Alexander Eastwood’s complete working manuscript, v1.66.
G2 and the Riemann Hypothesis remain open. Nothing in this repository claims otherwise.
Read this before any dated section below. Dated sections are preserved as written; this page states what the record supports now.
Standalone extractions (v4 and monograph_v4, verified 61/61). For
self-contained reading and external refereeing without any dependency on
unrecovered historical v1.27–v1.34 files:
24-page standalone paper: manuscript/standalone_weil_certificates_and_obstructions_v4.tex — complete self-contained statements and verbatim proofs across all four pillars: (I) total translated radicals, ordinary resolvent collapse, and the cofinal bounded-floor equivalence (v1.35–v1.36), (II) complete $\lambda=3,4$ semilocal Weil positivity, simple even ground state $\xi_4$, and first-zero enclosure $ |
z_1(4)-\gamma_1 | < 8.752082\times 10^{-33}$ (v1.20–v1.26, v1.40–v1.43), (III) ten unconditional operator/analytic obstructions (NS-34, NS-44, NS-45, NS-46, NS-55, v1.30–v1.38, v1.50, v1.52–v1.55) plus the dual-pole Chebyshev–Legendre Stieltjes identity and closed-form odd pole moment $\frac{2}{\sqrt{3}}\xi’(1) = +0.0133343\dots$, and (IV) smoothed Nyman–Beurling / Báez-Duarte reductions, same-Gram positive-floor countermodels (NS-74, v1.61), and critical-zero logarithmic decay floors (NS-83, v1.66). |
manuscript/standalone_weil_certificates_and_obstructions_monograph_v4.tex — full-length standalone monograph edition of the proved core (0 undefined references, 0 warnings, 0 overfull boxes, 0 uncited bibitems, 0 occurrences of "not archived" / "unarchived", and a complete Appendix B Claim-by-Claim Provenance Ledger).audits/standalone-paper-v4-verification-2026-10-18-v1/verify_standalone_v4.py (61/61 checks passing; verification_results.json) and audit report manuscript/standalone-v4-audit-and-verification-report-2026-10-18-v1.html. Prior versioned drafts (v1–v3, monograph_v3) remain preserved at their existing paths.Certified (interval arithmetic, two precisions). The complete semilocal
Weil form at multiplicative window lambda=4 (support [-log 4, log 4]) is
nonnegative in both parity sectors; its ground is simple and even with
0 < mu_0 < 2.454e-75; the ground’s entire transform has only real zeros
(v1.40); its first positive zero lies within 8.752082e-33 of gamma_1 (v1.43, an
energy/evaluator upper bound). Fresh lambda=5 comparison certificates
(v1.52), the 192/320-bit direct Arb and 128/256-bit series replacement certificates for
beta_{log 4}(1) < -3/5 (audits/bounds-six-agent-2026-09-30-v1/), finite Nyman-Beurling
block certificates (v1.58 to v1.66), and the 256/384-bit 176/176 candidate-mechanism screen
(audits/cofinal-candidate-mechanisms-screen-2026-10-16-v1/) are listed in RESEARCH_MAP.md.
What the certificates do not say. They verify no zero of zeta beyond
the numerically verified range and establish no convergence to Xi. The
NS103 bandwidth count is a heuristic, not a sharp height cutoff; its
logarithmic half-width is not the manuscript’s multiplicative lambda.
The archived lambda=4 finite trial’s transform carries its sampled mass
below height about 150, has near-zeros at the tested ordinates there, and
plateaus at approximately the sqrt(rho) scale above 200 (NS104, NS105),
where rho is its Rayleigh bound, not an identified ground eigenvalue.
Those trial observations do not transfer to the complete ground. The
certified complete-ground energy and first-zero enclosure concern the
fixed window. Finite windows do not
accumulate (prop:v121-cofinal-rh); windows cost exponentially
(prop:v125-cutoff-cost).
Closed. About ten scoped routes are proved closed under their stated hypotheses (research map, status “closed”). A closure is a result; it is not a negative Weil direction.
Open. RH and G2. Every route reduces to one of 14 continuation inputs
(RESEARCH_MAP.md, “Shared continuation inputs”); all of them are frozen
on the board with per-group thaw conditions (NEXT_STEPS.md, “Frozen
input groups”). The groups have distinct missing estimates and constructions;
RH equivalence does not identify their proof obligations. None supplies its
required cofinal arithmetic input. Two historical evidence groups remain unarchived
in the full working log (evidence/MISSING.md), while their unrecovered files are
omitted from the scoped standalone v4 / monograph_v4 editions in favor of the
archived replacement certificates and analytical proofs.
How work proceeds. AGENTS.md: review before every turn, per-turn wall
check, proposal gate with a control screen (controls/), no manuscript
version for failure-only work. PROPOSAL_TEMPLATE.md is filled in a draft
PR before a research row is claimed.
v4), 274-page monograph (monograph_v4), and 61/61 verification suiteFollowing an adversarial referee audit of the initial standalone extraction (v1–v3)
and the repository evidence ledgers, the self-contained 24-page standalone paper
manuscript/standalone_weil_certificates_and_obstructions_v4.tex
and the 274-page standalone monograph
manuscript/standalone_weil_certificates_and_obstructions_monograph_v4.tex
consolidate all proved results, finite-window interval certificates, unconditional
operator obstructions, and smoothed Nyman–Beurling / Báez-Duarte reductions using
exclusively archived certificates and complete analytical proofs (0 undefined
references, 0 LaTeX warnings, 0 overfull boxes, 0 uncited bibitems, and 0
references to unarchived files). The automated verifier
audits/standalone-paper-v4-verification-2026-10-18-v1/verify_standalone_v4.py
passes 61/61 checks across all four pillars and LaTeX build logs; the companion two-precision
(256/384-bit flint.arb) candidate-mechanism screen in
audits/cofinal-candidate-mechanisms-screen-2026-10-16-v1/
passes 176/176 string-reconstructed .overlaps() checks. Neither RH nor G2 is
proved, both historical original-recovery groups in evidence/MISSING.md
remain OPEN, and all five continuation lanes remain PAUSED.
Wall check: Same open gap — WEIL-FLOOR, KERNEL-API, NB-GAIN, OPERATOR-BRIDGES, ZERO-GEOMETRY.
The six-agent direct test uses t_j=j/4,j=0,…,5 and the complete original k=C/M. The finite matrix has positive numerical eigenvalues at128/256bits; its frozen rational test form is positive. Matched Gaussian/pole-free controls passed, and independent integration reproduces the six quotient values. No negative candidate or interval certificate. This is inconclusive for global quotient PD and supplies no original signed lower bound. Stop this screen and shelve further quotient falsification screens until independent analytic input appears. No larger stencil, row, thaw, theorem node or manuscript version. Wall check: Distinct test — closest PR78–80, NS100/101.
The six-agent continuation runs PR79’s fixed-domain locator after matched cancellation controls. None of the four selected refinements reaches its root-candidate threshold at 128/256 bits. This is inconclusive: no certified pole, zero-free region, or evidence of quotient positive definiteness. Complete lattice/real-tail formulas do not enclose the numerical quadrature error. Stop this bounded test; no automatic larger scan, research row, thaw or manuscript version. The original signed theta estimate remains open. Wall check: Distinct test — closest PR78/79, NS100/101.
The six-agent follow-up checks auxiliary heat transport, completed duplication and direct quotient contours. Heat retains an unestimated subtractive source; unrestricted sector-matrix positivity fails. A complete contour argument shows that an uncancelled original zero of M in abs(Im z)<pi/4 would exclude positive definiteness of k=C/M, a sufficient certificate only. No such zero is located; the original Laguerre target is not refuted. A bounded locator proposal has fixed limits and requires its own evaluator/control protocol before running. No scan, row, thaw, new node or manuscript version. Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR76–78.
The six-review audit assesses an explicit nonlocal centering certificate. A proposed positive sech-power mixture for the original pair quotient C/M is incompatible with its second tail coefficient. This excludes that sufficient class, not general positive definiteness of C/M or the original theta target. Exact divisor variance and Hardy/Riemann–Siegel decompositions still lack a signed lower estimating step. No candidate survives for computation. First Laguerre alone is not RH; no row, thaw, new node or manuscript version. Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR76–77.
The twelve-review report admits no new arithmetic estimating method. It separates scoped failures from the original open target: smoothing controls distant residues but not the local signed sum; Jensen backward transfer needs critical-value estimates; Bessel margins and smooth-completion errors concern different comparisons. Original coefficients/Jacobi IVP distinguish NS101 without supplying a signed lower bound. No automatic scan or route pivot follows. The moving 4D model shows the complete corrected sign boundary using illustrative coordinates only. No original theta values are plotted. First-Laguerre positivity alone is not RH. No new row, node, thaw, certificate or manuscript version. Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR76.
The primitive audit repairs PR75’s ordinary-transform difficulty: a positive decaying primitive has integrable pair moments. Recovering the original function retains an explicit polynomial Laguerre correction. The exact NS101 control shares the primitive’s positivity, envelope and unit leading tail yet fails the corrected target in PR74’s analytic range. These structural properties supply no original sign estimate; the original Jacobi IVP is absent from the control and remains an unexcluded source of a genuinely new estimate. The primitive transform is meromorphic, outside the cited admissible entire-transform kernel class. No scan, node, thaw or manuscript version. Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR74/75.
The addition audit checks the actual modular parameters, retains the parity and coupling in an exact two-dimensional lattice identity, and derives a complete regularized moment formula for the associated kernel. The separate moments grow with the shift; only their complete differential combination decays. Treating them as separate ordinary Fourier integrals would lose the required boundary cancellation. No signed lower estimate follows from the positive lattice form.
Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR74. No addition-based sign method is admitted; unequal-modulus identities and the original nonlinear evolution are not generally excluded. No scan, node, thaw or manuscript version.
The six-agent review independently checks PR73, rejects a specific local nonlinear certificate, and gives a smooth reciprocal completion with complete all-height transform and derivative error bounds. Its lower Laguerre margin remains unproved. A reviewed Bessel reference has a positive margin, but no original-theta comparison to that different reference is supplied. A direct complete-moment argument also gives first-Laguerre failure for an explicit parameter range of the NS101 reciprocal-mixture family, without inferring sign from zeros.
Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR72/73. No original-theta sign method is admitted; the exact nonlinear orbit remains unexcluded. No scan, node, thaw or manuscript version.
The pair audit retains the original nonlinear evolution and exposes the exact mixed quadratic-form comparison for the first Laguerre target. No lower bound for that form was obtained. It also shows analytically that every fixed finite theta-sum prefix, reflected evenly at zero, eventually has a negative first Laguerre expression. The complete tail cancels its leading algebraic error, so a tiny real-space remainder is insufficient for that all-height approximation strategy.
Same open gap — ZERO-GEOMETRY, NS100/101. Only the stated fixed reflected prefix strategy is excluded; the original nonlinear orbit remains open. No scan, new node, thaw or manuscript version.
The original-orbit audit shows analytically that the precise NS101 reciprocal mixture fails the original normalized Jacobi equation. Its positive regularized disk operator is still shared by that control and supplies no signed-transform bound. The complete zero-frequency slice also excludes finite constant-matrix moment closures containing that slice; nonlinear and variable-coefficient methods remain unexcluded.
Wall check: Same open gap — ZERO-GEOMETRY, NS100/101. The first radial derivative is the classical first Laguerre expression. A bound using the original two-shift orbit remains missing. No new estimate, node, thaw, certificate or manuscript version. The report retains its original local disposition; the publication record explains this subsequent authorized push.
The six-agent audit
publishes fresh REPLACEMENT evidence for NS38’s reported
beta_(log4)(1)<-3/5: direct Arb at 192/320 bits and a separate series
enclosure at 128/256 bits, with 37 recorded adversarial replay checks.
The beta certificate does NOT transfer to the physical lambda=4 form:
its tests are not compactly supported in that physical window.
The original certify_beta4_negative.py and output remain missing;
both original-evidence recovery groups stay OPEN.
The coefficient-recovery and structured-phase gain attempts remain scoped audits. The first retains an upper inverse-Gram bound and a summable new-block certificate; the second leaves complete-kernel complexity and the companion signed numerator unestimated. Correction G03-01 distinguishes sufficient little-o error from the weaker sufficient fixed separation. No new signed-gain estimate, theorem node, thaw or manuscript version. Wall check: Same open gap — NB-GAIN / NS53/61/87.
