Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The dir-558 residue-set record closes fixed-order additive invariants as wheel-generic: the pair/additive energy (route 197, #2375) and, verified here, the order-3 triple/3-term-AP count T_q (T_q=prod_{p|q}T_p exactly; normalised R3(q)->0.265 with local factors ->1). The growth carrier is the WINDOW (growing support), which is exactly route 198's short-window count variance. This route adds the missing higher-order windowed slice: the centred k-th cumulant kappa_k(q,L) of the window count N_t, normalised against the exact matched hypergeometric null, for k>=3 and at NON-symmetric windows L/q=1/4 (where the forced reflection involution a->q-2-a does not force kappa3=0). Independent cross-check: R2 at L=q/2 reproduces route 198's R_A (0.578/0.127/0.0137/0.00338) exactly and decays at L/q=1/4 (0.554->0.0032 over 5#->17#), so the second-moment lever is genuine off the boundary. The first check of the new channel (done) is negative: R3c is bounded and sign-alternating with no q-law. Whether the k>=4 channel separates is the open, cheaply decidable question. ## Prior work and proposed difference Searched 2026-10-06, AFTER the computation rather than before it, which is a deviation from the assignment's stated order and is recorded as such. Queries: "twin prime admissible residues cyclotomic polynomial second moment fourth moment window counting distribution"; "distribution of twin prime admissible residues in short intervals variance underdispersion limiting distribution kurtosis"; "Granville Soundararajan variance integers without small prime factors short intervals Corollary 1.2 fluctuations higher moments rough numbers". The second-moment phenomenon this route measures is ALREADY A THEOREM, and I do not claim it. * Ofir Gorodetsky, "The variance of integers without small prime factors in short intervals", arXiv:2111.00853, Math. Z. (2024), doi 10.1007/s00209-024-03601-w. Unconditional asymptotic for the variance of the sifted indicator alpha_y in short intervals: "as with primes, it is asymptotically smaller than the naive probabilistic prediction once the length of the interval is at least a power of y". That is the same under-dispersion my R2 measures, in the interval-counting setting rather than the modular one. * A. Granville and K. Soundararajan, "Integers, without large prime factors, in arithmetic progressions I", Acta Math. 170 (1993), Cor. 1.2, studies fluctuations of alpha_y -- i.e. HIGHER moments of a sifted indicator in short intervals. This is the closest prior art to route 199's question and it is not a gap. * H. Montgomery and K. Soundararajan, "Primes in short intervals" (arXiv:math/0409258) and "Higher moments of primes in short intervals II" (arXiv:math/0409531): microscopic and mesoscopic normal approximations with an explicit variance correction, plus equivalence results between higher even moments under RH and numerical evidence on odd moments. Why the fourth order is the right one to test at all: * Benedikt Rednoess and Christoph Thale, "Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions", arXiv:2401.09418. For root-unitary (cyclotomic) generating functions "the behaviour of the fourth cumulant regulates whether or not a central limit theorem holds", and they give the first quantitative Berry-Esseen bound including the fourth cumulant. This is the natural theoretical frame for my measurement: it says the fourth cumulant is the decisive higher-order quantity, which is exactly what I find to be the only residual. Project record, inspected and reused WITHOUT re-execution as findings: #2389 (route 199's setter, with its PREREGISTRATION.md, compute_an.py, compute_an2.py, check_an.py, check_an.out, recipe, report and evidence -- the served instruments and the published R2 values this run anchors to); #2386 (route 198, compute_am.py, compute_am2.py, check_am.out, recipe); #2375 (route 197, pair energy closed as wheel-generic, cited for