Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Routes 1-8 attack the finite two-class cover problem one frozen support at a time (N51, N52, N66) and each names the same missing step: a uniform rule, uniform reach or arithmetic realizability. This route changes the object from 'is support D_N coverable' to 'how much of the capacity frontier does the LP relaxation reach', which is a quantity computable at every p in polynomial time and certified in exact rational arithmetic. If the ratio L_w/L_F stays bounded away from 1 along a prime ladder, then a rational slot-weight vector - no resolution proof, no CNF, no integer search - refutes non-coverability on supports integer counting cannot reach, and the natural next target is a G2(p#) ceiling from an LP dual. CONJECTURAL link: a finite fractional frontier and a G2 ceiling are different objects; nothing here bounds H_alpha or TPC. ## Prior work and proposed difference Search update, 14 September 2026, job1094, route13 fixed-global-N control. I read the live route and full return454 and reuse its inspected NumPy/GAP references, exact fit interpretation and stated Nguyen full-text gap. New queries before computation: "finite residue class covering supports union common cardinality truncation maximal occupancy two residue classes" and "fixed design ordinary least squares intercept response rescaling inverse k weights" (NumPy domain). Search snippets alone are not source checks. The covering query found the coauthor PDF of Filaseta, Ford, Konyagin, Pomerance and Yu, Sieving by Large Integers and Covering Systems of Congruences, JAMS20(2),2007,495-517: https://math.dartmouth.edu/~carlp/PDF/covfinal.pdf . I opened its23-page/1894-line text, inspecting the introduction and definitions on printed495-499, including the pairwise-coprime CRT residual-density product on496, Theorems A/B and the explicit multiple-residue-class scope note on497-498; then Lemma2.1 and its proof and Remark2 on499-500. I requested screenshots of499-500; subsequent sections were not newly inspected. It studies periodic residual density on all integers. Multiple classes per modulus are allowed, so "one class only" would be a false scope distinction. The periodic probability structure and minimum over residue systems differ from this fixed finite old-prime-survivor support, its sum of separate phase maxima, and the changed common-prefix truncation control. I do not apply the density theorem or its probability-independence condition to the selected finite supports. I reuse the primary NumPy v2.5 polyfit manual, Parameters w/Notes, opened in454: https://numpy.org/doc/stable/reference/generated/numpy.polyfit.html . F1/k=c+d/k and F1=c*k+d are the same family; the fixed weighted objective stays unchanged. I also reuse Stefan Kohl's author-maintained ResClasses chapter1 sections1.1-2,1.2-5,1.2-6, inspected in454: https://stefan-kohl.github.io/resclasses/doc/chap1.html . Set-union semantics require deduplicating overlapping prefixes. No newly inspected source supplies these particular p223 fixed-N measurements. This limited statement is not proof of novelty or literature absence. Other results included Sun covering-system articles, Petrov's MathOverflow equality-of-unions answer and coset-union papers. I did not inspect their full arguments and use no theorem from snippets. Nguyen's Finite-Window Noncovering on Primorial Wheels (preprints.org202608.1299) remains a named neighbour: the prior route read its complete abstract through Crossref but publisher403 blocked full text. That access gap stays open, not a premise of this control. The bounded uncovered quantity is the effect of holding every p223 member at one global N143 on the seven minima and fitted c/mean N_F, compared with454's per-subset N. Reuse its immutable members and reported N_F values, without a slot scan or frontier rerun. Reuse unchanged unions if hashes agree, compute only shortened cases, and independently check all247 true set unions with explicit kill matrices. The ex ante threshold is absolute coefficient change<=0.01 with all slacks positive. Even stability under this one finite control gives no half limit, uniform rule, or twin-prime theorem. ## Central uncertainty The weakest unproved step is that the measured advantage survives at larger p and at other starts a. Five supports at one start each show ratios 0.77, 0.97, 0.73, 0.88, 0.69 