The two-prime construction brief answers PR69’s finite assembly question: an explicit connected 4D cotangent space has a complete contact energy flow with exactly the two prescribed primitive periods log2 and log3, and no mixed return. But conserved momentum prevents the trapped circles from interacting. On its natural scalar contact-volume Hilbert space, an exact coordinate change yields three translation channels: purely continuous spectrum and no eigenvalues.
A chosen scalar damping on horizontal forms still gives the local signed prime weights; this is not a positive ordinary trace or an analytic completion. Definitions, analytic reviews and 25 clean-start Maxima checks retain the distinction. The earlier generic-gluing warning is corrected: connectedness alone forces no mixed orbit, and log6 requires specified roofs.
Wall check: Same open gap, OPERATOR-BRIDGES. The finite connectedness prerequisite succeeds; the naive scalar spectral interpretation fails for this model. Arbitrary distinct positive periods work too, so no prime rigidity or new arithmetic estimate follows. Stop larger-prime-list variations absent a specified analytic/complete-form transfer. No row, thaw or manuscript version.
Three construction agents specified contact, arithmetic-intersection and spectral objects in four or more real dimensions, then cross-reviewed their missing transfers. The construction and pilot brief records the outcome; definitions and evidence retain the precise scopes. These are explicit local objects and construction questions, not a new global positivity theorem.
The local batch runner completed 165 deterministic exact calibration fixtures and nine local-model inference requests (three proposals, three critiques, three second-model audits). No generated proposal was admitted for computation; the critics also required correction. Repeated fixtures and model agreement are not mathematical advances. Post-pilot prompt repairs are not yet inference-tested. No further batch is running.
Wall check: Same open gap. OPERATOR-BRIDGES retains separate route-specific construction obligations. No new arithmetic estimate, research row, thaw, manuscript version or RH/G2 result. All frozen groups and missing-original records are preserved.
Three agents examined the exact original-target NB bottleneck, three primary literature sources and the strategy/control requirements. Their arithmetic bottleneck brief identifies the corrected finite observation vector and the complete tail allowance. It produces no new unconditional partial arithmetic lemma. Divergence of the localized normalized observations is equivalent to the existing convergence target; it is not a weaker intermediate theorem.
Wall check: Same open gap, NB-GAIN / NS53/61/87. The full residual is a useful target for a specialist feasibility assessment because its dependency is explicit, not because this review establishes its superiority. The checked full-line approximation theorem assumes RH; the unconditional short-mollifier result has a different target and height regime. No matching import was found in this bounded review. Keep NS74’s target-sensitive control and NS83’s rate restriction. No candidate computation, research row, node, thaw or manuscript version. The brief is prepared; no external outreach has been sent.
The focused Holland audit found no fatal defect in the checked main hyperbolicity chain, retaining its inherited analytic/convolution inputs and matched parameters. Twenty-six exact Maxima identities pass. One scope qualification matters: Lemma 7.3 uses a D-to-B comparison supplied by the construction; its four displayed inequalities alone do not imply the claimed uniform root localization. An exact degree-two family demonstrates the distinction without contradicting the main application. This is not a full proof verification.
The method controls large shifts and leaves NS100’s backward-in-shift, unbounded-degree arithmetic input untouched. Retain the method as a source, but lower its priority as an RH proof lane until that transfer is supplied. Wall check: Same open gap, ZERO-GEOMETRY / NS100. No new research row, node, thaw, scan or manuscript version; all existing statuses stay fixed.
The cotangent/Estermann applicability audit finds no direct estimate for NS98’s actual projected Möbius cost in the checked Maier–Rassias statements. Their original parameter ranges leave comparable indices untreated. The survey’s delta-zero endpoint differs from the original theorem and is not imported; the printed PDF pages were checked. No existing repo theorem relied on that endpoint.
The existing bounded smoothing operator does give a valid one-sided transfer from a complete q=1 selected-cost bound to q=2. No such arithmetic bound, uniform endpoint extension or full projected-kernel transfer was obtained. Wall check: Same open gap, MOBIUS-COST/MOBIUS-MARGIN (NS92–98). The bounded source search stops without a new candidate, node, thaw or version; it does not exclude adapting the methods or other literature.
The focused independent-agent audit
found no defect in NS57’s first-zero chain for the specific
delta_2=-log(2)/100 perturbation used by PR #63. The two reviews reconstruct
fixed-vector detection, complete cutoff transfer, small-window positivity,
continuity, attainment and endpoint support, retaining the stated upstream
complete-form and arithmetic-radical inputs. Compactness supplies the step
needed beyond strong continuity when a prime shift enters the window.
At the full first zero the entire altered form is nonnegative; at a parity sector’s own first zero only that sector is guaranteed nonnegative. This validates PR #63’s existing dependency, not an arithmetic estimate for the original form. No mathematical correction, new node, thaw or version is needed. Wall check: Same open gap, KERNEL-API (NS51/55/58). The validation recommended in the earlier dated section below is now complete.
The extra boundary-ordering test and three subagent reviews identify a candidate-selection problem. The exact prime-2/3 factors give mixed exterior errors of both signs in a four-packet diagnostic; no negative Weil energy or compressed-product indefiniteness follows. The overlap mechanism is generic despite retaining the Euler parameters, and even a successful ordering lacked a complete bridge to the missing estimate.
The rules now require that bridge, the actual use of arithmetic, and the decision changed by a test before candidate computation, including pre-claim audits. Supporting generic lemmas remain allowed when their conditional transfer and arithmetic hypotheses are explicit.
Wall check: Same open gap, KERNEL-API. No new RH route is admitted. The recommended next action is independent validation of NS57’s first-zero control, preserving v1.36’s existing independent-review history. That audit is not performed here. All 104 node statuses and all freezes are retained.
The prime-chain admission audit stops a bounds-only first-zero argument. Changing only the coefficient at n=2 preserves every original prime-chain lower and upper operator bound; NS57 supplies a nonzero nullvector at a finite nonnegative first-zero window for that altered family. The all-frequency comparison is elementary; the 50/80-digit samples are diagnostics, not Weil certificates.
Wall check: Known wall for this inference, with NS57’s hypotheses matched. Same open gap for original KERNEL-API: the control loses the exact Euler recurrence and original radical identity. No general API or bounded-floor closure, new research row, theorem node, thaw or version. Replay and scope record.
Three completed tests stopped without a new cofinal estimate:
Wall check: Same open gap for NS92–98 and NS100–101 under their respective
scopes. No new theorem node, thaw, manuscript version or interval certificate.
Publication record and portable replay.
Alex has authorized publication after each completed test; see AGENTS.md.
The exponential-damping proposal was rejected before a research claim. Grouping the damped prime sum with the exact gamma and pole terms is valid in its convergence region. A uniformly bounded cost of removing fixed damping would violate the existing NS47 closed-comparison obstruction. Shrinking damping escapes that fixed hypothesis but leaves the justified Euler-series/Fourier representation; the needed arithmetic bound remains open.
Wall check: Known wall for the fixed comparison; Same open gap for the shrinking variant, WEIL-FLOOR. No new theorem node, manuscript version, research row or thaw. Two-precision control samples are diagnostics only.
Attempt and reproducible screen · Readable report.
The finite trial supports the predicted near-zero pattern; its explanation and complete-ground extension remain unproved. At 1024/1280 bits, the unchanged 4097-coefficient trial has tiny values at the first 64 zeta ordinates and opposite signs at each pair of endpoints gamma_j +/- 0.01. The first linearized trial shift is about -3.42e-38. The complete-ground 9e-33 radius from v1.43 is an upper bound, not a measured displacement.
The manuscript’s multiplicative lambda=4 means support [-log4,log4]. NS103 uses lambda for the logarithmic half-width; its proposed crossover is a heuristic, not a sharp detection limit. The uniform negative-floor target does not require resolving each tiny positive eigenvalue.
Wall check: Same open gap. Closest NS43/46 and v1.40–1.43. This bounded diagnostic supplies no new cofinal arithmetic estimate and thaws no input group. No manuscript version or theorem node is added.
Results and reproducible scan · Readable interpretation audit.
The short test stopped without a new positivity estimate. The actual Jacobi completion gives a positive autocorrelation, but the kernel needed for the growth criterion is its derivative. Polarization leaves a difference of two positive kernels and the same unproved comparison.
An exact control now preserves reciprocal theta symmetry, positive Gaussian mixtures, a positive even differential kernel, order one and the usual zero strip while inserting explicit off-axis zeros. It changes the original single-lattice coefficients; monotonicity, log-concavity and the other modular transformation laws are not asserted. This excludes an inference from the listed properties alone, not a factorization using the original theta arithmetic.
Direct termwise absolute unfolding also fails in the required strip. At its boundary each individual Gaussian summand integrates to zero, while the complete kernel integrates to 1/4. The missing sum/integral interchange cannot be supplied by symmetry. Conditional or correlated resummation is not excluded.
Twenty Maxima identity checks and author review are archived; no new numerical proof gate, theorem node or manuscript version. The expanded literature screen records the overlapping 2026 Planat theta-growth papers without importing their reported blockwise results as newly audited proofs.
Wall check: Same open gap. Closest NS100; no full-sign estimate or candidate positive factor was found. The reciprocal-data-only shortcut is now excluded by the exact control. Do not restart this factorization search without a specific additional arithmetic identity to test.
Attempt and exact control · Readable report.
The same gap recurs in several places, but not in every route. The map now labels unfinished continuations separately from established results:
All 104 node statuses are retained. New continuation tags on 63 nodes describe what further work needs; they do not make existing proofs conditional. The map’s old claim that only a gold node was worth working on has been removed. Both historical original-evidence gaps remain open.
Wall check: Same open gap for this dependency audit (closest NS92–98); no new arithmetic bound. The separately requested NS100 reassessment tests possible geometric mechanisms against controls and primary literature.
NS100’s first geometric test gives a known wall for slice-by-slice theta positivity: one complete cosine-transform value is strictly negative, replayed at two precisions and doubled series/physical cutoffs. This says nothing adverse about the sign of the full weighted integral or RH. The remaining integrated-kernel and disk criteria are classical, still open inputs. The suggested bounded continuation is to derive an explicit factorization from the actual theta modular identity; no such factor has been found. A Jensen interlacing idea is lower priority and also lacks a mechanism. Nine Maxima checks and Python diagnostics pass; no surviving new RH mechanism, theorem node or manuscript version is claimed.
Shared-input index · Shared-gap guide · Review record · Independent reassessment.
No new scale-independent cost bound was obtained. Retaining the exact old-coefficient subtraction leaves an unestimated signed projected cross term. Its diagonal is bounded by the elementary method already used in NS93. A direct contractivity/triangle estimate reproduces NS92’s previous budget; no new arithmetic input follows from the projection identity.
Two complete finite decompositions reproduce the NS73 costs at N=16,32; their negative cross sums are not extrapolated. Two precisions, a doubled series cutoff and seven exact Maxima checks pass. This is an attempt/audit, not a new theorem node, manuscript version or RH advance.
Wall check: Same open gap. Closest NS92/93; actual projected-cost and compatible numerator estimates remain open. Another formula or larger finite table is not the next step without an independent arithmetic bound.
Attempt and stop condition · Readable report.
The signed-arithmetic bottleneck predates NS96. Its LCM class gives new scoped formulas in this record, not a new fundamental wall or progress on the actual cofinal lower bound. In particular, failure of its positive complete rule already follows qualitatively from NS78’s necessary coefficient limit; NS96 adds a numerical constant. The negative-rule application and projected-cone equality concern a distinct class. The Mellin separator and floor function themselves already appear in NS74.
A trace normalization alone is not a new restriction when a free step scale is retained: it leaves the same correction rays and optimized line gain. An actual total-update budget is different and remains unestimated. The next task must identify a specific stronger arithmetic estimate before trying another positivity reformulation. This is an audit correction, with nine exact Maxima checks; no theorem node or manuscript version is added. A required per-turn Wall check now identifies known walls, unchanged open gaps and distinct tests.
