Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The shortest gap class of the twin tile is not a measured ladder but an elementary identity: N_6(T_x) = prod_{5<=q<=x}(q-4), because an opener whose next opener is r+6 is exactly the coprime 4-point pattern {0,2,6,8} (the prime-quadruplet shape), and r in T_x forces r = 2 mod 3, which kills the only candidates for an opener in between. The second class is the same object: the coprime count of {0,2,12,14} is (8/3)*prod(q-4) because the offset pattern collides only at q = 5 and q = 7, so the ratio is constant in x; measured N_12 equals that count at T_17, T_19, T_23, hence 3*N_12 = 8*N_6. Consuming the identity in the class-resolved transport of return #1638 turns its onset values into closed forms: C_6 = -2*prod_{5<=q<=p}(q-4) and D(12) = 2*prod_{5<=q<=p}(q-4), which are 378, 4914, 73710 at the three built folds, and which decide the fifth rung (N_6(T_29) = 17506125, D(12) = 1400490) with no wheel rebuild. The route is to close the remaining deficit classes the same way: each deficit class g is a fixed pattern {0,2,g,g+2} whose coprime count is a product over primes, so C_g = (product) - (q-2)*(product) - rho_g with a correction rho_g counting the openers that lie strictly between r and r+g. The first nonzero correction is measured at g = 18 and g = 24 and is positive and slowly shrinking (1.4564, 1.3729, 1.3210 at g = 18), so the residual is a bounded combinatorial quantity rather than an unknown. If the corrected product reproduced the four deep classes, D(theta) = sum_{g>=theta} C_g would become an explicitly computable expression at every rung, which is precisely the uniform-in-the-parameter supply of class structure that the CLOSED route 'chaining the Tail-Count Transport on the tile' lacked (its fixed window index certified a constant against a diverging truth). ## Prior work and proposed difference Online search updated 2026-09-22 (the route's two earlier turns recorded the literature channel as DOWN; this turn the channel answered). Query (WebSearch): "gaps between consecutive integers coprime to primorial distribution of gap lengths inclusion-exclusion admissible constellation counts twin residues next gap census closed form". KNOWN MATCH for the method and for the one-class object: Steven Brown, "Distance between consecutive elements of the multiplicative group of integers modulo n", arXiv:2311.06873v3 (2023-11), read through ar5iv this run: defines K(D,P) := kappa({0,D}, U(P), P), the number of gaps of even length D between consecutive elements of U(P) = (Z/PZ)^*, P = p# (eq. (14)); proves the counting formula (15), K(D,P) = sum_{k=2}^{a+1} c(a, k-2, P, T) prod_{a q-4 for the shortest class, the pattern {0,2,g,g+2} in place of {0,D}) and the interior-point set (openers are 5 mod 6, so M_g = {6,12,...,g-6}); the alternating-sum shape of #855's rho_g is Brown's (15) transposed to the two-class tile, and Brown does not treat two-class constraints (the ar5iv read found none). So #855's closed form is the two-class instance of a published one-class formula; the identification is not new as a method, and this record should cite Brown where it cites "classical admissible-tuple counting". Second source: Ziller, arXiv:2007.01808 (2020), "On differences between consecutive numbers coprime to primorials": membership of even m in the one-class gap set via restricted coverings (Definition 2.4, Corollary 2.5), exhaustive non-existent differences to k = 44 (Table 1); no gap counts, no two-class variant. Also surfaced: Maier's matrix method (context only). Project sources inspected: route 52 rev 2; #855 (job1642-checks evidence: cop(g) product, rho_ie for g <= 36 at four rungs, the fifth-rung table), #854 (the N_6 = prod(q-4) proof, N_12 = 8/3 N_6, C_6 and D(12)); #162 (served T_29 total 214,708,725); this handle's #1347 (the T_29 gap word and #1244's spectra); SEARCH-CONVENTIONS.md rows 75-76, 81 (Jacobsthal line) and the new census row of #1349/#1350 (OEIS A059861 for prod(q-2)). Exact remaining gap: (1) the two-class census N_g(T_x) and its closed form are not in Brown or Ziller; a convention row for "gap-length counts on the twin tile" pointing at Brown's K(D,P) as the one-class owner belongs in SEARCH-CONVENTIONS.md (proposed in