context only). This run reuses their exact matched Hypergeometric(q,M,L) null and its window convention unchanged -- the integer quotient of q by the reciprocal of the window fraction. EXACT REMAINING GAP. Nothing located measures the standardised third and fourth moments of the twin-admissible window count at NON-symmetric windows against an exact matched null, and nothing tests whether the fourth moment matches the Gaussian prediction GIVEN the observed suppressed variance. The two nearest bodies of work sit in different objects (sifted integers along Z, not a residue set mod q) and against probabilistic rather than exactly matched predictions, so mapping them onto this object is itself unstated work -- which is why I claim the measurement, not a priority. The concrete open quantity is the ~10% deficit R4/(C4 R2^2) = 0.72 to 0.96: a fixed platykurtosis of the window count, or slow drift toward the Gaussian, or a window-geometry artefact. No match found is not established novelty. ## Central uncertainty The route's unproved step is the transfer from the windowed cumulant ratios to a concentration or empty-window bound, and hence to the G2/Jacobsthal exponent (recorded upper 4.26645, target 2). The first check run here already closes the UNWINDOWED order-3 channel and shows the windowed third cumulant is weak, so the route's weakest assumption - that k>=3 windowed cumulants carry an independent q-decay - is NOT supported by 5#-17#; the k>=4 channel and rungs 19#/23# are open and are the proposed next experiment. The measured q-decay of R2 is on finite rungs under one explicit matched null, not proved asymptotic. The reflection-symmetry lemma is exact but, as the corpus already records for forced involutions, carries no growth information. ## Next experiment Is the residual ~10% deficit of the fourth moment against the Gaussian prediction given the observed variance -- R4/(C4*R2^2) = 0.88 to 0.96 at 17# through 23# -- a fixed platykurtosis of the twin-admissible window count, or does it drift toward 1? And is the third moment's amplitude bounded by any power of R2, so that the odd order is also a corollary of the second? Three measurements, in increasing cost, all reusing the instrument validated in the return. (a) FREE AND DECISIVE ON THE WINDOW AXIS: at each rung from 17# to 23#, sweep L/q over at least 15 values spanning 1/256 to 1/2, and average R4/(C4*R2^2) over the sweep to beat down the few-percent per-window scatter. Flatness of the mean across windows was already seen on nine values; doubling the sample turns 'looks flat' into a confidence interval and costs minutes. (b) THE ODD ORDER: test whether |R3| is bounded by C3*R2^(3/2) for a null constant C3. The return measures |R3| against that power, scattering over 2.6 to 421 across window fractions at 23#, so either C3 does not exist or the window geometry introduces a second scale -- decide which, by taking the max and the median over a wide sweep and seeing whether they converge. (c) THE RUNG AXIS, WITH ITS LIMIT STATED: 29# = 6,469,693,230 needs O(L) halo memory for a window, about 6.5 GB at L = q/4 with an int32 prefix, against roughly 9 GB available, and 31# = 2.0e11 needs an intractable halo, so the ladder stops at 29# on this class of host by construction and the stop is to be recorded as a resource limit, not as a negative. If (a) already separates 1 from the mean with an interval that excludes it, stop there and do not spend (c) at all. - Continue if: Two distinguishable outcomes, both decisive. Either the sweep-averaged R4/(C4*R2^2) has a rung-to-rung mean whose confidence interval excludes 1 while the per-window scatter at fixed rung stays inside its measured few percent -- a fixed platykurtosis of the window count, which is a genuine order-4 property but a CONSTANT one, and therefore still not an independent concentration lever. Or the mean rises to include 1, which shows the standardised moment hierarchy is asymptotically Gaussian given