06. One-prime return and a faithful Banach completion
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-30. The completion report records work spanning 2026-09-29 to 2026-09-30 (JST).
Starting idea
If one arithmetic return operator grows subexponentially in both directions while retaining every zero evaluation, can it force every zero character to have zero real exponent?
What was tested
The work fixed the actual test quotient and constructed a normed completion from a specified quotient seminorm. It checked continuity of zero evaluations and finite jets, rather than assuming a Hilbert space or an exact spectrum.
What survived
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A two-sided subexponential return bound forces the real exponent of every retained character to vanish; the scalar implication was also formalized.
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The q₁ completion retains nonzero bounded evaluations at all relevant zeros, with the recorded bound |ℓρ| ≤ 4q₁. Descended finite jets have explicit bounds as well.
What failed or remains unproved
The available arithmetic bounds are exponential. No RH-independent subexponential estimate was proved. Banach completion does not establish Hilbert structure, absence of extra spectrum, a determinant formula or full topological equivalence to the original quotient.
Why the work moved on
The next experiment constructed representatives explicitly using Möbius inversion and a cutoff, so that any growth improvement would have a concrete arithmetic obligation.
Current scope
Faithful evaluation retention and an abstract sufficient lemma proved; its arithmetic growth hypothesis remains open.
“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.