bibkey: trmdy2026simplezeros authors: “Vivaswat Ojha; Tormod Haugland; contributors to trmdy/zeta-simple-zeros-673137” year: 2026 title: “Local zero-overlap certificates, finite trace-energy envelopes, and exact window-pressure assembly” doi: null url: https://github.com/trmdy/zeta-simple-zeros-673137/blob/1610b97b7895ff34982260f8dcaf04a0f7b82cf7/README.md claim: A fixed historical research revision supplies a nine-point kernel certificate and conditional consumers; the October 2026 primary-source comparison records stronger reported proportions with their explicit computational proof assumptions, while Section 21 gives a weaker elementary consumer of the unperturbed Montgomery-Taylor kernel. strata_touched: [] license: citation-only triage: anchor

Local overlap certificates and their exact assembly boundary

Version status, checked 8 October 2026. The pinned trmdy packet and the two conditional consumers below are retained with their original inputs. Knausgård’s 6 October preprint, arXiv:2610.08965v1, reports larger lower bounds for both corresponding counts. Consequently, the $67.3314101589\dots \%$ simple-critical and $83.7167490822\dots \%$ distinct-zero consumers below cannot be described as the current best known proportions. The exact comparison and the author’s computational proof scope are recorded in the final section of this note. Wang’s cited v1 is historical: v2 was withdrawn on 6 October.

The primary source is the trmdy repository README at the fixed revision. Every trmdy statement in this note refers to commit 1610b97b7895ff34982260f8dcaf04a0f7b82cf7. The source describes its assembled proportion as a research candidate. Its mathematical dependencies and finite computational inputs are distinct from journal acceptance or a complete formalization.

The paper source credits Vivaswat Ojha for the retuned seven-point contributor draft. The repository also preserves the previous contributions and numerical-certificate work. Its refined deduction explicitly attributes the finite-dimensional trace-energy envelope and window-in-frame pressure count to tawanerguo-cn, fixed revision 45149f6d403059a71be73c5e3f884cee7cd62b20. These ingredients are prior results, not new foundational claims of the present repository.

The common analytic interface

For an admissible profile $v$ on $[-1/2,1/2]$, let $$k_v(x)=\frac{{\int }_{-1/2}^{1/2}v(s)\cos (2\pi xs)ds}{{\int }_{-1/2}^{1/2}v(s)ds},w(x)=k_v(x{)}^2.$$ The proof outline imports the stability form $$S_T\ge H(v)N_T+\mathrm{tr}\Psi (M_T)-o(N_T),\Psi (t)=\{\begin{array}{ll}(t-1{)}^2,& 0\le t\le 2,\\ 2t-3,& t\ge 2.\end{array}$$ Here $S_T$ counts simple critical-line zeros in $(T,2T]$, $N_T$ counts all zeros there with multiplicity, and $M_T$ is the positive semidefinite Gram of simple-zero atoms. Its fixed-size, bounded-span blocks tend uniformly to the kernel Gram. The analytic source is the Alpöge–Furman framework together with the stability refinement in ainta’s full proof at 040c5e8. This note does not replace that analytic input with a finite matrix analogy.

The fixed trmdy profile is $$v(s)=\cos (\sqrt{2}s)+10^{-9}\sum _{j=1}^6{c}_j\cos (2\pi js),(c_1,\dots ,c_6)=(3322500,-7609135,1190194,-731476,-1680572,1141360).$$ The supplied window certificate gives $$H(v)\ge H_{\mathrm{cert}}=\frac{3362285207}{5000000000}=\mathrm{0.6724570414.}$$ This baseline belongs to this perturbed window. It cannot be replaced by the larger unperturbed Montgomery–Taylor baseline while retaining the trmdy kernel, local weights, or overlap certificate. The unperturbed window and its matching energy estimate are given directly in Lamzouri v2, Lemma 3.2.

The local certificate and successive consumers

For $q+1$ ordered real points $y_0,\dots ,y_q$, the local certificate has the form $$p(y_q-y_0)+\sum _{i<j}{a}_{ij}w(y_j-y_i)\ge \epsilon ,a_{ij}\ge 0,\sum _{i=0}^{q-r}{a}_{i,i+r}=2(1\le r\le q).$$ The exact nine-point weights and window coefficients are in the candidate data.

The complete profile can equivalently be recorded by its seven frequencies and numerators: $$({\omega }_0,\dots ,{\omega }_6)=(\sqrt{2},2\pi ,4\pi ,6\pi ,8\pi ,10\pi ,12\pi ),$$ $$v(s)=10^{-9}\sum _{j=0}^6{b}_j\cos ({\omega }_{j}s),(b_0,\dots ,b_6)=(1000000000,3322500,-7609135,1190194,-731476,-1680572,1141360).$$ The full 36 local pair weights are $a_{ij}=n_{ij}/10^7$, with numerators below. A dash marks an index outside $i<j$; an explicit zero is an actual zero weight.

All numerators are nonnegative. For every index separation $r=1,\dots ,8$, summing the displayed numerators on that diagonal gives $20000000$, so each span capacity is exactly two. These weights, the displayed profile, $p$, $\epsilon$, and $H_{\mathrm{cert}}$ form one fixed source packet.

