{"id":"c968a7a3-18f6-41e6-97bd-f5108e923629","arxiv_id":"2605.20224","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"High-precision numerical computations demonstrate that the zeros of the ground state of the truncated Weil form converge to the Riemann zeros with errors decreasing by over 100 orders of magnitude as the cutoff increases from 13 to 67.","lead":"This paper reports the first public numerical implementation of the Connes-van Suijlekom truncated Weil quadratic form at multiple cutoffs, showing extremely high precision approximations to the first Riemann zeros. Smart readers might be interested in seeing concrete computational evidence for a conjectural approach to understanding the Riemann hypothesis.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Unitary equivalence in CCM 2025 Lemma 5.1 is invoked for the finite-N Galerkin discretization, but its applicability to the truncated operator at finite c and N is not independently verified in the numerics.","rationale":"The reader's weakest_assumption already isolates the precise point on which the interpretation of all reported eigenvalues rests. No stronger internal inconsistency appears in the numerical claims themselves; the paper is explicit that it offers no proof and flags the c=100 caveat. The proposed test directly probes whether the external lemma transfers to the finite-N setting without requiring new theory.","tokens_in":2057,"tokens_out":408,"duration_ms":16896,"concrete_test":"For c=13 and N=100, compute the Galerkin matrix eigenvalues directly and also implement the CCM operator (via its explicit integral kernel or matrix elements) at the same finite cutoff; if the smallest positive even-sector eigenvalue differs by more than 10^{-50} between the two constructions, the equivalence does not hold at the discretization level used for the main tables.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline numerical claims (monotonic error reduction to 10^{-168} and eigenvector recovery to 300+ digits) are interpreted as eigenvalues of the CCM rank-one operator D_log^(λ,N) only via the unitary equivalence of CCM 2025 Lemma 5.1. The paper applies this to the concrete Galerkin matrices at N=100/250 and cutoffs up to c=100 (with T=800 archimedean cutoff), yet the lemma is formulated for the continuous setting. No direct check is reported that the discretized spectrum coincides with the continuous one at the reported precisions; the single noted caveat (negative eigenvalues at c=100) addresses only the archimedean cutoff, not the N-discretization error. If the equivalence fails at finite N, the observed small positive eigenvalues could be discretization artifacts rather than genuine approximations to Riemann zeros.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper implements the Galerkin discretization of the Connes-van Suijlekom truncated Weil quadratic form at cutoffs c=13--67 (N=100) and c=100 (N=250), reporting that the smallest positive even-sector eigenvalue decreases monotonically from ~2×10^{-55} to ~1.5×10^{-168} for the first zero (113 orders of magnitude) and reaches ~10^{-334} at c=100, N=250, with the associated eigenvector recovering the first ten Riemann zeros γ_1 to γ_10 to 307--329 matching digits. These eigenvalues are interpreted, under unitary equivalence from CCM 2025 Lemma 5.1, as those of the CCM rank-one operator D_log^{(λ,N)}, with an Aitken-Δ² extrapolation at c=100 yielding log_{10}|λ_∞^{even}| ≈ -536.76 and -533.70 (approaching the Connes 2026 heuristic of ≈-530.38) and an empirical power-law fit on c≤67 that is falsified at c=100. Negative eigenvalues at c=100 are attributed to the finite archimedean cutoff T=800 and vanish upon increasing T; no proof is claimed.","tokens_in":2264,"tokens_out":669,"duration_ms":27591,"significance":"If the numerical spectra and eigenvector recoveries are free of discretization artifacts, the results supply the first high-precision public evidence that the CvS truncated zeros approach the Riemann zeros, spanning 275 orders of magnitude in eigenvalue magnitude and recovering hundreds of digits. This directly tests the open convergence question in Connes 2026 and CCM 2025, supplies a falsifiable rate for finite-N behavior, and demonstrates that the method can produce machine-precision matches to known zeros without assuming the Riemann hypothesis.","major_comments":[{"comment":"The interpretation that the computed smallest-positive eigenvalues are those of the CCM rank-one operator (and hence genuine approximations to Riemann zeros) rests on the unitary equivalence of CCM 2025 Lemma 5.1. The lemma is formulated in the continuous setting; the manuscript applies it directly to the finite-N Galerkin matrices at N=100 and N=250 without an independent verification that the discretized spectrum coincides with the continuous spectrum to the reported precisions (hundreds of digits). The single caveat noted (negative eigenvalues at c=100) addresses only the archimedean cutoff T=800, not the N-discretization error. If the equivalence fails at finite N, the observed small positive eigenvalues could be discretization artifacts rather than approximations to the CCM operator spectrum. This is load-bearing for the central claim that the numerics approximate Riemann zeros via ","section":null}],"minor_comments":[{"comment":"The manuscript should state explicitly how the even-sector restriction is imposed on the Galerkin matrix and how the archimedean cutoff T=800 is chosen and increased to remove the negative eigenvalues.","section":null},{"comment":"Clarify whether the reported matching digits between eigenvector components and known Riemann zeros are obtained by direct comparison or via an auxiliary extraction step; include a brief description of the extraction procedure.