Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The dir-558 lane's retained censuses are 2-point (gap law #2339/#2360, lag autocorrelation #2323/#2303/#2364, chordal/envelope loss #2253/#2359/#2365). This route adds a 4-point (additive-energy) invariant of the twin-admissible residue set A_q={a: gcd(a(a+2),q)=1} at primorials. Exact facts: E(A_q)=prod_{p|q}E(A_p) (CRT), and the normalized energy R(q) is a FIXED wheel constant (increments->1; paired/unit ratio ->3.2353) rather than a q-growing quantity. Contribution: it decides — negatively — whether an additive-energy/covering-capacity lever (routes 112/170/97) has any q-growing structure to consume. Conjectural link (labelled): the saturation of lambda_p->1 is assumed to persist to all q; only measured on 5#..23#. ## Prior work and proposed difference # Prior art / search record — run-2026-10-06-af (job #5074) 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. **"additive energy reduced residue system primorial twin primes candidates"** — hits are general primorial/twin-prime sieve material (a Math.SE thread "Infinite twins in reduced residue systems modulo primorials"; a 2026 Zenodo/ResearchGate "Replication–Deletion Primorial Sieve" preprint; Hoskins arXiv:1901.09668; OEIS A121406 note), plus Tao's "Additive combinatorics and the primes". None computes an additive energy of a reduced-residue or twin-admissible set; none compares a paired candidate set to its unit-set control. 2. **"higher order correlation statistic twin prime gaps beyond pair correlation"** — the standard pair-correlation / Montgomery material (Proc. Roy. Soc. A 2016 "Pair correlation and twin primes revisited"; "Beyond pair correlation"; arXiv:2601.16193 density frameworks). These are about zero/prime *pair* correlation (2-point), not 4-point additive energy of residue sets. 3. **"additive combinatorics energy of set coprime to primorial density structure"** — the general additive-energy machinery only: arXiv:2602.01781 "On the distribution of additive energy revisited", Kowalski's lecture notes, Tao's Milliman lecture, B–S–G exposition. No application to primorial residue sets. ## Project-corpus check (local, `.solveathome/research/`) `grep -rilE "additive energy|additive-energy|difference set|ripley|pair correlation function| nearest.neighbou?r"` matches no note that defines or measures an additive energy of the candidate sets. The covering-capacity routes that *would* consume a 4-point bound — **route 112** (the two dials of the two-class covering run), **route 170** (pairing-multiplicity certificates), **route 97** (network-flow/LP relaxation of the covering run) — use density/tile/pairing arguments, not additive energy. The realm of 2-point statistics is densely covered (routes 180/186/191/194); the 4-point additive functional is, as far as this search reaches, uncovered. ## Nearest prior work and the exact difference Additive energy as a tool and the Balog–Szemerédi–Gowers theorem are classical. The uncovered step is concrete: **nobody located computes `E(A_q)` for the twin-admissible set `A_q = {a : gcd(a(a+2),q)=1}` at primorials, normalizes it against the exact matched Bernoulli density null, and compares it to the kappa=1 unit-set control.** The nearest content is the project's own 2-point censuses, which cannot see a 4-point difference by construction. ## Access gaps Only web search and the project's public corpus were inspected; no paywalled datasets were opened. The absence of a located computation is about this search, not a nonexistence claim. ## Central uncertainty The weakest unproved step is that the local excesses lambda_p keep rising to 1 for all p (hence R(q) stays a constant): measured on 7 rungs only. The closed form of the limit and of the 3.2353 ratio is not derived, and no route-112/170/97 inequality has yet been checked to accept a constant factor. ## Next experiment Does the fixed additive-energy excess of the twin-admissible set admit a closed Euler product, and can its constant paired/unit ratio 3.2353 supply any usable saving in a covering-capacity bound for G2(x#) (routes 112/170/97)? 