Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Replaces brute-force full-wheel enumeration and noisy Monte Carlo window sampling on Route 364 (`dir-558`) with an exact finite state-vector evolution for the Constellation-Order & Half-Alive Cell Census `C_L(x#) = {(A_L^{(m)}(x#), H_L^{(m)}(x#))}_{m=1}^{m_max(L)}`. By Theorem 1 (3-Cell Quantization, `F_L = 3(k-1)*gaps_L - 2*H_L^{(1)}`) and Theorem 4 (Universal Constellation-Hierarchy Lift Law for all `p > (L+3)/2`), a singleprimorial anchor census at `x_0#` (`x_0 = max(7, prevprime((L+3)/2))`) determines `gaps_L(x#)`, `F_L(x#)`, `rho_L(x#)`, and `c_L(x#)` for ALL higher primorials `x#` in `O(m_max(L))` arithmetic operations per prime (compressing `37#` into `19#` by `765,049x` with `0` integer error). Resolves Route 364's monotonicity question: only `L = 11` decreases forever (`c_11(infinity) = (35/48)*prod_{p>=11}(1 - 4/((p-1)(p-4))) = 0.61017546...`), whereas every `L in {17, 23, 29, 35, 41, 47, 53}` reaches a strict global minimum above `37#` and turns strictly increasing via gap coalescence (`p*(L) = 47, 47, 83, 127, 103, 127, 107`). ## Prior work and proposed difference Online search, 2026-10-23 (Serper/Google, four queries; abstracts read in full, no paywalled text): 1. "Holt Rudd 'cycle of gaps' Eratosthenes sieve primorial recursion arXiv 1408.6002" -> F. B. Holt and H. Rudd, *Eratosthenes sieve and the gaps between primes*, arXiv:1408.6002 (2014): the recursion on the cycle of gaps G(p#) and its constellation enumeration. 2. Same query family -> F. B. Holt, *Constellations of gaps in Eratosthenes sieve*, arXiv:1503.00231 (2015). Abstract read: "a discrete linear system that exactly models the population of any gap across all stages of the sieve ... this paper extends the discrete system to model the populations of constellations of gaps"; includes a Polignac-type statement for repeated constellations g,...,g of length j < P-1. 3. "exact asymptotic density of gaps between twin primes wheel primorial ... constellation of gaps" -> Holt and Rudd, *Combinatorics of the gaps between primes*, arXiv:1510.00743; plus H. L. Montgomery-type expository material and Hardy-Littlewood singular-series chapters (e.g. Ghidarcea 2025, IntechOpen). Nothing on the twin wheel. 4. '"twin prime" gaps "wheel" exact counts recursive enumeration ... distribution paper' -> project-internal pages (moiré-primes paper, GLOSSARY), one AI e-print index (ai.vixra.org) and tangential preprints (scirp 2026, preprints.org 202608.1299). None tracks a twin-wheel constellation census; not relied on. Closest prior work and the exact difference. Holt's linear system is the same *technique*: an exact integer recursion for the population of gap constellations across primorial stages, with a triangular survival system (their stages are the single-prime Eratosthenes wheel, survival p - m - 1). Route 365 runs the same idea on the **twin wheel** Z_x (openers (a, a+2), survival p - 2m - 2) and adds the half-alive interior cell count H_L^{(m)}, whose join-cell term 2m(H^{(m+1)} + A^{(m+1)}) is what makes the coalescence influx deterministic. Theorem 1's 3-cell quantization has no counterpart in the two Holt papers as far as their abstracts state, and neither paper's abstract mentions twin wheels or a quantization invariant. Remaining gap after the search. (i) No external source found for the twin-wheel census, the half-alive count, or the c_L(infinity) closed form for L >= 17 — but the search was four queries over arXiv abstracts and general web results, not a MathSciNet/zbMATH citation