Fixed-trial oscillation and the two remaining obligations

STATUS: RIEMANN HYPOTHESIS OPEN.

Research and public update: 2026-10-08, at the end of Cycle 17. This release summarizes established results from Cycles 6–17. It does not begin another research cycle.

What we learned

When the prime, Gamma and pole contributions are assembled, large terms cancel and a signed arithmetic residual remains. We showed that, for one fixed nonzero smooth endpoint profile, this residual cannot settle to either sign: positive and negative values occur arbitrarily far out as the window grows.

This rules out using that same paired even/odd trial profile to obtain a one-sided ordering for every sufficiently large window. Trial values are different from the true minima. The result does not prove crossings of the actual sector minima, or disprove CAP or PAR. It also does not exclude a chosen cofinal subsequence that visits only one sign.

What changed, and why

After CORE-S, the project examined commutators, background selection, prime signs, canonical ground directions and connections to existing transfer theorems. Prime-only minimization was shown incapable of capturing a fixed nonzero L² target. The question then shifted to what survives completion and cancellation. Rather than compute another smaller coefficient, Cycle 17 analyzed the transform of the full endpoint residual and settled the fixed-profile eventual-sign question.

This is an auxiliary consequence of existing unconditional theorems, derived and internally audited within the project. It does not close the adopted route. It is not presented as an externally reviewed discovery or a formally verified proof.

Current boundary and next question

The adopted central obligations remain capture in the full even ground space (CAP) and strict parity ordering (PAR) on the same resolving cofinal sequence. Both are unproved; the accounting is 2 → 2. This counts obligation types in the selected sufficient route, not distance to RH, completion percentage, or a universal minimum number of propositions.

CMP: NOT ESTABLISHED.
ES: NOT ESTABLISHED.
EVEN FULL-GROUND CAPTURE: NOT ESTABLISHED.
COFINAL COMPATIBILITY: NOT ESTABLISHED.
COFINAL INCOMPATIBILITY: NOT ESTABLISHED.

The remaining question is which actual arithmetic input can control capture and strict parity on one sequence, beyond comparisons of trial values. This publication task stops here: no Cycle 18, adaptive-profile investigation or new numerical work begins.

Mathematical details: the adopted route

$G_{+,j}$ is the physical L² orthogonal projector onto the entire even ground eigenspace; $$R_a$$ restricts support and $$P_N$$ is the canonical finite Fourier projection. The normalization remains $\widehat k=\Xi/4$ . The unproved requirement is existence of one and the same sequence satisfying:

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

\mathrm{CAP}:\quad\|(I-G_{+,j})P_{N_j}R_{a_j}k\|_2\to0,

\mathrm{PAR}:\quad e_-(a_j,N_j)>e_+(a_j,N_j)\quad\text{eventually}.

\mathrm{CAP}+\mathrm{PAR}\Longrightarrow\mathrm{CORE\text{-}S}/\mathrm{ES}\Longrightarrow\text{adopted transfer}\Longrightarrow\mathrm{RH}.

CORE-S is a finite auxiliary result: strict parity forces simplicity of the even ground. The separate $\Delta_+>0$ condition is not restored as an independent obligation. The adopted ES plus strong-L² joint transfer has additional real-zero hypotheses; ordinary L² proximity alone does not replace them. The earlier ES-independent CMP-R rate remains a separate sufficient condition.

CASE D — CMP OPEN + ES OPEN. In the earlier import audit, 12 of 19 items could be delegated to general theory or fixed-scope results. This does not mean that 12 RH obstacles were solved. Cycles 6–17 discharged 0 adopted CAP/PAR obligations.

Mathematical details: the Cycle 17 fixed-profile theorem

Fix $$b>0$$ and a nonzero real $h\in C_c^\infty(0,b)$ , and put $$J=h*h$$ . Neither nonnegativity nor unit norm is required. The following completed-Weil identity concerns smooth supported trials before finite Fourier projection:

f_{a,\sigma}(x)=\frac{h(a-x)+\sigma h(a+x)}{\sqrt2},\qquad \sigma\in\{+1,-1\},\quad a>2b,

Q(f_{a,\sigma})=Q(h)+\sigma Z_a[h].

\begin{aligned} Z_a[h]={}&-\int_0^{2b}\frac{e^{-5(2a-v)/2}}{1-e^{-2(2a-v)}}J(v)\,dv\\ &-\int_{(1,e^{2a}]}x^{-1/2}J(2a-\log x)\,dE(x),\qquad E(x)=\psi(x)-x. \end{aligned}

\forall A>2b\quad\exists a_+,a_->A:\qquad Z_{a_+}[h]>0,\quad Z_{a_-}[h]<0.

