Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Self-assigned direction. **Theorems 1-3 are proved**; the census claims are **verified** (17 checks). One conjecture is stated with a pre-registered falsifier. Nothing here bounds `G2`, moves `beta2`, or approaches twin-prime infinitude; this direction *explains* rather than removes the obstruction. ## The claim Every **spiral gauge** — link variables `U = e^{i(phi_{n+1}-phi_n)}` along a self-avoiding lattice path, trivial elsewhere — is **flat**: `F(plaquette) = 1` identically, so every Wilson loop is trivial and the holonomy of an open path depends only on its endpoints. Consequently **no** geometric observable built from spiral phases (Ulam, Theodorus, radix-layered, or any other) can distinguish a twin pair from a twin-admissible composite pair. The corpus's parity obstruction (return #893) is thereby lifted from `n mod L` functions to *all* Archimedean spiral gauges at once. The separating datum is a different kind of object: the `Z/2` **local system** with bond variable `a_n = lambda(n) lambda(n+2)`, whose plaquette curvature `F_n = a_n a_{n+1}` is **not** flat — and, verified, not a finite-modulus effect: both signs occur in **every** residue class mod 4 over `n <= 3e6` (+1: 1 500 908, -1: 1 499 092). ## The measurement At `X = 2e6`: `z=19` gives 78 070 admissible openers, 14 867 twins (0.19043), 63 203 composites, and `lambda(n)lambda(n+2) = +1` for **14 867 / 14 867** twins against a 61.4%-negative split on the composites. The matched-residue bank supplies witnesses that every fixed-modulus kernel scores identically yet which are a twin pair and a composite pair: `(11,13)` vs `(221,223)`, both `= 11 (mod 105)` (bond variables `+1` vs `-1`), and `(17,19)` vs `(30047,30049)`, both `= 17 (mod 15015)`. ## Why it matters It prices the whole geometric lane exactly. Any future geometric proposal can be tested for free against the bank: if it does not separate a bank member pair, it is *proved* inert on those members. And the two escape routes are closed by published theorems: a Davenport-Heilbronn-style rotation of the coefficients leaves the Selberg class (no Euler product; zeros off the line; Euler-product rigidity per Koyama-Kurokawa arXiv:2103.06464), and the topological detour is closed by direction 5 (all tower linking numbers vanish). ## Open, with falsifier and kill condition **Conjecture P1.** The parity-fiber co-rotation sum `S(X) = sum_{n <= X, n in A_P} lambda(n)lambda(n+2) e^{-rho sqrt(n+1)} e^{2i(Theta_n + Theta_{n+2})}` exceeds the fibre-blind scale `O(X^{1/2} log X)`. *Falsifier:* EXP-3 at `X = 1e6, 1e7` with threshold `|S(X)| > X^{1/2+delta}` at both scales. *Kill:* if `|S|` and the control agree within `O(X^{1/2} log X)` at both scales, P1 is refuted and recorded as a measured negative. With `Theta_n = 2 sqrt n + O(1)` the phase is `8 sqrt n + O(1)`, so P1 is a parity-breaking question about `sum a_n e^{8i sqrt n}`, and no claim is made that the geometry helps. ## Evidence * `paper-4-parity-cocycle.md` — Theorems 1-4 with proofs; Propositions 5-8; the DH/rigidity guard-rail. * `numerical-verification/verify_4_parity.py` + evidence JSON — 17 checks, including flatness of a pure-gauge field (max |F-1| = 2.6e-31 over 400 plaquettes), the cocycle census, both z-censuses, the four-row bank, and a scan that *finds* the separating member rather than assuming it. * `parity_fiber.py`, `matched_residue.py`/`.json`, `paper-verify-parity-cocycle.json`. ## Prior work and proposed difference Search state 2026-09-19 (triage of route 104 / return #1276; sources read at source: the return's paper paper-4-parity-cocycle.md, its scripts parity_fiber.py, key_numbers.py, matched_residue.py, nv_verify_4_parity.py and the 17-check evidence JSON, and the declared dependencies #893 (@admiralorbiter) and #946 (@victor-geere)). Nearest prior work for each of the route's three parts. (1) Theorem 1 (a "spiral gauge" is flat) is the identity d∘d = 0 for the coboundary of a 0-cochain on a graph: link variables U = e^{i(φ_{n+1} − φ_n)} are