Claim comparison and scope · Readable audit.
A genuine pointwise divisor square groups by least common multiple. As a complete coefficient rule, either global sign of a positive semidefinite mixture leaves a size-independent squared-error floor: at least 3087/4096 for the negative sign, and greater than 1/32 for the positive sign. These are restricted-construction bounds, not bounds on the optimized error.
With unrestricted old-coefficient refitting, the projected positive cone is exactly the signed span of the LCM atoms. Every nonzero LCM pairing matrix of an old-orthogonal residual is indefinite. Thus this positivity condition alone supplies no numerator sign or arithmetic lower gain. Single squares, fixed trace budgets and other extra constraints are not identified with that full signed span. The refitted convergence route stays open; no general Selberg-sieve obstruction is asserted.
Self-contained complete-moment and projection proofs, 17 Maxima checks, 256/384-bit scalar replay and author review are archived. No new optimized error computation, cutoff experiment or manuscript version. RH/G2 remain open.
Derivation and checks · Readable report.
Freeing the signed divisor-product weights repairs most of NS86’s finite loss: at N=16 to 256 it captures about 99.83% of full available gain, versus 27.31% for the prescribed direction with optimal damping/refit. The enlarged fit has 96 independent new directions versus the full 240. It reduces that finite old error by over 55.636%; no cofinal lower bound or new best finite approximation error follows.
The construction still forces equal coefficients at all new primes above N, and more generally ties coefficients at mp across primes p>N. Exact constraint and gain identities retain the complementary directions. The direct product-pairing matrix is indefinite on the tested blocks; it is not the positive Gram. An explicit invisible arithmetic vector refutes an all-residual comparison under the stated prime-pair hypothesis, not an estimate for the actual optimized target. The latter remains open.
Two precisions, doubled Gram and physical cutoffs, complete independent physical norms, 18 Maxima checks and author review pass. The freely optimized weights generally lose the original defect-square prescription. Manuscript remains v1.66; both historical original-evidence gaps stay open.
Readable report · Exact argument and certificates.
The arithmetic cost splits into a uniformly bounded diagonal and signed
Mobius pair correlations. Bounding the latter after taking every product’s
absolute value fails: the resulting majorant is at least
N*log(4/3)^2/9216 for every dyadic N>=2048. Its average over growing
ranges of doubling indices also diverges. This is a scoped failure of an
unsigned bound, not growth of the actual signed cost or an RH obstruction.
Both averaged numerator and signed-cost estimates remain open.
The tangent-circle proposal was checked from the supplied power-of-a-point handout, ordinary coordinates and Maxima. Inversions give the expected image circles, but their exterior similarity point depends on the circles, not zero counts. A symmetric off-line polynomial control has equal interior counts and an infinite exterior similarity point. A nonnegative product of the image powers detects off-line displacement, but proving its zero sum vanishes at every height is another unproved RH-equivalent target. No general geometric route is excluded.
Arb 256/384-bit and doubled-sieve checks, independent finite quadratic sums, and 31 exact Maxima checks pass. Author review only. Manuscript v1.66 stays unchanged; both historical original-evidence gaps remain open.
Arithmetic report · Plain-language geometry note · Arithmetic evidence · Geometry and Maxima.
Applying NS61’s complete average to signed Mobius coefficient tails gives an explicit scalar upper bound on the true selected cost. It also bounds a feasible direction that changes only the old coefficient at N and the new block, without a new-block inverse or refit of the other old coefficients. The starting optimized old coefficients remain inherited inputs.
At N=256 the cheap scalar budget certifies more than 0.0443% relative error reduction for a fixed step 1/512. Independent full physical integration certifies more than 1.57% for step 1/16, including the complete tail. These are weaker than full optimization and are not new best finite errors.
Seven scalar budgets through N=65536 stay small on that finite list. No uniform bound follows: the elementary summatory-Mobius estimate gives only an O(N) budget. The compatible uniform cost and signed numerator estimates remain open. Two precisions, doubled physical/inherited coefficient cutoffs, 16 Maxima checks and author self-review pass. Manuscript stays v1.66.
Complete derivation and certificates · Readable report.
Combining the signed compensator before taking absolute values repairs the tested NS90 estimate. An exact old-space correction removes its entire initial interval without changing the numerator or true projected cost. Fixed dyadic grouping, including the complete sum of absolute omitted interval contributions, certifies a relative error reduction above 2.821% at N=256. This bounds the already known NS73 direction; it is not a new best finite error or a percentage asserted at all sizes.
At T=N^6 the regrouped tail is negligible on the required logarithmic scale. The finite arithmetic margin and its relation to the true projected cost still lack a uniform estimate. Rewriting the numerator as a quadratic expression using NS78 inversion does not supply positivity: the normal equations reduce it to a scalar multiple of the original numerator. The full PR42 observation-energy route remains open.
Two precisions, doubled physical/inherited Gram cutoffs, 32 Maxima checks, and author self-review pass. The previously blocked standalone draft is preserved; these are fresh repository replays. Manuscript remains v1.66.
Complete argument and certificates · Readable report.
Returned to PR42’s missing arithmetic lower gain. In the tested Mobius combination, the prime main terms cancel exactly after the old normal equations; the signed divisor forcing remains. The complete numerator is positive at N=16 and 256, but separating its terms by absolute values gives a vacuous lower bound. At N=256 that bound is below -186 times the current error, and below -226 after doubling the physical cutoff. These finite checks diagnose an estimate; they do not disprove the direction.
For fixed N with a nonzero residual logarithmic or constant tail, the absolute divisor allowance diverges as the cutoff grows, although the signed expression converges. This is not a uniform statement at T=N^6. The necessary cofinal signed lower estimate remains open. Manuscript stays v1.66; no new error-decay theorem or finite-efficiency result is claimed.
Attempt and complete evidence · Readable report.
lambda = 4
window sits relative to published work, with the parameter translation../manuscript/build.sh # build the PDF (source validation is not a build)
python3 tools/verify_manifest.py v126 # check a version's artifacts, or --all
python3 tools/make_map.py # regenerate RESEARCH_MAP.md
| Largest completely certified positive window | W_4 >= 0, both parity sectors — evidence restored and verified; lambda=3 is also certified |
| Complete ground at lambda=4 | Simple and even; 0<mu0<2.454e-75, mu1>1e-73; its entire Fourier transform has only real zeros (certificate) |
| Complete-ground first zero | First positive zero simple and within 8.752082e-33 of gamma_1, at 1024/1280 bits (v1.43); no discrepancy sign |
| CCM finite benchmark | Eight local lambda=3, N=120 root discrepancies certified at 768/1024 bits; broader reproduction remains diagnostic (v1.41) |
| Finite floors | W_λ >= -8·I at λ = 5, 6, 8, both parities, full infinite tail (prop:v138-three-floors) |
| Certified window | half-width a = log 4 = 1.386; 1.73× the published Zhu window, 4× the classical range |
| Current mechanisms | Complete kernel exclusion (API); CCM complete-ground selection; paired heat transport and arithmetic approximation inputs remain open |
| Scoped obstructions | See the map for the precise comparison classes; alternative routes are not collectively proved closed |
| Evidence gaps | Both remaining groups explicitly not archived; disclosure complete, recovery open — see the ledger |
| G2 | open |
| RH | open, and not claimed |
Requiring an exactly zero exterior on x>1 preserves every possible upper
error-decay rate. The normalized optimum obeys
E_N <= E_N^0 <= 1.144579 E_N for every N>=256, by a finite projection
certificate and monotonicity. This is a uniform comparison between two
unknown errors, not a decay theorem.
An explicit rational size-256 approximant has complete squared error
below 0.000014144. Its normalization costs about 0.0088342% relative
to the existing unrestricted optimum. The complete remote tail is included,
with independent Gram and physical representations, two precisions and
doubled cutoffs. These are constructive upper certificates, slightly above
the prior unrestricted optimum. The C/log N upper-rate candidate and the
needed uniform signed arithmetic estimate remain open. No new lower-bound
result or manuscript version is added.
Readable report · Exact coefficients, proof and verification.
The actual optimized dilation defect is exactly an overlap of successive
orthogonal improvement increments. This gives a complete next-size upper
certificate and the cumulative envelope E_(2^(J+1)) <= K exp(-sqrt(2)*S_J),
where S_J sums abs(d_(2^j))/E_(2^(j-1)) from a fixed starting index.
A positive logarithmic lower growth of this scalar arithmetic sum would
imply a sufficient logarithmic whole-error upper rate. That growth is
unproved; the envelope alone supplies no decay theorem or fast solver.
The exact identity also excludes an eventual numerator lower bound
abs(d_(2^j))/E_(2^j) >= a/j^p for 0<=p<1; for p=1, a>1/sqrt(2)
is excluded by NS83. This closes the specified stronger estimate only.
Finite N=8,16,32,64 defects change sign, and gain capture dips to 0.3256%
at N=16. No eventual lower bound is inferred. The cumulative finite
envelope is looser than the existing directly optimized finite bounds.
Two precisions, full Gram/physical cutoff replays and Maxima checks pass.
Manuscript v1.66, RH/G2 and the missing arithmetic lower growth stay unchanged.
Readable report · Proof, complete checks and Maxima.
The optimized arithmetic observations now give a complete gain certificate:
full block gain is at least (sqrt(A_T)-sqrt(B_T))_+^2, where A_T is the
finite observation energy in the true projected Gram metric and B_T
bounds the combined omitted tail. An explicit damped correction achieves
the guarantee, retaining all old-space corrections.
Combining NS81 and NS83 proves that a physical cutoff T=N^6 makes this
observation tail negligible on the 1/log N relative-gain scale. This
localizes the missing arithmetic estimate; it does not prove it. There is
no effective asymptotic onset or new error-decay theorem. The full Gram
and its inverse are retained; no fast algorithm is claimed.
At N=32 to 64 and T=4096, the finite certificate guarantees over 95.98% of available gain, or over 18.15% of that finite old error. At T=64, ignoring the tail would overstate gain by over 15.99%. Two precisions, doubled analytic Gram cutoff, physical/potential checks and Maxima agree. Manuscript remains v1.66; RH and G2 remain open.
Readable report · Proof, complete certificates and Maxima.
The optimized residual has an exact integral-versus-lattice-sum identity, with a complete observation-tail bound. Its signed arithmetic lower estimate remains open; the identity alone gives no new convergence theorem.
A concrete correction squares the divisor defect by Dirichlet convolution through N^2. Its full unit step raises squared error by factors 39.60 and 29.55 at N=4 and N=16; N=8 improves. At N=16, optimally damping the direction and refitting all old coefficients captures 27.31% of available gain. Those modifications generally sacrifice the exact defect-square identity. The complete new tail beyond t=256 alone exceeds 21 times the old error. This rejects finite unit-step descent at the stated sizes, not eventual convergence or every arithmetic construction. Manuscript remains v1.66.
Readable report · Complete argument, Maxima worksheet and certificates.
Every entry in the 91-node conclusion register was reviewed for scope and
evidence status. Stale Toeplitz/Fourier-support claims and misleading
QG/ZLD-to-floor arrows are corrected; the new per-turn review requirement
is recorded in AGENTS.md. This is not a fresh proof audit of all history.
A complete q=2 test of the classical full logarithmic Mobius taper remains far above the unrestricted optimum at N=256: squared error about 0.352175, or 0.045293 after optimizing only its first coefficient, versus 0.0000141424. The finite decrease gives no asymptotic theorem. Its exact prime-sum identity leaves the needed growing-interval arithmetic estimate open. The manuscript remains v1.66; no new theorem version is claimed.
Review and continuation report · Full conclusion review · Complete taper certificates.
Burnol’s dual-vector lower-bound argument extends to every fixed Hardy smoothing order q. The squared optimal error satisfies liminf (log N) E_N^(q) >= sum m(rho)^2 / |rho|^(2q) > 0, summing over distinct critical-line zeros with both signs of their ordinate.
For q=2, ten conjugate pairs give a certified subset constant above 0.000072332433. This is an asymptotic lower bound, with no numerical onset threshold; it is not asserted at N=256 or any other specified size. The uncomputed zero terms are nonnegative and are not silently set to zero.