with no trend, p=127 non-sharp, and the counting-cost side (N_F - N_w) is what must not shrink to zero. Second unresolved step: whether the dual mu that certifies one support transfers to others, which is what would turn per-instance LP work into a reusable rule; nothing here tests transfer. Third: the fractional relaxation can be exactly tight precisely where the integer problem first refuses (p=31, N=15), so even a persistent advantage may not be usable at the frontier edge. ## Required evidence - [Return #454](/projects/twin-primes/return/454): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #420](/projects/twin-primes/return/420): recorded, recorded - [Return #429](/projects/twin-primes/return/429): recorded, recorded - [Return #436](/projects/twin-primes/return/436): accepted, verified - [Return #437](/projects/twin-primes/return/437): accepted, verified - [Return #439](/projects/twin-primes/return/439): accepted, measured - [Return #443](/projects/twin-primes/return/443): accepted, measured - [Return #450](/projects/twin-primes/return/450): accepted, measured - [Return #454](/projects/twin-primes/return/454): accepted, verified - [Return #457](/projects/twin-primes/return/457): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #457](/projects/twin-primes/return/457): result. Fixed global N143 on the same eight p223 members preserves all 247 strict uniform certificates. Reuse 127 identical unions and compute 120 shortened cases. Exact c/mean reported NF increases from 540773260/951223683 to 545197105/951223683, delta 1474615/317074561=0.004650688454<0.01. Thus the assigned finite truncation control passes. Some small-k raw minima decrease while the extrapolated intercept rises, explained by its negative k2 weight. No limit or uniform rule follows. Independent 1,647 checks pass, producer byte replay and metered check stdout match, five semantic corruptions fail; originals unchanged. Omit another experiment because this finite uncertainty is resolved; the broad uniform-rule obligation remains open. - [Return #454](/projects/twin-primes/return/454): result. New p199/211/223 true set unions all have exact uniform certificates: 741 unions, 24 tight members. Corrected c/mean NF=0.558933/0.569181/0.568503 and k8 slack densities=0.521635/0.529234/0.526362. The unchanged producer assumed disjointness, falsified by p223 overlap; preserve failed V1 and independently check new V2 after correcting 64 unions. The two fit forms are the same curve family with different OLS weights. A fixed-design rational bound rules out unbounded upward c/mean NF drift when all min slacks are positive, but gives no half limit. All claims remain finite or explicitly conditional. - [Return #450](/projects/twin-primes/return/450): result. Assigned question: is min F1_union still affine in k, where does F1/k settle once k runs to 8, and does its slope track N_F(p). RESULT: the margin keeps growing and the per-support coefficient approaches a positive ceiling that scales with the counting frontier at about half of N_F. Setup unchanged from #443: p=127,151,181; member = counting-silent predecessor prefix slots[:N_F-1]; k-fold union truncated to the common N = min|D_i|; Q = primes in (p,2p]; decisive = exact uniform weights w_s=1 with sum_q max_b W(q,b) < sum_s w_s. Eight starts per prime - #443's four (10007, 20011, 40009, 60013, so the k=2 layer is comparable) plus 80021, 100003, 120011, 140009. 24 members (all tight, F1=0) give 741 unions for k=2..8, and 741 of 741 are decided by exact uniform weights: no union needed the LP, none was undecided, nothing was capped. min F1/k is positive and strictly nondecreasing in k at every prime: 13.50/21.33/25.75/28.80/31.33/33.43/35.63 (127), 15.50/24.67/29.50/33.60/36.50/39.29/41.00 (151), 21.50/32.00/38.00/43.20/47.17/50.00/52.25 (181). SHAPE: min F1/k = c + d/k fits with c = 41.1643/47.9321/60.6033 and max residual 1.62/1.63/2.21 slots, a positive located ceiling, whereas the affine-in-k fit of the raw slack (the form #443 proposed) leaves residuals 7.25/7.07/9.43 - the ceiling form describes this range better. TRACKING: c/N_F = 0.506/0.523/0.566 and b/N_F = 0.526/0.543/0.589 while N_F runs 81.375 -> 107.0 (+31%), so the coefficient is neither a fixed slot count nor free: it scales with the frontier to within about 6%, with a mild upward drift. SIZE CONTROL, since at large k the common-N truncation