Continuity yields infinitely many alternations of sign. The theorem concerns a fixed $$h$$ as the continuous parameter $$a$$ grows. It does not assert alternation on every cofinal subsequence, uniformity for moving or adaptive profiles $$h_a$$ , the sign of a small projected residual, or ordering of the actual sector minima.

The same completed linear readout also oscillates for any fixed nonzero real kernel $J\in C_c^\infty(0,B)$ . An arbitrary such kernel is not asserted to have the endpoint-trial representation associated with $$h*h$$ .

Transform and proof dependencies

X=e^{2a},\qquad R_h(X)=Z_{\frac12\log X}[h],\qquad H_h(s)=\int_0^b h(u)e^{-su}\,du,

\widehat J(s)=\int_0^{2b}J(u)e^{-su}\,du=H_h(s)^2.

A_h(X)=-\int_{(1,X]}x^{-1/2}J(\log(X/x))\,dE(x).

A_h(X)=-X^{-1/2}\int_0^{2b}E(Xe^{-u})e^{u/2}\left(J'(u)+\frac12J(u)\right)du\qquad(X>e^{2b}).

\mathcal M_{X_0}R_h(s)=\int_{X_0}^{\infty}R_h(X)X^{-s-1}dX,\qquad X_0>e^{4b},\quad\alpha_n=\frac52+2n.

\begin{aligned} \mathcal M_{X_0}R_h(s)={}&H_h(s)^2\left[\frac{\zeta'}{\zeta}\left(s+\frac12\right)+\frac1{s-\frac12}\right]-C_{h,X_0}(s)\\ &-\sum_{n\ge0}\frac{H_h(-\alpha_n)^2X_0^{-(s+\alpha_n)}}{s+\alpha_n}, \end{aligned}

C_{h,X_0}(s)=\int_1^{X_0}A_h(X)X^{-s-1}dX.

\operatorname{Res}_{s=\rho-1/2}\mathcal M R_h(s)=m_\rho H_h(\rho-1/2)^2\qquad\bigl(H_h(\rho-1/2)\ne0\bigr).

R_h(X)=\sum_{\rho\ \mathrm{nontrivial}}^{\mathrm{distinct}}m_\rho H_h(\rho-1/2)^2X^{\rho-1/2}.

The transform identity initially holds in the absolute-convergence half-plane $\Re s>1/2$ . The finite-range correction $C_{h,X_0}$ is entire. The kernel vanishes near its support endpoints, so the integration-by-parts boundary terms vanish in the stated range. The convolution factor is the algebraic square $$H_h^2$$ , not an absolute square.

The zeta pole at 1 cancels against the density correction. The trivial-zero poles cancel against the displayed smooth completion. At a nontrivial zero $\rho$ of multiplicity $m_\rho$ , the logarithmic derivative has a simple pole; multiplicity enters its residue. If $H_h(\rho-1/2)=0$ , that pole is removable. Neither the smooth completion nor the finite-range correction cancels a surviving nonreal pole. An independent explicit-formula substitution confirms the final zero sum. For each fixed smooth kernel, that sum is absolutely convergent and locally uniform on compact real $$X$$ ranges.

The proof uses the following existing inputs:

Compact support and Jensen's formula give a multiplicity-counted zero bound $$O_h(T)$$ for $$H_h$$ .
Comparing that bound with the usual multiplicity-counted zeta zero count is insufficient. Conrey's unconditional positive proportion of simple critical-line zeros supplies $\gg T\log T$ distinct zeros; removing $$O_h(T)$$ kernel zeros leaves infinitely many genuine nonreal poles $s=i\gamma$ . Neither RH nor simplicity of all zeros is assumed.
PNT gives $R_h(X)=o(X^{1/2})$ . Under a hypothetical eventual-sign assumption, the nonnegative tail has convergence abscissa $0\le\sigma_c\le1/2$ . Yet the transform is holomorphic near every real point $$s>-1/2$$ .
A Landau-type lemma requires a real singularity at the finite convergence abscissa, a contradiction. Both eventual-sign assumptions are excluded.

No sign-change density or amplitude rate is proved. The general theorems are imported; the project verifies their dictionary and hypotheses for the actual residual.