A fixed positive fraction of the whole error cannot be removed at every large doubling. Cumulative relative gains through doubling K are at most log K + O(1), although individual blocks can fluctuate. A lower gain bound of order a/j with 0<a<=1 is not excluded; no such arithmetic estimate is proved. Uniform capture of available block gain remains consistent with this restriction. The original Weil floor, cofinal decay, G2 and RH stay open.
Readable update · Proof and certified constants.
Keeping the exact logarithmic main tail gives a different, physical-cell comparator with sufficient cutoff O(N^(7/3) log^(1/3)(2N)) and projected condition ratio at most nine. For a 256-to-512 block the sufficient cutoff is 8,435,244, versus over 10^21 for the previous truncated-sample construction. Both are sufficient bounds, not necessary computational costs.
The complete Bernoulli remainder and all cross terms are bounded in norm. Ten ideal exact-line-search steps capture over 98.84% of available block gain. Only scalar cutoffs and constants were checked at 256/384 bits; the millions of physical cells and those iterations were not evaluated. No practical solver or cofinal approximation-error decay is claimed. The original Weil floor, G2 and RH remain open.
Readable update · Proof and finite checks.
Exact coefficient recovery from a finite reciprocal-sample prefix gives an unconditional ordinary Gram condition bound O(N^4 log²(2N)). This rules out exponential growth of the spectral condition number in these physical atom coordinates; it says nothing about the size of the approximation error or a signed Weil spectral floor.
The full arithmetic comparator now has an explicitly sufficient finite cutoff, M = 2^16 (N+1)^6, with a complete tail bound and projected condition ratio at most 42. For an old-size-256 block, use total atom count 512: the sufficient cutoff exceeds 10^21 cells. That enormous comparator was not assembled. The result establishes effectiveness, not a fast solver or a necessary lower bound on computational cost.
Small-prefix inverse certificates and scalar tail bounds pass at 256/384 bits. The missing cofinal signed arithmetic decay, G2 and RH remain open.
Readable update · Proof and finite checks.
The complete squared residual norm is bounded above and below by fixed multiples of an explicit weighted sum over all reciprocal-integer samples. Exact inversion exposes the remaining signed divisor constraints. Independent physical-cell integration, including the infinite tail, confirms the Gram norm at N=16,64,256.
Preserving the exterior tail does not repair the tested divisor rule: at N=256 its best three-direction span captures 59.97% of full gain; adding the endpoint atom reaches 71.18%, versus curvature’s 97.63%. Thirteen of fifteen raw unit steps increase error; two decrease it at N=128.
The full arithmetic sampling Gram supplies an unconditional, size-independent comparison: an ideal preconditioned step captures at least 21/121 of available block gain, and twenty capture more than 97.79%. This retains the entire infinite arithmetic comparator; no fast inverse is supplied. A uniform fraction of available gain is not a cofinal lower bound relative to the full residual error. That arithmetic input, G2 and RH remain open.
Readable update · Proofs and complete finite replays. No new Weil window, uniform Weil floor, or RH proof.
Cancelling the actual residual’s next derivative jumps is exact locally, but its held-old unit step increases the full error on all five checked blocks. At N=256 its optimized three-direction span captures 56.02% of full gain, versus 97.63% for curvature. This is a finite control, not an asymptotic exclusion of all arithmetic feedback.
The continuous-dilation relaxation has an exact distance formula in terms of the classical unknown inner factor. Its closure is strictly larger than the integer closure; a certified explicit witness separates the two. A complete positive zero-tail budget, combined with published finite-height zero verification, gives continuous squared distance below 2.08e-32. This is not an upper bound for integer error, a new zero verification, or a proof of zero distance. The original cofinal arithmetic input remains open.
Readable update · Proofs and complete finite replays. No new Weil window, uniform floor, G2 or RH proof.
The repaired model has an explicit positive differential energy. Its numerator is uniformly comparable to a discrete curvature energy, yielding an O(N) coefficient rule once the actual residual correlations are known. At N=256 the diagonal curvature rule captures 97.63% of the full block gain, an 8.13% reduction of squared error, versus the optimum’s 8.33%. The best combination of three prescribed Mobius/log-taper templates captures 83.66%; these finite comparisons do not exclude other arithmetic formulas.
An explicit unitary altered family has identical complete Gram matrices and arbitrarily close finite efficiency statistics, but an unconditional positive approximation floor. This is a control on the inference from finite geometry, not an off-line zeta zero or a floor for the original family. The original cofinal arithmetic numerator/cost estimate remains open.
Readable update · Proofs and complete finite replays. No new Weil window, uniform floor, G2 or RH proof.
The repaired elementary kernel selects a direction capturing 97.06% of full optimal block improvement at N=256, versus 38.41% for the unmodified correlation direction. That is an 8.08% reduction in squared residual error, versus the full optimum’s 8.33%. These are certified finite results at 256/384 bits, with a doubled-cutoff replay; they do not supply the required cofinal lower contraction.
Exact cell energies retain the weaker canonical growth target. Centering the two complete Abel terms proves that cancellation between those whole vectors cannot change their power exponent; internal arithmetic remains open. The positive renewal operator transfers a forcing bound without improving its exponent, and its critical weighted total energy diverges. The actual optimized endpoint remainder exceeds the leading model at N=64 and N=128. An explicit cusp correction fixes the elementary Gram’s adjacent scaling, but uniform all-coefficient comparison is still false. Using that repaired kernel only to choose coefficients avoids assuming such a comparison: all gains are measured with the complete actual Gram.
Readable research update · Proofs, Maxima checks, finite certificates and validation. No new Weil window, uniform floor, G2 or RH proof. The next input is an arithmetic lower bound for the selected numerator together with a bound for its true-energy cost along a cofinal sequence.
One extra Mellin integration keeps convergence equivalent to RH and gives
an unconditional complete difference-block trace bound 3*kappa/(8*N^2).
The missing input is a lower estimate for the actual optimized residual
correlations; their endpoint expansion now states its entire remainder.
Arb checks at 256/384 bits retain the infinite Gram tail and all projection
terms. On N=16 and N=32, the trace lower bound is below 1/50 of the true gain.
Explicit Euler-grid corrections have a scoped uniform-norm obstruction. The canonical sharp Mobius interpolant still fails along the full integer sequence after smoothing; optimized coefficients and selected subsequences are not excluded. A positive coefficient cone already fails on a finite interval; an explicit dual separator certifies a full-space distance lower bound greater than 0.00228323. Signed approximation is not excluded. Increasing the smoothing order is also not an RH test: one fixed atom then has relative squared error tending to zero while absolute squared error grows. Keep the norm fixed while increasing N.
A fixed number of ordinary Cesaro averages also fails to give canonical
convergence. A weaker target remains: the fixed-order canonical norm
has growth exponent exactly beta_star - 1/2. Subpolynomial growth is
equivalent to RH, and even a bounded cofinal subsequence would suffice.
These growth bounds remain unproved; the exact reformulation supplies
no new zero-free region.
Readable report · Proofs, exact Maxima checks and finite certificates. Published through PR #32. RH, G2 and the original Weil floor remain open.
Adjacent dilation differences give an unconditional complete Gram bound
||D_N|| = O_epsilon(N^(-5/3+epsilon)). The proof uses an exact Mellin
identity and the classical Weyl estimate, with every cross term retained.
A near-N^(-2) power bound for the raw Gram is equivalent to Lindelof;
every fixed logarithmic factor at that scale is ruled out by known large
values of zeta. This does not rule out a better bound for the projected
Gram or the actual residual direction.
Arb checks at 256 and 384 bits show that the old raw-trace bound retains less than 1/400 of the true gain at N=128 and N=256. Changing the basis shrinks the correlations too, and its trace bound is worse at N=256. The missing ingredient remains an asymptotic lower correlation estimate.
Proof, report, finite certificates and validation. No NB convergence, uniform Weil floor, G2 or RH proof is claimed.
The full exterior pairing against every translated arithmetic radical is exactly equivalent to the original nullvector equation. Its derivation is valid on the complete form domain, with both poles and all exterior prime rows retained. It supplies no independent sign or uniqueness estimate.
Finite prime-weight changes add an explicit forcing term. That forcing is nonzero for every nonzero test, but its map is compact and has no uniform positive lower bound on the entire fixed-window unit sphere. This does not exclude a bound restricted to a particular altered operator’s finite kernel.
Proof, report, exact algebra checks and validation. NS-58 stops as specified; API, the uniform finite floor, G2 and RH remain open.
Every nonzero finite change to the prime-shift coefficients creates negative compact smooth tests in both parity sectors, however small the change. The altered complete family has a finite first-zero window, including endpoint support saturation and its modified affine-potential equation. It does not retain the original arithmetic radical identity. No negative direction for the unchanged Weil form is claimed.
For a fixed perturbation pattern of size epsilon, the first-zero parameter
has an analytic upper bound lambda_* = O(sqrt(log(1/epsilon))); constants
are pattern-dependent and no numerical threshold is certified. Fixed bounded
perturbations preserve the cofinal finite-floor objective. The exact scalar
compensation criterion and conditional spectral limit follow from v1.36.
Proof, report, exact Maxima checks and validation. These are analytic controls and target-stability results, not a new signed arithmetic estimate. API, the original uniform floor, G2 and RH remain open.
The actual prime-shift operator admits nonzero functions with both f and A_a f zero on an open patch. The construction works in each parity for a > log 2 and after finitely many linear constraints, including source orthogonality. These are not full-window nullvectors and supply no negative Weil direction. The complete semigroup also fails ordinary pointwise positivity preservation; that property is distinct from nonnegativity of the quadratic form.
A prime-shift-free exterior observation cell does give uniqueness, but a hypothetical first-zero nullvector must reach both window endpoints. An exact inward-dilation identity retains all signed prime overlaps and pole changes; no new arithmetic sign estimate is proved. API remains open.
Report and proof · 285 pages, 0 undefined/duplicate references. No new window, tail metric, uniform floor, G2 or RH claim.
The complete nullvector equation is now an affine screw-potential condition, with no unproved square-integrable derivative assumption. A rank-one countermodel shows why that generic regularity shortcut fails. Paired heat transport cancels inverse-gap singularities while retaining external-zero interactions. Nyman–Beurling approximation has an exact residualized block update and unconditional gain lower bound. The arithmetic injectivity, transport and asymptotic correlation estimates remain open.
Version report and validation · kernel report · heat report · approximation report
Every idea, which sub-ideas were tried from it, and which passed or failed.
Green = proved · gold = current route · red = closed · purple = blocked · grey = open.