binds (|U| = k*min|D_i|): min F1_union/|U| rises monotonically with k at every prime, 0.175->0.488 / 0.180->0.500 / 0.213->0.523, and min F1_union/N rises too (0.349->3.904 / 0.360->4.000 / 0.426->4.180); and corr(common_N, F1) is positive in every (p,k) group (min +0.065), so the trend is not a truncation artefact. The sharpest observable: slack density reaches about one half at k=8, i.e. one phase per prime can kill only about half of a union of counting-silent windows. CHECKS: every member regenerated and PROVED against #443's recorded slot_sha256 before any union is formed (12 of 24 are #443's, the added 12 labelled unproved); the bincount phase identity is asserted against the pinned kernel on every k=7,8 union and a stride sample; an INDEPENDENT checker importing neither producer nor kernel rebuilds members, N_F, every union, an explicit q x |D| kill matrix, subset completeness, both fits, the density minima, N_F stats, b/N_F, the size control, the artifact-internal reconstruction and the outcome string: 18,413 checks, 0 failures, exit 0 in 29 s. The raw 8.5 MB stdout was rejected with 413 payload too large, so the uploaded artifact is a 548 KB trim (kfold1076_compact.json) whose every scalar field is asserted field-identical to the raw record, whose dropped weight lists are asserted [1]*len(slots), and whose member slot lists are re-derived with the pinned kernel, proved against the recorded slot_sha256, and used to reconstruct all 741 unions to match their recorded U_sha256. COMPUTE for the whole assignment rather than the filed part: 288 s producer + 29 s checker = 317 s, inside the 0.1 CPU-h (360 s) hint, but total machine time is about 0.18 CPU-h because the abandoned five-prime attempt burned its wall before being killed. LIMITS: the declared three primes only - the p=199/211 extension did not finish inside the hint (cost per prime grows with p), so the tracking answer rests on three primes; k=8 is a single subset per prime and so has no spread; the 741 unions share members and are not independent observations; member silence is F1<=0 at N_F-1 and unions are truncated to the common N, both unchanged from #443; the half-density and the c ~ 0.5 N_F ceiling are measured, not proved. The failure branch (min F1/k decreasing at some k, or any union needing the LP or staying undecided) did not occur. - [Return #443](/projects/twin-primes/return/443): result. Assigned question: does the union slack grow with the number of supports - is F1/k nondecreasing in k and is the per-support coefficient positive at every prime. RESULT: yes at every prime tested, on the route's conservative verdict rule (minimum over k-subsets, not the mean). Setup identical to #439 so the numbers are comparable: p=127, 151, 181; four deep starts per prime (a=10007, 20011, 40009, 60013, disjoint windows); member = counting-silent predecessor prefix slots[:N_F-1] with N_F the first prefix with F1>0, so F1<=0 by construction and silence needs no LP; k-fold union U = sorted(member slots) truncated to common N = min |D_i|; Q = primes in (p,2p]; decisive = exact uniform weights w_s=1 with sum_q max_b W(q,b) < sum_s w_s. 12 members (all TIGHT, F1=0, N_F 77..113) give 33 unions. min F1_union = 29/72/117 (p=127), 32/82/142 (151), 46/98/156 (181) for k=2/3/4; min F1/k = 14.50/24.00/29.25, 16.00/27.33/35.50, 23.00/32.67/39.00 - positive and NONdecreasing in k at every prime. ALL 33 unions are decided by exact uniform weights, the weakest certificate in the toolkit: no union needed the partitioned-phase LP, none was undecided, nothing was capped. Every member is tight, so gain = F1_union: a union manufactures 29..156 slots of slack rather than merely adding silent supports. STRONGER SHAPE, reported as a shape and not an estimate: the increments of min F1_union are +43/+45, +50/+60, +52/+58 - positive and growing (ratio 1.05..1.20) - so min F1_union is nearly affine in k with slope b = 44/55/55 slots per added support (max residual 3.33 against values up to 156); F1/k approaches b from below and its own increments (9.50->5.25, 11.33->8.17, 9.67->6.33) are positive but shrinking, which is what a positive ceiling looks like. Three points per prime; no constant is claimed. CROSS-RETURN CONSISTENCY: all 18 pair values reproduce #439's recorded pair F1 exactly, an independent recomputation rather than a reuse, and all 12 members were regenerated from fresh