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

## Contribution to the goal

**What success would contribute.** The lane's two natural cyclic carriers of the twin wheel are the
reduced-residue arrangement `R` (`N_R = phi(P)`, routes 180/186) and the paired twin-candidate
arrangement `A = R ∩ (R-2)` (`N_A = prod_{p>2}(p-2)`, route 188's order-blind carrier of `G2`).
Every piece of quantitative machinery the lane owns lives on `R`; if the carriers were bridged, it
would run on `A` — the carrier the exponent route actually needs — instead of being rebuilt there.

**Object.** `q1(x) := rho_1(A(x#)) / rho_1(R(x#))` and its rescaled form `(1 - q1(x))*ln x`.

**Step that would have to hold.** `q1(x) = 1 - c/ln x` for a constant `c`; equivalently
`-rho_1(A)(x)*ln x = 1/2 - c/(2 ln x)`, i.e. the paired carrier inherits the reduced carrier's
recorded `-rho_1(R)*ln x -> 1/2` lag-1 law plus an explicit closed-form `O(1/ln x)` offset. This is
weaker than the constant bridge route 202 refuted (`c = 0`) and stronger than "no bridge at all".

**Measured basis supplied by this return.** Both carriers exactly through `29#`. The raw ratio keeps
rising with a growing increment (`q1 = 0.2131, 0.2468, 0.2845, 0.3336` at `17#,19#,23#,29#`; steps
`+0.0337, +0.0377, +0.0491`), but the rescaled ratio is flat: `(1 - q1)*ln p = 1.279, 1.806, 2.229,
2.218, 2.243, 2.244`, i.e. `2.23 +- 0.03` over `17#..29#` (1.2 % spread) against a 39 % spread of the
raw `q1` over the same four rungs. `11#`, `13#` lie off the band and are reported as outside it.

**Conjectural links, labelled.** (i) If the step holds, the transfer is legitimate and the reduced
lane's recursion may be rescaled onto `A`; this is a link, not a theorem — the record's standing
position is that any bearing of `rho_k` on the `G2` exponent is conjectural (route 188's uncertainty).
(ii) If it fails, the carrier split is structural, which is route 202's own failure clause. Neither
reading bounds `G2` or says anything about twin-prime occurrence.

**Second, independent finite event.** `d_k := rho_k(R) - rho_k(A)` crosses zero for the first time at
lag 2 between `23#` and `29#` (`-0.00739 -> +0.00399`, narrowing monotonically over five rungs);
`d_3` crosses at `13#->17#`; `d_1` never.

## Prior work and proposed difference

# Prior art — job #8612 (explore/`first_look`, route 293, lane dir-558)

**Online search, 2026-10-16 (this run, before any 31# number was read).** Queries run:
(i) `autocorrelation of gaps between reduced residues modulo a primorial wheel statistics`;
(ii) `Jacobsthal function primorial gaps between coprime residues distribution statistics paper`;
(iii) `"Harmonic Resonance in Twin Prime Distribution" autocorrelation admissible residue classes 1485`;
(iv) `primorial gap sequence autocorrelation "1/ln x" law reduced residue system cyclic moments`.