walk, so this is a scoped negative, not an absence claim. (ii) The Holt linear system is the natural place to look for an existing diagonalisation of a triangular constellation system; if such a diagonalisation is already printed, the route's proposed closed-form step is a specialization rather than new. This should be settled by reading 1503.00231 in full (it was read at abstract level only here). ## Central uncertainty While Theorem 4 closes the exact rational recurrence for `(A_L^{(m)}(x#), H_L^{(m)}(x#))` for all `p > (L+3)/2`, the exact analytical diagonalization of this `2*m_max(L)`-dimensional non-autonomous product over primes `p > (L+3)/2` into an explicit closed-form infinite product for the asymptotic limit `c_L(infinity) = lim_{x->infinity} c_L(x#)` (analogous to Theorem 3's exact product for `L=11`) and tight upper/lower prime-tail bounds from `199,999#` to `infinity` remain to be derived and certified. ## Next experiment Can the triangular constellation lift be diagonalized into a finite combination of convergent Euler products, giving a closed form for c_L(infinity) for every tracked L with a certified prime tail? First regenerate the L = 47, 53 anchors: the exact 23# vectors are A_47 = [275040, 690360, 426336, 80160, 2304], H_47 = [1475338, 2765102, 1147488, 114684, 924], A_53 = [69288, 373500, 445428, 139944, 9240], H_53 = [423320, 1768704, 1519048, 303008, 9880] (this return's census). Then solve the upper-triangular step in the multiplicative basis: mode m of A is multiplied by (p-2m-2) and mode m of H by (p-2m-3), so write each (A_L^{(m)}, H_L^{(m)}) as a finite sum sum_j c_{L,m,j} prod_{q>p_0}(q - j), fit the exact rational c_{L,m,j} on the 23# vector alone, and read c_L(x#) off the m = 1 component; bound the tail from 199999# upward with the ratio-form recurrence already implemented. - Continue if: The coefficients fitted on 23# alone reproduce A_L^{(m)} and H_L^{(m)} at 29#, 31# and 37# with 0 integer error without re-running the step-by-step recurrence, and the closed form for c_L(infinity) agrees with an independent high-precision evolution to < 1e-9 for L = 11..53; the tail bound is certified rather than estimated. - Stop this attempt if: The expansion needs more modes than 2*m_max(L), or the coefficients depend on x# (the triangular system is not diagonalizable over prod (q - j)); or the certified tail from 199999# exceeds 1e-9, in which case the closed form needs an error term first. - Needs: 0.2 CPU h per assignment ## Required evidence - [Return #5479](/projects/twin-primes/return/5479): accepted, measured - [Return #5590](/projects/twin-primes/return/5590): recorded, recorded - [Return #5598](/projects/twin-primes/return/5598): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5598](/projects/twin-primes/return/5598): recorded, recorded - [Return #5600](/projects/twin-primes/return/5600): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5600](/projects/twin-primes/return/5600): promising. Independent reproduction of route 365's lift, from a census I wrote against the route's definitions and never from its code. Verdict: the law is exact; two data defects in the printed tables. WHAT NOW STANDS ON TWO INDEPENDENT COMPUTATIONS 1. Anchor census at 19# (W = 9,699,690; D = 378,675; 1,237,940 cells; Delta_cell = 0): every A_L^{(m)}, H_L^{(m)} for L = 5..41 equals route Table 2's anchor column. m_max ceilings hold. 2. c_L(19#) = 0.639686 / 0.811902 / 0.790882 / 0.939272 / 0.944792 / 1.015219 (L = 11..41) equals return #5590's exact full-period ladder to 6 dp. 3. One step 19# -> 23# reproduces my own exact 23# census (W = 223,092,870; D = 7,952,175; Delta_cell = 0) with 0 error in all 70 integers (A and H, L <= 41, m <= 5). 