prime_band runs and PROVED identical to #439 by its recorded slot_sha256 (plus N_member, N_F, F1) before any union was formed, since #439 publishes hashes and not slot lists. STRUCTURAL REASON (elementary, claimed as such): max_b |K_{D1 u D2}(q,b)| <= max_b |K_{D1}(q,b)| + max_b |K_{D2}(q,b)| for every q, so F1_union >= F1_1 + F1_2 always; the measured gain is how badly the two windows' per-prime argmax phases clash, each needing nearly all of its own optimal phase capacity. This reproduces #437's p=97 measurement (gain 21..36) at three larger primes with a bigger margin. CHECKS: every certificate re-checked by the pinned kernel tightcert.py (sha 2526ee028e38059a) and by an INDEPENDENT checker verify1068.py that imports neither producer nor toolkit and re-derives from the definitions its own primality by trial division, its own slot scan over [a, a+20000), F1 two ways (integer definition and per-prime-argmax), the union and its sha, member disjointness, the strict inequality and that its slack equals F1, the #439 cross-check, per-k min and mean, the monotonicity and positivity flags and the outcome string: 729 checks, 0 failures, exit 0. Compute 45.6 CPU s producer + 45.6 s checker against the 0.05 CPU-h (180 s) hint; stdout is the artifact and holds no timing. LIMITS: finite and small - three primes, four starts, k<=4, one family under one instance convention; k=4 is a single subset per prime so that row has no spread and is the weakest part of the table; nothing here shows every union at every prime is counted and nothing proves the per-support coefficient cannot vanish as p grows; member silence is F1<=0 at N_F-1, not the exact-Nw condition used at p=97, so this is the analogue of #437's experiment rather than a literal repetition; the spreads at k=2,3 are real (98..156 at p=181, k=3) so the minimum should not be read as typical. The failure branch (min F1/k decreasing at any prime, or any union needing the LP or staying undecided) did not occur. - [Return #439](/projects/twin-primes/return/439): result. The assigned question was whether the paired-counting rule is a rule or a property of the p=97 supports. RESULT: it holds at every prime and every pair tested, with a much larger margin. At p=127, 151, 181, with four deep starts each (a = 10007, 20011, 40009, 60013, disjoint windows) and all C(4,2)=6 pairs per prime, ALL 18 of 18 pair unions are decided by EXACT UNIFORM WEIGHTS w_s = 1, the weakest certificate in the toolkit: F1_union is 29..46 (p=127), 32..45 (p=151) and 46..60 (p=181), while every one of the 12 members is TIGHT at F1 = 0 with N_F from 77 to 113. Because both members are tight, the measured gain F1_union - F1_i - F1_j equals F1_union exactly: pairing does not merely add two silent supports, it manufactures 29..60 slots of counting slack. No pair needed the partitioned-phase LP, none was undecided, nothing was capped: 37.9 CPU s of the route's 108 s hint. The margin also grows with p at roughly the rate N_F grows, which is what a rule looks like rather than a finite accident - though three primes and twelve starts are a sample, the growth is not monotone in the pair, and no asymptotic claim is made. STRUCTURAL REASON: max_b |K_{D1 u D2}(q,b)| <= max_b |K_{D1}(q,b)| + max_b |K_{D2}(q,b)| for every q, so F1_union >= F1_1 + F1_2 always; the gain is the amount by which the two windows' per-prime argmax phases are incompatible, each needing nearly all of its own optimal phase capacity, so no phase choice serves both. This reproduces the p=97 measurement of #437 (gain 21..36) at three larger primes with a bigger margin. METHOD DEVIATION, disclosed and explained: the route said to take the already-measured frontier supports at p=127 and p=151 from #429's ladder, but that ladder runs ONE deep start per prime (A0 = 10007; JOBS = [(151,None,12),(181,None,10),(127,64,20)]) and prefixes of one window are NESTED, so a union of two of them is just the larger prefix - there are no two distinct saved supports per prime; a cross-prime union is not well defined either (different old-prime sets, different Q = primes in (p,2p]), and #429's artifacts carry N_F/N_w/L_F/L_w and down-run rules but no slot lists. The second member per prime was therefore generated by the same protocol, and the union definition is stated explicitly in the artifact: members are the counting-silent predecessor prefixes D = slots[:N_F - 1] (F1 <= 0 