Investigating provability — from arithmetization to prime-support normalization

Public update: 2026-10-09. Research dates appear in the stage table below.

STATUS: RIEMANN HYPOTHESIS OPEN. B PROGRAM: CONTINUE.

L7 complete. L8 not started.

L7 restricts the form of integers appearing in a fixed arithmetic sentence equivalent to RH, while controlling the precision and proof cost needed to return to the original finite comparison. The result is arithmetic normalization in usual ZFC and a length upper bound in the same proof system. No new proof of RH, its negation, either side of ZFC unprovability, or full independence has been obtained. Both B obligations remain OPEN: 2 → 2.

Research originals remain privately preserved. This article publishes selected editorial summaries and minimal theorem statements. Complete new proof manuscripts, internal audits and code are not published. The article is neither a complete proof nor the full evidence required for third-party reproduction of every claim.

How the question changed

Earlier work investigated an analytic route through finite Weil forms and approximating functions, seeking to restrict every nontrivial zero to the critical line. The limited results of the joint transfer, CORE-S, and Cycles 6–17 are retained.

Along that route, finite checks and conditional implications still left a bridge: all necessary conditions had to hold on the same sequence. Closing a universal statement required something beyond the individual results. This gap is the starting point for understanding the present state.

The researcher chose B to investigate not only fixed RH but also the axioms from which it can be proved. B is a continuing investigation of ZFC independence; it does not establish that independence is promising. The finite-Weil route is HOLD, with its CAP/PAR obligations preserved. No impossibility of that route has been proved.

From arithmetization to L7

Dates follow research manuscripts and completion records, not Git timestamps. The number of stages and the volume of work do not measure progress.

Recorded research date	Stage	Result and boundary
2026-10-08	Preflight / P1 series	Examined conditional routes, premises and arithmetic constructions. Failures of specified inferences or connections do not establish impossibility of the entire earlier route
2026-10-08	L0–L2	Organized arithmetic RH representations and the two independence obligations. Specified constructions and extraction attempts supplied no new independence bridge
2026-10-08	L2-F → L2-D	Found fixed certification of the original certificate format unestablished. Then explicitly defined a different method and established arithmetic equivalence in PA and RH equivalence in ZFC through usual mathematics; no retroactive certification of the old format
2026-10-08	L3–L4	Examined counterexample-integer amplification, model transfer with a consistency premise, and finite proof transformations requiring additional premise proofs. No new connection to fixed RH independence
Begun 2026-10-08; completion checks 2026-10-09	L5	Rejected some large prime-power inputs uniformly and constructed short proof families. Refuted the specified policy that every proof must literally contain huge checking outputs. No lower bound for all proofs
2026-10-09	L6	Scoped audit and adoption of the external 7/8 nonvanishing theorem; not counted as our proof of RH
2026-10-09	L7	Prime-support normalization for the fixed comparison, explicit precision bound and transfer with costs in the same full-strength proof system. No new independence consequence

Fixed target and two obligations

$T=\mathrm{ZFC}$ ; $\rho$ is the arithmetic RH sentence fixed in L2-D 1.0.0. It expresses absence of finite arithmetic violations using a fixed finite rational comparison. Usual mathematical reasoning establishes its RH equivalence inside the same ZFC. L6 and L7 do not replace this canonical sentence.

Independence means that neither a statement nor its negation can be proved from the specified axioms. This program records that goal as two consistency obligations. $\operatorname{Con}$ means that the indicated theory has no finite proof of contradiction.

I:\operatorname{Con}(T+\neg\rho)\;\text{— OPEN},\qquad II:\operatorname{Con}(T+\rho)\;\text{— OPEN}.

The 2 → 2 count classifies research obligations, not a minimum problem count in mathematics or an independence probability. General incompleteness alone does not establish RH's independence. Finite-Weil CAP/PAR also remain OPEN, 2 → 2, in a separate account; they are neither renamed I/II nor combined into four obligations.

L6: the level of external quasi-RH adoption

L6 adopted OpenAI's public quasi-RH result as an external theorem. The sources are the 7/8 main manuscript dated 2026-09-30, the 11/12 separate manuscript dated 2026-10-05, the corresponding public Lean theorem, and an external rechecking report. Manuscripts and formalization refer to official revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb; the report is fixed at 3f11b2203b76902f1b6b234115437302a94b79c3.

The adopted ordinary analytic nonvanishing statement is, for all complex $$s$$ ,

\Re(s)>7/8,\quad s\ne1\quad\Longrightarrow\quad\zeta(s)\ne0.

It does not include the boundary $$7/8$$ , and it excludes the ordinary analytic pole $$s=1$$ . With the functional equation, the remaining region for nontrivial zeros is the closed band $$[1/8,7/8]$$ . The result was connected to an equivalent auxiliary sentence requiring all nontrivial zeros in that band to lie on the critical line; RH inside the band has not been proved.

The checks examined external Lean/kernel success raw logs, compared target sources and settings across versions, and checked selected local arithmetic and analytic steps. The upstream revision checked by the external report was adc7f1241b42e322a6451854ab7e4b4c146bf78a, distinct from the adopted official revision. Static comparison of the target proof sources and settings listed in that report associated its evidence with the adopted version. Our own Lean/kernel/comparator was not run. Full reproof of the paper, faithful correspondence of every paper and Lean lemma, machine translation from Lean to ZFC and external peer review are not certified. Lean's totalized value at $$s=1$$ is not interpreted as removal of the analytic pole.

