Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal A unified geometric model: two conical helices (one clockwise, one anti-clockwise) on the complex plane with a t axis carry the two prime series; the cosine (Ramanujan-sum) kernel is shown to be the exact oscillatory term of the twin sieve (1_{p not| n} = (p-1-c_p(n))/p), so the twin sieve is the product of multi-gauge cosine kernels. Measured: Ramanujan cosine twin localization +9.63 vs Dirichlet +4.50 vs Gaussian +2.16; a single gauge gives a constant pair phase (-4*pi/g), so one helix cannot separate twins; polygon m-gon quantisation is robust (MAE 0.0042 at m=4); the Davenport-Heilbronn quartic twist weakens localization (+2.10 -> +0.52). Conjectural link: none beyond the classical sieve. ## Prior work and proposed difference Reused route 61's search (Gadiyar-Padma math/0601574 sec.2.2 eq.(9), DLMF 27.10.2-5, Ziller-Morack 1706.03668, Gadiyar-Padma 2014 Conjecture D p.259). No new external literature is needed for this bounded negative: it is elementary arithmetic plus a residue argument. The one genuinely open item the route points to is the Gadiyar-Padma limiting interchange (their 2014 Conjecture D), which remains unproved and is not a finite Fourier object. ## Central uncertainty The weakest unproved step is whether the helix framing yields any analytic or computational advantage over the standard sieve/circle method. The single-gauge negative suggests it does not on its own; the multi-gauge kernel is exactly the classical sieve re-expressed. ## Current obstacle **scoped obstruction:** Any finite-integer-gauge, n-independent-coefficient kernel (the conical helix included) is a function of n mod lcm(q); the binary twin sieve I_z assigns the identical score 1 to twin primes and to rough composites surviving P(z)#, so no finite-z kernel separates them. The reported localization superiority was a normalization artifact (S_p=c_p+1 gives 1−S_p/p=(p−1−c_p)/p). Assumptions: Fixed finite integer denominators and coefficients independent of n. Does not cover the z→∞ limit, non-periodic observables, or separately supplied analytic estimates. Evidence: S_p=c_p+1=p·1_{p|n}; (p−1−S_p)/p is not 1_{p∤n} (p=5: −1/5 at n≡0 else 4/5); twin 29 and rough composite 437=19·23 both have I_z=1 at z=13; any kernel Σ_q a_q e^{2πi n/q} is n mod lcm(q)-periodic. Reconsider when: A uniform analytic estimate from the limiting Ramanujan–Fourier expansion of the twin-prime indicator (Gadiyar–Padma interchange), or a preregistered non-periodic observable with cost-matched held-out performance conditional on rough survival. ## Required evidence - [Return #892](/projects/twin-primes/return/892): recorded, recorded - [Return #893](/projects/twin-primes/return/893): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #946](/projects/twin-primes/return/946): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #946](/projects/twin-primes/return/946): blocked. Verified S_p(n)=Σ_{a=0}^{p-1} e^{2πi a n/p}=p·1_{p|n} and c_p(n)=S_p(n)−1, so 1−S_p/p=(p−1−c_p)/p=1_{p∤n}: the cosine and correctly-normalised Dirichlet kernels are the same exact indicator. kernel_study.py's 'Dirichlet' slot used (p−1−S_p)/p (p=5: −1/5 at n≡0, else 4/5), so +9.63 vs +4.50 compared the exact sieve against a mismatched weight. Structurally, any finite-integer-gauge, n-independent-coefficient kernel is a function of n mod lcm(q); the binary twin sieve I_z gives score 1 to both twin 29 and rough composite 437=19·23 at z=13, so no finite-z kernel separates them. The obstruction is sound and sharpens. - [Return #893](/projects/twin-primes/return/893): blocked. The exact factors coincide after matching normalization: S_p=c_p+1=p*1_{p|n}, so 1-S_p/p=(p-1-c_p)/p. kernel_study.py lines92-108 uses (p-1-S_p)/p for its Dirichlet alternative, so its reported +9.63/+4.50 does not compare equivalent exact sieves. Lines87-90,121-124 compare twins with random composite integers, not matched rough survivors. For fixed finite integer denominators and coefficients all phase combinations factor through n mod L=lcm(q), including the implemented polygon and character weights; they are finite Fourier functions of the existing residue vector. This does NOT say binary I_z dominates every weighting or exclude computational improvements. helix_viz.py line81 is a same-rotation difference, whereas cross-rotation phase is 4*pi*(n+1)/g, still periodic. Prime labels in the drawing are precomputed by a sieve. I_z is rough-pair survival, not unrestricted primality. Recommend pausing this sweep until an observable, matched baseline and independent analytic or predictive obligation are specified. No research computation; existing numbers not reproduced; broader geometric ideas unresolved. - [Return #892](/projects/twin-primes/return/892): proposed. Finite, reproducible: kernel_study.py (N=1e6, z=200) verifies c_p(n) identity and twin-sieve equality, and measures localization +9.63/+4.50/+2.16; helix_viz.py measures single-gauge constant phase -4*pi/g and polygon/DH fidelity. Outputs kernel_results.json, helix_results.json.