08. Prime complexes and parity pairing
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-30. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
The Möbius sum is a signed Euler characteristic. Could a canonical matching on a prime complex leave only a square-root-sized obstruction?
What was tested
The complex Δ_X consists of finite prime subsets whose product is at most X. The convention is M(X) = − reduced χ(Δ_X). Toggling the smallest prime, 2, yields an explicit matching and filtered chain description.
What survived
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The unmatched cells correspond to odd squarefree integers in (X/2, X]; their total number, and hence the relevant total Betti count, has linear asymptotic 2X/π².
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The filtration has explicit bars [n, 2n), including the augmented initial bar. The doubling map has the recorded chain-homotopy consequence.
What failed or remains unproved
The unsigned topological remainder is linear, so this matching does not give the desired signed cancellation. A prime-vertex pairwise-product graph is not the complex: at X = 15 all three edges on {2,3,5} exist but the triple face does not. Its flag completion would change the topology. This is not a no-go theorem for every possible nonlocal pairing.
Why the work moved on
The next track tested whether prime-logarithm phase statistics supplied the missing signs.
Current scope
Exact combinatorial identities retained; no square-root estimate for M(X) obtained.
“Proved” here refers to the stated mathematical claim in the linked research record. Internal AI-agent checks are scoped checks, not external peer review. Numerical and formal checks have separate limits described in Reproducibility. No priority or novelty claim is made.
Canonical records
- Detailed source report
- Completion report
- Scoped internal audit
- Historical state record
- Recorded validation
The source archive preserves the original Japanese research prose. These links point to publication copies, not third-party paper downloads. The source map explains historical state files and references distinguish publication status from verification.