The pressure and threshold parameters are $$q=8,p=\frac{1}{2500},\epsilon =\frac{15211}{2500000}.$$ The nine-point deduction identifies the final candidate with the split certificate log. That log reports two disjoint shard ranges covering all 96 initial shards, matching table hashes, and a combined 116,272,426 visited nodes. It is an external computational input; this note supplies no independent replay of that interval certificate. The data file’s earlier discovery flags are not themselves a proof; the separate source report links the final certificate to this candidate.

Write $${\Phi }_m(E)=\{\begin{array}{ll}E,& E\le m/(m-1),\\ 2\sqrt{(m-1)E/m}-1+E/m,& E\ge m/(m-1).\end{array}$$ For a unit-diagonal positive semidefinite $m$-point Gram, $E=\mathrm{tr}(G-I{)}^2$ and $D=\mathrm{tr}\Psi (G)$ satisfy $D\ge {\Phi }_m(E)$. Summing local windows in an $m$-point block gives $E+P\ge A=\epsilon (m-q)$. Exact containment counts charge averaged pressure $(m-q)qp/m$ per unit total normalized span. The fixed source uses $D+P\ge {\Phi }_m(A)$ and obtains $$\frac{m{H}_{\mathrm{cert}}-(m-q)qp}{m-{\Phi }_m(A)}.$$ Its final nine-point choice $m=177$ yields $0.673312742272245998143847403168\dots$. The older paper source concerns a seven-point retuning; the subsequent seven-point and nine-point refinements are stated in the linked repository documents, not attributed to that older paper theorem.

The earlier seven-point certificate has $q=6$, $p=1/2736$, $\epsilon =891/200000$. The same refined formula at $m=235$ gives $0.67324258935589670294\dots$. The source explicitly retains the analytic interface and local certificate as dependencies.

Scope of the earlier three-point and seven-point work

These are different consumers, windows, and counting functions. The table is a comparison of identified primary sources, not an assertion of a worldwide record or an exhaustive priority search.

The v2 history of arXiv:2609.33043 records a revision on 7 October 2026; its v2 text retains the distinct-zero constant $0.83699288145242\dots$. The fixed v1 arguments used below remain identified by their own version. The separate arXiv:2610.08965v1 paper supplies the stronger October comparison.

Knausgård’s reference [9] identifies the two-certificate source, whose manuscript is signed Yuhang Shi, with repository revision 1aeda8e. Its stated simple-critical proportion, $67.3316977\%$, is higher than the $67.3314101589\dots \%$ single-certificate consumer below. Thus that consumer is a comparison with the pinned trmdy assembly, not a claim to the best known simple-critical proportion.

The full Shi manuscript at that revision, Section 2, already states the scaled finite-dimensional envelope $D+[{\Phi }_m(A)/A]P\ge {\Phi }_m(A)$ for its stated range ${\Phi }_m(A)<2$, and explicitly carries the smaller pressure coefficient into the shifted-block deduction. Its following remark also proves the coefficient’s optimality in the abstract $E+P\ge A$ interface. Consequently, the single-certificate chord-plus-counting combination and that optimality argument have a direct predecessor; they are not new claims here. The full-range projection proof provides a unified presentation, while the separately stated $\tau =12/5$ mixed-multiplicity consumer uses a different counting interface.

Relation to the finite Gram section and the coupled consumer

The three-point spectral bound in Section 17 of the finite-structure theory volume is exactly the existing envelope under a change of normalization: $$F(e)=2e-(2\sqrt{e/3}-1{)}_{+}^2={\Phi }_3(2e).$$ Here $e={\sum }_{i<j}|G_{ij}{|}^2$ and the envelope’s energy is $E=2e$. The finite three-point formula is therefore an attributed specialization, not a newly discovered universal spectral bound. Its use for a particular kernel perturbation or cyclic phase remains a separate consumer.

Section 19 of the same volume applies the already available scaled-pressure principle to the envelope’s chord coefficient $\eta ={\Phi }_m(A)/A$ to the exact window-pressure count. This gives $$D+\eta P\ge {\Phi }_m(A),\frac{m{H}_{\mathrm{cert}}-\eta (m-q)qp}{m-{\Phi }_m(A)}.$$ For the same nine-point certificate, $m=189$ gives $0.673314101589328672733916842945\dots >6733141015/10^{10}$. The finite coupling is an attributed use of the scaled-pressure principle with a full-range projection proof, and the rational arithmetic is explicit; the zero-proportion statement is a conditional consumer of the supplied local interval certificate and the same-window analytic interface. This is a comparison with the fixed trmdy source’s stated assembly, without a novelty claim for the single-certificate combination. A positive proportion estimate does not exclude every exceptional off-line zero and does not prove RH.