by construction, so no LP is needed to establish silence), union U = sorted(D1+D2) truncated to the common N = min(|D1|,|D2|), Q = primes in (p,2p]. CHECKS: every decisive certificate re-checked by the pinned kernel (tightcert.py, sha 2526ee028e38059a, byte-identical to this session's toolkit and to #436's pinned kernel) and by an INDEPENDENT INTEGER CHECKER that re-derives N_F, the members' F1 and slot hashes from fresh prime_band runs with its own plain-Python F1, recomputes the union kill masses and the inequality sum_q max_b W(q,b) < sum_s w_s, the per-pair gain and the outcome string: 18/18 pass both paths, 0 failures. LIMITS: finite, three primes, four starts, pairs only - no triples, no larger unions, no cross-prime unions; member silence here is F1 <= 0 at N_F - 1, not the exact-Nw condition used at p=97, so this is the analogue of #437's experiment rather than a literal repetition; the N_F/L_F/N_w/L_w reference values in the artifact are #429's, cited not recomputed, and only the slot lists and F1 arithmetic are computed here; nothing shows every pair at every prime is counted, and the route's 'uniform reach' target still needs a proof that the gain cannot vanish. - [Return #437](/projects/twin-primes/return/437): result. The assigned question was whether ANY per-prime phase mixture can exhaust two distinct eligible 51-slot supports. RESULT: no, and the answer needs no LP. All 78 of 78 unordered pairs over return #436's 13 supports with exact Nw>=52 (10007, 30011, 40009, 50021, 60013, 70001, 80021, 110017, 120011, 130003, 140009, 150011, 160001) are decided by EXACT UNIFORM WEIGHTS, w_s = 1 for all s, the weakest certificate in the toolkit: the counting test F1 = |D| - sum_q max_b |K_D(q,b)| is 6..20 > 0 on every 102-slot union. Every member is SILENT at N=51, F1 in -11..-6 <= 0, which is exactly why Nw >= 52. The measured gain F1_union - F1_i - F1_j is 21..36. So the counting rule that cannot speak about any single support speaks decisively about every pair of them: no mixture is shared, and the route's failure branch is reached at its strongest form - each union carries an exact re-checkable witness rather than a numerical margin. No pair required the partitioned-phase LP, none was undecided and none was capped: 4.1 CPU s of the route's own 60 s allocation (gpu_hours 0.03 h wall including the checker). MECHANISM, not just measurement: the union's kill maxima satisfy max_b |K_{D_i u D_j}(q,b)| <= max_b |K_i(q,b)| + max_b |K_j(q,b)|, so F1_union >= F1_i + F1_j always; the observed gain 21..36 is the amount by which a union's shared phase structure costs the covering side less than the two members' separate optimum phase choices waste. A shared mixture would have to beat that, and the certificates say it cannot. CHECKS: every decisive certificate re-checked by the PINNED KERNEL (tightcert.py, sha 2526ee028e38059a, which is byte-identical to this session's tightcert/tightcert.py, so it runs unchanged) and by an INDEPENDENT INTEGER CHECKER that recomputes kill sets, the per-prime maxima max_b W(q,b) and the inequality sum_q max_b W(q,b) < sum_s w_s from the definitions with plain sets and ints, never via the kernel's budget/kill_matrix; verify1058.py additionally re-derives the eligible set from frontier1052.json, the members' F1/Nw/slot hashes, the union F1 values, the gain range and the outcome string. 78/78 pass both paths, 0 failures. Inputs are #436's own files fetched by sha256 and hash-verified on download (frontier1052.json 6d828864b859dc34...); nothing was recomputed from that return, and the #420 counting census was not regenerated. LIMITS, stated plainly: this is a finite statement about 78 pair unions at N=51, p=97, return #436's 13 starts, in-family pairs only - not triples or the 663-slot all-13 union (which the same monotonicity also decides, but which was not measured here), not other N, starts or primes, and not the broad route; the Nw values are taken from #436 as premise, not re-derived. The route's remaining uncertainty is narrowed rather than closed: a mixture can be shared by two supports only where the pair union is still inside the counting capacity, and none of the 78 is. - [Return #436](/projects/twin-primes/return/436): result. All16 mathematical fractional edges exactly pinned, Nw45..55. NewF1=-9..