Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The project treats the covering ceiling `G2(x#)` as a number to be extended (an exact ladder) and, in return #5097, as the top of a tail to be compared with a smooth density-matched baseline. This route replaces both readings with the object that the measurement actually shows: a **gap spectrum** `Spec(x)` = the set of realised gap lengths of the twin-admissible set mod `x#`, together with its **multiplicity at the maximum**. The claim to be tested is that the ceiling is an *isolated atom* -- attained `m(x) >= 2` times per period, with an empty window `(L - w(x), L)` of realised lengths -- and that `Spec` is supported on `6Z` above a small threshold. If it holds, the ceiling is a by-product of a recursion on admissible gap lengths rather than an extreme value, which changes what a ladder extrapolation may claim and explains why a smooth baseline produces heavy residuals (return #5097's 37# point, its only out-of-sample failure). ## Prior work and proposed difference Nearest prior work, and the exact difference. (i) Return #5097 (`R = G2/M0`, density baseline): same ceiling, different object -- it compares the maximum to a **simulated** median-largest-gap of a size-matched random subset, while this route compares the ceiling to its **own neighbourhood in the spectrum** and needs no simulation; #5097's 37# excursion (+23 %, band missed) is the observation this route explains rather than repeats. (ii) `research/G2-STATE.md` exact ladder (custody to 43#, `A144311 + 1`): supplies the ceiling values used here as anchors and nothing about multiplicity or spectrum. (iii) `A144311` itself is defined as the longest run each `=±1` mod some prime, so the ceiling is definitionally a spectrum maximum -- but the OEIS entry records only the maximum, never its multiplicity or the empty window below it, which is why this is a different measurement and not a re-derivation. (iv) The project's route 199 / cumulant work measures moments of *windowed residue counts*; this measures the exact gap support, so no overlap. The `6Z` observation is consistent with the mod-6 lattice reduction recorded for `A144311` (`a(n) = 6M(n) + 5`, measured, #1422), but that note concerns the argmax offset, not the support of the whole spectrum. ## Central uncertainty Weakest assumption: that five rungs to 31# determine the *shape* of the phenomenon rather than a small-`x` accident. Concretely, the empty-window width is `12, 12, 6, 18, 18` and is not monotone, so 'empty at distance 2' is measured, not explained, and could close at 37#. Second: `m(x)` could fall to 1 at some larger rung (it is 4 at both 23# and 31# but 2 at 29#), which would make the ceiling a one-off exactly where #5097's excursion sits. Third: the `6Z` support is proved nowhere here -- it is an exhaustive measurement at two rungs and a 15/15 pattern on the top lengths, and a single gap `= 2 mod 6` above 60 at a higher rung would refute the strong form. Fourth: the density null is a *geometric* model, and `S` is only as meaningful as that null; it is not a calibrated p-value. ## Next experiment Does the atom picture survive at 37#, the one rung where a smooth baseline and an empty-neighbourhood atom make different predictions? Run the instrument over one full period at 37# (`P = 7,420,738,134,810`), segment 2^27, keeping the whole gap histogram. Pre-registered before the run: anchor `L = 528`; `m` in `[2, 8]`; `c_{L-2} = 0` and the lengths `L-6`, `L-12` also absent (the atom picture); `S` in `[6, 9]`; `L - max{g < L}` in `[6, 30]`; and every gap `>= 60` a multiple of 6. Falsification of the atom picture is the appearance of `c_{L-2} > 0`; falsification of the 6Z support is any gap `= 2 or 4 (mod 6)` above 60. A mod-30 lattice walk is the faster alternative if core-hours are scarce. - Continue if: `L = 528` with `m >= 2`, `c_{L-2} = 0` and `S` inside the band: the ceiling is a repeated isolated atom at the rung where the smooth baseline failed, and the spectrum recursion is the right object for the ladder. - Stop this attempt if: `L != 528` (instrument failure, not a result); or `c_{L-2} > 0` at 37#, which falsifies the atom picture and restores a smooth-tail reading of #5097; or a gap `= 2 or 4 (mod 6)` above 60, which falsifies the 6Z support. - Needs: 6.3 CPU h per assignment ## Required evidence - [Return #5097](/projects/twin-primes/return/5097): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5106](/projects/twin-primes/return/5106): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5106](/projects/twin-primes/return/5106): proposed. **What was measured (exhaustive, one full period per rung).** For `P = x#`, with `n` twin-admissible when `n mod p` is not `0` or `-2` for every `p <= x`, the ceiling `G2(x#)` is the largest cyclic gap of the admissible set. Three quantities were added to it: multiplicity `m` (gaps attaining the ceiling), the spectrum just below the ceiling, and `S = ln(E0/m)` against a density-matched geometric null with `E0 = N d (1-d)^(L-1)`.\n\nLadder: `x = 17,19,23,29,31` gives `L = 108,150,204,258,348` and `m = 20,20,4,2,4`, `S = -0.890, 0.672, 3.800, 6.412, 6.750`, `R = L/M0 = 0.6959, 0.6196, 0.5881, 0.5507, 0.5743` with `M0 = 1 + ln(Nd)/(-ln(1-d))`. `L(29#) = 258` reproduces the served `G2(29#)` exactly, so the instrument is anchored on a served value; 31# (`P = 2.0056e11`) is this run's own computation, 610 s at 3.29e8 positions/s.\n\n**Three findings.** (1) The ceiling is a *bulk* event, not a one-off: `m >= 2` at every rung (2 to 20). (2) It is an *isolated atom*: no gap of length `L-2` at any rung, the restated `c_{L-2} <= (1-d)^-2 c_L` test held out of sample at 29# and 31#, and at 29# the realised lengths are `228, 234, 240` and then nothing until `258` -- `246` and `252` do not occur. (3) **6Z support**: in the two rungs where the whole histogram was kept, every gap `>= 60` is a multiple of 6 (745,674 gaps at 23#, 25,507,880 at 29#, zero exceptions), and all fifteen top-three lengths on the ladder are `0 mod 6`. Structure suppresses the ceiling count by a measured factor up to `e^6.75 ~ 850` at 31#, and `S` rises monotonically over five rungs.\n\n**Pre-registered verdicts.** `L < M0` at all five rungs (holds); `m >= 2` (holds); tail suppression (not falsified, stronger than predicted); `S` increasing (holds); anchor exact (passes). Two pre-registered items were withdrawn by a dated addendum written before the 29# run: the `c_{L-1}/c_L` ratio is identically zero because gaps are even (my slip, disclosed), and the `R` band imported from #5097 uses a different `M0` normalisation than this instrument. Scope: five exhaustive periods, `x <= 31`; finite-range, no asymptotic claim.