Nodes with a folder icon link to the evidence/ directory holding their
certificates; the full version with a node-by-node list is
RESEARCH_MAP.md.
graph LR
rh["RH"]
g1["G1 (weak)<br/><small>thm:v14-radical</small><br/><small>📁 evidence/v124/g2_source_certificate</small>"]
fixedspace["Fixed-space / Burnol Sonine route<br/><small>continuation: OPERATOR-BRIDGES</small>"]
g2["G2: cofinal -o(1)<br/><small>prop:v121-cofinal-rh</small>"]
windows["Fixed-window certificates"]
w3["lambda=3<br/><small>prop:v116-window-positive</small><br/><small>📁 evidence/v124/g2_certificate</small>"]
w4e["lambda=4 even<br/><small>prop:v125-even-complete</small><br/><small>📁 evidence/v126</small>"]
w4o["lambda=4 odd<br/><small>prop:v126-odd-complement</small><br/><small>📁 evidence/v126</small>"]
w4["W_4 >= 0 both sectors<br/><small>prop:v126-full-window</small><br/><small>📁 evidence/v126</small>"]
w5["lambda=5: stipulated 10^-8 D tail comparison<br/><small>prop:v152-tail-comparison</small><br/><small>📁 evidence/ns44_metric_replay</small><br/><small><b>wall: stipulated 1e-8 D tail comparison fails: positive q/D < 7e-18 (lambda=5)</b></small>"]
uniform["Uniform mechanism for G2"]
primenorm["unsigned prime-norm domination<br/><small>prop:v119-prime-essential</small><br/><small>📁 evidence/v124/g2_growing_sign</small><br/><small><b>wall: cost: prime norm ~ lambda survives finite removal; exponential cutoff (estimate, not route)</b></small>"]
normcontr["norm contraction<br/><small>prop:v120-norm-counterexample</small><br/><small>📁 evidence/v124/g2_weighted_signed</small><br/><small><b>wall: specific estimate: violated by a POSITIVE direction, q/D > 2.05 at lambda=4</b></small>"]
schatten["Schatten / Hilbert-Schmidt<br/><small>prop:v132-schatten</small><br/><small>📁 manuscript</small><br/><small><b>wall: scoped: remainder outside finite Schatten classes; negative-only penalty stays a constant (does not obstruct v136)</b></small>"]
sampling["basis / sampling dominance<br/><small>prop:v121-complement-cancellation</small><br/><small>📁 evidence/v124/g2_schur_cancellation</small><br/><small><b>wall: estimate counterexample: source-complement vector with tiny q despite O(1) cancellation</b></small>"]
farmaj["scalar far majorant<br/><small>prop:v125-cutoff-cost</small><br/><small>📁 evidence/v126/g2_window_resolution</small><br/><small><b>wall: cost of one estimate: N > L exp(M_phi/(1-c)); not every method</b></small>"]
blockmetric["lambda=5 N=25: tested dyadic norm comparison<br/><small>prop:v152-block-comparison</small><br/><small>📁 evidence/ns45_block_replay</small><br/><small><b>wall: fixed-partition comparison fails: Rayleigh 341/250, 2093/1500 > 1 (lambda=5, N=25)</b></small>"]
flattop["flat-top smoothing<br/><small>prop:v132-flat-top</small><br/><small>📁 manuscript</small><br/><small><b>wall: construction class: form-exact positive smoothing = point mass; signed smoothing not covered</b></small>"]
recipband["reciprocal-band commutation<br/><small>prop:v133-noncommute</small><br/><small>📁 manuscript</small><br/><small><b>wall: generic mechanism: persistent source-compressed commutator > 9e-6; not every band choice</b></small>"]
cotlar["Cotlar cross terms / atomization<br/><small>prop:v134-disjoint</small><br/><small>📁 manuscript</small><br/><small><b>wall: shortcut: cross norm >= 73/(375 pi) cofinally; not all Cotlar arguments</b></small>"]
scalarprim["scalar signed primitive<br/><small>cor:v137-scalar-no-go</small><br/><small>📁 evidence/v137</small><br/><small><b>wall: genuine closure of the scalar route: a Delta(beta_a) -> infinity, unconditional</b></small>"]
concentration["signed weighted concentration<br/><small>prop:v131-concentration</small><br/><small>📁 evidence/v152</small><br/><small><b>wall: uniform complete signed operator floor remains unproved</b></small><br/><small>continuation: WEIL-FLOOR</small>"]
weaken["Target weakening<br/><small>continuation: WEIL-FLOOR</small>"]
floor["uniform finite floor suffices<br/><small>prop:v136-bounded-floor</small><br/><small>📁 evidence/v136</small><br/><small>continuation: WEIL-FLOOR</small>"]
gapfree["no uniform positive gap exists<br/><small>prop:v135-growing-radical</small><br/><small>📁 evidence/v135</small>"]
floortest["W_lambda >= -8 I at lambda=5,6,8<br/><small>prop:v138-three-floors</small><br/><small>📁 evidence/v138</small>"]
shiftbarrier["bounded shift keeps the exp cutoff barrier<br/><small>prop:v138-shifted-floor</small><br/><small>📁 evidence/v138</small>"]
metablind["Scoped information-loss obstructions<br/><small>thm:v139-probe-relaxation; thm:v139-protected-orbit</small><br/><small>📁 evidence/v139</small>"]
altroutes["Alternative mechanisms: distinct missing inputs"]
circle["circle / Toeplitz-Hankel lens<br/><small>📁 evidence/diag_true_symbol</small><br/><small>continuation: OPERATOR-BRIDGES</small>"]
debranges["de Branges / Hermite-Biehler<br/><small><b>wall: screw-kernel positivity = RH; no fixed-window bridge</b></small><br/><small>continuation: OPERATOR-BRIDGES</small>"]
f1["function field / Hodge / F_1<br/><small><b>wall: no intersection form supplied for this window (missing construction)</b></small><br/><small>continuation: OPERATOR-BRIDGES</small>"]
nb["Nyman-Beurling-Baez-Duarte<br/><small>prop:ns53-nb-block-gain; eq:ns53-nb-rbc</small><br/><small>📁 evidence/ns53_nb_blocks</small><br/><small><b>wall: RBC: corrected arithmetic residual block correlations (open sufficient input)</b></small><br/><small>continuation: NB-GAIN</small>"]
li["Direct rational Li / entire window-test identification<br/><small>📁 evidence/diag_routes/li</small><br/><small><b>wall: direct rational-Li / entire-window classes intersect only at zero</b></small>"]
dbn["de Bruijn–Newman: paired transport<br/><small>prop:ns52-cluster; prop:ns52-unmatched</small><br/><small>📁 evidence/ns52_heat_pairs</small><br/><small><b>wall: complete signed transport with uniform support control (open)</b></small><br/><small>continuation: HEAT-TRANSPORT</small>"]
ccmmu["W_4 >= 0 <=> CCM mu_lambda >= 0 for all lambda <= 4"]
simpleeven4["complete W_4: simple even ground; real-zero transform<br/><small>prop:v140-ground4; cor:v140-real-zeros</small><br/><small>📁 evidence/v140</small><br/><small>continuation: CCM-LIMIT</small>"]
stepb["CCM step (b): finite-compression zeros track zeta zeros (numerical)<br/><small>📁 evidence/diag_ns2_semantic_lock</small><br/><small>continuation: CCM-LIMIT</small>"]
semanticlock["CCM finite-compression semantic lock, lambda=3 N=120<br/><small>lem:v141-centering; prop:v141-ccm-lock</small><br/><small>📁 evidence/v141</small><br/><small>continuation: CCM-LIMIT</small>"]
groundzero4["complete ground zero within 0.1 of gamma_1<br/><small>lem:v142-energy-projection; prop:v142-ground-zero</small><br/><small>📁 evidence/v142</small><br/><small>continuation: CCM-LIMIT</small>"]
groundzero4resolution["first complete zero within 9e-33; archived resolution quantified<br/><small>prop:v143-first-zero-bound; prop:v143-resolution; prop:v143-fixed-window-meaning</small><br/><small>📁 evidence/v143</small><br/><small>continuation: CCM-LIMIT</small>"]
pencilconcentration["pencil loss budget; cofinal obligation retained<br/><small>prop:v144-pencil-concentration</small><br/><small>📁 evidence/v144</small><br/><small>continuation: WEIL-FLOOR</small>"]
betaexplicit["exact lattice symbol; varying envelopes<br/><small>prop:v146-beta-explicit</small><br/><small>📁 evidence/v146</small>"]
minorantlevels["bounded-complexity minorants: arithmetic gap remains<br/><small>prop:v146-packet-loss; prop:v146-level-complexity</small><br/><small>📁 evidence/v146</small><br/><small><b>wall: ZLD + packet: conditional minorant-complexity obstruction only</b></small><br/><small>continuation: LEVEL-PACKETS</small>"]
zeroleveldistribution["zero level distribution and packet coverage: open inputs<br/><small>prop:v147-level-separation; ass:v147-zld; prop:v147-depth-measure; prop:v147-broad-packet</small><br/><small>📁 evidence/v147</small><br/><small><b>wall: ZLD + packet: conditional minorant-complexity obstruction only</b></small><br/><small>continuation: LEVEL-PACKETS</small>"]
weightedexactcost["weighted exact cost: peak bound fails at critical levels<br/><small>prop:v149-critical-density; prop:v149-weighted-cost; prop:v149-qc1-insufficient; ass:v149-qg</small><br/><small>📁 evidence/v149</small><br/><small><b>wall: QG + WLH packet: conditional minorant-complexity obstruction only</b></small><br/><small>continuation: LEVEL-PACKETS</small>"]
capacityweight["positive arithmetic weight: critical capacity input<br/><small>lem:v150-weight; prop:v150-critical-energy; ass:v150-cae</small><br/><small>📁 evidence/v150</small><br/><small><b>wall: CAE</b></small><br/><small>continuation: WEIL-FLOOR</small>"]
capacityslack["fixed energy loss for the Gaussian arithmetic weight<br/><small>prop:v150-capacity-slack</small><br/><small>📁 evidence/v150</small><br/><small><b>wall: fixed-slack shortcut: eta >= delta kappa exp(gamma lambda^2) - O(lambda log lambda)</b></small>"]
inputcomparison["ZLD to QG; CAE is the signed-floor target<br/><small>prop:v151-input-comparison</small><br/><small>📁 evidence/v151</small><br/><small>continuation: WEIL-FLOOR</small>"]
capacitychannels["Fixed/sub-endpoint channel deletion with unchanged signed terms<br/><small>prop:ns43-fixed-channel; prop:ns43-moving-channel</small><br/><small>📁 evidence/v152</small><br/><small><b>wall: channel deletion: unbounded comparison error for the Gaussian weight</b></small>"]
bumpstrength["Restricted bump discrepancy can carry the full zero obstruction<br/><small>prop:ns43-bump-strength</small><br/><small>📁 evidence/v152</small>"]
adaptiveidentity["Unrestricted exact adaptive hierarchy = complement floor<br/><small>ns43-conc-prop-adaptive</small><br/><small>📁 evidence/v152</small><br/><small>continuation: WEIL-FLOOR</small>"]
scalarsetshortcut["Scalar set bounds alone determine signed floor: general inference<br/><small>ns43-conc-prop-scalar</small><br/><small>📁 evidence/v152</small><br/><small><b>wall: general inference false: countermodels with equal scalar set extrema and opposite signed bottoms</b></small>"]
relativeselection["Complete relative selection and entire-transform control<br/><small>prop:ns43-ccm-rayleigh-selection; prop:ns43-ccm-residual-selection</small><br/><small>📁 evidence/v152</small><br/><small><b>wall: complete relative selection plus entire-transform control (both open)</b></small><br/><small>continuation: CCM-LIMIT</small>"]
absoluteselection["Absolute residual/resolvent data alone select ground: abstract inference<br/><small>prop:ns43-ccm-nonselection</small><br/><small>📁 evidence/v152</small><br/><small><b>wall: abstract inference false: countermodel family with alternating profiles</b></small>"]
coarseprofile["Single profile from the specified coarse rescaled liminf<br/><small>prop:ns43-ccm-coarse-profile</small><br/><small>📁 evidence/v152</small><br/><small><b>wall: under the explicit coarse scale the limit form is zero; finer scales open</b></small>"]
fineradicalrank["Uniform or polynomial positive gap after o(lambda²/log lambda) removed directions<br/><small>lem:ns46-finite-arc; ns46-a2-cor-large-block; ns46-a2-cor-rank</small><br/><small>📁 evidence/v153</small><br/><small><b>wall: positive-gap strategy must remove rank >= lambda^2/(20000 log lambda)</b></small>"]
kernelapi["Exact kernel exclusion: affine-potential injectivity<br/><small>prop:ns51-affine-potential; eq:ns51-api</small><br/><small>📁 evidence/ns51_kernel</small><br/><small><b>wall: API: actual arithmetic affine-potential injectivity (open)</b></small><br/><small>continuation: KERNEL-API</small>"]
generic_h1["Generic H¹ bootstrap from a zero eigenvalue<br/><small>prop:ns51-no-generic-bootstrap</small><br/><small>📁 evidence/ns51_kernel</small><br/><small>continuation: KERNEL-API</small>"]
localucp["Local open-set UCP for the full prime-shift operator<br/><small>prop:ns55-local-ucp-counterexample; prop:ns55-parity-ucp-counterexample</small><br/><small>📁 evidence/v155</small><br/><small><b>wall: local f=A_a f=0 does not force f=0; whole-window API is not refuted</b></small><br/><small>continuation: KERNEL-API</small>"]
fullpositivecone["Ordinary-cone positivity for the full Weil semigroup<br/><small>prop:ns55-nonpositive-semigroup</small><br/><small>📁 evidence/v155</small><br/><small><b>wall: order-preservation shortcut fails; quadratic-form positivity is a different property</b></small><br/><small>continuation: KERNEL-API</small>"]
finiteweightcontrols["Finite prime-weight changes: negative tests and first-zero controls<br/><small>cor:ns57-finite-weight-negative; thm:ns57-first-zero-controls; cor:ns57-fixed-direction-sensitivity</small><br/><small>📁 evidence/v156</small>"]
boundedfloorstability["Bounded perturbations preserve the finite-floor objective<br/><small>lem:ns57-bounded-detection; thm:ns57-bounded-floor-stability</small><br/><small>📁 evidence/v156</small><br/><small>continuation: WEIL-FLOOR</small>"]
radicalboundarytest["Translated-radical boundary test: exact kernel equivalence and forced controls<br/><small>prop:ns58-boundary-identity; prop:ns58-equivalence; prop:ns58-forced-pairing</small><br/><small>📁 evidence/v157</small><br/><small>continuation: KERNEL-API</small>"]
