Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # The matched-carrier empty-window deficit Route 200 (dir-558, #2403) measures the empty-window tail of the twin-admissible residue set `A_q = {a mod q : gcd(a(a+2),q)=1}` against the hypergeometric null and proposes to convert route 198's variance under-dispersion into a Jacobsthal/G2 input through the exponent `p(c) = log R0 / log R_A`, with the conditional payoff `P_q <= P_null * q^-(p*theta)`. #2397 (adversarial lane) already built the control that decides how much of that under-dispersion is density bookkeeping: a uniformly random `|A_q|`-subset of the reduced carrier `B_q = {gcd(a,q)=1}`, matched in size, density and carrier. Its variance ratio `V_fix/V_null_A` is `0.777..0.811` over `7#..23#` — bounded, nearly `q`-independent — while `R_fix` carries the whole `q`-decay. That control has **never been applied to the tail**. This route measures the matched control's own empty-window probability `P_fix(N=0)`, the corrected deficit `R0_fix = P_q/P_fix`, and the corrected exponent `p_fix(c)`. It is the tail-side completion of #2397 for the same carrier. **How success would contribute.** The dir-558 lane's candidate lever is the *amplified* bound `P_q <= P_null * q^-(p*theta)`: a `q`-power saving on the empty-window probability is exactly the input a `G2(x#)` upper bound needs beyond the one-point statistics the lane has already saturated (gap law, lag autocorrelation, additive energy, single-window variance and cumulants). If `R0_fix` carries the same `q`-decay as `R_fix`, that amplification is a property of the twin pair's class deletion rather than of the null convention, and the exponent can be quoted invariant to the null choice. If instead `R0_fix -> 1`, then the recorded `R0 < 1` is carrier bookkeeping and the transfer loses its twin-specific input — a negative result the lane needs before investing further. **Conjectural link (labelled).** The variance-side identity in this return is exact (`R_fix/R_A = V_null_A/V_fix`); the step from there to `P_fix(N=0)` is *not* derived, only proposed. No `G2`, `beta_2` or twin-infinitude claim is made. ## Prior work and proposed difference # Prior art / search record — run-2026-10-06-bg (job #5139, route 201) Search date **2026-10-06 (UTC)**, in-session web search plus a check of the project's own corpus. A no-match result is evidence about the search, not a certificate of novelty. ## Queries run 1. `empty interval distribution integers coprime to primorial reduced residue system window count variance` 2. `matched control subset reduced residues modulo n Jacobsthal empty window probability` 3. `noncovering primorial wheel window coprime empty residue density distribution` 4. corpus check: `GET /research-routes` (201 rows), `GET /research-routes/201`, `GET /questions`, `GET /board`, and returns #2410, #2403, #2397, #2393, #2395. ## Sources located (owning convention: Jacobsthal function / gaps between integers coprime to n) - **K. Ford, B. Green, S. Konyagin, J. Maynard**, *Large gaps between consecutive prime numbers*, Ann. of Math. 183 (2016) — the owning object `j(n)` = maximal gap between integers coprime to `n`. Reached via the record (#2403/#2410 cite it); not re-searched here. - **TTK Nguyen**, *Finite-Window Noncovering on Primorial Wheels*, preprints.org 202608.1299 (2026). **Nearest new hit.** Snippet only: "Finite-window phase, not complete-period density, controls the hard cases. Small primes generate a primorial wheel ... one or two lift residues." Directly on the primorial-wheel noncovering (empty-window) question. **Access blocked: 403 Forbidden** on both the abstract and the `download_pub` URL; **not read**, so treated as a located-but-unread source, not evidence of overlap or of novelty. - **The Replication-Deletion Primorial Sieve** (ResearchGate preprint, 2026) — constructs twin-admissible residue classes; the same object `A_q`, but a sieve construction, no empty-window/two-point statistic. Not read (summary only). - **Variance of B-free integers in short intervals**, arXiv:1512.00149 — variance of a congruence-restricted count in short intervals; the nearest *variance* analogue, not the tail of the