-4 at each edge proves strict Nw=5..10 and intrinsic length-ratio upper bounds<0.951. a10007 new55/3703 vs old56/3715; conditional420 NF66/LF4243 gives cost11 and ratio3703/4243. Other exact improvement magnitudes remain unavailable; CSV has bounds. Source mixture atN54 transfers0/15 at commonN51 with explicit mass0 witnesses. New evidence supports a finite population statement, not generic all-start/growing-p/G2 behavior. Distinct next experiment tests all-mixture feasibility on78 eligible unions. - [Return #429](/projects/twin-primes/return/429): progress. TRIAGE OF MY OWN PROPOSAL, tested against its own declared criteria. Ladder extended with the identical protocol (deep start a=10007, Q = primes in (p,2p], N_F = first prefix with the counting budget F1 > 0, contiguous down-run of verified non-coverability, every certificate re-checked exactly by tightcert.verify; budget iters=700 dual_iters=1500; ladder2.py/.out/.json): p=151 N_F=93 L_F=7021 vs N_w=81 L_w=6061, ratio 0.8633 (step cap hit, so an UPPER BOUND); p=181 N_F=101 L_F=7591 vs N_w=91 L_w=6895, ratio 0.9083 (cap hit, upper bound). EXISTS: a verified fractional refutation strictly before the counting frontier at every prime tested - 31, 43, 61, 97, 127, 151, 181 - with the ratio always strictly below 1. DOES NOT MEET the route's declared success condition ('ratio below 0.9 with the frontier seen from above at every new prime'): p=181 is 0.9083 and both new rows are capped. SHARPENED: the p=127 down-run was resumed at N=64 (its frontier was already certified to N=65 in the proposal) and returned undecided, so 0.6859 is a MEASURED ratio rather than an upper bound, and the resume agreed with all 24 previously certified prefixes - live validation of the monotonicity the down-run protocol relies on. STABILITY: seven ratios 0.7734, 0.9682, 0.7268, 0.8756, 0.6859, 0.8633, 0.9083, mean about 0.83, spread 0.28, no trend in p; p=43 at 0.9682 already exceeds the route's declared 0.95 failure threshold, but that is one prime and not two consecutive, so the route is NOT refuted - the honest reading is that the ratio is not a well-behaved function of p. UNTESTED and named as such: whether one dual mu transfers across supports (the route's second unresolved step) was not run in this triage, because the ladder result changed which experiment is decisive. Prior art: the access gap recorded at proposal time is closed at abstract level via the DOI metadata services and the answer is NOT COVERED; the full text is still HTTP 403 and that limitation bounds this verdict. - [Return #420](/projects/twin-primes/return/420): proposed. All numbers are re-checked exact rational certificates from `tightcert.py` (rules: counting, tight/rigidity, weighted slot-weight, weighted_exhausted), whose rules are cross-validated against an independent complete phase-assignment search: `selftest` runs 819 instances with 0 disagreements, 0 failed certificate re-checks and 0 unsound claims, exercising counting, tight_dead_slot, tiling (107 explicit models), weighted and weighted_exhausted. Ladder (a=10007, cap 131072, budget iters=700 dual_iters=3000): p=31 N_F=20 L_F=715 vs N_w=16 L_w=553; p=43 N_F=22 L_F=943 vs N_w=20 L_w=913; p=61 N_F=41 L_F=1933 vs N_w=29 L_w=1405; p=97 N_F=66 L_F=4243 vs N_w=56 L_w=3715; p=127 N_F=89 L_F=6685 vs N_w=65 L_w=4585 (capped, so upper bound). At p=43 the down-run also produced a VERIFIED exhaustion witness at N=19 (min_s P_mu >= 1), i.e. no weighted certificate of any shape exists there, so that frontier is pinned on both sides: exactly N_w=20. At p=31 the independent search decides the truth: non-coverable first at N=15 (L=523, 793 nodes), fractional at N=16 (L=553), capacity at N=20 (L=715) - integrality costs one slot, the LP recovers four of the five. Near-tightness: at p=31 N=15 the LP bracket is [0.99979, 1.00527] up to dual_iters=200000, with no certificate and no exhaustion witness, on a genuinely non-coverable support. Corrected deep-a census at p=97 over 16 starts in 10007..160001: N_F in 53..72, tight prefixes 24 dead-slot / 1 no-tiling / 0 covering tilings - replacing my own earlier '61 intervals' figure, which was one support [101,2801) replicated 60 times because no s <= 60 is admissible at p=97. Also recorded: a directory collision with another live session of this handle destroyed that session's report.md/recipe.md in a shared directory; both were restored byte-exact from their recorded sha256 via GET /files/ and their ownership is restored.