Section 19.5 separately consumes Knausgård’s mixed-multiplicity theorem and Section 3, equation (1). That source uses the identical window and energy baseline, but counts all distinct zeros with $0<\gamma \le T$. With $\tau =12/5$, $c=17/5$, the same nine-point certificate, and $m=972$, the scaled-pressure calculation gives the conditional lower proportion $$\underset{T\to \infty }{\mathrm{liminf}}\frac{N_d(T)}{N(T)}\ge 0.837167490822524242030905102366\dots >\frac{8371674908}{10^{10}}.$$ The deduction retains the source’s counts of high-multiplicity on-line points and off-line pairs, checks their remaining coefficients are nonnegative, and takes the limit in $T$ before removing the smoothing. It compares with the stated Knausgård v1 proportion $0.83699288145242\dots$. The mixed-multiplicity theorem remains an attributed input; the nine-point interval certificate remains external. This distinct-zero consumer is different from the simple-critical comparison above and is not an assertion about every zero’s location.

October 2026 primary-source comparison and the elementary consumer

Kristian Muri Knausgård, More than 83.9% of the zeros of the Riemann zeta function are distinct and more than 67.35% are simple and on the critical line, arXiv:2610.08965v1, was submitted on 6 October 2026. Theorems 1.1 and 1.3 state the following unconditional lower limiting proportions. Here $N(T)$ counts all nontrivial zeros with $0<\Im \rho \le T$ with multiplicity, $N_d(T)$ counts distinct zeros, and $N_0^s(T)$ counts zeros that are both simple and on the critical line.

Both new numbers exceed the corresponding earlier consumers. Lower bounds on these proportions become stronger when they increase. The two rows concern different counting functions; neither the distinct-zero ratio nor a boundary for a zero-free half-plane can be substituted for the simple-critical ratio. This comparison identifies stronger results reported in a primary preprint; it does not assert a worldwide ranking, completed peer review, or an independent verification of that preprint by this repository.

The accompanying source is fixed at kristianmk/839, commit 0e961ea199b425010843bc8044676d4ad29fe031. Its Lean README and axiom report give the following scope:

• distinct_zeros_839 depends on the standard axioms propext, Classical.choice, Quot.sound, and Distinct839.Local.check_passes._native.native_decide.ax_1_1.
• simple_critical_zeros_673 depends on the same three standard axioms and Simple673.Local.check_passes._native.native_decide.ax_1_1.

Each extra axiom records that the corresponding compiled exhaustive checker returned true. According to the author, Lean checks the checker and kernel-table soundness theorems, the analytic interfaces, and the final counting implication; evaluating the full search additionally trusts the Lean compiler, interpreter, and runtime. The author’s supporting computations report 60,467,309 visited boxes for the first search and 535,332,163 for the second, with matching counts in reference implementations. The small kernel-evaluated spot checks do not replay either complete search. This note has not rebuilt the external Lean project or rerun these searches.

The development extends anthropics/formal-math/zeta23 at fbdc36bbf17d20af3fd0447c6d1a8a02773c9844 with the documented 14-file relaxation of window-moment hypotheses. Its Challenge*.lean files deliberately state challenges using sorry; the corresponding Solution*.lean files and comparison files carry the claimed solutions. The reported theorem-specific axiom closures, rather than the presence of a challenge file, determine the stated formal proof scope.

The mathematical improvement over the earlier unlabelled clipped consumer has a concrete source. Sections 2–3 of the new paper use a maximal matching of close pairs, a Fourier majorant for the separated remainder, and local inequalities retaining the multiplicity factors $d_i{d}_j$. Corrections telescope to a boundary term. These mechanisms preserve more pair energy and are prior methods to cite when extending the present consumer. Their quantitative certificates and windows belong to that paper’s own source packet.

The Section 21 consumer of the unperturbed kernel

Section 21 of the finite-structure volume independently proves the finite numerator inequality $$F_a(u{)}^2+F_a(v{)}^2+F_a(u+v{)}^2\ge \frac{3}{4},F_a(t)=at\sin (\pi t)-\cos (\pi t),$$ for all real $a,u,v$. It then uses the specified Montgomery–Taylor kernel identity, the finite clipped-counting proof, and Lamzouri v2, Lemma 3.2 and equations (3.3)–(3.4) to obtain the paper-level conditional consumer $$r\ge C_0+\frac{r^5}{2(8{\pi }^2-r^2{)}^2},C_0=\frac{3}{2}-\frac{1}{\sqrt{2}}\cot (1/\sqrt{2}),$$ where $r$ is an accumulation point of $N_0^s(T)/N(T)$. Its unique threshold is $r_{\mathrm{geom}}=0.672511864084414704\dots$, hence a $67.2511864\dots \%$ simple-critical lower bound under the stated analytic input, with the strict rational gain $r_{\mathrm{geom}}>C_0+11/10^6$. This number is lower than the nine-point conditional consumer above. The finite argument needs no local interval certificate; the zero-counting conclusion still uses the matching analytic energy and the stated order of limits.

Section 21 also proves that the bounds obtained from its uniform numerator energy and this particular sliding-window support-line construction decrease when the window point count increases beyond three. This is a comparison inside that specified family. It does not cover the weighted multi-point certificates in this note or the October paper. Classical linear algebra and the earlier three-point nonvanishing argument remain attributed inputs; no worldwide priority claim, new Lean verification, RH proof, or prime-triplet asymptotic is asserted.