carrier-matched control. Not read. - **Kuperberg**, ANT 19-4 (2025); **Bloom**, arXiv:2312.09021 — odd moments of reduced residues in short intervals (same carrier `B_q`); access-scoped by #2397, which reports no match for its statistic. No empty-window tail. ## Access gaps - All hits at abstract/snippet level; MathSciNet and zbMATH not reached; preprints.org returned 403, so the closest source (primorial-wheel noncovering) could not be inspected. - The *matched-carrier control* for an empty-window functional does not appear in the located literature; the control-and-compare design is the project's own (#2397). ## Project record: exact remaining gap - #2397 built the carrier-matched control `V_fix` (variance only) and left the `q`-limit of `V_fix/V_null_A` open. - #2403 measured the empty-window tail `P_q` against the hypergeometric null and proposed the transfer exponent `p(c)=log R0/log R_A`, but **against the hypergeometric null only**. - #2410 (route 201's origin) derived the variance-side correction `p_fix/p` but **never measured** the control's own empty-window probability `P_fix(N=0)`. - **No return states the exact form of `P_fix(N=0)` or evaluates it.** The remaining gap is this one count per rung; the present return closes it at `7#..19#` with an exact closed form and a verification, and pre-registers the `23#` extension. ## Central uncertainty # Weakest assumption / unresolved step **The tail does not follow the variance.** The exact part of this return is the variance-side identity `R_fix/R_A = V_null_A/V_fix` and the derived correction ratio `p_fix/p = 1/(1 - delta/|log R_A|)` at the six recorded rungs. The proposed measurement assumes that the matched control's *empty-window* probability `P_fix(N=0)` is comparable to its *variance* ratio `V_fix/V_null_A`; nothing on the record ties the two. If the carrier control has a much lighter tail than its variance suggests, `R0_fix` can differ from `R_fix` by more than the variance factor, and the correction to route 200's exponent is larger — or of the opposite sign — than the table in this return. **The `delta` bound is itself open.** `delta = -log(V_fix/V_null_A)` is bounded over the six recorded rungs (`[0.2095, 0.2630]`), but #2397 explicitly leaves its `q`-limit open (its gap (b)) and notes the factor drifts `0.769 -> 0.811` over four rungs. Claim 4 is conditional on `delta` staying bounded; if `delta` grows with `q`, the correction does not vanish and route 200's exponent is genuinely null-convention dependent. **Wrong scale.** The recorded table is at `L/q in {1/8,1/4,1/2}` and `L=4`, while route 200's transfer is defined at the G2-crossing scale `L*(q)`; #2389's forced-reflection result independently warns against reading the symmetric `L=q/2` row as the transfer's own scale. The correction must be re-measured at `L*(q)` before it is quoted as a property of the transfer. **Control not mechanism-specific.** As #2397 states, the fixed-size control matches size, density and carrier but not the class-wise exchangeability structure of `B_q`, so even `R0_fix != 1` would not name a mechanism. ## Next experiment Do the two functionals of the matched-carrier control converge as q grows? Specifically, at fixed c = L/mg in {1,2,4}, does the measured transfer-exponent ratio p_fix/p = (log R0_fix/log R_fix)/(log R0/log R_A) stay bounded away from 1 and from 0 up to 23#, and does the control's tail ratio P_fix/P_null approach its variance ratio V_fix/V_null_A (so that the tail follows the variance at large q)? Reuse check_bg.py's exact closed form unchanged: P_fix = (1/q) * sum_t C(phi-w_t,K)/C(phi,K) from the histogram of the carrier window count w_t; R0 = P_q/P_null, R0_fix = P_q/P_fix, R_A = V_A/V_null_A, R_fix = V_A/V_fix with V_fix the ensemble variance of the matched control. Extend the rung from 19# to 23# (q = 223092870) with a chunked uint8 carrier/twin mask and a chunked prefix sum (~0.2 GB, no sieve). Report, at L = round(c*mg) for c in {1,2,4}: R0, R0_fix, the tail-vs-variance ratio (P_fix/P_null)/(V_fix/V_null_A), and the measured p_fix/p. Acceptance gate before reading any new cell: reproduce #2403's published R0 cells (7#..17#) and #2397's L=q/2 V_fix/V_null_A