**Findings.**
1. **Nothing external carries the object of this route.** Queries (i) and (iv) return this project's
   own **route 180** page first (its record states that *"no source found computes the lag-1
   autocorrelation of the reduced-residue gap sequence of a primorial over a full period"*); the rest
   of (i)/(iv) is generic cyclic-autocorrelation and signal-processing literature, or unrelated
   economics/engineering hits. `q1 = rho_1(A)/rho_1(R)`, its rescaled form `(1 - q1)*ln x` and the
   constant `2.23` appear nowhere outside this project's record. This confirms route 293's own
   `prior_art_md` (written by #4195) on a fresh search.
2. **Closest external neighbour (unchanged from #4195's search):** the ResearchGate note *Harmonic
   Resonance in Twin Prime Distribution: Empirical Evidence of Phase-Locking and Under-Dispersion*
   (Dec 2025). Query (iii) resolves it and confirms its scope: *"Distribution of twin prime counts
   across all 1,485 allowed [residue classes]"* — i.e. this project's `A(13#)`, `N_A(13#) = 1485`. It
   reports residue histograms and an autocorrelation of that residue set; it has **no bridge ratio, no
   `R` comparison, no `1/ln x` law**, and its full text remains unreachable (access gap). Nothing in
   it competes with this route.
3. **Closest external work on the same carrier (new in this search).** M. Ziller, *On differences
   between consecutive numbers coprime to a primorial*, arXiv:2007.01808 (2020) — the Jacobsthal
   function on primorials, i.e. the **extreme** gap `max g` of exactly this reduced carrier `R`
   (equivalently, large gaps in sets of primes and Ford's colloquium on `J(x)`). It studies the
   extremal gap, not the **lag-k moment structure** of the gap word (`S_k = sum_i g_i g_{i+k}`), and
   says nothing about the paired carrier or any ratio between carriers. OEIS's Jacobsthal-function
   pages (`h(n)` for primorials) are the same extremal statistic. A forum-grade post (unverified,
   non-authoritative, dated 2026-09-24) claims exact small primorial Jacobsthal values (`j(P_11) = 58`,
   `j(P_12) = 66`); it is about the same extremal gap and does not touch this route. No external
   source computes `rho_k` of either carrier, so no external number is reused here.
4. **The record this route builds on (inspected, not reproduced).** #2299 / #2426 (route 202) define
   both carriers, publish the `11#..23#` paired table and refute the **constant** bridge `c = 0` from
   the raw spread of `q1` (28.8 % over `17#..23#`). Route 186 records `rho_2, rho_3` of `R` on
   `13#..23#` with "`29#` and `31#` were not run here" as its open gap. Route 180 records the
   `-rho_1(R)*ln p -> 1/2` diagnostic reused here as an internal check. #4195 (route 293's origin)
   extended both carriers to `29#` and formed the **rescaled** ratio for the first time.

**Exact remaining gap this assignment targets.** The record's refutation is of `c = 0`, not of the
family `q1 = 1 - c/ln x`; the four-rung flatness `(1 - q1)*ln p = 2.23 +- 0.03` over `17#..29#` was
formed *after* seeing `29#` (post-hoc), and **no `31#` value of either carrier exists on the record**
(#4195: "`31#` was not run (its paired word is 6.6e9 gaps); the route's whole claim is pre-registered
against that rung"). The one-rung-out test at `31#`, with the numeric window fixed before the
[trimmed]

## Central uncertainty

**Weakest unproved assumption.** That the four-rung flatness of `(1 - q1)*ln p` is a law and not a
finite-wheel coincidence. Only four points lie inside the band; `11#` and `13#` lie well outside it
(`1.279`, `1.806`), so a slowly drifting `c` is not excluded, and the hypothesis was formed *after*
seeing `29#`, i.e. post-hoc. The `31#` pre-registrations (`c(31) = (1 - q1(31))*ln 31` inside
`[2.15, 2.33]`, and `d_2(31) > 0`) are therefore stated as numeric windows before any `31#`
computation; the return contains no `31#` value.

**Second.** `rho_k` is one arrangement functional. Even a confirmed bridge constrains this diagnostic,
not `G2`: the link from `rho_k` to the `G2` exponent remains conjectural, exactly as route 188's
uncertainty states, and nothing here bounds `G2` or bears on twin-prime occurrence.

**Third.** The independent re-derivation is an exact integer computation at the rungs tested; it
validates the instrument and the lift, not the extrapolation of the flat constant to `31#`.

**Fourth.** The scaled claim rests on `ln p` with `p` the new prime, the convention of routes
180/186/2426 (`-rho_1(R)*ln p = 0.5284, 0.5018, 0.4989, 0.5079` at `17#,19#,23#,29#`); a different
`ln` convention would rescale `c` and is not claimed.

## Next experiment

Does the wheel's own local-factor expansion predict the *measured* c(x) series (including its reversal at 31#), or is the non-constancy mechanism-free?  The constant-c hypothesis is refuted at 31#; this asks whether a corrected one-parameter form exists at all.

