Certified Arithmetic Trace Reconstruction Beyond Finite Matrices: Effective Weil Dictionaries, Relative Operator Limits, and Optical Boundary-State Benchmarks

The paper establishes that finite computational models are structurally incapable of matching this arithmetic reality, and it provides an alternative path using infinite-dimensional operators and verified interval arithmetic.

We formulate a certificate-driven program for realizing arithmetic explicit-formula functionals without importing
zeta-zero ordinates into the construction. A finite-dimensional no-go theorem first excludes ordinary
matrix traces: a finite analytic exponential sum cannot equal the infinite prime-power distribution on the
compactly supported smooth test-function space, and assigning that distribution to an unrestricted counterterm
makes the operator representation vacuous. The admissible route therefore separates four tasks. First, an
explicit rational-parameter spline–mollifier dictionary is constructed in logarithmic half-density coordinates
and proved dense in the LF topology, with quantitative derivative-seminorm certificate requirements. Second,
every Weil Gram entry is reduced to a finite prime-power ledger and a certified archimedean enclosure
using directed interval arithmetic, exact support metadata, Hermitian intersection, and symbolic singularstage
compatibility. Third, the operator problem is reformulated on compatible infinite-dimensional semilocal
Hilbert spaces with trace-class relative resolvent control and a restricted counterterm class. Fourth, fibreoptical
analogue-horizon experiments are typed as boundary-state benchmarks for nonlinear mode conversion,
backreaction, metrology, and conservation accounting. The optical systems provide a valuable architecture
for testing infinite-dimensional field models, but their measured spectra are not operator traces and supply no
evidence for an arithmetic trace law. The result is a falsifiable advancement protocol in which mathematical,
numerical, operator-theoretic, and experimental claims receive separate admission statuses.

The arithmetic explicit formula is an identity of distributions. Its prime side is an infinite, locally finite atomic
measure in logarithmic scale, while its spectral side is organized by the nontrivial zeros of the zeta function.
This distinction sharply limits what can be learned from finite matrix fitting. Any finite self-adjoint matrix
produces a finite analytic exponential sum under unitary evolution; it cannot reproduce the atomic prime-power
distribution on the full test-function space. The obstruction is structural, not numerical.
This paper turns that obstruction into a design boundary and reorganizes the research program around four
objects that can be stated and audited independently:

1. an explicit countable spline–mollifier dictionary that is dense in the correct LF topology;
2. a primary Weil Gram engine whose entries are certified entirely from the arithmetic side;
3. a no-go gate excluding finite-dimensional raw trace models and unconstrained counterterms; and
4. an infinite-dimensional semilocal operator target with trace-class resolvent control, compatible prime adjunction,
and a restricted renormalization law.

A fifth component is deliberately kept separate. Recent fibre-optical analogue-horizon experiments provide
a useful reference specimen for infinite-dimensional field dynamics, mode conversion, active backreaction, and
conservation accounting (Procopio et al. 2026; Felipe-Elizarraras et al. 2026). They do not produce an arithmetic
trace formula. Their role here is to sharpen system typing, measurement discipline, and the boundary between
an observed spectral channel and a distributional operator trace.
The central mathematical result of the paper is the finite-dimensional no-go theorem in Section 3. Section 4
gives a concrete dense-dictionary theorem and the associated certificate estimates. Section 5 specifies the interval
Gram engine, including the singular Schur stage. Section 6 states an infinite-dimensional prime-local construction
as a theorem specification rather than a claimed realization. Section 7 evaluates the optical reference specimen.
Sections 8 and 9 collect the minimum benchmark suite and research sequence.

Keywords: Weil explicit formula; LF test-function spaces; certified interval arithmetic; prime-power distributions;
relative trace formulas; trace-class resolvents; analogue Hawking radiation; nonlinear fibre optics; quantum
backreaction.