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for noting the potential significance of the results. We respond below to the major comment on the application of the unitary equivalence.","responses":[{"response":"We agree that Lemma 5.1 is formulated in the continuous setting and that the manuscript does not contain an independent, rigorous verification that the finite-N Galerkin spectra coincide with the continuous spectra to hundreds of digits. Such a verification would require a complete a priori error analysis of the Galerkin method for this operator, which is not currently available. The existing caveat addresses only the archimedean cutoff. We will revise the manuscript to state explicitly that the lemma is applied heuristically to the discretized matrices and that the primary numerical claims (monotonic eigenvalue decrease, 113–275 orders of magnitude improvement, and 307–329 digit recovery of known zeros) stand on their own as empirical evidence. We will add a short discussion of why the observed stability under N-increase from 100 to 250 makes discretization artifacts unlikely for the reported small positive eigenvalues.","revision_made":"partial","referee_comment":"The interpretation that the computed smallest-positive eigenvalues are those of the CCM rank-one operator (and hence genuine approximations to Riemann zeros) rests on the unitary equivalence of CCM 2025 Lemma 5.1. The lemma is formulated in the continuous setting; the manuscript applies it directly to the finite-N Galerkin matrices at N=100 and N=250 without an independent verification that the discretized spectrum coincides with the continuous spectrum to the reported precisions (hundreds of digits). The single caveat noted (negative eigenvalues at c=100) addresses only the archimedean cutoff T=800, not the N-discretization error. If the equivalence fails at finite N, the observed small positive eigenvalues could be discretization artifacts rather than approximations to the CCM operator spectrum. This is load-bearing for the central claim that the numerics approximate Riemann zeros via "}],"tokens_in":1870,"tokens_out":442,"duration_ms":30291,"standing_objections":["Rigorous independent verification that the discretized spectrum coincides with the continuous spectrum to hundreds of digits"]},"desk_editor":{"model":"grok-4.3","letter":"The main new thing is the concrete numerical output: first public CvS Galerkin matrices at sixteen cutoffs through c=100, with the first-zero error falling monotonically from roughly 2e-55 at c=13 to 1.5e-168 at c=67 (N=100), and at c=100, N=250 the smallest positive even eigenvalue around 1e-334 whose vector recovers the first ten Riemann zeros to 307-329 digits. They also run an N-sweep at c=100 that extrapolates close to the Connes heuristic continuum value. That level of reported precision and the monotonic trend across cutoffs had not appeared in the cited literature.\n\nThe work handles the practical side cleanly. It flags the block of negative eigenvalues at c=100 as an artifact of the finite archimedean cutoff T=800 and states they disappear when T increases. The Aitken extrapolation and the explicit statement that no proof is claimed keep the claims proportionate. The authors also report that their own power-law fit on smaller c fails badly at c=100, which is useful to see.\n\nThe soft spot is the interpretation step. The small eigenvalues are read as those of the CCM rank-one operator only through unitary equivalence in CCM 2025 Lemma 5.1. The lemma is formulated for the continuous operator; the numerics use a finite-N Galerkin discretization at N=100 and 250. No test is described that confirms the discretized spectrum matches the continuous one at the claimed precisions, and the single caveat mentioned covers only the archimedean cutoff, not the N-discretization error. If the equivalence does not carry over tightly, those tiny positive eigenvalues could be discretization effects rather than approximations to the Riemann zeros. The fit breakdown at higher c already shows finite-N effects are still large.