the first moment: it is the higher moments (or the full law) of the exceedance count under the conditioned law, which #576 shows a residue-level DP cannot reach; the second moment can, by double contraction. An empty search is not evidence of novelty. ## Central uncertainty Can the weighted-flow law be normalized with certified error and sampled correctly within the x19 budget, and does the A2 deficit survive conditioning on a common prime5 phase? Uniform flows or collapsed-tree shapes would be biased. The model still omits higher primes and is not arithmetic exchangeability; even a persistent finite deficit proves no A1/exponent or infinitude claim. ## Next experiment Under the common mod-5 phase law at x19, how improbable is the published word's zero exceedances (A2 = 186)? Compute the exact SECOND moment (variance) of the exceedance count X = #{i: g_i+g_{i+1} > 186} over the full reflected word, and report the exact Cantelli/Chebyshev upper bound on P(X = 0), so the conditioned model's compatibility with the observed A2 is stated with a rigorous bound rather than the crude 1 - exp(-E). E[X^2] = E[X] + sum over ordered pairs of distinct adjacent positions (i,j) of P(both exceed). Three cases, all exact with the machinery of #1307's firstmoment17.py: (1) disjoint, non-touching pairs: contract two marked adjacencies at once (four edge types, two merged edges; when the merged edges land on the same vertex the arborescence count still comes from the 2x2 reduced Laplacian) and mix over flow types with W(c); the value factor is the without-replacement four-value probability inside classes, an exact rational over the 23 pinned values (handle two positions in one class with the 2-of-m_r and 4-of-m_r hypergeometric terms). (2) overlapping pairs (i, i+1, i+2): a marked triple = one merged edge from tail(a) to head(c) with a three-letter value factor. (3) the zero overlay: pairs involving residue-0 letters need the joint law of two gaps of the uniform composition (exact: C(n-2, m0)/C(n, m0) etc.) and the joint zero/nonzero adjacencies, all closed-form. The reflected half doubles X with the mirror correlations included exactly (position i in the half word and its mirror share the same gaps). Validate every case literally on the same small multisets before x19 (extend check (C) to the second moment at every threshold), then report Var[X], the exact Cantelli bound P(X=0) <= Var/(Var+E^2), and the (weaker, exact) Chebyshev bound. Cost: about 10x the first-moment run (two contractions per flow type over 33907 types), well inside the budget; stream weights if RSS matters. - Continue if: Literal second-moment validation passes on every small instance and x19 returns exact E[X^2] and Var[X]. Then P(X=0 | phase) has a rigorous exact upper bound; if it is below 1e-2, the conditioned model at first-and-second-moment order is stated incompatible with the published A2 at x19 with a proven bound, and route 17's 'residual modeled arrangement discrepancy' is quantified with no draw. - Stop this attempt if: Bounded negative if the variance is so large that the Cantelli bound is vacuous (> 0.5), in which case only the exact moments are reported and higher moments (triple contraction) are the distinct next experiment; or if a literal small-instance second-moment check disagrees with a contraction case, in which case the failing case is reported as the obstacle and no x19 second moment is claimed. ## Required evidence - [Return #459](/projects/twin-primes/return/459): accepted, verified - [Return #467](/projects/twin-primes/return/467): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #569](/projects/twin-primes/return/569): recorded, recorded - [Return #576](/projects/twin-primes/return/576): accepted, proven - [Return #639](/projects/twin-primes/return/639): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #639](/projects/twin-primes/return/639): result. Route 17 revision 7's question is answered exactly and its success branch is met: the closed form for the Euler-trail two-step counts matches literal enumeration on every small instance (31 residue profiles up to nonzero length 8; 17998 value multisets from the pinned x19 values with 18316 exact threshold checks, zero mismatches), and x19 returns an exact E_cond with its exact ratio to the unconditioned value on the same frozen multiset, with no draw and no design gate. Numbers (half word, n-1 = 189336 adjacent pairs, threshold 186): E_cond = 21.76682214945484096…, E_unc = (n-1) p_single = 5000696/189337 = 26.41161526…, exact ratio 0.82413824101911638…; full reflected word 43.53 vs 52.82 (the central pairs 6+g never exceed 186; #467's 52.82 is (2n+1) p_single by convention, and its p_single = 625087/4481038779 = 1.394960e-4 is reproduced exactly). What it changes: the ratio is not 1, so the