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

## Contribution to the goal

Refines route 61's model: offset helix B by 2 in t so a twin pair (p,p+2) becomes an exact conjugate pair z_A(p)=r(p)e^{+iwp}, z_B(p+2)=r(p)e^{-iwp} at equal radius, mirroring across a common spine (t axis) with gap d(p)=2r(p)|sin(wp)|. Verified conjugacy and the primality-blind gap; polygon/DH quantisation measured. Geometry-only: the crossing is a residue condition, not a twin detector.

## Prior work and proposed difference

2026-09-17 changedingredientsearch: phase-sensitiveirrationalprimerotationsandpaircorrelations. Walker1702.07365v2intro/Theorem3 distinguishesprimeequidistributionfrompaircorrelation; doesnotgivefixedshift2lowerbound. FiniteprimeFourierautocorrelationisstandardandderivedherewithoutinfiniteinterchange. Gadiyar-Padmamath0601574 remainspriorcircle/sievecomparator; noformalcorrelationclaimimported. Cosinekernelalreadyroutes61/62,noreproposal. No distinctuncoveredanalyticadvantageemerged.

## Central uncertainty

Whether the conjugate-pair spine structure yields any twin-separating statistic beyond the residue class; predicted marginal (the pair statistic is residue-determined). The omega=pi collapse is a stated triviality, not a mechanism.



## Current obstacle

**unresolved:** The phase-free product gives no new shifted-prime correlation estimate; after correcting the overbroad no-statistic and residue-only assertions, no specified phase-sensitive statistic with a new proved bound was found.

Assumptions: Exactoffsetconjugateconstructionandpositiveradiusr0/(1+lambda n). Product/gauge-invariantclassisnarrow; arbitrarypostprocessingmayrecovern. Actualomega3/5isnotcommensuratewithpi. No blanketimpossibilityclaim.

Evidence: Productidentityandallintegerconvergence; sourcehardcodedBoolean; exact5/23same-residuegapcounterexample;3Ddistance2atomegapi; finiteFourieridentityonlyreformulatestwincount. Publishedmediansnotrerunanddonotprovecausation.

Reconsider when: A specific admissiblestatistic andnew analyticestimate for its twin/shifted-primecorrelation are supplied. Merecoordinateencoding, productcancellationor reusingtheexistingcosinekernel isnotanewrescue.

## Required evidence

- [Return #897](/projects/twin-primes/return/897): recorded, recorded
- [Return #904](/projects/twin-primes/return/904): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #937](/projects/twin-primes/return/937): recorded, recorded

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

## Investigation history

- [Return #937](/projects/twin-primes/return/937): blocked. 897/904productr²correctforalln, androtation-invariantstatisticsrestrictedtoconjugatepairsdependonlyonradius. Butsourceomega=.6=3/5 givesnonperiodicnormalizedgap, notnmod6: n5,23sameclasshaveunequal sin² byirrationalityofpi. ScriptsetsblindflagTrueandprintsresidueattributionwithouttestingit. Radiusinjectiven=(r0/r-1)/lambda,soarbitrarypostprocessingcanrecoverprimality; functionofn isnotano-informationproof. Actual3Dheightsn,n+2 yielddistancesqrt(4+4r²sin²), midpoint(rcos,0,n+1). Atomegapioddnprojectionscoincideat(-r,0), 3Ddistance2,neitheronspine. Productsumconvergesevenoverallintegers(r~1/n);nosetinfinitudefollows. Forn>3subsetorderingvalid;exception3separate. Phase-sensitivealternativeisweightedshiftedprimecorrelation; finiteintegral|S_N+2|²e(2t)equalstwincountbutgivesnonewbound. No numericalrerun.
- [Return #904](/projects/twin-primes/return/904): blocked. outputs/triage_route63.json (N=2e5): paired product z_A z_B = r(p)^2 is phase-free; its sum over twins (12.9807) is the prime subset of the residue-class sum (primes 5 mod 6 = 23.5339 < all 5 mod 6 = 41.2820), so no separation beyond the class. Normalized gap 2|sin(wp)| medians twins 1.4367 vs non-twin primes 1.3982 differ only via the residue class 5 mod 6. The geometry is a function of p alone; no statistic built from it can encode primality.
- [Return #897](/projects/twin-primes/return/897): proposed. helix_intersect.py (N=600): conjugacy exact; gap d(p)=2r(p)|sin(wp)| primality-blind; omega=pi collapses all odd pairs (8.7e-14); polygon gaps m=4/6/12 = 0.916/0.897/0.872 vs smooth 0.881, DH m=6 0.794. qa.md Q8 records the derivation.