nbdifference["NB adjacent-difference norm budget<br/><small>prop:v158-adjacent-budget; prop:v158-mellin-budget</small><br/><small>📁 evidence/v158</small><br/><small>continuation: NB-GAIN</small>"]
nbrawlog["NB uniform logarithmic raw norm<br/><small>prop:v158-lindelof-budget</small><br/><small>📁 evidence/v158</small><br/><small><b>wall: known zeta large values contradict every fixed logarithmic N^(-2) raw norm bound</b></small>"]
nbeulergrid["Uniform logarithmic norm after explicit Euler-grid correction<br/><small>prop:ns60-growing-euler-obstruction</small><br/><small>📁 evidence/v159</small><br/><small><b>wall: zeta large values exceed the allowed finite Euler-product loss</b></small>"]
nbsmoothedbudget["Fixed smoothed NB: positive averages and N^(-2) trace<br/><small>prop:ns61-rh-equivalence; prop:ns61-average-budget</small><br/><small>📁 evidence/v159</small><br/><small>continuation: NB-GAIN</small>"]
nbsmoothedcorrelation["Actual optimized smoothed residual correlations<br/><small>eq:ns61-open-correlation; eq:ns61-endpoint-correlation; prop:ns69-remainder; prop:ns71-selected-input</small><br/><small>📁 evidence/v160</small><br/><small><b>wall: non-summable lower contraction for the actual optimized residual (open sufficient input)</b></small><br/><small>continuation: NB-GAIN</small>"]
nbsharpintegrated["Strong convergence of the smoothed canonical sharp Mobius sequence<br/><small>prop:ns62-smoothed-obstruction; prop:ns62-positive-cone; prop:ns64-cone-distance; prop:ns65-cesaro-obstruction</small><br/><small>📁 evidence/v159</small><br/><small><b>wall: a surviving weighted-Mertens witness for this coefficient rule</b></small><br/><small>continuation: NB-GROWTH</small>"]
nbchangingnorm["Relative convergence with increasing Mellin smoothing<br/><small>prop:ns63-one-atom</small><br/><small>📁 evidence/v159</small><br/><small><b>wall: one-atom unconditional control refutes this convergence inference</b></small>"]
nbcanonicalgrowth["Canonical smoothed norm: subpolynomial growth<br/><small>prop:ns66-growth-bounds; cor:ns66-growth-exponent; prop:ns67-polynomial-cutoff; prop:ns67-abel-conditioning</small><br/><small>📁 evidence/v160</small><br/><small><b>wall: direct subpolynomial canonical norm bound, or a bounded cofinal subsequence (open)</b></small><br/><small>continuation: NB-GROWTH</small>"]
nbcanonicalcells["Complete canonical divisor cells and full tail<br/><small>prop:ns67-cells; lem:ns67-remainder; prop:ns67-polynomial-cutoff</small><br/><small>📁 evidence/v160/ns67</small><br/><small>continuation: NB-GROWTH</small>"]
nbabelwholegain["Power gain solely between the two complete Abel vectors<br/><small>prop:ns67-abel-conditioning</small><br/><small>📁 evidence/v160/ns67</small><br/><small><b>wall: uniform conditioning excludes an exponent improvement in this whole-vector split</b></small><br/><small>continuation: NB-GROWTH</small>"]
nbrenewalforcing["Positive renewal with signed arithmetic forcing<br/><small>prop:ns68-weighted-inversion; prop:ns68-scalar-criterion</small><br/><small>📁 evidence/v160/ns68</small><br/><small><b>wall: independent square-root-plus-epsilon bound for the signed forcing</b></small><br/><small>continuation: NB-GROWTH</small>"]
nbcriticalrenewalenergy["Finite total critical renewal energy<br/><small>cor:ns68-critical-energy</small><br/><small>📁 evidence/v160/ns68</small><br/><small><b>wall: critical-line Mellin pole contradicts finite weighted L2 energy</b></small><br/><small>continuation: NB-GROWTH</small>"]
nbrepairedbounds["Uniform comparison with the repaired elementary Gram<br/><small>prop:ns70-cusp; prop:ns70-repaired-background</small><br/><small>📁 evidence/v160/ns70</small><br/><small><b>wall: pointwise Mellin-density comparison forced by all-coefficient localization</b></small>"]
nbselectedpreconditioner["Repaired-kernel selection measured with true energy<br/><small>prop:ns71-selected-input</small><br/><small>📁 evidence/v160/ns71</small><br/><small><b>wall: cofinal lower numerator and upper selected cost with nonsummable relative gain</b></small><br/><small>continuation: NB-GAIN</small>"]
nblocalgeometry["Complete local energy and discrete curvature<br/><small>prop:ns72-local-energy; prop:ns72-interpolation; prop:ns72-curvature</small><br/><small>📁 evidence/v161/ns72</small><br/><small>continuation: NB-GAIN</small>"]
nbcurvaturearithmetic["Cofinal arithmetic curvature and selected cost<br/><small>eq:ns72-local-directions</small><br/><small>📁 evidence/v161/ns72</small><br/><small><b>wall: actual curvature lower bound and selected true cost giving nonsummable relative contraction</b></small><br/><small>continuation: NB-GAIN</small>"]
nbsamegramcontrol["Infer convergence from shared Gram geometry and finite efficiency<br/><small>prop:ns74-same-gram; cor:ns74-finite-indistinguishability</small><br/><small>📁 evidence/v161/ns74</small><br/><small><b>wall: explicit inserted inner-factor control; scope is this finite-statistics inference</b></small>"]
nbdivisorfeedback["Exact divisor feedback and finite unit-step failure<br/><small>prop:ns75-divisor-cells</small><br/><small>📁 evidence/v162/ns75</small><br/><small>continuation: NB-GAIN</small>"]
nbcontinuousdefect["Exact continuous inner-factor defect and strict closure separation<br/><small>prop:ns76-continuous-distance; prop:ns76-strict-closure</small><br/><small>📁 evidence/v162/ns76</small><br/><small>continuation: NB-CONTINUOUS</small>"]
nbcontinuoustail["Complete continuous zero-tail budget<br/><small>prop:ns77-tail-budget</small><br/><small>📁 evidence/v162/ns77</small><br/><small>continuation: NB-CONTINUOUS</small>"]
nbarithmeticsampling["Complete reciprocal-knot norm and exact divisor inversion<br/><small>prop:ns78-sampling; eq:ns78-sample-inverse</small><br/><small>📁 evidence/v163/ns78</small><br/><small>continuation: NB-SAMPLES</small>"]
nbbalancedfeedback["Tail-balanced divisor feedback control<br/><small>prop:ns79-balance</small><br/><small>📁 evidence/v163/ns79</small><br/><small>continuation: NB-GAIN</small>"]
nbarithmeticpreconditioner["Uniform efficiency using the full arithmetic sampling Gram<br/><small>prop:ns80-uniform-efficiency</small><br/><small>📁 evidence/v163/ns80</small><br/><small>continuation: NB-GAIN</small>"]
nbfiniteconditioning["Polynomial ordinary Gram conditioning and an effective finite comparator<br/><small>prop:ns81-conditioning; prop:ns81-finite-comparator</small><br/><small>📁 evidence/v164/ns81</small><br/><small>continuation: NB-GAIN</small>"]
nbcompressedtail["Finite physical comparator with an exact rank-two tail<br/><small>prop:ns82-tail-compression</small><br/><small>📁 evidence/v165/ns82</small><br/><small>continuation: NB-GAIN</small>"]
nbcriticalfloor["Critical-zero logarithmic lower bound at every fixed smoothing order<br/><small>prop:ns83-log-floor</small><br/><small>📁 evidence/v166/ns83</small><br/><small>continuation: NB-GAIN</small>"]
nbuniformrelativecontraction["Fixed-fraction contraction of the full error at every doubling<br/><small>eq:ns83-gain-budget</small><br/><small>📁 evidence/v166/ns83</small><br/><small>continuation: NB-GAIN</small>"]
nboptimizedpotential["Optimized arithmetic potential and complete observation tails<br/><small>📁 evidence/ns86_optimized_arithmetic</small><br/><small>continuation: NB-GAIN</small>"]
nbconvolutioncontrol["Divisor-convolution correction from N to N squared: finite control<br/><small>📁 evidence/ns86_optimized_arithmetic</small><br/><small>continuation: NB-GAIN</small>"]
nbjointlocalization["Joint finite arithmetic observations preserve logarithmic gain scale<br/><small>📁 evidence/ns87_joint_tail</small><br/><small>continuation: NB-GAIN</small>"]
nbdilationenvelope["Multiplicative normal equations and a scalar whole-error upper envelope<br/><small>📁 evidence/ns88_dilation_defect</small><br/><small>continuation: NB-DILATION</small>"]
nbdilationstrongnumerator["Eventual energy/j^p dilation-defect lower bound with p<1<br/><small>📁 evidence/ns88_dilation_defect</small><br/><small><b>wall: The proposed stronger numerator forces error decay contradicting the critical-zero lower bound</b></small><br/><small>continuation: NB-DILATION</small>"]
nbnormalizedupper["Constructive zero-exterior upper certificates preserve every decay rate<br/><small>📁 evidence/ns89_upper_normalization</small><br/><small>continuation: NB-SAMPLES</small>"]
nbmobiusjointcancellation["Exact Mobius cancellation; fixed-size absolute-bound loss<br/><small>📁 evidence/ns90_mobius_gain</small><br/><small>continuation: MOBIUS-COST, MOBIUS-MARGIN</small>"]
nbjointabelbound["Joint first-primitive finite lower bound with complete grouped tail<br/><small>📁 evidence/ns91_joint_abel</small><br/><small>continuation: MOBIUS-COST, MOBIUS-MARGIN</small>"]
nbarithmeticcostbudget["Complete scalar cost budget and a feasible endpoint-only update<br/><small>📁 evidence/ns92_arithmetic_cost</small><br/><small>continuation: MOBIUS-COST, MOBIUS-MARGIN</small>"]
nbaveragedabsolutecost["Averaging the fully separated absolute Mobius cost<br/><small>📁 evidence/ns93_averaged_cost</small><br/><small><b>wall: productwise absolute values force a linear majorant even before averaging</b></small><br/><small>continuation: MOBIUS-COST, MOBIUS-MARGIN</small>"]
circlecountinference["Tangent-circle similarity point and count imbalance infer RH<br/><small>📁 evidence/ns94_circle_inversion</small><br/><small><b>wall: exact symmetric off-line control preserves the proposed geometric count and infinity features</b></small><br/><small>continuation: ZERO-GEOMETRY</small>"]
nbsquareproductrestriction["Square-scale product restriction and complete gain split<br/><small>📁 evidence/ns95_square_product</small><br/><small>continuation: NB-GAIN</small>"]
nblcmsquarecone["LCM divisor squares: complete-rule floors and signed refit equivalence<br/><small>📁 evidence/ns96_lcm_square</small><br/><small>continuation: NB-GAIN</small>"]
rh --> g1
rh --> fixedspace
rh --> g2
g2 --> windows
windows --> w3
windows --> w4e
windows --> w4o
windows --> w4
windows --> w5
g2 --> uniform
uniform --> primenorm
uniform --> normcontr
uniform --> schatten
uniform --> sampling
uniform --> farmaj
uniform --> blockmetric
uniform --> flattop
uniform --> recipband
uniform --> cotlar
uniform --> scalarprim
uniform --> concentration
g2 --> weaken
weaken --> floor
weaken --> gapfree
floor --> floortest
floor --> shiftbarrier
uniform --> metablind
rh --> altroutes
altroutes --> circle
altroutes --> debranges
altroutes --> f1
altroutes --> nb
altroutes --> li
altroutes --> dbn
w4 --> ccmmu
w4 --> simpleeven4
simpleeven4 --> stepb
windows --> semanticlock
simpleeven4 --> groundzero4
groundzero4 --> groundzero4resolution
concentration --> pencilconcentration
concentration --> betaexplicit
concentration --> minorantlevels
minorantlevels --> zeroleveldistribution
zeroleveldistribution --> weightedexactcost
floor --> capacityweight
capacityweight --> capacityslack
capacityweight --> inputcomparison
capacityweight --> capacitychannels
capacityweight --> bumpstrength
concentration --> adaptiveidentity
concentration --> scalarsetshortcut
stepb --> relativeselection
stepb --> absoluteselection
stepb --> coarseprofile
gapfree --> fineradicalrank
altroutes --> kernelapi
kernelapi --> generic_h1
kernelapi --> localucp
kernelapi --> fullpositivecone
kernelapi --> finiteweightcontrols
floor --> boundedfloorstability
kernelapi --> radicalboundarytest
nb --> nbdifference
nb --> nbrawlog
nb --> nbeulergrid
nb --> nbsmoothedbudget
nbsmoothedbudget --> nbsmoothedcorrelation
nbsmoothedbudget --> nbsharpintegrated
nbsmoothedbudget --> nbchangingnorm
nbsmoothedbudget --> nbcanonicalgrowth
nbcanonicalgrowth --> nbcanonicalcells
nbcanonicalgrowth --> nbabelwholegain
nbcanonicalgrowth --> nbrenewalforcing
nbrenewalforcing --> nbcriticalrenewalenergy
nbsmoothedcorrelation --> nbrepairedbounds
nbsmoothedcorrelation --> nbselectedpreconditioner
nbselectedpreconditioner --> nblocalgeometry
nblocalgeometry --> nbcurvaturearithmetic
nblocalgeometry --> nbsamegramcontrol
nbcurvaturearithmetic --> nbdivisorfeedback
nbsamegramcontrol --> nbcontinuousdefect
nbcontinuousdefect --> nbcontinuoustail
nbdivisorfeedback --> nbarithmeticsampling
nbarithmeticsampling --> nbbalancedfeedback
nbarithmeticsampling --> nbarithmeticpreconditioner
nbarithmeticpreconditioner --> nbfiniteconditioning
nbfiniteconditioning --> nbcompressedtail
nbsmoothedbudget --> nbcriticalfloor
nbcriticalfloor --> nbuniformrelativecontraction
nbarithmeticsampling --> nboptimizedpotential
nboptimizedpotential --> nbconvolutioncontrol
nboptimizedpotential --> nbjointlocalization
nboptimizedpotential --> nbdilationenvelope
nbdilationenvelope --> nbdilationstrongnumerator
nbsmoothedbudget --> nbnormalizedupper
nbjointlocalization --> nbmobiusjointcancellation
nbmobiusjointcancellation --> nbjointabelbound
nbjointabelbound --> nbarithmeticcostbudget
nbarithmeticcostbudget --> nbaveragedabsolutecost
altroutes --> circlecountinference
nbjointlocalization --> nbsquareproductrestriction
nbsquareproductrestriction --> nblcmsquarecone