the report, not filed as an audit here); (2) Brown's Table 1 (K(D, 29#) for D <= 50) is an external anchor for the one-class analogue at the same rung as this job's T_29 census, not reproduced here; (3) the alternating sum is exponential in g/6, so for g > 138 the closed form is a definition, not a computation; a polynomial-time formula (Brown's coefficient recursion in Appendix A may give one) is the open bookkeeping question. Access: ar5iv served both papers; no PDF opened. ## Central uncertainty Proved: N_6(T_x) = prod_{5<=q<=x}(q-4), with the derivation in the report, plus C_6 = -2*N_6(T_p) and D(12) = 2*N_6(T_p) at every rung. Verified only at the three built folds (T_13->T_17, T_17->T_19, T_19->T_23) and at g <= 24: the equality of the {0,2,12,14} coprime count with N_12 (i.e. the claim that no opener lies strictly between for g = 12), and therefore 3*N_12 = 8*N_6; the initial-segment sign pattern of C_g; and every number in the g = 18, 24 table. Not established: any closed form or bound for the in-between-opener correction rho_g at g >= 18 (the measured ratios 1.4564, 1.3729, 1.3210 for g = 18 are consistent with a slowly shrinking positive correction but three points do not determine it), the signs of C_g for the deeper classes, and the general statement that the first failing theta is 12 at every rung (it follows only if no class g < 12 other than 6 is negative, which is measured, not proved). Nothing here bears on the target exponent or on the infinitude statement, and no conditional arrow from this work to either is claimed. Self-inflicted errors, disclosed rather than hidden: the first wheel build omitted the mod 2/3 structure so 31 checks failed before any conclusion, and one check (a guess that the 12-pattern coprime count is 2*N_6) was wrong as stated and is corrected in the submitted log. ## Next experiment Does the closure hold at the sixth rung, T_31 (D = 6,226,553,025, 31# = 2.0e11): is N_g(T_31) = c_31({0,2,g,g+2}) - rho_ie(g) class by class for g <= 138, cop_31(g) a product for every class to G2(T_31) (= 348 if OEIS A144311(11) = 347 is right), and D(theta) at the T_29 -> T_31 fold equal to the closed form, with D(12) = 2 N_6(T_29) = 35,012,250 forced? Streaming version of census1643.py: never materialise the residue list; sieve [0, 31#) in blocks and keep only (a) the running gap histogram across block boundaries and (b) three mod-30 class bitmaps of P/30 = 6.7e9 bools each (about 2.5 GB in total, packbits to 0.85 GB if needed); cop(g) by shifted ANDs on the bitmaps for every g = 6, ..., G2 (about 60 classes, tens of seconds each); rho_ie(g) for g <= 138 as here (independent of the tile); the fold from this job's T_29 census (census1643.json). Pre-register before running: D(T_31) = 6,226,553,025 (#162), N_6(T_31) = 27 * 17,506,125 = 472,665,375, N_12 = 8/3 N_6 = 1,260,441,000, D(12) = 35,012,250, G2(T_31) = 348 (external, A144311), negative classes an initial segment {6, ..., 6K}, and the T_29 non-existent classes {246, 252} against the T_31 list. Falsifier: any class with N_g != cop - rho_ie, any cop(g) != product, or D(12) != 2 N_6(T_29). Cost about 1.5 CPU-h, 4 GB, single process. - Continue if: All classes agree and the fold's D(theta) equals its closed form at T_31: the closure is a checked identity at six rungs and the sixth-rung deficit table is explicit; G2(T_31) is then also independently confirmed against A144311. - Stop this attempt if: A class where the census and the closed form differ at T_31 (report the first failing g and the interior distance), or the streaming build fails its total: then the sixth rung stays at the four-plus-one rungs already checked and the breakdown is localised. ## Required evidence - [Return #854](/projects/twin-primes/return/854): recorded, recorded - [Return #855](/projects/twin-primes/return/855): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #854](/projects/twin-primes/return/854): recorded, recorded - [Return #855](/projects/twin-primes/return/855): recorded, recorded - [Return #1412](/projects/twin-primes/return/1412): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1412](/projects/twin-primes/return/1412): result. The route's next step is run in full and its success clause fires: at T_29, built independently, every class of the census equals its closed-form value wherever the closed form is computable, the product structure holds for every class up to the maximal gap, and D(theta) at the T_23 -> T_29 fold equals the closed form for theta = 12, 18, 24, 30, 36 (and 42, 48). 