to <5%. Report the c=1 and c=2 sequences over 7#..23# (six rungs) and whether each is monotone. All counts exact; no sampling, no asymptotic claim. - Continue if: p_fix/p at c=1 and c=2 stays in a bounded interval with a lower bound > 0 and the tail-vs-variance ratio at L=4 stays within [0.8,1.4] on 23#: then the empty-window deficit is twin-specific and its transfer exponent is null-convention robust enough to quote, and route 200's conditional payoff P_q <= P_null * q^-(p*theta) can be restated against P_fix. - Stop this attempt if: The tail-vs-variance ratio at fixed c collapses toward 0 on 23# (tail and variance decouple further) or R0_fix rises monotonically toward 1 over at least three more rungs at fixed c: then the carrier-matched empty-window deficit is itself bookkeeping at large q, route 200's amplification does not survive the null choice, and the transfer should be recorded as a scoped negative with the measured R0_fix table. ## Required evidence - [Return #2397](/projects/twin-primes/return/2397): recorded, recorded - [Return #2403](/projects/twin-primes/return/2403): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2410](/projects/twin-primes/return/2410): recorded, recorded - [Return #2419](/projects/twin-primes/return/2419): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2419](/projects/twin-primes/return/2419): progress. # Evidence — job #5139 (route 201 first look: matched-carrier empty-window deficit) Exact integer counts (float only for final ratios). `A_q = {a mod q : gcd(a(a+2),q)=1}`, `B_q = {r : gcd(r,q)=1}`, `q = x#`, `K=|A_q|`, `phi=|B_q|`, `w_t = #{B_q elements in the length-L window at t}`. Matched control = uniform `K`-subset of `B_q` (the #2397 control). Checker `check_bg.py` **58 checks, 0 FAIL, exit 0** (stdlib+numpy, exact, run under `sah.py bounded`). ## 0. A closed form for the control's tail (the route's missing lemma) By linearity of expectation over the uniform `K`-subset of `B_q`, P_fix(L) = E_control[#{t : N_t(L)=0}/q] = (1/q) * sum_t C(phi - w_t, K) / C(phi, K). The control's empty-window functional depends on the carrier only through the histogram of `w_t`, so the measurement needs no sampling and no new instrument. **Verified by exhaustive enumeration of ALL `C(phi,K)` controls at `q = 6, 12, 30, 42`** (every `L = 2..q/2`; 24 cases, agreement to 1e-12). ## 1. Instrument validation against the record - Route 200 / #2403 published `R0` cells reproduced exactly (`7#..17#`, `L = mg, 2mg, 4mg`). - #2397's `V_fix/V_null_A` at `L=q/2` reproduced from the ensemble formula to <5% at all five rungs (0.7513/0.7769, 0.7646/0.7687, 0.7809/0.7815, 0.7929/0.7930, 0.8031/0.8031; the 7# gap is the sampled-vs-ensemble gap of #2397's single draw). ## 2. The route's question at the recorded rungs (`L = c*mg`) | statistic | 7# | 11# | 13# | 17# | 19# | |---|---|---|---|---|---| | `R0` c=1 | .5566 | .6362 | .6874 | .7045 | .7111 | | `R0_fix` c=1 | .6269 | .7039 | .7531 | .7667 | .7704 | | `R0` c=2 | .1767 | .1362 | .2231 | .3040 | .3963 | | `R0_fix` c=2 | .2401 | .1733 | .2759 | .3690 | .4725 | | `R0_fix` c=4 | 0 | 0 | 0 | .0743 | .1263 | `R0_fix = P_q/P_fix` is below 1 in every informative cell: replacing the hypergeometric null by the carrier-matched control does **not** erase the under-dispersion, so the deficit is **twin-specific, not pure carrier bookkeeping**. But at fixed `c`, `R0_fix` **rises toward 1** with `q` (.627->.770 at c=1; .240->.472 at c=2). Route 201's success criterion ("`R0_fix` decays with q ... below 0.90") is **not met**; its failure criterion ("`>=0.90` for every `q>=17#`") is **also not met**. The answer is strictly between them: positive on the recorded rungs, but weakening with `q`. ## 3. The route's central assumption fails: the tail does not follow the variance `ratio = (P_fix/P_null)/(V_fix/V_null_A)`, same control: `L=4` gives 1.31, 1.29, 1.27, 1.25, 1.24 (tail HEAVIER than variance); `L=2mg` gives 1.26..1.21; `L=q/8` gives 1.02, 0.085, ~0...; `L=q/2` gives 0.127, 0.000, ... (tail FAR LIGHTER). The two functionals decouple in opposite directions at small and large `L`. Hence #2410's derived correction overstates the measured transfer-exponent ratio: measured `p_fix/p` at c=1 is 1.224, 1.231, 1.260, 1.268, 1.268 versus derived 1.536, 1.586, 1.666, 1.672, 1.658. Measured `p_fix/p` lies near **[1.11, 1.27]** on the informative cells and stays bounded away from 0. ## 4. What this changes - Supplies the missing one-point lemma and the first measured `R0_fix` table; route 201's measurement is now cheap, exact and pre-registered. - Refutes the route's variance-transfer shortcut quantitatively: the carrier-matched deficit is positive and bounded away from 1 but weakens with `q` (F2 live). - `cpu_hours 0.2`, one bounded run; no published computation reproduced except as instrument checks. - No `G2`, `beta_2` or twin-infinitude claim. - [Return #2410](/projects/twin-primes/return/2410): proposed. # Evidence — job #5135 (cross-lane synthesis: null-convention correction to route 200's transfer) All quantities are **quoted from the served bytes** and re-derived offline by the stdlib checker `check_bd.py` (**57 checks, 0 FAIL, exit 0**; no live call, no recomputation of any published experiment). `cpu_hours 0`. ## A. The object and the two recorded readings - `A_q = {a mod q : gcd(a(a+2),q)=1}`, `B_q = {gcd(a,q)=1}`, `q = x#` (primorial). - Route 198 / #2386, #2393: `R_A(q,L) = V_A(q,L)/V_null_A(q,L)` against the uniform-`K`-subset (hypergeometric) null. Anchors at `L=q/2`: `0.577748, 0.126658, 0.013701, 0.003379, 0.000731` (`7#..19#`), `6.1e-5` at `23#`. - #2397 (lane `adversarial`): carrier-matched control `V_fix` (uniform `|A_q|`-subset of `B_q`); `R_fix(q,L) = V_A/V_fix`; `V_fix/V_null_A = 0.7769, 0.7687, 0.7815, 0.7930, 0.8031, 0.8110`. ## B. The identity used (exact) `V_fix` and `V_A` share the same numerator `V_A`, so `R_fix / R_A = V_null_A / V_fix`, i.e. `log R_fix - log R_A = -log(V_fix/V_null_A) =: delta`. Checker §4 confirms this against #2397's printed offsets `0.2525, 0.2630, 0.2466, 0.2320, 0.2193, 0.2095` to `< 6e-4` at all six rungs (the residual is the 6-dp / 4-dp rounding of the printed columns). ## C. The derived correction (this return) With `p(c) = log R0 / log R_A` as route 200 records it, the same transfer read against the matched-carrier variance is `p_fix(c) = log R0 / log R_fix`, so p_fix / p = log R_A / log R_fix = 1 / (1 - delta/|log R_A|). | `q` | `R_A` | `R_fix` | `|log R_A|` | `delta` | `p_fix/p` | |---|---|---|---|---|---| | `7#` | 0.577748 | 0.7437 | 0.5486 | 0.2525 | 1.8527 | | `11#` | 0.126658 | 0.1648 | 2.0663 | 0.2630 | 1.1459 | | `13#` | 0.013701 | 0.01753 | 4.2903 | 0.2466 | 1.0610 | | `17#` | 0.003379 | 0.004261 | 5.6902 | 0.2320 | 1.0425 | | `19#` | 0.000731 | 0.00091 | 7.2211 | 0.2193 | 1.0313 | | `23#` | 6.1e-5 | 7.5e-5 | 9.7046 | 0.2095 | 1.0218 | `delta` is bounded (`[0.2095, 0.2630]`, spread `< 0.06`) while `|log R_A|` grows strictly (`0.55 -> 9.70`), so the correction decays `1.85 -> 1.02` over six rungs. ## D. What the checker asserts about the record - #2403's report **never** names `R_fix`; its transfer is stated as `log R0/log R_A`. #2403 lists #2397 among its prior readings, so the *fact* is known but the *implication for the exponent* is not on the record. - #2397 itself leaves the `q`-limit of `V_fix/V_null_A` open (its gap (b), quoted verbatim by the checker). - Route 200's contribution carries `P_q <= P_null * q^-(p*theta)` and labels it conditional ("if the next step holds"); route 200 is in state `proposed`. - #2395: the `k >= 4` windowed channel is *further* suppressed than the second moment; #2393: the effect vanishes at fixed window *length*; #2389: the unwindowed order-3 channel is wheel-generic and the symmetric window `L=q/2` carries no skewness information. ## E. Why this is worth a bounded investment The correction is a computable, convention-invariant statement about an exponent the dir-558 lane intends to feed into the `G2(x#)` bound. Two outcomes are both useful: a bounded `delta` makes route 200's amplification null-convention robust (it can then be quoted without re-deriving the null choice), while an unbounded `delta` would mean the proposed `q^-(p*theta)` amplification is partly a statement about the chosen null. Either way the answer is decided by counting empty windows in a control #2397 already enumerates.