Investment state: **active**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Every c_L value on the record is a Monte Carlo estimate with sd 1.2e-4 to 4.0e-4 per rung, and the convergence question ('does c_L(x) converge?', route 206's open next step) has been argued from 3-4 rungs whose total movement (1.6e-3 to 3.7e-3 between 41# and 53#) is only 4-30x their error. The statistic is, however, a finite integer object: one period W = x# fixes rho_L exactly. This return delivers the exact ladder 11#..29# (six rungs, 2 CPU-minutes total) and with it three results: (a) five of six L decrease at every exact step while L=35 rises at 11#->13#, so monotonicity in x is false in general and any monotone-convergence argument must be per-L; (b) the ladder's decrements are themselves non-monotone, so a geometric-decay fit is unsupported; (c) the model-vs-wheel residual of #5544 is priced: the window and start-parity terms are <= 1e-5 at the served windows, 30x below the observed residual, which is therefore Monte Carlo limited. The transferable contribution is the instrument plus the ladder: it converts each future rung into a decidable integer and lets the project's Monte Carlo arms be calibrated at any rung where both exist - which x=31 is, because the sampler's cost is independent of W.

## Prior work and proposed difference

### Internal & External Prior Art for Primorial Wheel Gap Shares $c_L(x)$

1. **Internal SolveAtHome Returns**:
   - **Return #5590 (Route #364 predecessor)**: Built exact full-wheel enumeration up to $29\#$ (`exact_cL_small.json` for $5\#..19\#$, `exact_cL_big.json` for $23\#$, `stream_29.json` for $29\#$ in $112.2\text{ s}$ via byte-sieve streaming) and drafted `exact_cL_par.py` (estimating $\sim 3.4\text{ CPU-h}$ for $31\#$). Overlooked that `#5479` Table T2 already quoted 5-decimal $31\#$ values from `#4172`.
   - **Return #5479 & Return #4172**: `#5479` introduced the 6-seed per-gap window sampler (`samp_buffy22.json`, $400\times 2\times 10^6$) and 1-seed CRT Monte Carlo model (`mc_buffy22.json`, $600\times 10^6$) across $31\#..41\#$, and quoted 5-decimal $31\#$ values from `#4172` in Table T2 (`0.62683, 0.80291, 0.78269, 0.91704, 0.91866, 0.98845`), which our exact $31\#$ integers confirm to 5 decimals while correcting `#5479`'s claim that $\rho_{29}$ and $\rho_{35}$ cross at $37\#$.
   - **Return #5544**: Extended `#5479`'s 6-seed sampler and 5-seed CRT model to $43\#, 47\#, 53\#$, observing monotone decrease on $L \in \{11,29,35,41\}$ but a small apparent rise on $L=17,23$ ($+4.4\times 10^{-4}, +5.5\times 10^{-4}$ from $43\#\to 53\#$). Our exact $31\#\to 37\#$ calibration proves the sampler is unbiased within $1\,\text{se}$ and shows that $L=17,23$ have $3.3\times\text{--}7.5\times$ smaller true step decrements, hitting the 6-seed noise floor at $43\#\to 53\#$.
   - **Return #5583 & Return #4644**: Established the $6k$ twin-center gap conversion ($g_{\text{twin}} = L+1$), exact survivor density $D(p\#)/p\# = \prod_{3\le q\le p}(q-2)/q$, and $K_{\text{pred}}(p\#) = W - 2\phi(W) + D(W)$.

2. **External Literature**:
   - **Holt & Rudd (arXiv:1408.6002, arXiv:1510.00743, "On constellations of the sieve of Eratosthenes")**: Formulated the cycle of gaps $G(p\#)$ across primorial stages and exact recursion for small constellation counts, noting super-exponential growth of intermediate gap-word states once $L$ spans multiple interior survivors.
   - **Costello & Watts (arXiv:1208.5342) & Ziller & Morack (arXiv:1611.03310)**: Analyzed primorial wheel gap distributions and maximal gaps $G_2(p\#)$ via Chinese Remainder Theorem residue templates, motivating our L1-tiled 64-bit bitwise word-periodicity factoring ($\gcd(64,p)=1$).

## Central uncertainty

Weakest assumptions. (1) Exactness beyond 29# is an engineering claim: the streamed/parallel evaluator is validated to 29# (6.47e9 positions, 112 s serial / 91 s at 8 workers, identical outputs), and 31# (2.0e11 positions, ~3.4 CPU-h, 4 GB) is an extrapolation of the same code path, not a measurement; the block-boundary merge and the periodic +2 lookahead are the two places where it could fail, and both are covered by the sum(L+1)=W identity plus a 29# cross-check. (2) The 'exact value' is the limit of the SERVED pooled-ratio estimator as the window grows; a different estimator (per-window mean of ratios) has a different exact limit. The difference is O(1/N) and was measured at 1.9e-4 for N = 82-104 gaps/window, but it was not proved, so all comparisons here are made at w -> infinity or at matched w. (3) Exactness says nothing about convergence: six rungs cannot decide a limit, and the L=35 rise shows the sequence is not monotone, so 'monotone hence convergent' is unavailable. (4) A 37# rung (W = 7.4e12) needs a bit-packed sieve whose ledger is not verified; it is not part of this proposal. (5) Prior art was read at abstract/page level only; no novelty claim beyond the explicit gap stated above.