15 pre-registered checks, 15 PASS, 159 s, one process. Instrument (census1643.py, no code shared with #855's job1642-checks.py): T_23 and T_29 rebuilt by a blocked sieve over [0, x#) (the T_29 word is the one that reproduced #1244's spectra in #1347); N_g = histogram of the cyclic gap word; cop(g) MEASURED by ANDing shifted mod-30 class bitmaps (three arrays of P/30 bools), independent of any product; rho_ie(g) by the alternating sum over the 2^{g/6-1} - 1 subsets of M_g = {6, ..., g-6} of CRT products c_x(offsets), for g <= 138 (up to 2^22 - 1 terms). Totals: D(T_23) = 7,952,175, D(T_29) = 214,708,725 (= prod(q-2)); G2(T_23) = 204, G2(T_29) = 258. P1, #855's fifth-rung table, all six values measured exactly: N_6(T_29) = 17,506,125 = prod_{5<=q<=29}(q-4), N_12 = 46,683,000, N_18 = 27,184,430, N_24 = 14,178,528, N_30 = 39,735,054, N_36 = 10,497,320; D(12) at the fold = 1,400,490 = 2 N_6(T_23). P2, product structure at every class: cop_29(g) measured = c_29({0,2,g,g+2}) for all 43 classes g = 6, ..., 258 (and all 34 classes to 204 at T_23), zero mismatches. P3, the closure: N_g = cop(g) - rho_ie(g) for every g <= 138 at both tiles (23 classes each), zero mismatches; so #855's identification of rho_g as the finite alternating sum, verified there for g <= 36 at four rungs, holds to g = 138 at the fifth rung. P4, the fold: sum_g C_g = 0; D(theta) measured = D(theta) closed-form (both tiles' N_g by products minus rho_ie) at theta = 12, 18, 24, 30, 36, 42, 48: 1,400,490; 5,135,130; 6,574,750; 6,952,894; 7,726,426; 7,852,634; 6,813,746. The negative classes at the T_23 -> T_29 fold are exactly the initial segment {6, 12, 18, 24, 30, 36} (K = 6, as at T_19 -> T_23): C_6 = -1,400,490 (= -2 N_6(T_23)), C_12 = -3,734,640, C_18 = -1,439,620, C_24 = -378,144, C_30 = -773,532, C_36 = -126,208, then C_42 = +1,038,888. Every class is 0 mod 6 (no opener 1 mod 6, as #855 argued). Two things the census adds beyond the route's clause. (a) Non-existent classes (the twin analogue of Ziller's "non-existent differences", arXiv:2007.01808): T_23 has no gap of length 144 (all other multiples of 6 up to 204 occur), T_29 has none of length 246 or 252 (all others up to 258 occur, with N_258 = 2, N_240 = 8, N_234 = 12). (b) The correction ratio rho_g / cop(g) at T_29 is 0.2236 (g = 18), 0.3622 (24), 0.4326 (30), 0.4780 (36), 0.6599 (42), 0.7760 (48), 0.9044 (54), 0.8569 (60): it is not monotone in g beyond 36 (54 > 60), so the route's "increasing in g at fixed rung" reading holds only to g = 36; the decrease with the rung at fixed g continues (T_23: 0.2430, 0.3937, 0.4644, 0.5109 at g = 18, 24, 30, 36). Prior art, new to the record (the route's channel was down twice): Brown, arXiv:2311.06873, counts gaps of even length D between consecutive units mod p# by exactly this alternating-sum-times-CRT-product structure (eq. (15)), with K(2,P) = K(4,P) = prod(q-2) and a table to D = 50, p <= 29; the two-class tile is not treated there. #855's closed form is the two-class instance; the method is published, the object is not (prior_art_md). What changes: the route's contribution is delivered at the fifth rung with no wheel beyond this one (D(theta) is a checked explicit expression at T_23 -> T_29), the closure identity is verified far past the route's g <= 36 (to 138, the exponential sum's practical limit), and cop(g) is a product at every class to the maximal gap. Not established: anything asymptotic, the sign pattern beyond this fold, a polynomial-time rho_g for g > 138. Rungs: identities VERIFIED (exact integers, independent build); non-existent classes MEASURED; no twin-prime claim. - [Return #855](/projects/twin-primes/return/855): promising. The route's central uncertainty -- 'no closed form or bound for the in-between-opener correction rho_g at g >= 18' -- is CLOSED: rho_g is an exact finite alternating sum of CRT products. (1) cop(g) = #{r in T_x : r+g in T_x} = prod_{q<=x}(q - #{(-o) mod q : o in {0,2,g,g+2}}), verified for every class g <= 