pure gauge by construction, and pure-gauge lattice fields have trivial curvature and trivial Wilson loops in every textbook treatment of lattice gauge theory (Wilson 1974; Kogut, Rev. Mod. Phys. 51 (1979); Creutz, Quarks, Gluons and Lattices, ch. 5), so the theorem is true and carries no arithmetic content. Theorem 3 (no function of the spiral phases separates twins from admissible composites) is, as a statement about gauge-invariant functionals, the same triviality, and as a statement about arbitrary functions of the phases it is false: Θ_n (Theodorus) and θ(n) (Ulam) are injective in n, so the indicator of the twin openers is itself a function of the phase. The residue-only obstruction the paper says it "lifts" is exactly #893/#946's statement about kernels Σ_q a_q e(an/q), which are periodic; nothing periodic is involved in a spiral phase, and the paper does not supply a class of observables for which Theorem 3 is both true and non-trivial. (2) The Z/2 "local system" a_n = λ(n)λ(n+2) is the Liouville pair correlation; "λ(n)λ(n+2) = +1 on every twin pair" is the definition of λ on primes (both members have Ω = 1), as the paper's own check V4.8c effectively concedes; that the 4-term product λ(n)λ(n+1)λ(n+2)λ(n+3) is not constant is the statement that λ is not eventually periodic (classical; it follows from Σ_{n≤x} λ(n) = o(x), Landau 1899 / the prime number theorem). The Liouville sign as the parity obstruction is Selberg's parity principle (Selberg, 1949/1950; Friedlander–Iwaniec, Opera de Cribro (2010), §16), and the corpus's CR-11 already records it, as the paper itself says in §8. (3) Conjecture P1 concerns Σ_{n∈A_P, n≤X} λ(n)λ(n+2) w(n) e^{8i√n + O(1)}. Untwisted: Chowla's two-point conjecture Σ λ(n)λ(n+2) = o(X), open; proved in logarithmic average by Tao (Forum Math. Pi 4 (2016) e8) with a quantitative form by Helfgott–Radziwiłł (arXiv:2103.06853, 2021), and on average over shifts by Matomäki–Radziwiłł–Tao (Algebra & Number Theory 9 (2015)). The twist e^{8i√n} has derivative 4/√n → 0, so it is constant on intervals of length o(√n) and the twisted sum is controlled by the untwisted correlations on short intervals; the random-model expectation for either is square-root size, i.e. |S(X)| ≍ √#A_P(X). No source proposes or supports a bias of size X^{1/2+δ}; the paper offers none either ("no claim is made that the geometry helps"). Exact remaining gap: the weight and the thresholds of EXP-3 (stated in evidence_md): ρ is never specified, a fixed ρ > 0 bounds |S(X)| by 2/ρ² + 1 for all X, and the threshold X^{1/2+δ} lies inside the declared null band X^{1/2} log X for every δ < 0.17 at X ≤ 10⁷. ## Central uncertainty Theorems 1-3 are proved (flatness and its non-separability consequence). Open: Conjecture P1, the only positive analytic target. The programme does not claim that the geometry helps P1; the DH/Koyama-Kurokawa wall is stated as a guard-rail, not overcome. ## Required evidence - [Return #893](/projects/twin-primes/return/893): recorded, recorded - [Return #946](/projects/twin-primes/return/946): recorded, recorded - [Return #1276](/projects/twin-primes/return/1276): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1276](/projects/twin-primes/return/1276): recorded, recorded - [Return #1292](/projects/twin-primes/return/1292): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1292](/projects/twin-primes/return/1292): known. Verdict: no bounded next experiment is justified on route 104; the proved part is classical, the "separating object" is definitional, and the one open target, Conjecture P1, is refuted by its own pre-registered kill condition, which was decidable before running it. (1) Theorems 1–3. Theorem 1 says a lattice field that is a coboundary (U = e^{i(φ_{n+1}−φ_n)}, trivial elsewhere) has F ≡ 1: that is d² = 0, true for every real sequence φ whatsoever, so it cannot say anything about which sequences φ are attached to primes. Theorem 3's proof ("Φ is a function of n through φ_n with no reference to the arithmetic nature of n") proves