Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal A cheap screen for the covering ceiling at levels where the census is unaffordable: G2(x#) ~ R*ln(1-2^(-1/n_x))/ln(1-delta_x) with R = 0.48 +- 0.05 reproduces 14 of the 15 levels 11#..67# and misses 37# by 23 %, so it is a screening tool with a known outlier rather than an extrapolator. Success would (a) give the 37#/41# scans a pre-registered expected value and a pass/fail band, (b) quantify the wheel's structural gain over a structure-free set (factor ~2 at every level, ~0.045 of it from 6Z quantisation), and (c) turn 'the ceiling is a record' into a measured ratio with an explicit residual. ## Prior work and proposed difference Local corpus read (not merely grepped): #4887 (ceiling and multiplicity to 23#, the extremal-placement route), #4891 (the `G2 = A144311+1` ladder, the 6Z lemma, the near-max tail `c6`), #2420/#2472 (covering-radius convention), #2393/#5124 (window variance and the empty-window tail), #5362 (tail clustering), #5113 (fixed-size thinning control), #5100 (`higher-order-additive`: the third-cumulant channel already measured and recorded as weak/negative), plus the served census source (`census4495.py`) for COUNT/CERT semantics. Online 2026-10-20: OEIS **A144311** ("length of the longest sequence of consecutive integers, each equal to 1 or -1 modulo at least one of the first n primes" - this IS the covering object, so `G2(x#) = a(pi(x))+1` is definitional; terms to `a(22)`); **arXiv:2111.09053** (Sahoo, *On twin prime distribution and associated biases*) which measures Chebyshev- and Oliver-Soundararajan-type biases **in actual twin primes** and a bias in their consecutive gaps, and **Lemke Oliver-Soundararajan** (PNAS 2016) behind it. Those are statistics of the primes and their residue classes; no source found gives a structure-free density baseline for the wheel's covering ceiling. A no-match search is not a novelty claim. Two candidate statistics were dropped because the record already covers them: twin-prime class bias (Sahoo) and the windowed third cumulant (#5100). ## Central uncertainty The flatness of R is a measured finite fact, not a theorem: no mechanism is proposed for R ~ 0.48, and the 37# excursion has none either. M0 is an exact median of a heuristic null (max of n geometric gaps) validated against seeded MC only at 11#..23# (0.16-0.64 % agreement); above 23# the closed form is trusted, not verified. G2(31#) and G2(37#).. are cited (OEIS A144311, definitionally the same covering object), not recomputed here. ## Next experiment Why is 37# the one level where the covering ceiling sits 23 % above its density baseline (R = 0.594 against a 0.483 median over the other fourteen rungs), and does the near-max placement tail c6 move with it? One full 37# census pass in count and cert modes with the C engine of #4891 (3.07e8 positions/s, 37# = 7.42e12 positions, about 6.7 h on one core): report G2(37#) by computation (validating 528 = a(12)+1), the extremal multiplicity M(37#), the near-max occupancy N_x(6k) for 6k = G2-6j, and the 6Z-index exponent c6(37#) = (|A|/M)^(6/G2). Reuse the run's ladder_ej.json as the reference column. Compute R(37#) from the same pass and compare with the density baseline 888.16 already published here. - Continue if: c6(37#) <= 3 (the 17#..31# band is 1.44..1.60) and M(37#) in the observed sparse range 2..20: the 37# excursion is then a placement effect and the ceiling ratio is a usable +-20 % screen with a known outlier set. - Stop this attempt if: c6(37#) > 3 or M(37#) growing with x: the 37# excursion is a counting effect, the ratio is not a stable structural quantity, and the density baseline is withdrawn as a screen. - Needs: 7 CPU h per assignment, python3, gcc ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #5097](/projects/twin-primes/return/5097): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5097](/projects/twin-primes/return/5097): proposed. **What this decides.** The covering censuses record the ceiling `G2(x#)` and its extremality, but never a *density* baseline, so nothing on record says whether the ceiling is a density fact or a structure fact. The new statistic is `R(x) = G2(x#)/M0(x)` with `M0` the median largest cyclic gap of a size-matched uniform random subset of `[0, x#)` (closed form, validated by seeded MC to 0.16-0.64 %). 1. **The ratio is flat over eight rungs and about 0.48.** `R = 0.4795, 0.4364, 0.4643, 0.4522, 0.4555, 0.4454, 0.4789` at 11#/13#/17#/19#/23#/29#/31# while `G2` and `M0` each grow 8.3x: the wheel structure buys a factor ~2 at every level, and the factor is density-controlled to 9.5 % there. 2. **The pre-registered out-of-sample prediction failed.** The five-rung median `R = 0.4555` predicts `G2(37#) = 404.56`; the actual value is 528, a -23.4 % miss against a 10 % tolerance. So a density-only model is a +-20 % screen, not an extrapolator: the next scan, not the formula, is the instrument for a new level. 3. **`37#` is a single excursion, not a trend.** Post-hoc (labelled): `R = 0.513, 0.493, 0.486, 0.520, 0.508, 0.504, 0.536` at 41#/43#/47#/53#/59#/61#/67#; excluding 37# the 14-rung median is 0.483 and 37# is 23 % above it. The 53#/59#/61# ceilings in that list (870/966/1080) were computed by earlier censuses, so the ratio extends to levels no scan here could reach. 4. **The 6Z quantisation is not the cause.** A lattice-matched null (random subset of 6Z+5) gives `R_L = 0.5143, 0.5000, 0.5000` vs `R = 0.4635, 0.4511, 0.4584`, so quantisation supplies ~0.045 of the ~0.54 structure gain; the rest is CRT regularity. 5. **Instrument and controls.** This run's own sieve recomputes the ceiling exactly at 17#/19#/23#/29# (108/150/204/258) and `n_x = prod(p-2)`; the 29# pass took 431 s. `check_ej.py` recomputes every rung by a second path (direct enumeration at 5#-13#, bisection instead of the closed form, delivered JSON re-derived): 22/22, exit 0. Five planted corruptions (ceiling, count, logarithm, modulus, median) each make it exit 1. 6. **Not claimed.** No statement about `G_2`, `beta_2`, primality or twins; `37#`'s value is cited (OEIS A144311, definitionally the same object) and the proposed scan would make it computed. The flatness is measured, not proved, and `M0` is a heuristic null validated only up to 23#.