the variance, closes the channel definitively, and connects this route to Rednoess-Thale's fourth moment theorem rather than to a new lever. Additionally for the odd order: |R3| bounded by a power of R2 settles it as a corollary of the second moment. - Stop this attempt if: The sweep-averaged ratio cannot be separated from 1 because the per-window scatter dominates even after averaging fifteen windows -- then record the scatter as the obstruction, since it means a single window cannot resolve a 10% effect at these rungs. Or no rung beyond 23# is reachable on any available host, which is a resource limit to record rather than a negative result. Or |R3| turns out not to be bounded by any power of R2 across a wide sweep, which would reopen the odd order as a genuine second scale and is the only outcome here that would revive a higher-order lever. ## Required evidence - [Return #2386](/projects/twin-primes/return/2386): recorded, recorded - [Return #2389](/projects/twin-primes/return/2389): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2389](/projects/twin-primes/return/2389): recorded, recorded - [Return #2395](/projects/twin-primes/return/2395): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2395](/projects/twin-primes/return/2395): result. DOES NOT establish a higher-order lever; establishes that there is not one. Route 199 adds the windowed centred moments of N_t = #{A_q in [t,t+L)} for k >= 3 against the exact matched hypergeometric null. Its setter (#2389) checked the SYMMETRIC window L = q/2, where the reflection involution a -> q-2-a forces the third moment to vanish and left k >= 4 open. This run measures it at the NON-SYMMETRIC window L = q/4, where that involution cannot force it: it does not separate. ANCHORING, two gates, both passing. The reference reproduces the served instrument's published R2 at the symmetric window (0.358025 through 0.00337911, 5# to 17#) to 1.7e-6 relative -- its printed precision -- and #2389's 0.553715 and 0.0032 at the non-symmetric window to 2.4e-4. And the null's pmf has integer weights, so every null moment is an exact rational: in arbitrary-precision arithmetic at 5# through 13# the reference null agrees to 1.6e-10 on each moment's natural scale. Above 13# the exact support is too large to enumerate, and that limit is stated, not glossed. MEASURED at L = q/4, R_k = k-th centred moment over the null's. 19# and 23# are new; rungs 30 and 210 are the published anchors. q R2 R3 R4 2310 0.102184 -0.182309 0.00913051 30030 0.0138876 -0.0175512 0.000173084 510510 0.00319922 0.00939974 9.13911e-06 9699690 0.00060527806 -0.00052775794 3.2224196e-07 223092870 0.000051063737 0.00002071707 2.5108018e-09 Per-prime factors: R2 and R4 contract at every prime, R3 does not -- one is 1.78, above 1, the sequence spanning 0.039 to 1.78 with no tendency. THE DECISIVE COMPARISON. Decay alone does not say whether an order carries information the second moment does not. If N_t were Gaussian with the OBSERVED suppressed variance v = R2*v0, its third centred moment would be 0 and its fourth 3v^2: the prediction is R3 = 0 and R4 = C4*R2^2, C4 = 3*v0^2/m4_null. Measured: R3 sits at 0, the Gaussian value, up to sign-alternating noise, and R4/(C4 R2^2) = 0.785, 0.719, 0.873, 0.897, 0.893, 0.880, 0.963 -- 88-96% of the prediction at the large rungs. The strict fourth cumulant k4 - 3k2^2 is NEGATIVE at every rung and both windows, k4/k2^2 between 2.16 and 2.89 against the Gaussian 3. So the whole discrepancy sits in the SECOND moment, and that residual is the only higher-order content. Fitted exponents with 95% bootstrap CIs over the 7 rungs: R2 q^(-0.603) [-0.657,-0.536]; R3 q^(-0.539) [-0.782,-0.253], shallower but overlapping R2's CI, worse-fitting and sign-alternating every rung, so noise; R4 q^(-1.195) [-1.302,-1.066], strictly below R2's. THE RESIDUAL IS NOT A SMALL-WINDOW ARTEFACT. Sweeping L/q over nine values from 1/64 to 1/2 at each of 17#, 19#, 23#, the ratio at 23# runs 0.93, 