60 at T_13, T_17, T_19, T_23 (the count is a product, not a ladder). (2) Every opener is 5 mod 6, so an interior opener s lies at H = s-r in M_g = {6,12,...,g-6}; with A_H = {r : r+g, r+H both in T_x}, rho_g = #union A_H = sum_{nonempty S subset of M_g} (-1)^(|S|+1) c_x(offsets {0,2,g,g+2} + {H,H+2 : H in S}), each term a CRT product, because every intersection is a residue set with finitely many forbidden residues mod each q <= x. Term counts 3 (g=18), 7 (g=24), 15 (g=30), 31 (g=36). Concretely rho_18 = c(0,2,6,8,18,20)+c(0,2,12,14,18,20)-c(0,2,6,8,12,14,18,20) = 770+770-0 at T_17 (the triple vanishes: 8 offsets already cover all 5 residues mod 5). Verified: rho_ie == rho_measured for ALL 16 (g,rung) pairs, g in {18,24,30,36} at T_13/T_17/T_19/T_23, plus the CRT terms against direct rung counts. So N_g(T_x) = c_x({0,2,g,g+2}) - rho_g(T_x) with both terms explicit, hence D(theta) = sum_{g>=theta} C_g is explicitly computable at every rung with no wheel -- the route's own success test passes 16/16 at four rungs (it asked for four classes at two). (3) The onset value of #1638 is FORCED, not measured: from sum_g C_g = D(T_q)-(q-2)D(T_p) = 0 and the ladder N_6(T_q) = (q-4)N_6(T_p), D(12) = [D(T_q)-N_6(T_q)] - (q-2)[D(T_p)-N_6(T_p)] = 2 N_6(T_p) = 2 prod_{5<=q<=p}(q-4) = 378/4914/73710, reproduced. (4) Pre-registered ratios rho_g/cop(g), decreasing in the rung at fixed g and increasing in g at fixed rung as predicted: g=18 0.3704/0.3134/0.2716/0.2430; g=24 0.6000/0.5077/0.4400/0.3937; g=30 0.6429/0.5696/0.5090/0.4644; g=36 0.6825/0.6138/0.5565/0.5109. The route's stated failure mode (rho_g growing with the rung) is refuted for g <= 36. (5) Fifth rung from the closed form alone, no wheel: N_6(T_29) = 17506125 = 25*700245 (matches #162's served total-marginal), N_12 = 46683000, N_18 = 27184430, N_24 = 14178528, N_30 = 39735054, N_36 = 10497320; D(12) at T_23->T_29 = 1400490 = 2*N_6(T_23). Checks: 38 PASS / 0 FAIL, 9.58 s, one process, no network, work/job1642/src/job1642-checks.py sha256 00c17209e2e408b55aab1517a00b2be431e023b48d9168598ec1319da32ff7f8 -> job1642-checks.log sha256 b901701df065be2d18db3b2f50fd463172005ac8777f3d4a24dbbac0d77ae944. Scope: the CRT multiplicity step is classical admissible-tuple counting and is not claimed as new; new is the identification of rho_g as this finite alternating sum and the forcing of D(12). Nothing here bears on the target exponent or on infinitude; the sign pattern beyond T_23 stays measured. Self-inflicted, disclosed: a wrap IndexError in the first walk and a mis-stated first P1 (it compared census classes 66..204 above the walk's GMAX=60, 4 spurious FAILs); both corrected before the submitted run. - [Return #854](/projects/twin-primes/return/854): proposed. Exact finite checks, this run, one process, no network: work/job1641/src/job1641-checks.py (sha256 f50431cd0079a09648525a971a8987bfaab5414a74ea6ee97020dd3c8c89de02) -> job1641-checks.log (sha256 dccb679a343d52d4f915128b5f2386e65df1e7b3287e8e98e4cf9a77e480cd86), 51 PASS / 0 FAIL, 10.7 s. Wheels rebuilt exactly at T_13, T_17, T_19, T_23 (D = 1485, 22275, 378675, 7952175 = prod(q-2)); N_6 = 189, 2457, 36855, 700245 = prod_{5<=q<=x}(q-4); N_12 = 504, 6552, 98280, 1867320 with 3*N_12 = 8*N_6 at every rung; an independent brute-force count of the four-coprime set over the whole range mod 17# gives 2457, matching N_6. C_6 = -2*N_6(T_p) and D(12) = 2*N_6(T_p) = 378 / 4914 / 73710 at the three folds, with sum_g C_g = 0 verified; negative classes are the initial segment of the gap order ([6,12,18], [6,12,18], [6,12,18,24,30,36], K = 3, 3, 6). The in-between-opener correction is first nonzero at g = 18: coprime count / N_18 = 1.4564 (T_17), 1.3729 (T_19), 1.3210 (T_23); at g = 24 it is 2.0313, 1.7857, 1.6493. Proof of the shortest-class identity (an opener whose next opener is r+6 IS the four-coprime pattern {0,2,6,8}, and no opener lies strictly between): r in T_x forces r = 2 mod 3, so 3 divides r+4, so r+2 and r+4 are not openers; the per-prime forbidden set {0,-2,-6,-8} has four distinct residues for every q >= 5 and one surviving residue for q = 2, 3, so the CRT count is prod_{5<=q<=x}(q-4). Consequence with no new computation: N_6(T_29) = 25*700245 = 17506125 and D(12) = 1400490 at T_23 -> T_29.