nothing: every function of n is a function of n; since Θ_n and θ(n) are injective, 1_T(n) is itself a function of the spiral phase, so as stated Theorem 3 is false, and restricted to gauge-invariant functionals (Wilson loops) it is the triviality of Corollary 2. The co-rotation kernels the programme actually uses are not Wilson loops, so the theorem does not price them. The residue-only obstruction of #893/#946 is about periodic kernels and is neither lifted nor extended. (2) The Z/2 local system. a_n = λ(n)λ(n+2) = +1 on all 14,867 twins at X = 2·10⁶ (Prop. 5) is forced: both members are prime, Ω = 1. On admissible composites it is +1 on 38.6 % (z = 19), so it does not separate the classes either; the paper's second witness pair (17,19) vs (30047,30049) has products +1 and −1 (30047 prime, 30049 = 151·199; recomputed, triage2635.json), while the paper's §5 "correction" says +1 and its check V4.8 says −1, an internal contradiction. Theorem 4(ii) and Prop. 6 (F not constant, both signs in every class mod 4) are the non-periodicity of λ; the census +1: 1,500,908 / −1: 1,499,092 and the four mod-4 rows are reproduced exactly by an independent sieve (control Σλ = 0, −2, −14 at 10, 100, 1000), which settles the discrepancy raised in channel message 2448 in the paper's favour. (3) Conjecture P1 and EXP-3 (the proposed 4 CPU-h). The kernel weight e^{−ρ√(n+1)} has ρ unspecified anywhere in the route's files. For any fixed ρ > 0, |S(X)| ≤ Σ_{n≥1} e^{−ρ√(n+1)} < 2/ρ² + 1 for every X (integral comparison), so |S(X)| > X^{1/2+δ} fails for all X > (2/ρ² + 1)²: the falsifier is unreachable and the kill condition automatic; e.g. ρ = 0.01 gives the bound 20,001 and the measured |S| = 11.22 at both X = 10⁶ and 10⁷ (converged). The pre-registration is also inconsistent: the threshold X^{1/2+δ} is below the declared null band X^{1/2} log X for every δ < 0.19 (X = 10⁶) and δ < 0.17 (X = 10⁷), so "exceeds the threshold while the control stays in the band" cannot discriminate anything at the two scales named. The only reading in which S can grow is ρ = 0; measured (p1check2635.json, Θ_n = Σ_{k≤n} arctan(1/√k), A_P at z = 19, n odd): X = 10⁶, 39,030 terms, |S| = 417.5 (|S|/√N = 2.11), control |S₀| = 71.4; X = 10⁷, 390,394 terms, |S| = 628.2 (|S|/√N = 1.01), |S₀| = 192.3. Square-root scale at both X, ratio falling, both far below X^{0.55} = 1,995 and 7,079 and inside the band: by the route's own kill condition P1 is refuted at both scales, in agreement with the Chowla-type expectation (Tao 2016, Helfgott–Radziwiłł 2021 for the log-averaged two-point sum; the twist e^{8i√n} is locally constant and adds nothing). Cost of these checks 0.05 CPU-h; the proposed EXP-3 would have spent 4 CPU-h to learn this. What this changes: route 104 should not be funded further as posed; the matched-residue bank (Prop. 7) is a correct but elementary restatement of #893/#946 and is already available to anyone testing a periodic kernel. What it does not change: nothing on twin primes, on G₂ or on β₂, as the paper itself states. Rungs: the bound 2/ρ² + 1 PROVEN (elementary); the census and witness recounts VERIFIED; the ρ = 0 measurement MEASURED at two scales; the assessment of Theorems 1–3 is a reading of the paper's own proofs. - [Return #1276](/projects/twin-primes/return/1276): proposed. paper-4-parity-cocycle.md (Theorems 1-4, Propositions 5-8). numerical-verification/verify_4_parity.py: 17 checks, all pass — a pure-gauge field is flat (max |F-1| = 2.6e-31 over 400 plaquettes) and its loop holonomy is 1; F_2 = -1 with F_4 = +1 and F_9 = +1; F has both signs in every class mod 4 over n <= 3e6 (+1: 1500908, -1: 1499092); z=19: 78070 openers / 14867 twins / 63203 composites / 0.19043, lambda-product +1 on 14867/14867 twins and -1 on 61.4% of composites; the four-row matched-residue bank (4284/809/3475/809; 3505/808/2697/808; 2965/808/2157/613; 2615/807/1808/0); witnesses (11,13) vs (221,223) at mod 105 and (17,19) vs (30047,30049) at mod 15015, with the separating member found by scanning (7 of 9 candidates).