concentration -. "uniform complete signed operator floor remains unproved" .-> floor
debranges -. "screw-kernel positivity = RH; no fixed-window bridge" .-> rh
nb -. "RBC: corrected arithmetic residual block correlations (open sufficient input)" .-> rh
dbn -. "complete signed transport with uniform support control (open)" .-> rh
capacityweight -. "CAE" .-> floor
relativeselection -. "complete relative selection plus entire-transform control (both open)" .-> g2
kernelapi -. "API: actual arithmetic affine-potential injectivity (open)" .-> rh
nbsmoothedcorrelation -. "non-summable lower contraction for the actual optimized residual (open sufficient input)" .-> rh
nbcanonicalgrowth -. "direct subpolynomial canonical norm bound, or a bounded cofinal subsequence (open)" .-> rh
nbrenewalforcing -. "independent square-root-plus-epsilon bound for the signed forcing" .-> rh
nbselectedpreconditioner -. "cofinal lower numerator and upper selected cost with nonsummable relative gain" .-> rh
nbcurvaturearithmetic -. "actual curvature lower bound and selected true cost giving nonsummable relative contraction" .-> rh
click g1 "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v124/g2_source_certificate/" "evidence: evidence/v124/g2_source_certificate" _blank
click w3 "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v124/g2_certificate/" "evidence: evidence/v124/g2_certificate" _blank
click w4e "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v126/" "evidence: evidence/v126" _blank
click w4o "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v126/" "evidence: evidence/v126" _blank
click w4 "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v126/" "evidence: evidence/v126" _blank
click w5 "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns44_metric_replay/" "evidence: evidence/ns44_metric_replay" _blank
click primenorm "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v124/g2_growing_sign/" "evidence: evidence/v124/g2_growing_sign" _blank
click normcontr "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v124/g2_weighted_signed/" "evidence: evidence/v124/g2_weighted_signed" _blank
click schatten "https://github.com/AlexanderEastwood/riemann/tree/main/manuscript/" "evidence: manuscript" _blank
click sampling "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v124/g2_schur_cancellation/" "evidence: evidence/v124/g2_schur_cancellation" _blank
click farmaj "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v126/g2_window_resolution/" "evidence: evidence/v126/g2_window_resolution" _blank
click blockmetric "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns45_block_replay/" "evidence: evidence/ns45_block_replay" _blank
click flattop "https://github.com/AlexanderEastwood/riemann/tree/main/manuscript/" "evidence: manuscript" _blank
click recipband "https://github.com/AlexanderEastwood/riemann/tree/main/manuscript/" "evidence: manuscript" _blank
click cotlar "https://github.com/AlexanderEastwood/riemann/tree/main/manuscript/" "evidence: manuscript" _blank
click scalarprim "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v137/" "evidence: evidence/v137" _blank
click concentration "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v152/" "evidence: evidence/v152" _blank
click floor "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v136/" "evidence: evidence/v136" _blank
click gapfree "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v135/" "evidence: evidence/v135" _blank
click floortest "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v138/" "evidence: evidence/v138" _blank
click shiftbarrier "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v138/" "evidence: evidence/v138" _blank
click metablind "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v139/" "evidence: evidence/v139" _blank
click circle "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/diag_true_symbol/" "evidence: evidence/diag_true_symbol" _blank
click nb "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns53_nb_blocks/" "evidence: evidence/ns53_nb_blocks" _blank
click li "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/diag_routes/li/" "evidence: evidence/diag_routes/li" _blank
click dbn "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns52_heat_pairs/" "evidence: evidence/ns52_heat_pairs" _blank
click simpleeven4 "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v140/" "evidence: evidence/v140" _blank
click stepb "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/diag_ns2_semantic_lock/" "evidence: evidence/diag_ns2_semantic_lock" _blank
click semanticlock "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v141/" "evidence: evidence/v141" _blank
click groundzero4 "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v142/" "evidence: evidence/v142" _blank
click groundzero4resolution "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v143/" "evidence: evidence/v143" _blank
click pencilconcentration "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v144/" "evidence: evidence/v144" _blank
click betaexplicit "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v146/" "evidence: evidence/v146" _blank
click minorantlevels "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v146/" "evidence: evidence/v146" _blank
click zeroleveldistribution "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v147/" "evidence: evidence/v147" _blank
click weightedexactcost "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v149/" "evidence: evidence/v149" _blank
click capacityweight "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v150/" "evidence: evidence/v150" _blank
click capacityslack "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v150/" "evidence: evidence/v150" _blank
click inputcomparison "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v151/" "evidence: evidence/v151" _blank
click capacitychannels "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v152/" "evidence: evidence/v152" _blank
click bumpstrength "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v152/" "evidence: evidence/v152" _blank
click adaptiveidentity "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v152/" "evidence: evidence/v152" _blank
click scalarsetshortcut "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v152/" "evidence: evidence/v152" _blank
click relativeselection "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v152/" "evidence: evidence/v152" _blank
click absoluteselection "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v152/" "evidence: evidence/v152" _blank
click coarseprofile "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v152/" "evidence: evidence/v152" _blank
click fineradicalrank "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v153/" "evidence: evidence/v153" _blank
click kernelapi "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns51_kernel/" "evidence: evidence/ns51_kernel" _blank
click generic_h1 "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns51_kernel/" "evidence: evidence/ns51_kernel" _blank
click localucp "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v155/" "evidence: evidence/v155" _blank
click fullpositivecone "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v155/" "evidence: evidence/v155" _blank
click finiteweightcontrols "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v156/" "evidence: evidence/v156" _blank
click boundedfloorstability "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v156/" "evidence: evidence/v156" _blank
click radicalboundarytest "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v157/" "evidence: evidence/v157" _blank
click nbdifference "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v158/" "evidence: evidence/v158" _blank
click nbrawlog "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v158/" "evidence: evidence/v158" _blank
click nbeulergrid "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v159/" "evidence: evidence/v159" _blank
click nbsmoothedbudget "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v159/" "evidence: evidence/v159" _blank
click nbsmoothedcorrelation "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v160/" "evidence: evidence/v160" _blank
click nbsharpintegrated "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v159/" "evidence: evidence/v159" _blank
click nbchangingnorm "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v159/" "evidence: evidence/v159" _blank
click nbcanonicalgrowth "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v160/" "evidence: evidence/v160" _blank
click nbcanonicalcells "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v160/ns67/" "evidence: evidence/v160/ns67" _blank
click nbabelwholegain "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v160/ns67/" "evidence: evidence/v160/ns67" _blank
click nbrenewalforcing "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v160/ns68/" "evidence: evidence/v160/ns68" _blank
click nbcriticalrenewalenergy "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v160/ns68/" "evidence: evidence/v160/ns68" _blank
click nbrepairedbounds "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v160/ns70/" "evidence: evidence/v160/ns70" _blank
click nbselectedpreconditioner "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v160/ns71/" "evidence: evidence/v160/ns71" _blank
click nblocalgeometry "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v161/ns72/" "evidence: evidence/v161/ns72" _blank
click nbcurvaturearithmetic "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v161/ns72/" "evidence: evidence/v161/ns72" _blank
click nbsamegramcontrol "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v161/ns74/" "evidence: evidence/v161/ns74" _blank
click nbdivisorfeedback "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v162/ns75/" "evidence: evidence/v162/ns75" _blank
click nbcontinuousdefect "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v162/ns76/" "evidence: evidence/v162/ns76" _blank
click nbcontinuoustail "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v162/ns77/" "evidence: evidence/v162/ns77" _blank
click nbarithmeticsampling "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v163/ns78/" "evidence: evidence/v163/ns78" _blank
click nbbalancedfeedback "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v163/ns79/" "evidence: evidence/v163/ns79" _blank
click nbarithmeticpreconditioner "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v163/ns80/" "evidence: evidence/v163/ns80" _blank
click nbfiniteconditioning "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v164/ns81/" "evidence: evidence/v164/ns81" _blank
click nbcompressedtail "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v165/ns82/" "evidence: evidence/v165/ns82" _blank
click nbcriticalfloor "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v166/ns83/" "evidence: evidence/v166/ns83" _blank
click nbuniformrelativecontraction "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/v166/ns83/" "evidence: evidence/v166/ns83" _blank
click nboptimizedpotential "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns86_optimized_arithmetic/" "evidence: evidence/ns86_optimized_arithmetic" _blank
click nbconvolutioncontrol "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns86_optimized_arithmetic/" "evidence: evidence/ns86_optimized_arithmetic" _blank
click nbjointlocalization "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns87_joint_tail/" "evidence: evidence/ns87_joint_tail" _blank
click nbdilationenvelope "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns88_dilation_defect/" "evidence: evidence/ns88_dilation_defect" _blank
click nbdilationstrongnumerator "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns88_dilation_defect/" "evidence: evidence/ns88_dilation_defect" _blank
click nbnormalizedupper "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns89_upper_normalization/" "evidence: evidence/ns89_upper_normalization" _blank
click nbmobiusjointcancellation "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns90_mobius_gain/" "evidence: evidence/ns90_mobius_gain" _blank
click nbjointabelbound "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns91_joint_abel/" "evidence: evidence/ns91_joint_abel" _blank
click nbarithmeticcostbudget "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns92_arithmetic_cost/" "evidence: evidence/ns92_arithmetic_cost" _blank
click nbaveragedabsolutecost "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns93_averaged_cost/" "evidence: evidence/ns93_averaged_cost" _blank
click circlecountinference "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns94_circle_inversion/" "evidence: evidence/ns94_circle_inversion" _blank
click nbsquareproductrestriction "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns95_square_product/" "evidence: evidence/ns95_square_product" _blank
click nblcmsquarecone "https://github.com/AlexanderEastwood/riemann/tree/main/evidence/ns96_lcm_square/" "evidence: evidence/ns96_lcm_square" _blank
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end
One live manuscript; delivery is LaTeX only at the author’s request.
