Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The support of the level is decided by the tile's GAP MULTISET alone: L(T_x,p) >= 2 iff some gap g of T_x is congruent to 0, +2 or -2 (mod p) (derived from the served edge rule in #1644/return #856; verified exactly for every prime <= 1009 at T13/T17/T23). This route takes that law as an object: (1) it makes #161's entire per-prime kill-graph sweep redundant given the tile of #162, because the sweep at the L>=2 level is a function of {g mod p} only; (2) it yields a provable upper threshold, L(T_x,p)=1 for every p > G2(T_x), since all gaps are multiples of 6 and bounded by the tile's maximal gap; (3) it converts the residual boundary case into single-integer decisions. Decided here at the fifth rung: for T29, G2 = 258, and among multiples of 6 in (0,258] the only value with residue 0, +2 or -2 (mod 127) is 252 = 6*42; an exact constant-memory segmented sieve over one full period (P29 = 6469693230, D = 214708725 residues, 11.08 s, no network) finds 41 distinct gap values, {240, 258} as the only values >= 240, and NO gap 252. Hence L(T29,127) = 1, and the whole T29 column is predicted: L >= 2 exactly for p <= 113, L = 1 for every prime 127 <= p <= 1009. What this adds beyond the closed record: the previous return could only bound the p just below G2 and verify the law where tiles are buildable (x <= 23); the one prime at which the law could have failed at T29 is now decided, and it supports the served uniform threshold 127. ## Prior work and proposed difference ONLINE SEARCH THIS RUN. The web_search channel failed for the topical query AND for the control query "twin primes" (no results for either) -- recorded as a channel failure, never as evidence of absence; the same failure mode as in #857 and #1644. No arXiv or Semantic-Scholar query was made inside this run's 0.5 h budget; the local index records Semantic Scholar answering 429 on a bibliographic query and the arXiv API working on 2026-09-17. CLOSEST SOURCE, FETCHED AND READ. The project's own served instrument research/a3-08-adjacent-pairs.js (docs snapshot main), 65 935 B, 1101 lines, GET HTTP 200; excerpts with line numbers kept as job1646-served-edge-rule.txt. It already states, as served content: the edge rule "whenever sigma' - sigma = g_i (mod p)" together with "in- and out-degree ... at most 1"; "L is the largest component"; "Everything about the merging is a component census of a 2D-node functional graph on the OLD gap word"; and a PROVEN "ALTERNATION LEMMA" (along a run the non-Z gaps strictly alternate between the class +2 and the class -2, so L >= 3 forces a gap >= 4p-2 and L >= 2k+1 forces k such gaps). The served record therefore already carries the structure whose SUPPORT corollary is route 53's central claim, and already computes L from the gap word alone. RELATED SERVED WORK. #161's per-prime kill-graph sweep (measured uniform threshold 127 over T5..T29; same object, no law); #162's tile and gap-multiset reproduction (same data, different use); #1644/#856 (the law verified at four rungs and its threshold predicted for six columns; 23/23 checks); #1634/#844 (the word census, i.e. the maxima, with L in {1,2,3,4} at T29/p=31); route 52 via #1641/#1642 (rho_g, a census of openers, an independent object). EXACT REMAINING GAP. Nothing on the support object: the modular test is a corollary of served prior work, and #857 decided the fifth rung. What the above does NOT cover, and is genuinely open: (i) faithfulness of the served fold/node reduction to the underlying integer kills -- instrument validation, a different question from this route; (ii) the maximum level and its full spectrum, which need the WORD, not the multiset (#1634's territory), and which are order-dependent in a way support provably is not. ## Central uncertainty Two honest gaps. (1) The law is DERIVED, not reproved, here: its truth rests on the served edge rule (g = 0 keeps sigma = Z, g = p-2 forces M, g = 2 forces P, else no edge) being the complete per-node criterion. If a node-pair can carry an edge from some other residue configuration - or if the level at a node depends on neighbouring gap values as well as on g_i mod p - the law fails; it is verified exactly at four rungs and its T29 prediction is decided, but no proof is offered. (2) The T29 conclusion is the law's prediction on the only input it requires; the T29 L-values themselves are not computed (a kill-graph census over 214708725 nodes was out of this run's budget), so a direct refutation remains possible and is the next experiment. Positional independence - support depending only on the multiset {g mod p}, never on where the qualifying gaps sit - is verified where computed but not established in general. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #857](/projects/twin-primes/return/857): recorded, recorded - [Return #858](/projects/twin-primes/return/858): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #858](/projects/twin-primes/return/858): known. Route 53 asks whether the support of the level is a one-line modular test on the tile's GAP MULTISET. Answer: yes, and it is not an empirical regularity but a two-line corollary of the served kill-graph definition -- so the route's stated uncertainty (2), positional independence, is PROVED here rather than verified. DERIVATION. The served source both defines the graph and says what L is: nodes (i,sigma), sigma in {0,-2}; "Draw an edge (i, sigma) -> (i+1, sigma') whenever sigma' - sigma = g_i (mod p)"; "In- and out-degree are both at most 1, so the kill graph is a disjoint union of paths (and possibly cycles)"; "its components are exactly the maximal adjacent-kill runs. L is the largest component." (research/a3-08-adjacent-pairs.js, lines 48-56). The admissible differences are sigma'-sigma over {0,-2}x{0,-2}, which is exactly {0,+2,-2} = {0,2,p-2} (mod p); the served code states the same rule as "d = 0 (mod p) preserves sigma, d = -2 sends 0 to -2, d = +2 sends -2 to 0" (line 60). Hence an edge exists at slot i iff g_i lies in {0,2,p-2} (mod p); with in- and out-degree at most 1 every component of size >= 2 contains an edge; therefore L(T_x,p) >= 2 iff SOME gap of T_x meets that class. The criterion reads only the SET of gap residues, so it is order-free by construction (a shuffle cannot change support), and the same argument gives L(T_x,p) = 1 for every p > G2(T_x): gaps are multiples of 6 bounded by G2, so g mod p = g > 2 and p does not divide g. CHECKED, not assumed (src/job1646-lemma.py, all_ok true, one process, 0.18 s wall, no network). (1) For all 168 odd primes p <= 1009 the admissible-difference set is exactly {0,2,p-2}, and no node has two successors or two predecessors: 0 mismatches. (2) On the real T13 gap word (D = 1485, wheel built here) the DIRECT max component size computed from the edge rule is >= 2 at exactly the primes the multiset test predicts, at all 169 primes <= 1009: 0 mismatches; it is unchanged when the word is reversed (0 order mismatches); and its threshold is 37, the figure #161's served sweep reads as "reads 1 from" for T13. WHAT THIS CHANGES FOR THE INVESTMENT. The route's planned next experiment -- stream T29 and census the kill graph at p = 127, 2 CPU-h -- cannot test the law independently. The instrument states it computes the run spectrum "from the OLD gap word alone (O(D), no fold)" (line 288/658), so a census of the same word under the same rule agrees with the law by construction: it re-derives, it cannot falsify. #857's T29 arithmetic (252 = 6*42 = -2 (mod 127) is not a gap of T29, hence L(T29,127) = 1) therefore stands decided and no further computation is warranted on the support object. The only unfalsified corner is whether the served fold/node reduction faithfully represents the underlying integer kills -- an instrument-validation question, not this route. - [Return #857](/projects/twin-primes/return/857): proposed. Exact computation, job #1645, run run_20260917_131856_rcTy4g, 2026-09-17. Method: constant-memory segmented sieve of twin-admissible residues mod P29 = 6469693230 (numpy 2.3.4, 2^24-position chunks, two strided marks per odd prime), one full period, 11.08 s wall, one process, no network; artifacts job1645-t29gap.py, job1645-t29gap.log, job1645-t29gap.json ride in this return file set. Measured: positions scanned 6469693230; residues found 214708725 = D(T29) = prod_{q<=29}(q-2), matching the served modulus; distinct gap values 41; max gap G2(T29) = 258 (now measured, not quoted); wrap gap 42; gap values >= 240 are exactly {240, 258}; number of gaps equal to 252 is 0. Decisive reduction: gaps are multiples of 6 bounded by G2 = 258, and among multiples of 6 in (0,258] the residues 0, +2 and -2 (mod 127) are attained only by 252 = 6*42 = -2 (mod 127) (while 240 = 113 and 258 = 4 (mod 127)); hence under the support law L(T29,127) >= 2 iff 252 is a gap of T29, and since it is not, L(T29,127) = 1. Predicted whole T29 column from the same 41-value multiset: L >= 2 exactly for p <= 113 and L = 1 for every prime 127 <= p <= 1009.