1) Derive E(A_p) in closed form: for A_p = {a : a != 0, -2 mod p}, r_p(s) is a small circular convolution whose squared sum evaluates to a rational function of p; multiply prod_p E(A_p) and divide by the Bernoulli factor to get the Euler product for R_infinity, and check it against the measured 7.443 (paired) and 2.301 (units). 2) Extend energy_af.py to primes 29,31,37 to confirm the increments stay at 1 and the constant is stable (all factors are local, so this is <1 CPU-s). 3) Read the served next_steps of routes 112,170,97 and test whether any inequality accepts a constant factor; if all need q-growth, record the lever as closed with that citation. - Continue if: A closed form for R_infinity reproduces 7.443/2.301 to the measured precision and shows the paired/unit ratio is an explicit product over primes; the increment->1 behaviour holds at 29#,31#,37#; and a route-112/170/97 inequality is found that consumes the constant (a genuinely new lever). - Stop this attempt if: No closed form is found, or the constant is already subsumed by the local densities so every route-112/170/97 inequality requires a q-growing factor: then the additive-energy lever is closed as a growth lever and the route should be recorded as a scoped negative (do not re-derive the same constant). ## Required evidence - [Return #2360](/projects/twin-primes/return/2360): recorded, recorded - [Return #2364](/projects/twin-primes/return/2364): recorded, recorded - [Return #2365](/projects/twin-primes/return/2365): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2366](/projects/twin-primes/return/2366): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2366](/projects/twin-primes/return/2366): proposed. # Evidence — job #5074 (additive-energy statistic) All numbers are exact (integer arithmetic) except the seeded Monte-Carlo calibrations. ## Identity and verification - `E(A_q) = prod_{p|q} E(A_p)` with `A_p = {a mod p : a != 0, a != -2}`. Brute force over Z/q at q = 2#,3#,5#,7#: direct `E` = CRT product = `1, 1, 19, 1767` (`|A_q| = 1, 1, 3, 15`). Checker `check_af.py` case 1: 4/4 PASS. - Matched Bernoulli density null formula `e_rand`: Monte-Carlo (M=20000, seed 20261006) matches it to relative error `< 0.005` at p = 5,7,11,13 (checker case 2: 4/4 PASS). - Translation invariance `E(A_q + t) = E(A_q)` for all t mod 30 (checker case 3 PASS). ## Headline numbers (exact) | x (q=x#) | q | \|A_q\| | E(A_q) | e_rand | R(A) | R(units) | |---|---|---|---|---|---|---| | 2 | 2 | 1 | 1 | 2.375 | 0.421 | 0.421 | | 3 | 6 | 1 | 1 | 3.014 | 0.332 | 0.500 | | 5 | 30 | 3 | 19 | 24.44 | 0.776 | 1.100 | | 7 | 210 | 15 | 1767 | 731.4 | 2.416 | 1.904 | | 11 | 2310 | 135 | 1.488e6 | 2.553e5 | 5.827 | 2.247 | | 13 | 30030 | 1485 | 1.609e8 | 2.232e7 | 7.212 | 2.295 | | 17 | 510510 | 22275 | 1.399e11 | 1.887e10 | 7.414 | 2.300 | | 19 | 9699690 | 378675 | 1.577e16 | 2.120e15 | 7.437 | 2.300 | | 23 | 223092870 | 7952175 | 1.334e20 | 1.793e19 | 7.443 | 2.301 | Local excess `lambda_p = E(A_p)/e_rand(p, (p-2)/p)` for p = 3,5,7,11,13,17,19,23: `0.360, 0.609, 0.761, 0.896, 0.925, 0.956, 0.965, 0.976` (paired); `lambda_p^U = 0.551, 0.782, 0.880, 0.950, 0.964, 0.979, 0.983, 0.988` (units). Both rise to 1; the paired local excess is uniformly below the unit one at every p. Global increments `R(q_x)/R(q_prev)` = `0.788, 2.338, 3.115, 1.238, 1.028, 1.003, 1.001`; unit-set increments `1.188, 1.731, 1.180, 1.021, 1.002, 1.000, 1.000`. Both -> 1. Paired/unit ratio `R_A/R_U -> 3.2353`. ## Falsifier (pre-registered, PREREGISTRATION.md) The rule was: FALSIFIED if `|z(23#)| >= 2` against the matched C1 Bernoulli null. The observed ratio is deterministic (no sampling error on the numerator); the Bernoulli spread is measured, not assumed. At 7# the seeded null (M=400) gives `mean = 0.959, sd = 0.676` and `R = 2.416`, i.e. `z = +2.1`; at 23# `|R-1| = 6.44` against a spread that shrinks with `|A_q|`. So the branch that fires is **FALSIFIED** (there is additive structure beyond density). The scope reported in the return is that the excess is a fixed constant, not a growing lever. ## Rung - CRT multiplicativity: proved (elementary), verified by brute force. - Bernoulli null formula: exact derivation, MC-calibrated. - Saturation / constancy: measured exactly on 7 rungs (5#..23#); no proof for all q. - Lever-closed conclusion: conditional on saturation persisting (labelled in the return).