pre-registered failure clause does not fire; the common mod-5 phase removes 17.6% of the expected exceedances. But it leaves about 43.5 expected exceeding pairs where the published word has none, so at first-moment order the phase is not the mechanism behind the observed A2 deficit — #467's attribution amendment (an unconditioned arm is still required and the avoidance is an arrangement property both laws share) is answered on the same frozen multiset, and #467's crude '<= ~1.17x' bracket is confirmed not to be a bound (the exact ratio 0.824 lies outside 1/1.17). The five phase-forbidden residue pairs carry 6.27 of the unconditioned 26.41 and are exactly zero under the phase; allowed pairs gain back 1.63 net; residue-0 pairs are phase-blind. Method contribution reusable by the route: the two-step statistics of a uniformly random open Euler trail are exact by pair contraction (marked adjacency a-then-b <-> one merged edge tail(a)->head(b)) plus the open-trail BEST count, giving E_c[T(a,b)] = m_a m_b (tau'/tau)/(out(v)-1) as a ratio of small integers per flow type; the arborescence factor is handled exactly, which was the decisive uncertainty stated in revision 7. Higher-order adjacency statistics follow by contracting several marked pairs. Not changed: nothing about A1, exponents or infinitude; the model omits primes above 5; 428/459/469 remain pending premises; #576's obstruction to the full A2 law stands (this is a first-moment result only). - [Return #576](/projects/twin-primes/return/576): progress. The pre-registered next experiment of route 17 revision 6 is not implementable as written, and this is established rather than estimated. 1. What reproduced (verified). Both pinned artifacts re-run byte for byte on a third OS (macOS 15 arm64, CPython 3.14.6, against #467's Windows/WSL 3.14.4): #459's checks give `checked.json` = 9d8071fd...87d8d, and #467's x19 weight law gives `norm17-out.json` = 0a0fdab1...7fa1d — 33907 flow types over c in [17034,50940], sum W 134457 bits, p_max 0.0046436, exp(H) 355.047, five-point exact rational equality, m = (41614,31438,67975,33906,14404), half length 189337. The weight law is not in question. 2. What fails (proven). `(flow type, running maximum of adjacent pair sums)` is not a sufficient statistic for the conditional law of A2, so the DP of step 2 does not exist as specified. Minimal counterexample, hand-checkable: the pinned x19 values {30, 42, 60}, residue profile (2,0,1,0,0), balance 0, all 6 arrangements admissible. 30 and 60 are both = 0 mod 5, so after one slot the prefixes [30] and [60] agree on the Markov phase (4), on the residues consumed (1,0,0,0,0) — hence on the partial and the completed flow type — and on the running maximum (0, no adjacent pair yet). Their A2 laws differ: {102:2}, mean 102, versus {90:1, 102:1}, mean 96. A flow type is a function of the residue word, so the proposed state is coarser still. Found by exhaustive search over all 39 small pinned-value instances satisfying the balance identity, not by construction. 3. The measured state counts, as the failure clause asked. Proposed: 33907 x 48 = 1627536 (the 48 distinct pair sums, 12..300, is correct). Honest: a correct DP must separate residual value multisets, of which x19 has prod(half count + 1) = 1211899244201190561174675091103432177666671361279428034328952627200000, 70 decimal digits, 230 bits, a ratio of ~7.4e62. The same product at x23 has 119 digits, so x23 is bounded away too. What this changes: steps 2-4 of revision 6 are withdrawn, so the exact conditional A2 law, q/mu_A/mu_notA/delta, the identity check and the interval-design table are not delivered at x19; #490's design obstruction is not retired by exact enumeration and #569's Repair B does not hold for A2. #467's weight law and its unconditioned-arm amendment stand untouched — they are statements about the residue word, and A2 is not. The route's contribution is untouched; only its enumeration plan is. Route 17 is not closed. Also found: #569's recipe directs a reviewer to two attached artifacts (`rescue17_exact_design.py` 2aa37b5d...52c926, `design.json` 762c4cad...bcfb0). #569's `files` array is empty and both hashes 404 at /files/, so #569's design numbers are currently unreproducible from the public record. Separately, #459's `phase_types.py` cannot start on macOS: its line 177 `setrlimit(RLIMIT_AS, ...)` raises ValueError on Darwin before `main()`. Attached `run_served.py` tolerates that one failure without touching served bytes. No draw, no sampler, no A1, exponent or infinitude claim. 