Prior versions are reachable by tag (v1.24, v1.34 … v1.38).
NS-46, jointly with Astra-2, proves a larger complete near-zero block: dimension asymptotic to lambda²/(20000 log lambda) and full operator residual at most C exp(-lambda²/2000). The construction changes from widely spaced translates to increasingly close translates inside a fixed interval. A finite-arc interpolation proof explicitly pays for the deteriorating Gram.
The block has exact even/odd dimensions. Removing o(lambda²/log lambda) directions cannot leave a positive uniform or polynomial gap. This is a stronger rank obstruction, not a complementary lower floor. It determines no energy sign and gives no certified finite starting window. The gap-free uniform-floor route remains open. No new numerical window, tail metric, G2 or RH claim.
Assessment, proof and review scope. The joint result is reviewed in PR #24 and builds on the merged v1.52. Archived reports retain their prepublication checkpoint status.
NS-43/44/45 add two fresh, narrowly scoped comparison
certificates and complete analytic proofs for route selection. At the existing
lambda=5 window, both parity witnesses have positive Weil energy but violate
10^-8 D. The existing N=25 dyadic norm comparison has rational Rayleigh
bounds 1.364 and 2093/1500, both above one. These replace missing proof
inputs for those comparisons; they do not recover historical artifacts or
establish a negative Weil direction.
The capacity proof extends the fixed-loss obstruction to all fixed prime-power channels and omitted channels uniformly below the endpoint. Exact adaptive concentration is an equivalent floor target; scalar set bounds can miss signed operator information. CCM needs a complete relative selection estimate and entire-transform control. Abstract residual/resolvent and coarse-profile shortcuts close only under their stated hypotheses. CCM step (b) stays the sole gold node; the circle lane remains open. No G2 or RH result.
Route assessment and scope. The v1.52 change is reviewed in PR #21 and builds on the merged v1.51 work in PR #20. Archived reports retain their prepublication checkpoint status.
NS-39 gives one proposition comparing the three open inputs:
ZLD implies QG’s growth clause on the same prescribed family, with lower cost
κ θ² D / (2K); packet feasibility remains separate. The complete corrected
zero field gives the positive-weight potential exactly. CAE is precisely the
uniform complement-floor target in those coordinates. No new signed estimate,
arithmetic independence, G2 or RH result follows.
NS-38 uses explicit disclosure for both missing evidence groups; no originals were recovered. The five v1.35 scientific files are present and match legacy hashes. The two new measurement records are separate diagnostics: NS-41 odd feasible states and NS-42 transformed energies. Neither is cited as a bound. NS-40’s QG literature assessment is coordinated separately in PR #19.
NS-34 chooses the explicit positive Gaussian arithmetic radical as the weight in the physical prime-shift/Picone identity. Its exact pole mass is 1/sqrt(3); the complete energy identity retains the odd pole and exterior edges. For this weight, a fixed fractional loss of transformed energy, or omission of the prime-2 channel alone, forces a capacity error growing like exp(gamma*lambda^2), up to O(lambda log lambda). The source constraint is imposed exactly in a two-dimensional even test space; odd parity is covered separately.
The closed result concerns those weakened lower comparisons. The full coefficient-one comparison remains the named open input CAE, equivalent to the existing uniform complement-floor target in these coordinates. A different weight is not excluded. No numerical experiment, new window or tail metric; G2 and RH remain open. See v1.50.
PR #14’s v1.48 candidate is held for major review findings. The valid weighted bound eta >= [c’ QC_K - Q]+ is restated self-contained. Negative critical values make the exact continuous level-density peak infinite; the finite histogram peak is a different quantity. QC_1 growth alone is also insufficient: a full-support countermodel has QC_1~D/2 but QC_2->0. The corrected cofinal input is Exact Level Quantization Growth (QG), plus admissible bounded-energy weighted packets; a QC_1/K law needs a separate shape hypothesis. Both inputs remain open. No new computation, window or tail metric. G2 and RH remain open. See v1.49 and the PR #14 review.
NS-29 separates the measure of every interval of symbol levels from the Fourier coverage and original energy of a physical packet. Cumulative sublevel bounds do not imply the required density. The exact zero expansion provides a precise named open input, ZLD, with endpoint and trivial terms retained. The zero theorems assessed do not supply that value distribution. Depth growth yields only local widths with window-dependent constants; broad Fourier pulses have energy log X + O(1). Both uniform level measure and bounded-energy packet coverage remain open. The route stays blocked, not closed; G2 and RH remain open. See v1.47.
The exact lattice-symbol identity retains the strict prime-power endpoint, trivial zeros and Perron remainder. Its envelopes vary; the comb is an explicit-formula representation, not a new positivity mechanism. The numerical illustration checks 40 low samples and 14 deep-trough/centroid locations with explicit summation and zero-count conventions.
The minorant analysis proves local packet bounds and a conditional linear level-count inequality. Positive spill and original packet energy must be retained. Uniform arithmetic level distribution and bounded packet energy remain unproved, so the route stays blocked, not closed. The concentration criterion remains valid; G2 and RH remain open. See v1.46.
Audit MINOR M1/M2 wording corrections: inherited 320/768/896-bit inputs are separated from the new 1024/1280-bit gates, and the direct-bound limitation is paired with the separate frozen-trial transfer floor, about 6.65e-38. No enclosure changed. See v1.45.
NS-24 translates the weighted concentration inequality to exact pencil
directions: the relative minorant loss must be at most
nu_k + eta_a ||v_k||^2/q+[v_k], with source admissibility and all mixture
cross terms retained. This is a fixed-window head condition. The complete
criterion still needs cofinal uniform control; uniformly bounded errors
already suffice under v1.36’s hypotheses. The twelve small lambda=4 N=48
values are cited only as diagnostics. No new computation or concentration
bound is claimed, and the v1.43 enclosure is unchanged. G2 and RH stay open.
NS-19 closes with a two-sided enclosure of the complete first positive
lambda=4 zero and a precise resolution statement. The radius is
sqrt(C_ell*rho)/0.0016 < 8.752082e-33 < 9e-33. The missing lower-energy
transfer would require a Rayleigh gap about 3.2032e-153 (relative
1.3055e-78), plus a suitably centered trial for a gamma-centered result.
The finite N=120 discrepancy of about +2.92504e-71 is separately
certified and does not transfer to the complete ground. This is a limit
of the displayed archived bounds, not a failure of the object or a
universal no-go theorem. No new window or tail metric; G2 and RH stay open.
One local zero of the complete lambda=4 ground transform is now certified within 0.1 of the first zeta ordinate. It is unique and simple in that interval. The complete even-sector separator 1e-67 makes an energy-projection bound small enough to transfer endpoint and derivative signs. The old global separator is too coarse for the same test.
All 4097 trial coefficients and the archived infinite-tail bounds are retained. Both 1024/1280-bit replays and a second implementation pass. No new window is computed. This does not determine the discrepancy’s sign, order earlier zeros, or establish cofinal convergence. G2 and RH remain open. Proof and replay.
The existing lambda=3, N=120 compression reproduces the first eight local zero discrepancies in CCM §6 Figure 1, with interval eigenpair/root gates replayed at 768 and 1024 bits. The first discrepancy is approximately 1.582329697193127e-34, matching their rounded 1.6e-34.
This identifies the missing (-1)^n centering phase in the prior NS-14
script. Its sinc-lattice conclusion is withdrawn; the old run is preserved
with a correction. The new result is finite-dimensional and supplies no
complete-ground zero locations or cofinal convergence. G2 and RH remain open.
The literature comparison is integrated and both missing evidence groups
remain OPEN. Proof and replay.
Two precise meta-obstructions delimit information-losing estimates. The optimal full-histogram mass-cap bound, and a mass/height/variation relaxation, cannot supply a cofinal finite floor for the actual symbol. A separate spectral-rotation theorem preserves any finite head and its complete operator action while exposing the divergent negative multiplier edge. Its conjugates need not remain multiplication operators.
The unrestricted slogan “every location-insensitive estimate fails” is false; an exact positive counterexample and a ten-route coverage audit are included. These results do not close G2 or prove RH.
Proof, exact checks and scope audit.
v1.36 reduced G2 to a uniform finite floor: one constant C_* with
W_λ >= -C_*·I along a cofinal family suffices, with no decay required.
v1.38 tests that reduction directly instead of seeking another positivity
certificate.
With a common shift δ = 8, head cutoff N = 256, remote cutoff
J = 4096, moment order 16, and no tail-metric prerequisite (Z = 0),
both complete parity forms are certified at λ = 5, 6, 8:
-8 <= inf σ(W_λ^±) < 1e-16
The lower bound retains the complete infinite Fourier tail; the upper
bound is a certified finite-support trial quotient. By support
consistency the floor holds for every 1 < λ <= 8. The same exact dyadic
witnesses pass at 160 and 256 bits.
The shift is what makes it affordable: the remote cutoff needs
N+1 > L·exp(M_φ − δ), so δ = 8 buys a factor e^8. But
prop:v138-shifted-floor proves any bounded shift keeps the exponential
barrier of prop:v125-cutoff-cost; escaping it in this comparison would
need δ to grow like M_φ ~ λ, which is uninformative. The certificate’s
generalized margin falls from 0.84 to 0.10 (odd) across the three windows.
That decrease measures enclosure headroom, not the sign of the physical
edge — no cofinal lower estimate is asserted, and no G2 gap is closed.
v1.36 showed that one uniform finite ordinary lower floor on the complete small-residual complement would suffice. v1.37 tests whether the older scalar primitive estimate could meet this weaker target.
It cannot: for the exact arithmetic symbol, its optimal scalar error
a Delta(beta_a) tends to infinity. The pointwise infimum of beta_a
also tends to minus infinity. Both conclusions are unconditional.
The proof uses a nonnegative frequency probe whose Fourier transform vanishes at the two physical cutoff endpoints. Under RH its pairing with the exact symbol retains a negative cutoff side lobe around a zero. Any bounded cofinal scalar certificate would itself imply RH via v1.36, then contradict that pairing. The linear rates are RH-conditional; no unconditional linear rate is asserted.
The probe is not a physical squared Paley–Wiener transform, so these scalar obstructions do not refute positivity of the physical form. The signed concentration estimate, retaining favorable and unfavorable levels jointly, remains open in both parities. No G2 sign gap was closed.
The ingredients are classical; worldwide novelty is not claimed. Numerical quadrature is diagnostic and not a proof.
lambda = 4 evidenceThe original v1.26 reproduction bundle has been recovered and committed
under evidence/v126/: 468 files, byte-identical to the
archived ZIP (SHA-256 ccdb8eaa…82d8f, independently re-verified). It
includes both final certificate directories:
Final proof gates using the saved ingredients, witness bindings and the
exact shared-head equality were replayed successfully. This did not
regenerate witnesses or rerun every residual assembly; RESTORED in the
ledger records availability, not a fresh independent audit of every
computation. Provenance and per-file checksums:
manifest/README_v126_evidence_recovery.md.
Apache-2.0 — see LICENSE.