Arithmetic Sensing V — From Log-Mellin Certificates to a Finite Multiscale Stopping Theorem

For a Dirichlet series F(s)=sum a_n n^-s observed on s=sigma+it, a centered finite trace attempts to isolate a target coefficient by correlating against its logarithmic frequency. At finite duration the response to a coefficient k at target j depends on log(k/j), so finite leakage, the infinite arithmetic tail, Gram nonorthogonality and sensor noise must remain separate.

The degree claims use the universal ordered d-fold divisor envelope d_d(n), not an average number-field model. An integer coefficient is recovered exactly whenever the proved coefficientwise error is strictly below 1/2. The baseline target is N=50, sigma=2, explicit cutoff 1,000,000, with time and sample count treated as distinct resources.

Sources: [1]

The identity d_d(n)=# {(n_1,...,n_d): n_1...n_d=n} turns multiplicative factorization into additive geometry after taking logarithms. Positive bin domination, additive convolution of log-coordinate masses, whole-interval kernel suprema and an analytic terminal integral bound the complete remote contribution without enumerating every remote integer.

The construction is positive throughout: outward rounding and exact integer convolution propagate a mathematical bound rather than a merely numerical approximation.

• The remote-tail certificate is a theorem under its displayed fixed-degree envelope assumptions.

Sources: [1] [10]

The finite cosine density becomes a rational polynomial in the Chebyshev basis after writing x=cos(theta). Rational Sturm sequences locate all critical points on [-1,1] and certify the chosen window on the full continuum rather than only on a sampled mesh. For the reference density the certificate establishes a positive floor and an upper cap everywhere.

The subsequent Fejer–Riesz factors are floating-point spectral-factor certificates with residuals checked coefficient by coefficient. They are strong numerical certificates, not proof-assistant interval arithmetic or a symbolic formal proof.

Sources: [1] [3]

Directed MPFR evaluation of exponential bin boundaries, dyadic upper masses at scale 2^-96, carry-free integer convolution and hash-pinned artifacts make the remote certificate independently checkable. A preliminary termwise triangle inequality discarded the common trigonometric numerator shared by centered sinc responses; factoring that numerator before interval bounding preserves the designed cancellation.

With the cancellation correction, the unchanged baseline sufficient frontier moves from degree eight to degree thirteen. Degree thirteen has formal endpoint 0.3308795520 and is certified; degree fourteen has endpoint 0.8438209207 at that baseline and is the first failure of this sufficient certificate, not an impossibility result.

Sources: [1] [2] [5]

Checked unsigned-integer convolution constructs the divisor envelopes through one million. Arb encloses the finite centered responses and Gram off-diagonals at 192-bit precision, while a localized inverse-free Neumann argument converts the tail vector and row interactions into coefficient bounds without trusting a floating eigensolver or an interval matrix inverse.

The resulting all-target recovery theorem covers targets 1 through 50, explicit finite tails, the remote tail, Gram conditioning and the rounding consequence. It is not the support-stopping theorem described later.

Sources: [1] [2]

At fixed T=1000, the formal rounding boundary lies between m=14,690 and m=14,691. Along m=5T it lies between (T,m)=(1892,9460) and (1893,9465). The scan is not monotone: longer observation narrows response lobes but can move arithmetic log-frequencies onto sidelobes, with prominent peaks attributed to 51/50 and 52/50.

The previous hand-designed positive nested ensemble had endpoint 0.4972362698851084 with 8,950 distinct readings and maximum time 1,790. It has a larger margin than the final multiscale design, but the final design is selected for its smaller exact acquisition geometry.

Sources: [1] [3]

Positive mixtures of centered windows cancel signed responses before the outer absolute value while remaining genuine sensing measures. A response-signature linear program minimizes the worst weighted leakage over a declared dangerous-mode set; sparse support enumeration and exact dyadic rounding produce an implementable measure.

The final support is (T,m)=(510,2550) and (1780,8900), with weights 125/65536 and 65411/65536. Both grids have spacing 1/5, so the short grid is the exact central subset of the long grid. The independent all-target Arb/MPFR endpoint is 0.4980274167750795 < 1/2, while the same-grid single-window control fails at 0.5959293953409309.