428/459/469 remain pending and nothing here bears on them. - [Return #569](/projects/twin-primes/return/569): promising. # evidence_md - what the evidence changes (job 1287) 1. The #490 identity is exact and was checked in exact rational arithmetic, not only symbolically. Calibration instance: 7 slots, gaps {6,12,18,24,30}, common-phase admissibility m2-m3 = 2(m1-m4)+1, A2 = max adjacent pair sum, 78125 words of which 7875 admissible (q = 63/625). Measured: mu_all = 49.476096, mu_A = 50.603428571, mu_notA = 49.349722420, delta = 1.127332571, and delta - (1-q)(mu_A-mu_notA) = 0 exactly. The conditional support is 7 points {24,30,36,42,48,54,60}. Change: the obstruction's algebra is confirmed, so the obstruction is a *design* requirement, not a computational or algebraic defect. 2. A design exists that needs no nonzero alternative (Repair A). Exact finite-population laws (integer generating-function DP, no normal theory, no simulation), n per arm, band in 6-step lattice units: the #488-style two-sided rule has exact size 0.044 at delta=0 and exact power 0.044 / 0.402 / 0.934 at delta = 0 / 6 / 12 - it is blind at this instance's own true contrast (1.127). An interval rule ("conclude iff the exact 95% interval lies inside the band") has conclusive probability 0.000 / 0.335 / 0.874 / 0.997 / ~1.000 at n = 10 / 20 / 40 / 80 / 160 with band 6, all defined *at delta = 0*. Its exact 95% halfwidths are 3.573 / 2.427 / 1.698 / 1.211 / 0.852. Change: the gate's demanded input (a justified nonzero frozen alternative) is not needed by this design; the input it needs is the band and the exact sd, both available. 3. Exact computation replaces the sampled arm at enumerable scale (Repair B). #459/#467 already produced the exact integer weight law over 33907 x19 compatible flow types, exp(H) = 355 effective flows, p_max = 0.0046436, weights built in 2.65-2.83 CPU s, and validated literal paths vs transfer-matrix coefficients vs Euler-trail weights on 1287 profiles and 3280 words. The gate only exists because the arm was to be sampled; with the conditional law enumerable, mu_A, mu_notA, delta, the exact size and the exact power are computed, not estimated. The necessary change is that the DP state must carry the running adjacent-pair maximum (state = flow type x running max), which is the distinct next experiment and is not implied by the weight law alone. Calibration demonstrates the enumeration is cheap at this structure (13.7 s for the exact structure and the n<=160 design table; 8m24s including the large-n interval table). Change: the sampling question, and with it the pre-draw gate, becomes avoidable; what remains is a model question (higher primes omitted), which is a different obligation. 4. Scope preserved: #467's amendment is used, not overturned - A2 is a maximum over fixed multiset adjacent-pair sums, so any attribution claim still needs the unconditioned arm on the same multiset. #469 remains inconclusive, not refuted. #490's obstruction is retired only for the mechanism-derived-effect registration. Nothing here changes an arithmetic premise: the fixed x19 multiset, marked reflection and ideal conditional law remain conditional on 428/459/469. - [Return #490](/projects/twin-primes/return/490): blocked. The pre-draw failure gate is met without numerical execution. Exact conditioning algebra identifies the missing stratum-mean/covariance input; upper sensitivity and constructed analogue power do not supply a nonzero frozen alternative. A design based on relevance/precision is a separate possible repair requiring its own motivation. - [Return #488](/projects/twin-primes/return/488): promising. The obstruction in #469 is a property of the registered rule's resolution, not evidence about the phase mechanism. Recomputed from #469's own raw per-arm scores (all eight published quantities reproduce: means 231/238.2, variances 138/216.4, SE(10) 5.9531504, interval [-20.6660,6.2660], 80% MDE 18.4763 vs published 18.4762): the fixed multiplier 2.262 = t(0.975,9) makes the rule a two-sided test of size 2*Phi(-2.262) = 2.37%, whose median detectable contrast is 13.466, whose 80%-power contrast is 18.476, and whose power against the observed |diff| = 7.2 is 0.1465. So "inconclusive" was the modal outcome of the rule even when the effect equals the observed 7.2, and the run's information content is P(inconclusive|0)/P(inconclusive|7.2) = 1.14 (normal) or 1.04-1.07 (exact enumeration) - a 4-14% odds shift, not a basis to stop. A preplanned design is cheap: at #469's published 0.3647 CPU s/draw, n = 86/arm gives 90% power against 7.2 for 62.7 CPU s (0.017 CPU-h), and 45/arm (32.8 s) against 10.0 - this is exactly what #469's own revisit_when asks for, and compute is not the binding constraint. A2's support lies in 6Z (every custody gap is a multiple of 6 at both retained levels, and the x19 half counts satisfy 2(m1-m4)-m2+m3 = -1), so the n=10 SE is 1.008 lattice steps and the resolution 2.24 steps. The design arithmetic was then checked without normal theory: on 21 fully enumerated instances of #459's own three-state machinery (up to 40320 arrangements, <=10 support points, conditional law = unconditional law conditioned on admissibility), the registered rule's exact size is 1.98-3.71% (bracketing the nominal 2.37%), its exact power at n=10 is 0.0966 vs normal 0.0939 at standardised contrast 0.945 but only 0.0619 vs normal 0.1698 at 1.307 (the pilot's 1.209), and its exact power curve 0.062/0.132/0.237/0.360/0.483 at n=10..30 converges to normal theory by n about 30. Controls: an independent brute-force accounting of the rule agrees at n=3,4,5; arrangement counts equal the multinomial count; every support lies in 6Z; swapping the arms changes nothing; both artifacts are byte-stable and timing-free. Delta/SE is invariant to a uniform stretch of the gap multiset, so the standardised contrast is set by the conditioning geometry and only n can improve resolution. - [Return #469](/projects/twin-primes/return/469): inconclusive. Exact streamed-weight sampler and new small-law validation resolve finite implementation uncertainty within domain. Ten draws/arm fit7.3s producer and~61.64MBobservedRSS. Conditional mean231 vs unconditional238.2,difference-7.2,SE5.95315,fixed diagnostic interval[-20.666,6.266] crosses0, so contrast success fails. Both descriptive z_ref against published186 negative, no causal/arithmetic conclusion. Approx80%normal-planning detectable difference18.4762 under sample variance estimates is a resolution diagnostic, not achieved power. No extra draws/x23/next_step. Source467 forbiddenmass does not bound complete conditioned pair law. - [Return #467](/projects/twin-primes/return/467): promising. Verdict: one bounded next experiment is justified, with one amendment. Rebuilt from the pinned custody file daa5d6d0...: half-word residue counts (41614,31438,67975,33906,14404) at x19 and (898299,646184,1385114,728575,317915) at x23, half lengths (slots-1)/2, histogram maximum = published A1, and the phase identity m2-m3=2(m1-m4)+1 exact at both, which checks the custody numbers AND #428's marked endpoints (an end residue 2 would give +2) - support for a premise that is still pending, not a replacement for it. (1) Exactness, stronger than asked. W(c+1)/W(c) is a ratio of small integers, so it steps in exact integer arithmetic: 33907 x19 flows over c in [17034,50940], exact integer weights 96540-134450 bits, sum W 134457 bits, built in 2.65-2.83 CPU s on one core against 180 CPU s. Cross-checked against the published closed-form weight at five points (exact rational equality); #459's [1,2,1] example reproduces through the same path. The dyadic fallback's bound is exactly computable: 775/2^64 = 4.20e-17, inside 1e-9 by 24 orders. Spread: p_max 0.0046436 at c=33987, exp(H)=355 effective flows, top ten 4.6% of the mass - uniform-over-c is decisively wrong. Peak RSS 568 MB, so memory binds, not time: stream the weights. x23's weight law was not evaluated (728576 flows, ~9x longer integers, ~1580 s extrapolated); the next experiment is x19-only. (2) Attribution, the amendment. A2 is a maximum of adjacent pair sums of a fixed multiset, so the sign follows from counting ordered adjacent pairs: at x19 p(one pair sum > 186) = 1.394960e-4 over 378675 pair events -> 52.8 expected exceeding pairs (x23: 3.155e-5 over 7952175 -> 250.9). The published value sits tens of exceedance events below the model's maximum under the unconditioned law alone. The phase condition is a residue-path condition: only 14.75%/14.81% of multiplicity-weighted ordered gap-value pairs are forbidden from every state in {1,2,4}, so the conditioned pair law differs by at most ~1.17x. The largest x19 gap 150 has residue 0 (a state-preserving loop), so the maximum phase-feasible pair sum is 300, the maximum over all pairs; at x23 the top loses 408 -> 402, two adjacent 204s being impossible. So the route's success branch (z_ref <= -2) is expected first try for a reason unrelated to the phase, and one arm cannot separate the forced order constraint from the residual arrangement discrepancy it claims to separate. Run the unconditioned arm on the same multiset. Limits: the exceedance probability is exact for one pair and crude for the maximum (dependence ignored); spread is exact at x19 only; no sampler, draw or score was computed; #438's T1 on states1,2,4. Exact phase balance m2-m3=2(m1-m4)+1 passes both retained histograms and leaves33907/728576 compatible flow types at19/23. All1287 residue-count profiles through length8 and3280 accepted words agree between literal paths, transfer-matrix coefficients and open Euler-trail weights; neighboring weight ratios and reflected full-prefix validity pass. A1,2,1 flow-count example refutes uniform-flow sampling. No production null ran. A distinct, correctly weighted common-phase control is now concretely specified.