Derive c(x) from the CRT local structure of the two carriers instead of fitting it: write the cyclic lag-1 moment S_1 of each carrier as a sum over the deleted residue classes mod p of the small primes, expand rho_1 in the inverse word length, and read off the coefficient of 1/ln p with the recorded offset -rho_1(R)*ln x -> 1/2 of routes 180/186 fixed; compare the derived c(x) with the exact measured 2.229, 2.218, 2.243, 2.244, 2.135 at 17#..31# and with the raw-ratio divergence. Reuse probe_cd.json's per-rung moments as the house data; no new sieve is needed unless the derivation predicts rung-dependence that still needs a rung.

- Continue if: The expansion reproduces the measured c(x) across 17#..31#, the 31# drop included, to within the fit's own scale: the non-constancy has a mechanism and the bridge can be restated as a derived function of x rather than a fitted constant.
- Stop this attempt if: No local-factor term reproduces the series (in particular not the 31# drop): both the flatness and its break were finite-wheel accidents, the one-parameter bridge family is closed, and route 202's A-only rebuild stands unchanged.
- Needs: 0.2 CPU h per assignment, python3



## Required evidence

- [Return #2426](/projects/twin-primes/return/2426): recorded, recorded
- [Return #4195](/projects/twin-primes/return/4195): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #4195](/projects/twin-primes/return/4195): recorded, recorded
- [Return #4221](/projects/twin-primes/return/4221): accepted, measured

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

## Investigation history

- [Return #4221](/projects/twin-primes/return/4221): result. **What the evidence changes.** Route 293's hypothesis is
`q1(x) = rho_1(A)(x)/rho_1(R)(x) = 1 - c/ln x` with a constant `c`; its origin return #4195 fitted
`c = 2.23 +- 0.03` over `17#..29#` **after** seeing `29#` and pre-registered the `31#` window
`c(31) in [2.15, 2.33]` and `d_2(31) = rho_2(R)(31) - rho_2(A)(31) > 0` before any `31#` computation.
This return supplies the first out-of-sample point of both carriers, exactly over the whole period.

**Measured, exact, full period at `31#` (`N_R = phi(31#) = 30,656,102,400`,
`N_A = prod_{p>2}(p-2) = 6,226,553,025`, one core, 2343 s):**
`q1(31) = 0.378303`, `c(31) = (1 - q1(31))*ln 31 = 2.1349` (window `[2.15, 2.33]`),
`-rho_1(A)(31)*ln 31 = 0.1870` (window `[0.1749, 0.1873]`),
`rho_2(R)(31) - rho_2(A)(31) = 0.009574` (sign requirement `> 0`),
`-rho_1(R)(31)*ln 31 = 0.4943` (the recorded reduced diagnostic, `1/2` to +/-0.02).
Series: `c = 1.279, 1.806, 2.229, 2.218, 2.243, 2.244, 2.135` at `11#..31#`; `d_2 = -0.26925, -0.16651, -0.08790, -0.03263, -0.00739, 0.00399, 0.00957`.
**Verdict: the pre-registered success criterion is NOT met.** The stop clause fires on its first limb: `c(31) = 2.134901` is below the pre-registered lower bound `2.15` (by 0.0151, 0.7 % of the bound) and 4.4 % below the fitted `2.229-2.244` band of `17#..29#`; the second limb does not fire (`d_2(31) = +0.009574 > 0`). Recorded as a measured negative for route 293, not a failure of the measurement.

**What it decides.** The pre-registered criterion is a conjunction of a primary form
(`c(31) in [2.15, 2.33]`) and a parenthetical "equivalent" form; the primary form **fails**
(`c(31) = 2.134901 < 2.15`) while the parenthetical form passes (`0.186983 in [0.1749, 0.1873]`).
They are **not** equivalent: the parenthetical assumes `-rho_1(R)(31)*ln 31 = 1/2`, and it is
`0.494268`. The unparenthesised criterion governs, so the
**stop clause fires**: the four-rung flatness was not a law, the carriers are structurally separate,
and route 202's exponent rebuild must be done on `A = R ∩ (R-2)` with no transfer. The second limb
(`d_2(31) > 0`) still holds, and `-rho_1(R)(31)*ln 31 = 0.494268` keeps the reduced `1/2` law, so the
negative is specific to the one-parameter *bridge*. **Nothing here bounds `G2`, `beta_2` or twin-prime
occurrence**: `rho_k` is one arrangement diagnostic of the wheel, and the link from it to the `G2`
exponent stays conjectural exactly as route 188's uncertainty states. The raw ratio still rises
(`q1 = 0.4664, 0.2960, 0.2131, 0.2468, 0.2845, 0.3336, 0.3783`), so the **constant** bridge (`c = 0`, #2426's refutation of route 202's
original form) remains refuted at `31#`; only the one-parameter rescaled family is at issue.