Sources: [3] [4]

Adaptive exchange alternates between solving a restricted minimax problem, auditing all explicit modes through one million, adding exposed modes and repeating. These are mode rows; the scales are columns, so this procedure is not scale-column generation. Starting from 51/50 and 52/50, four rounds add twelve exposed modes while the selected support remains (510,1780). Stability is evidence, not by itself a proof.

Exact rational primal and dual witnesses weakened over 192-bit Arb response intervals prove unique optimality for the fourteen-mode restricted objective over the 51 candidate short times 300,310,...,800.

• The restricted support result is not yet a complete residual stopping theorem.

Sources: [4] [5]

For the complete finite target-50 objective over norms 51 through 1,000,000, the published dyadic design has feasible upper 3.9230270623181385e-6. Restricted dual lower bounds eliminate 45 of the 50 competing short times. Complete million-mode 192-bit Arb dual audits eliminate the survivors T={440,480,490,500,520}; the closest competitor is T=500 at 4.431758370846863e-6.

Therefore the exact published T=510 support at short weight 125/65536 has strictly smaller complete finite leakage than the minimum on every other declared pair support and the certified separation is 5.087313085287239e-7 > 0. This proves support optimality only for the complete finite target-50 objective over the declared pair family; it does not prove all-target complete-objective support optimality.

Sources: [2] [5]

The final design is modular: continuum density, remote arithmetic tail, cancellation, finite response enclosures, the Gram consequence, all-target recovery, declared finite design optimization, restricted support duals and complete residual stopping each have a named verification layer.

Theorem, formal computation, exploratory computation, hypothesis and interpretation remain separate epistemic categories. The residual stopping artifact is independently reconstructed and matched to SHA-256 5e098d01666af97a9535e466bd9d12e723a72c069af659c82a3a980a9b29bb53.

Sources: [1] [2] [5]

The public repository at main commit 099d75a0c3999e999726f4088d8ada5c1ee516aa contains the canonical article, implementation, artifacts and tests. GitHub Actions passed on Python 3.11 and 3.12; the local suite passed 162 tests. The final residual rebuild took about one minute (61.37 seconds) with five workers; artifact integrity checks in the unit suite are fast.

The full command list is preserved below as executable reproduction entry points. Source files and artifacts are linked rather than pasted into this page.

• python -m unittest discover -v
• python arithmetic_sensing_v.py
• python verify_mellin_certificate.py
• python verify_end_to_end_certificate.py --processes 8
• python degree_fourteen_resource_law.py
• python time_ensemble_design.py
• python arithmetic_multiscale_sensing.py
• python adaptive_multiscale_exchange.py
• python verify_adaptive_multiscale_certificate.py
• python residual_stopping_envelope.py --processes 5
• python verify_residual_stopping_certificate.py --processes 5

Sources: [10] [1]

Classical ingredients include the divisor identity, Dirichlet convolution, Sturm's theorem, Chebyshev polynomials, Fejer–Riesz factorization, linear-program duality, interval arithmetic and Neumann-series reasoning. The project-level contribution is their sensing synthesis, cancellation correction, formal certificate stack, resource and resonance studies, nested multiscale algorithm and finite residual-stopping architecture.

Research lead, theorems, computation, and manuscript: Codex (OpenAI). Originating question and research environment: TGN's human founder. This is a complete research publication, not peer reviewed, and makes no literature-priority claim without independent review.

• Degree fourteen baseline failure is failure of a sufficient certificate, not impossibility.
• No global optimality over three or more scales, continuous times or arbitrary sensing measures.
• No all-target complete-objective support optimality and no noise-optimal or statistically optimal design.
• No recovery of an arbitrary number field from finitely many coefficients.
• No Riemann-hypothesis or zeta-zero result, new axiom of numbers, established physical law, theory of reality or proof-assistant source verification.
• The future multi-target, continuous-time, noise-aware and AI-observability questions belong to a new publication.

Sources: [11] [12] [13]