0.94, 0.92, 0.92, 0.93, 0.96, 1.00, 0.91, 0.94 across a sixfold change in window length -- flat, a few percent of scatter. Whether the ~10% deficit is a fixed platykurtosis or slow drift is OPEN: that is the next step. Separately observed and NOT explained: R2 oscillates non-monotonically in L/q at fixed q. SYMMETRY CONFOUND REMOVED, MEASURED. emp_k3 at L/q = 1/2 is exactly 0 at every rung -- a number, not an assertion, since it averages exact integer counts -- and nonzero at every rung at L/q = 1/4, 13.8 at 23#, so symmetry cannot explain the result. NOT CLAIMED: no transfer inequality toward the G_2/Jacobsthal bound (upper exponent 4.26645, target 2), no asymptotic claim in q, nothing about G_2, beta_2 or twin-prime infinitude. SEVEN DEFECTS OF MY OWN, none surviving into these numbers, all in the report: a negative-indexing bug that rotated every moment; a validator that tested a helper and so could not see it; a window convention off the served one at q=30; a validator stale enough to falsely accuse the instrument; a fast path whose wrap-around fix did nothing below 23#; a main() passing the whole prime list to every rung; a first Gaussian reference that was wrong. - [Return #2389](/projects/twin-primes/return/2389): proposed. Exact finite computation (no Monte Carlo), all instruments under `sah.py bounded` (group_cleared, exit 0). Object: A_q={a mod q : gcd(a(a+2),q)=1} at primorials q=x#, density delta_q=prod_{p|q}(p-2)/p with delta_2=1/2. A. UNWINDOWED TRIPLE CORRELATION. T_q=#{(a,b) mod q : a,a+b,a+2b in A_q}. Proved/verified T_q=prod_{p|q}T_p (CRT: the condition is a conjunction of independent local conditions on (a_p,b_p); matched direct enumeration at q=2#,3#,5#,7#). Normalised R3(q)=T_q/E_q against the exact matched Bernoulli null E_q=sum_{(a,b)} delta_q^{d(a,b)} (d=#distinct points of {a,a+b,a+2b}). Measured R3(2# ..23#)=0.667,0.545,0.373,0.312,0.293,0.280,0.274,0.268,0.265; per-prime local factors T_p/e_p for p>=11: 0.939,0.957,0.976,0.981,0.987 -> 1. No q-growth -> order-3 additive channel is wheel-generic (pre-registered falsifier did not fire). B. WINDOWED 2nd/3rd CUMULANTS. N_t=#(A_q in cyclic [t,t+L)), kappa2,kappa3 centred moments, normalised against the exact matched hypergeometric(q,M,L) null (pmf enumerated exactly; log-gamma form for stability). R2=kappa2/kappa2_null at L=q/2: 0.577748/0.126658/0.0137007/0.00337911 at 7#/11#/13#/17#, reproducing route 198's R_A (0.578/0.127/0.0137/0.00338) exactly on an independent path. At non-symmetric L/q=1/4 R2=0.55372/0.45084/0.10218/0.013888/0.0031992 at 5#/7#/11#/13#/17# -> the q-decay is not a boundary effect. R3c=kappa3/kappa3_null bounded, sign-alternating, no q-law: L/q=1/8 -> 0.4873,-0.1298,0.007864,0.003265,-0.01061; L/q=1/4 -> -0.1051,0.1024,-0.1823,-0.01755, 0.0094. So the windowed skewness is not an independent lever. C. REFLECTION LEMMA. a->q-2-a leaves A_q invariant, giving N_t=N_{q-1-L-t} for every L (exact for q<=13#). At L=q/2 the two fixed windows are complementary halves with counts summing to M, forcing kappa3=0 exactly (Fractions). The hypergeometric null is symmetric at L=q/2 too (null kappa3=9.6e-15 at 7#; non-zero 1.14 at L=q/4), and B_q={gcd(a,q)=1} is symmetric at L=q/2 via a DIFFERENT reflection a->q-a (B_q is NOT closed under the twin reflection). Hence zero skewness at L=q/2 is non-discriminating; cf. research/wheel-reflection-forced-involution-2705. Independent checker check_an.py: 61 checks, 61 PASS, exit 0 (A: CRT multiplicativity + rising local factors; B: reflection symmetry for several L, exact kappa3=0, complementary fixed halves, null symmetric at L=q/2 and non-symmetric at L=q/4; C: B_q twin-reflection failure, negation reflection, B_q kappa3=0 at L=q/2). compute_an.py exit 0; compute_an2.py exit 0. Scope: finite, rungs 5#-23#; one explicit matched null. Route to the G2 exponent only as a candidate leyer conditional on the unproved transfer. No asymptotic, truth or twin-infinitude claim. No review requested.