07. Dyadic arithmetic reduction
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-30. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
Can the representative of a class be pushed into one side of the logarithmic line with a quantitatively small norm after repeated dilation by 2?
What was tested
The construction used full integer Möbius inversion of the arithmetic sum, a smooth cutoff and a moment correction. Sampling the flow at powers of 2 does not restrict the arithmetic input to powers of 2.
What survived
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For arbitrary admissible input, an explicit representative cancels the right tail. An unconditional upper bound has rate 2ⁿᐟ²(1 + n).
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A conditional Mertens estimate M(X) = O(X^θ) improves the same construction to rate 2⁽θ⁻¹ᐟ²⁾ⁿ. The endpoint family of subexponential estimates, with the reverse implication established in the notes, is equivalent to RH.
What failed or remains unproved
The construction supplies no independent endpoint cancellation. The unconditional exponent 1/2 was not proved optimal. A compactly supported folding argument is obstructed by the actual arithmetic subspace, rather than repaired by the cutoff.
Why the work moved on
The inquiry moved to the signed combinatorics of Möbius cancellation: first a prime complex, then prime phases and a weighted halfspace.
Current scope
Explicit reduction and unconditional rate retained; the desired endpoint is an equivalent reformulation, not progress toward proving RH.
“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.