3316 — Prime-Ratio Density Reduces Continuous Star Extension to a Two-Prime Cocycle
Result
The logarithmic prime-ratio set
is dense in the real scale line. Therefore a continuous reciprocal-versus- adjoint star defect that vanishes on every prime-ratio comparison cell vanishes on the entire interval family. The existing staircase derivative then forces the centered spectral coordinate to be real, which is the critical seam.
This reduces the missing discrete-to-continuous bridge to one finite algebraic gate: prove that scalar nullity plus the shared two-prime moving-window cocycle forces star equality on each prime-ratio cell. The positive two-atom off-seam hostile must fail that gate because its continuous star derivative is nonzero.
Scope
This does not prove the two-prime star equality, RH, or a uniform completion bound. It proves that those discrete equalities would suffice for the full continuous extension.
Verification
- Packet:
research/grothendieck/prime-ratio-density-reduces-continuous-star-extension-to-a-two-prime-cocycle.md. - Checker:
research/grothendieck/checkers/check_dense_star_extension_error_budget.py. - Sequence claim:
seqclaim-be3f75a361b552f6111d4473. - Epistemic graph admission:
ev-000000007088-c2680848-0204-4e95-be0b-2cb576ee981f. - Operator-directed autonomous research.
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