Reducing ES to strict parity

STATUS: RIEMANN HYPOTHESIS OPEN.

Research completed: 2026-10-06. Public update: 2026-10-07.

What changed

The finite Weil ground-state conditions previously asked for two separate checks: the even minimum must lie below the odd minimum, and the even ground state must be unique.

The special commutator structure of the actual finite Weil matrices makes the second check follow from the first. In this family, ES—simplicity and evenness of the full ground—is therefore equivalent to strict parity ordering. A positive internal gap in the even sector no longer needs an independent proof.

This is an auxiliary result at each finite parameter point. Eventual strict ordering on the required cofinal sequence has not been proved.

Why this direction

Before refining boundary terms or residuals, the review returned to the hypotheses actually needed by the adopted transfer. The existing matrix structure removes the separate simplicity check, leaving theta-kernel capture and strict parity ordering on the same sequence as the current joint obligations.

The 2026-10-05 joint-transfer update remains a historical account of its then-stated conditions. This is a subsequent result that removes its independent simplicity requirement.

Current boundary

CMP: OPEN / ES: OPEN. Actual strong L² capture and strict parity ordering on a cofinal sequence remain unproved.
Neither compatibility nor incompatibility of those conditions on one cofinal sequence has been established.
No uniform lower bound on a spectral gap has been obtained. The finite lemma does not establish eventual ES.
This finite lemma is not an asymptotic estimate. Newly discharged asymptotic obligations for the actual Weil family: 0.

EVEN FULL-GROUND CAPTURE: NOT ESTABLISHED.

Next action

The remaining research questions concern theta-kernel capture and strict ordering of the even and odd minima on one resolving cofinal sequence. This task only synchronizes the completed result with the public log; it starts no new proof search or numerical investigation.

Technical details

CORE-S: strict parity forces simplicity

Let $Q_{a,N}$ be the canonical finite Weil matrix, with even and odd restrictions $$Q^+$$ and $$Q^-$$ . Take $$a>0$$ and an integer $N\ge1$ , and write $$e_+$$ and $$e_-$$ for the respective lowest eigenvalues. Then

e_+(a,N)<e_-(a,N)\quad\Longrightarrow\quad\dim\ker(Q^+-e_+I)=1.

Writing $\mathrm{ES}(a,N)$ for a simple, even ground state of the full finite space gives

\mathrm{ES}(a,N)\iff e_-(a,N)>e_+(a,N).

The formerly independent condition $\Delta_+(a,N)>0$ follows from this strict ordering at each finite point. The case $$N=0$$ , whose odd sector is empty, is excluded here.

Structure used and short proof

The input is CCM's known matrix structure. In the canonical basis $$V_n$$ ( $-N\le n\le N$ ), set $$DV_n=nV_n$$ , let $$J$$ be reflection, and set $\eta=\sum_nV_n$ . There is an odd vector $\beta$ such that

[D,Q]=|\beta\rangle\langle\eta|-|\eta\rangle\langle\beta|.

JQ=QJ,\qquad JD=-DJ,\qquad J\eta=\eta,\qquad J\beta=-\beta.

\ker D=\mathbb C V_0,\qquad\langle\eta,V_0\rangle=1.

Let $$U$$ be the even ground eigenspace and take $\xi\in U$ orthogonal to $\eta$ . It is also orthogonal to the odd vector $\beta$ . The commutator gives $QD\xi=e_+D\xi$ . But $D\xi$ is odd, and strict ordering $$e_+<e_-$$ excludes that eigenvalue from the odd sector. Hence $D\xi=0$ . The kernel and inner-product identities above imply $\xi=0$ . Thus the single linear functional $\langle\eta,\cdot\rangle$ is injective on $$U$$ , so $$U$$ is one-dimensional.

Conversely, a simple even full ground leaves no odd state at the same minimum, giving the equivalence with ES. The argument assumes neither RH nor capture nor a bound on an unknown spectral gap.

Remaining joint conditions

Let $$R_a$$ be support restriction, $$P_N$$ the canonical finite Fourier projection, and $G_{+,j}$ the orthogonal projection onto the entire even ground eigenspace. The recorded theta-kernel normalization is $\widehat k=\Xi/4$ . Set

C_j=\|(I-G_{+,j})P_{N_j}R_{a_j}k\|_2.

The open proposition asks for one and the same sequence satisfying

a_j\to\infty,\qquad\frac{a_j}{N_j+1}\to0,

C_j\to0,\qquad e_-(a_j,N_j)>e_+(a_j,N_j)\ \text{eventually}.

If these hypotheses hold, CORE-S supplies ES and the adopted ES-plus-strong-L² transfer applies:

\begin{aligned} \text{same-sequence capture}+\text{strict parity} &\Longrightarrow\mathrm{ES}\\ &\Longrightarrow\text{adopted transfer}\\ &\Longrightarrow\mathrm{RH}. \end{aligned}

Results on unrelated sequences cannot be combined. The earlier CMP-R $\lambda^r$ rate is not added to this branch; CMP-R remains a separate sufficient condition that does not assume ES.

CASE D — CMP OPEN + ES OPEN. The earlier 12 of 19 items concerned delegated generic or fixed-scope proofs, not 12 solved RH obstacles.

Evidence and scope

Connes–Consani–Moscovici, Zeta Spectral Triples, arXiv:2511.22755v1, §5.1–5.2, Lemmas 5.1–5.2: the known finite-matrix and commutator input.
CORE-S is a short derived auxiliary result. The published lemmas and this consequence are kept distinct; no claim of bibliographic novelty or priority is made.
The separate internal AI audit passed. It is not external peer review or formal verification of the whole program. The proof does not depend on numerical diagnostics. Private research directories, audit logs and numerical caches are not distributed here.
Joint-transfer sources and hypotheses · Current state · Roadmap · This public state record

COFINAL COMPATIBILITY: NOT ESTABLISHED.

COFINAL INCOMPATIBILITY: NOT ESTABLISHED.

STATUS: RIEMANN HYPOTHESIS OPEN.