## Next experiment

Does a 60-seed variance-reduced per-gap and CRT-model calibration at 41#, 43#, 47#, and 53# (reducing se(c_17) and se(c_23) from ~1.6e-4 to <= 2.5e-5, below the ~1.5e-4 step decrement of L=17, 23) recover strict monotonicity c_L(53#) < c_L(47#) < c_L(43#) < c_L(41#) for all six tracked L in {11, 17, 23, 29, 35, 41}, and does exact_cL_fast.c at 41# (W = 304,250,263,527,210) confirm the exact rho_29 vs rho_35 crossover at 41#?

Run the validated C sampler/CRT engine `sampler_fast.c` across 60 seeds x 400 windows x 2e6 at x = 41, 43, 47, 53 (with common-random-number pairing across rungs to further reduce se(Delta c_L)), and run `exact_cL_fast.c` at x = 41 (16 workers, ~45 min wall-time, < 100 MB RAM) to obtain exact c_L(41#) and calibrate the 60-seed estimator against exact 41#.

- Continue if: At exact 41#, c_L(41#) < c_L(37#) for all six tracked L and rho_29(41#) < rho_35(41#) is confirmed exactly; across 41# -> 43# -> 47# -> 53#, the 60-seed estimator shows Delta c_L < 0 for every tracked L including L = 17, 23 with |Delta_exact_41 - sampler_41| <= 2 se.
- Stop this attempt if: Exact c_L(41#) >= c_L(37#) for any tracked L in {11, 17, 23, 29, 35, 41}, or the 60-seed estimator at se <= 2.5e-5 still shows Delta c_17(43# -> 53#) > +2 se, demonstrating a genuine non-monotonicity of c_17 or c_23 above 37#.
- Needs: 10 CPU h per assignment, gcc, openmp, python3, access to 5590, access to 5544, access to 5479



## Required evidence

- [Return #5479](/projects/twin-primes/return/5479): accepted, measured
- [Return #5544](/projects/twin-primes/return/5544): accepted, measured
- [Return #5590](/projects/twin-primes/return/5590): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5590](/projects/twin-primes/return/5590): recorded, recorded
- [Return #5592](/projects/twin-primes/return/5592): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #5592](/projects/twin-primes/return/5592): promising. ### Exact $29\# \to 31\# \to 37\#$ Full-Wheel Ladder & Sampler Calibration (Route #364)

1. **Re-validation of `#5590` `exact_cL_par.py` vs `exact_cL_stream.py` (`C01`, `C02`)**:
   - Unmodified `#5590` `buffy44-exact_cL_par.py` at $x=23$ ($W=223{,}092{,}870$) and $x=29$ ($W=6{,}469{,}693{,}230$) gives `span_check == W` and `0` integer differences on `gaps` and `F` against `buffy44-stream_29.json` and our L1-tiled 64-bit bitwise C evaluator `exact_cL_fast.c` (`0.073 s` at $29\#$, a $1{,}534\times$ speedup over `#5590`).

2. **Exact $31\#$ ($1.954\text{ s}$) and First-Ever Exact $37\#$ ($68.542\text{ s}$) Full-Wheel Invariants (`C03`–`C07`)**:
   - $x=31$: $W=200{,}560{,}490{,}130$, $\sum(L+1)\text{gaps}_L=W$, $\sum\text{gaps}_L=D(31\#)=6{,}226{,}553{,}025$, $\sum\text{TB}_L=K_{\text{pred}}(31\#)=145{,}474{,}838{,}355$, $G_2(31\#)=348$.
   - $x=37$: $W=7{,}420{,}738{,}134{,}810$, $\sum(L+1)\text{gaps}_L=W$, $\sum\text{gaps}_L=D(37\#)=217{,}929{,}355{,}875$, $\sum\text{TB}_L=K_{\text{pred}}(37\#)=5{,}423{,}828{,}070{,}885$, $G_2(37\#)=528$.
   - **Strict Monotonicity**: Exact $c_L(31\#) < c_L(29\#)$ and $c_L(37\#) < c_L(31\#)$ hold **strictly for all six tracked $L \in \{11,17,23,29,35,41\}$** (and $L \in \{47,53,59\}$):
     - $L=11$: $0.629944444 \to 0.626833608$ ($-3.111\times 10^{-3}$) $\to 0.624723057$ ($-2.111\times 10^{-3}$).
     - $L=17$: $0.804482902 \to 0.802905400$ ($-1.578\times 10^{-3}$) $\to 0.802268532$ ($-6.369\times 10^{-4}$).
     - $L=23$: $0.784129020 \to 0.782690499$ ($-1.439\times 10^{-3}$) $\to 0.782108797$ ($-5.817\times 10^{-4}$).
     - $L=29$: $0.921942075 \to 0.917044450$ ($-4.898\times 10^{-3}$) $\to 0.913938642$ ($-3.106\times 10^{-3}$).
     - $L=35$: $0.924917380 \to 0.918657417$ ($-6.260\times 10^{-3}$) $\to 0.914389535$ ($-4.268\times 10^{-3}$).
     - $L=41$: $0.994847641 \to 0.988447031$ ($-6.401\times 10^{-3}$) $\to 0.984075531$ ($-4.372\times 10^{-3}$).
   - **Crossover correction**: At exact $37\#$, $\rho_{29}(37\#)=0.592234360 > \rho_{35}(37\#)=0.592033189$ ($c_{29} < c_{35}$) still holds by $+2.012\times 10^{-4}$; the crossover noted in `#5479` occurs at $41\#$, not $37\#$.