Mathematical details: the Cycle 15 prime-only obstruction

$G_j^{\rm ar}$ projects onto the ground eigenspace of the prime-only operator on the canonical finite even space, not the completed Weil ground. Fix a nonnegative unit $h\in C_c^\infty(0,b)$ as an auxiliary endpoint profile. On every resolving cofinal sequence, for every sequence of unit vectors $t_j\in E^+_{N_j}$ in the canonical finite even heads converging in ordinary L² to a fixed even unit state $$f$$ , the project obtained:

H=\int_0^b h(u)e^{-u/2}du>0,\qquad \liminf_j\|(I-G_j^{\rm ar})t_j\|_2^2\ge\frac{H^2}{4+H^2}>0.

Positive affine changes or isometric phase representations preserving the same physical minimizers do not remove this obstruction. This is a scoped exclusion of prime-only selection; it does not disprove capture for the completed form. The original arithmetic energy is linear in prime data and has no cross terms between amplitudes of distinct primes. Coupling occurs among state frequencies and through the subsequent optimization.

Where the finite A1 result fits

Let $$K$$ be the full lowest eigenspace of the actual canonical finite Weil matrix, and $\eta$ its boundary vector. If

P_K\eta\ne0,\qquad u=\frac{P_K\eta}{\|P_K\eta\|}

then the Fourier transform of this canonical direction has only real zeros. This conditional finite bridge allows ground multiplicity and an even/odd tie. It is not a conclusion for every ground vector. Actual cofinal visibility and target convergence of that direction remain unproved, so the result has not shortened the adopted CAP/PAR route. Nor is future shortening proved impossible.

The path through Cycles 6–17

The research records date Cycles 6–14 to 2026-10-07 and Cycles 15–17 to 2026-10-08. These later cycle numbers are distinct from the earlier numbered tracks. The chronological record has been added to the timeline.

Cycle	Retained result	Boundary
6	Finite scalar root equation and state reconstruction from the CORE-S commutator	Root existence and target capture remain separate inputs
7	Arithmetic comparison of capture's mixed evaluation and parity's quadratic evaluation	A common prime error does not merge the two requirements
8	Obstructions to background-only comparisons with theta and same-coefficient parity trials	Not a disproof of background sector-minimum ordering or full Weil ordering
9	Finite conditional parity gate retaining the constant even component	No universal sign from individual primes or powers of one prime
10	Background competitors orthogonal to theta and finite-source obstructions	A small source does not identify the ground target
11	A1: finite real-zero bridge for the boundary-visible direction in the lowest space	Cofinal visibility and canonical-direction convergence remain open
12	A1 allows ties and multiplicity at finite level, but no actual shortening was obtained	Frozen as an auxiliary theorem; future shortening is not proved impossible
13	Search for a global arithmetic selection law; scoped rejection of specified Euler-kernel and bounded-mass connections	No exclusion of the whole finite-Weil route or every selection law
14	Outcome F; specified F-RM and C-QP connections rejected	B-PO remains an unproved parked candidate; no positive cofinal theorem
15	Exact finite prime-power phases, prime-only cofinal no-go, leading prime/pole cancellation	Completed-form CAP/PAR remain open
16	Exact completed endpoint residual	No law transferring trial signs to the true minima; Outcome F
17	SUCCESS A: auxiliary fixed-profile oscillation; extension to fixed linear kernels	Excludes only fixed-trial ordering for all large continuous parameters; 2 → 2

Primary sources and audit scope

Connes–Consani–Moscovici, Zeta Spectral Triples, arXiv:2511.22755v1, §3: Weil explicit formula and test-function conventions. §5 supplies the known finite matrix input.
Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line (1989), Theorem 1, definitions (11)–(12), pp.3–4: author PDF · DOI. Only the positive proportion of simple critical-line zeros is used.
Fesenko–Ricotta–Suzuki, Mean-periodicity and zeta functions (2012), published Lemma 4.12, p.1855: journal PDF. The author's preprint uses Lemma 4.11.
Each cycle's original research, independent audits and state records was reread for this editorial release. The public state and source hashes identify read-only inputs without distributing private research bundles or internal audit logs.
Mathematics and site audits are separate checks. An internal AI audit PASS is not external peer review, formal proof verification or academic approval. No bibliographic novelty or priority is claimed. Numerical diagnostics are not used as proof.

Current state · Roadmap · CORE-S public article

STATUS: RIEMANN HYPOTHESIS OPEN.