\n\nThis is for people working on numerical or operator methods in the Connes program. The computations are specific enough to be checked, and the limitations are stated, so it deserves a serious referee to examine the code and the applicability of the lemma to the discrete setting.","headline":"The paper gives the first public high-precision runs of the CvS Galerkin matrices up to c=100, with reported error drops of over 100 orders of magnitude on the first zero and eigenvector matches to 300+ digits, but the link to the CCM operator rests on applying a continuous lemma to finite-N matrices without a direct check.","tokens_in":2714,"tokens_out":540,"would_cite":false,"duration_ms":31971,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The truncated Weil quadratic form at rising cutoffs matches the first Riemann zero to 168 decimal places.","keywords":["Riemann zeros","truncated Weil quadratic form","Galerkin matrix","critical line","eigenvalue spectrum","prime cutoff","numerical approximation","even sector"],"falsifier":"A run at cutoff 200 with basis size 300 in which the first-zero error stops decreasing or begins to increase would contradict the reported monotonic improvement.","tokens_in":2954,"feed_emoji":"","tokens_out":786,"duration_ms":33521,"temperature":0.7,"pith_summary":"The paper constructs the Galerkin matrix for the truncated Weil quadratic form at cutoffs c from 13 to 67 and at c=100. It shows that the first positive zero extracted from the ground-state eigenvector converges to the first Riemann zero, with the absolute error dropping monotonically by 113 orders of magnitude as c increases. At c=100 the same eigenvector recovers the first ten Riemann zeros to 307-329 matching digits while the smallest even-sector eigenvalue reaches 10 to the minus 334. The results rely on an assumed unitary equivalence that lets the discrete spectrum be read as that of the underlying rank-one operator.","feed_headline":"Truncated Weil form matches first Riemann zero to 168 digits","feed_subtitle":"Error falls monotonically by 113 orders of magnitude from cutoff 13 to 67; first ten zeros recovered to 300+ digits at cutoff 100.","key_machinery":"The Galerkin matrix of the truncated Weil quadratic form indexed by cutoff c, whose ground-state eigenvector encodes the Fourier-Mellin zeros that approximate the Riemann zeros.","core_discovery":"The central claim is that the Fourier-Mellin zeros of the ground state of the truncated Weil quadratic form, obtained from its finite Galerkin matrix at cutoff c and basis size N, lie on the critical line and approach the Riemann zeros as c grows. Explicit computation yields a first-zero error of roughly 1.5 times 10 to the minus 168 at c=67 with N=100, and recovers the first ten zeros to hundreds of digits at c=100 with N=250. Aitken extrapolation of the smallest even-sector eigenvalue at the largest cutoff approaches the continuum heuristic value, while finite-cutoff artifacts in the spectrum are shown to vanish when the archimedean cutoff is enlarged.","pith_inferences":["If the trend persists, the same discretization could compute additional Riemann zeros beyond currently tabulated lists by scaling c and N together.","The power-law fit observed up to c=67 is already falsified at c=100, suggesting the asymptotic rate with cutoff may differ from the moderate-cutoff regime.","Extending the archimedean cutoff further at c=100 would test whether positivity of the spectrum is preserved in the continuum limit.","The same matrix construction might be applied to related operators whose spectra are conjectured to encode other L-function zeros."],"forward_implications":["Raising the cutoff from 13 to 67 reduces the first-zero error by more than 100 orders of magnitude at fixed basis size.","The eigenvector at cutoff 100 recovers the first ten Riemann zeros to at least 307 matching digits.","Aitken extrapolation on the basis-size sweep at cutoff 100 predicts a limiting eigenvalue close to the continuum heuristic.","Negative eigenvalues present at finite archimedean cutoff disappear when that cutoff is enlarged, leaving the smallest positive eigenvalue as the genuine ground state."],"fun_headline_variants":["168-digit agreement between Weil form and first Riemann zero","300-digit recovery of first ten Riemann zeros at Weil c=100","Truncated Weil form approximates first Riemann zero at 168 digits","Error to Riemann zero falls 113 orders from c=13 to c=67"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The unitary equivalence between the finite discretized matrix and the continuous truncated operator continues to hold at the cutoffs and basis sizes used.","fun_headline_variants_meta":{"raw":{"variants":["168-digit agreement between Weil form and first Riemann zero","300-digit recovery of first ten Riemann zeros at Weil c=100","Truncated Weil form approximates first Riemann zero at 168 digits","Error to Riemann zero falls 113 orders from c=13 to c=67"]},"model":"grok-4.3","cost_usd":0.007272,"raw_usage":{"total_tokens":3527,"prompt_tokens":1021,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":72724500,"prompt_tokens_details":{"text_tokens":1021,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2434,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1021,"tokens_out":72,"duration_ms":21337,"temperature":1.0,"reasoning_tokens":2434,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T21:09:30.084241+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A run at cutoff 200 with basis size 300 in which the first-zero error stops decreasing or begins to increase would contradict the reported monotonic improvement.","supporting_citations":[],"review_version":2}