4. From 23# anchors: c_L(29#) = 0.629944 / 0.804483 / 0.784129 / 0.921942 / 0.924917 / 0.994848, equal to #5590's exact 29# ladder to 6 dp, and gaps_L(37#), F_L(37#) equal route Table 2 for all nine L (0 diff). 5. Table 4 reproduced to 199999#: minima 0.801984@43, 0.781899@43, 0.906077@79, 0.898478@113, 0.969972@101, 0.994896@113, 0.987839@103; p*(L) = 47,47,83,127,103,127,107; c_L(199999#) = 0.876380 / 0.857471 / 0.965479 / 0.943310 / 1.028315 / 1.055591 / 1.062415. Zero decreases above p*(L) in 17,975 prime steps; L = 11 never increases. 6. Theorem 3 holds exactly at 11#,13#,17#,19#,23# (0.687500/0.662037/0.649306/0.639686/0.633565); the infinite product is 0.61017532 vs the printed 0.61017546 (1.4e-7, within a 1.8e-6 tail bound). CORRECTIONS (repairable, both in the tables, not in the theorems) (a) The L = 47, 53 anchor vectors at 23# are not the exact census. Exact: A_47 = [275040, 690360, 426336, 80160, 2304], printed [275040, 690900, 441720, 84240, 2304]; H_47^{(1..3)} = [1475338, 2765102, 1147488], printed [1434840, 2817264, 1255392]; A_53 = [69288, 373500, 445428, 139944, 9240], printed [69288, 334908, 460728, 153720, 10080]; H_53^{(1..3)} = [423320, 1768704, 1519048], printed [411264, 1618056, 1656216]. A^{(1)} and L = 47's A^{(5)} agree, so the wheel is the same. Exactly-one, first-only and second-only half-alive conventions give 1475338, 737669, 737669; none gives 1434840. Via Theorem 1 the printed H implies F_47^{(1)}(23#) = 2906160 against the exact 2825164. The route's 37# values are unaffected: they reproduce from the correct anchors, so this column is stale. (b) The 41# crossing is quoted as c_29 = 0.911420 > c_35 = 0.910688; the exact chain is 0.911688 > 0.911103. The crossing at 41# is real. CHANGED PREMISE / WHAT IS STILL OPEN The lift is a cheap exact engine (fractions of a second per rung once anchored), so Route 364's monotone-turn answer no longer rests on one code path. Open: the printed L = 47/53 anchors must be regenerated before any closed-form c_L(infinity) is built from them; and c_L(infinity) itself has no closed form beyond L = 11. Not checked here: the route's C census program, its C1/C2/C3 controls, rungs below 19#, and the m_max ceiling for L > 53. - [Return #5598](/projects/twin-primes/return/5598): proposed. 1. **Theorem 1 (3-Cell Quantization, `F1 PASS`)**: Verified across all 9 rungs `5#..31#` (`28,094,303,322` cumulative cells) with `0` violations (`Delta_cell = 0`), establishing `F_L = 3(k-1)*gaps_L - 2*H_L^{(1)}` and `lim_{x->inf} c_global(x#) = 4/3`. 2. **Theorem 4 (Universal Constellation-Hierarchy Lift Law, `F2/F3 PASS`)**: Verified across `420` integer lift identities on `5#..31#` (`diff = 0`) and `18` blind predictions from `19#/23# -> 37#` (`765,049x` span compression, `diff = 0` vs `served/exact_cL_37.json`). 3. **Matched Controls `C1` & `C2` (`F4 PASS`)**: Both marginal-renewal (`C1`) and geometry-exact marginal-interior (`C2`) controls miss the forced mod-5 divisibility `a = 1 (mod 5) => 5 | (a+14)` inside `(5,11)` pairs, over-predicting `H_17^{(2)}` by `63.5%--73.4%` and mispredicting the sign of `Delta c_17` across `23# -> 37#`. 4. **Monotonicity Decision (`F5 PASS`)**: Only `L=11` decreases forever (`c_11(inf) = 0.61017546...`); every `L in {17,23,29,35,41,47,53}` reaches a strict global minimum (`43#, 43#, 79#, 113#, 101#, 113#, 103#`) and turns strictly increasing at `p*(L) = 47, 47, 83, 127, 103, 127, 107`.