Using a theorem of the same ZFC does not add new proof power to the axioms. Narrowing the region has not supplied a new independence input closing I/II.

L7: prime-support normalization and precision bounds

This technical summary omits the complete proof. $$S(n)$$ is the sum of positive divisors of a positive integer $$n$$ , and $$H_n$$ is its harmonic number. $$q(n,k)$$ is the finite rational upper bound for $H_n+\exp(H_n)\log H_n$ fixed in L2-D 1.0.0. It uses the original finite exponential and logarithmic sums and rational tail terms; $$k$$ is the precision input. It has not been replaced with another bound or a new reduced-fraction computation.

For valid inputs $n,k\ge1$ , violation is the strict rational comparison

V(n,k)\quad\Longleftrightarrow\quad S(n)>q(n,k).

The original guards make it false at $$n=0$$ or $$k=0$$ ; $$n=1$$ gives equality and no violation. The transfer is conditional on a violation, without assuming an accepted violation exists.

One replacement step

Assume primes $$p<r$$ , $a,e,k\ge1$ and $\gcd(a,pr)=1$ , and set

R=ar^e,\qquad z=ap^e,\qquad K=z(5R+7).

Usual ZFC reasoning establishes, for the same fixed $$q$$ ,

\frac{q(z,K)}{z}<\frac{q(R,k)}{R},\qquad V(R,k)\Longrightarrow V(z,K).

This accounts for a positive margin in the comparison upper bound and the finite-precision error, beyond the increase in divisor-sum ratio when moving to a smaller prime. $$K$$ is independent of the original $$k$$ ; it need not increase $$k$$ .

Removing prime-support gaps

$$C$$ is the family of positive integers whose use of a prime implies use of every smaller prime. It includes $$1$$ . It does not require nonincreasing exponents and is not the family of superabundant numbers.

Iterating replacements of missing small primes moves a violation into $$C$$ . Under the premise $$V(n,k)$$ , the output $$(n_*,k_*)$$ satisfies

C(n_*),\qquad 1\le n_*\le n,\qquad V(n_*,k_*),\qquad k_*\le\max(k,12n^2).

Finite recursion on the input defines a total normalization; no polynomial-time algorithm is claimed. Global violation preservation is an ordinary argument in ZFC. Its global PA-provability and the corresponding PA equivalence are not certified here.

Proof transfer with costs in the same system

Natural numbers include $$0$$ and $m\ge0$ . Both the integer and precision inputs are bounded in the finite sections

G(m)=\forall n,k<2^m\,\neg V(n,k),\qquad G_C(m)=\forall n,k<2^m\,[C(n)\to\neg V(n,k)].

The precision bound gives, in usual ZFC,

G_C(2m+4)\Longrightarrow G(m).

This is not an equivalence at the same $$m$$ in the reverse direction.

$$P$$ is the Hilbert proof-sequence system of the conservative definitional extension of full-strength ZFC fixed in L5. It permits ordinary auxiliary lemmas, cuts and all ZFC axiom schemata. Length $|\pi|$ counts every character in formulas, numerals, inference annotations and newlines. References use binary backward distances; numerals are arithmetic terms based on binary digits. It does not count lines or checking time.

Appending the fixed internal implication and binary boundary evaluation transfers any correct $$P$$ proof $\pi$ of $$G_C(2m+4)$$ to a proof $\tau_m(\pi)$ of $$G(m)$$ . With $\ell(m)=\max(1,\operatorname{bitlen}(m))$ ,

|\tau_m(\pi)|\le|\pi|+A\ell(m)^2+B.

$$A,B$$ are fixed independently of $m,\pi$ ; their numerical values have not been measured. Large values are not counted as free one-character constants, and evaluation of the boundary $$2m+4$$ is included.

The level is usual mathematical reasoning inside ZFC and a syntactic length upper bound. This does not mean a complete Hilbert string or implemented transformer was generated and machine-certified. The ability to transfer proofs differs from establishing that no proof exists. No lower bound for all $$P$$ proofs, new reduction to finite consistency, or new relative-consistency input has been obtained.

Prior work and limits of the checks

Extremization of the divisor-sum ratio through smaller-prime replacement has prior literature. Section 5 of Choie–Lichiardopol–Moree–Solé (2007) treats Hardy–Ramanujan integers, which also impose exponent ordering. Sections 2–3 of MacArevey, arXiv:2602.15905v2 discuss monotonicity of the normalized Lagarias upper value and superabundance of a least counterexample. The families, comparisons and finite-precision conditions cannot simply be identified. No academic novelty is claimed in this publication.

Original small-range rational interval checks and separate internal AI review are distinguished from the usual-mathematics global proof. Preliminary comparisons accepted no input violation and are not presented as execution successes for violation preservation. Internal AI review is neither external peer review nor a machine proof. This publication edit did not rerun research code or mathematical experiments.

L7 complete; RH OPEN; B CONTINUE. Both I/II OPEN, 2 → 2. Finite Weil HOLD; CAP/PAR separately OPEN. L8 and Cycle 18 not started.

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