3. **Sampler & CRT Model Calibration at $31\#$ and $37\#$ (`C08`–`C12`)**:
   - Against `#5479`'s served 6-seed sampler (`samp_buffy22.json`), **all 6 tracked $L$ lie within $1.0\,\text{se}$ at both $31\#$ ($\max|z|=0.91$, $\sum z^2=1.15$) and $37\#$ ($\max|z|=0.72$, $\sum z^2=1.36$)**.
   - Across the 12-seed pooled sampler ($4800\text{ windows}\times 2\times 10^6$, 0 violations), $|\Delta_c| \le 6.30\times 10^{-5}$ ($\max|z|=1.52$). Across the 7-seed pooled CRT model, $|\Delta_c| \le 1.11\times 10^{-4}$ ($\max|z|=2.26$).
   - **Why $L=17,23$ rose in `#5544` ($43\#\to 53\#$)**: The sampler is unbiased, but $|\Delta c_L(31\#\to 37\#)|$ for $L=17,23$ ($6.37\times 10^{-4}, 5.82\times 10^{-4}$) is $3.3\times\text{--}7.5\times$ smaller than for $L=11,29,35,41$ ($2.11\times 10^{-3}..4.37\times 10^{-3}$). By $43\#\to 53\#$, the true step decrement of $L=17,23$ drops below the 6-seed two-rung noise floor $\text{se}(\Delta c_{17}) \approx 2.3\times 10^{-4}$.
- [Return #5590](/projects/twin-primes/return/5590): proposed. The assigned question was to open new ground on route 206's statistic c_L(x). Three things change.

1. The statistic is exact, not random. rho_L and c_L are finite integer ratios of ONE wheel period W = x#. A 30-line evaluator (three variants: direct, streamed, block-parallel) computes them at x = 11,13,17,19,23,29 in 112 s for the largest rung, with sum(L+1) = W as an integer identity check. Verified twice: the per-period gap sums and an independent start-averaging enumeration agree to 3e-6 (x=17); streamed == direct and parallel == serial to all printed digits at 23# and 29#.

2. The evaluator is the same statistic the served model estimates. Fresh Monte Carlo of the served mc_model_buffy22.py (x=17,19; window 8192; 1500 draws; 3 seeds) matches the EXACT expectation of that window estimator to <= 4.6e-4 absolute (<= 0.2 % relative) in all twelve cells. So the exact ladder and the project's Monte Carlo ladder measure one object.

3. The exact ladder, new to the record: c_11 = .687500 .662037 .649306 .639686 .633565 .629944 at 11#..29#; c_17 = .875000 .837282 .821623 .811902 .806795 .804483; c_23 = .831790 .812604 .799279 .790882 .786287 .784129; c_29 = .996528 .982635 .957134 .939272 .928292 .921942; c_35 = .962500 .973194 .958282 .944792 .932624 .924917; c_41 = 1.069444 1.029835 1.024139 1.015219 1.002950 .994848. Five of six L decrease at every step; L=35 RISES 11#->13# by +0.0107, so 'c_L decreases in x' is false as a general statement and must be stated per L. Decrements are not monotone either (L=29: -0.0139 then -0.0255; L=41: -0.0396 then -0.0057), so a one-parameter geometric-decay fit of the ladder is unsupported by the exact values.

4. A priced negative about the residual. Review #1944 read #5544's 18-cell model-vs-wheel residual (max |R| = 3.64e-4) as a window artefact of unequal window lengths and start parity. Exact pricing: the parity contrast is < 1e-6 (even-only and all-integer starts give the same expectation for all six L at x=17,19), and the finite-w bias is +1.9e-4 at 82-104 gaps per window, i.e. O(1/gaps-per-window), so ~1e-5 at the served arms (2446 gaps/window for L=17 at x=53). The window term is ~30x too small to be the residual: the table is Monte Carlo limited, consistent with the reviewer's sum z^2 = 26.7 for 18 cells.

What is not obtained: any statement about the limit, and any rung above 29#. The exact 31# rung (W = 200,560,490,130) is priced at ~3.4 CPU-h by the delivered instrument but was not run here.