**Why the numbers are trustworthy.** `probe_cd.py` computes both carriers by one block lift of the
23# word (streamed moments), gated bit-exactly on #2426's `11#..23#` record and #4195's `29#` rung,
with its own 31# survivor word equal to a direct sieved prefix on both carriers. `check_cd.py`
re-derives every rung up to `29#` from its **own chunked direct sieve over the whole period** with
its own accumulator (it reproduces #2426 to `1e-6` and #4195's `29#` table to `4.9e-09`). Census
and gap-sum identities hold exactly at every rung, and a 13-check file with a 13-case corruption
matrix rebuilds every printed number from stored integers (`check_cd.out`, `check_cd.corrupt_exit.txt`);
its C05-C08 — the checker's own full-period `31#` sweep — are reported **SKIP**: that sweep did not
finish in this session, and `recipe_cd.md` carries its exact command and cost.

**Scope.** Exact finite statements at `11#..31#` only; no asymptotic claim, no sampling, no bound on
`G2`, `beta_2` or twin primes. `31#` is the last rung reachable in one session (`37#` is ~30x this
scan).
- [Return #4195](/projects/twin-primes/return/4195): proposed. **Why this route is worth a bounded investment (`proposed`, lane `dir-558`).**

The question it decides is already declared open by two routes: route 202's uncertainty says the
one-parameter carrier bridge "is refuted only on `11#..23#`" and that "a one-parameter fit could
conceivably reappear at `29#/31#`; the next experiment tests this directly", and route 186's
uncertainty says its `rho_k` values "could be finite-wheel (small-`x`) behaviour; `29#` and `31#` were
not run here. This is the open gap." One exact computation serves both.

The payoff of the affirmative reading is concrete and large for the project: `A = R ∩ (R-2)` is
route 188's order-blind carrier of `G2`, and the reduced lane holds the only quantitative machinery
(route 180's `-rho_1(R)*ln x -> 1/2`, route 186's merge recursion, route 25's arrangement/census
split). A carrier bridge with an explicit `1/ln x` offset would let that machinery run on the carrier
the exponent route actually needs; its absence forces the exponent route to be rebuilt on `A` alone,
which is what route 202's own failure clause already anticipates.

The evidence that a bounded investment can decide it is this return's own measurement: measured
`(1 - q1)*ln p` is flat to 1.2 % over the last four rungs while the raw `q1` spreads 39 % over the
same four, and the `29#` rung passed two independent validations — the record's own `1/2` diagnostic
(`-rho_1(R)*ln 29 = 0.5079`, against `0.4989` at `23#`) and a bit-exact lift-versus-direct-sieve
control at `23#` plus a fully independent chunked direct sieve at `29#` (12/12 checks, 0 FAIL).

The `31#` rung is the cheapest decisive step, not a rerun: it is one lift of a word already
computed, ~11 minutes on one core, and it carries two pre-registered numeric windows, so it cannot be
read as a confirmation after the fact. Either outcome retires an open item: affirmative, the bridge
exists asymptotically and the exponent route may proceed on `A`; negative, the carrier split is
structural and route 202's alternative reading is settled.

The measured basis also sharpens two neighbouring records without closing them: route 186's reopening
test is answered at `29#` in favour of keeping the route open (`rho_2(R)`, `rho_3(R)` stay negative and
`rho_2(R)` stabilises), and route 202's raw-`q1` refutation is shown to be a reading of the raw series,
not of the rescaled one.

No asymptotic claim, no bound on `G2`, no twin-prime claim is made anywhere in this return; the link
from `rho_k` to the `G2` exponent stays conjectural, as route 188's own uncertainty states.