TNFR–Riemann: Current Evidence and Scope

Status: secondary mathematical comparison program; no proof or disproof of RH, no derived Hilbert–Pólya bridge, and no autonomous arithmetic NFR mechanism. Review: 2026-09-19. Documentation/source audit; historical measurements were not rerun for this consolidation. Authority: this note owns the current scope of the arithmetic comparison. The active research queue remains in FIVE_STAGE_EXECUTION_PLAN.md.

1. Current conclusion

The repository contains finite arithmetic constructions, classical analytic identities, numerical comparisons and conditional symmetry arguments. They are useful as reproducibility baselines and examples of information retained or discarded by an observation. They do not derive prime labels, logarithmic frequencies, arithmetic carriers or a physical particle ontology from the nodal equation.

The identity \(\dot x_i=\nu_i p_i\) determines a form rate after the capacity and pressure are specified. It does not select the arithmetic assignment \(\nu_i=\log i\), the phase law \(\dot\theta_i=-\log i\), a Hamiltonian, or a schedule that produces a prime ladder. Positive capacity is not proof of an oscillator. The current foundations and joint scale obligations are maintained in NODAL_PARAMETER_FOUNDATIONS.md and TNFR_SCALE_GEOMETRY_AND_BRIDGE.md.

RH concerns the real parts of all nontrivial zeta zeros. Finite critical-line scans, agreement at known zeros, a target spectrum placed on a diagonal, or a small residual against those targets do not settle that assertion. The historic label G4 denotes this unresolved external target; it is not a quantified fraction of the remaining TNFR development.

2. Reusable construction and evidence inventory

Owner	What is supplied	Supported result	Boundary
von_mangoldt.py	Prime labels, integer ladder depth, weights \(\log p\)	Finite weighted trace; classical infinite Dirichlet identity on \(\Re s>1\)	Finite cutoff is not the full analytic function
prime_ladder_hamiltonian.py	Frequencies \(k\log p\), selected Hamiltonian coefficients	Decoupled real diagonal spectrum and finite trace readback	Frequencies are assigned; ladder edges are not an executed REMESH history
analytic_continuation.py	Classical zeta evaluator and known-zero comparisons	Numerical evaluation of the established meromorphic continuation	Does not derive continuation or establish that every zero is on the critical line
weil_explicit_formula.py	Test functions, cutoffs and prime/zero data	Finite residual comparison for a classical explicit formula	A reported small residual is not a uniform remainder estimate or a new proof
li_keiper.py	Finite supplied critical-line zeros or line-restricted peak coordinates	Truncated conjugate-pair sums	No certified zero-sum tail; signs are not an independent zero-location test
structural_zero_density.py	Classical Riemann–Siegel theta function	Smooth counting targets from a declared scalar inversion	Target construction does not emerge from nodal evolution
admissible_rescaling.py	Positive source and target spectra, eigenvectors	Finite spectral congruence identity	Does not independently predict targets or establish an analytic range/kernel decomposition
nodal_pulse.py	Integer labels, amplitudes, logarithmic phase rates, cutoff	Finite interference sum and known-ordinate comparisons	No certified critical-line Dirichlet-series limit or autonomous graph evolution
pulse_coherence.py	Finite pulse or classical analytic zeta evaluation	Phase/counting and rectified-phase diagnostics	Pulse, analytic evaluator and supplied comparison oracle must be distinguished
dirichlet_l.py and twisted modules	Explicit Dirichlet characters and arithmetic data	Analogous finite/classical GL(1) comparisons	No GRH result or elliptic-curve GL(2) construction follows

Reproduction entry points

The example catalog owns runnable usage. These finite comparisons exercise the constructions above under their declared inputs:

Example	Comparison
Von Mangoldt trace	Truncated prime-ladder sum against the classical evaluator on its declared domain
Analytic continuation	Selected known critical-line ordinates and poles of the supplied classical function
Prime-ladder Hamiltonian	Eigenvalues against the spectrum assigned to the diagonal
Weil explicit formula	Finite residuals under chosen test functions and cutoffs
Li–Keiper sums	Truncated sums from supplied zeros, with the input-dependence boundary below
Spectral rescaling	Supplied smooth targets and the finite congruence residual

Record the environment, source, inputs, cutoffs and residuals for each run. Historical tolerances are not guarantees for a new configuration; no examples were rerun for this consolidation. These checks provide comparison baselines, not an exhaustion of possible nodal constructions or a novelty claim for classical identities. The old atlas remains recoverable through the retirement record.

3. Exact arithmetic identities and their domains

The classical identity is

\[ -\frac{\zeta'(s)}{\zeta(s)} =\sum_{n\ge1}\Lambda(n)n^{-s} =\sum_{p}\sum_{k\ge1}\log(p)\,e^{-sk\log p}, \qquad \Re s>1. \]

The implemented finite trace uses a finite prime set and ladder depth. Convergence of the infinite series in this half-plane is distinct from the meromorphic continuation outside it. At a zero \(\rho\) of multiplicity \(m_\rho\), the continued function \(-\zeta'/\zeta\) has residue \(-m_\rho\); at the pole \(s=1\) of zeta its residue is \(+1\). Writing an arbitrary nontrivial zero as \(\rho=1/2+it\) would assume RH. Critical-line examples only use selected zeros already known or supplied on that line.

The decoupled P14 finite matrix with diagonal \(k\log p\) has that spectrum by construction. With diagonal trace weights \(\log p\), its finite exponential trace is precisely the finite sum above. This is a useful consistency identity; reading assigned entries back as eigenvalues is not independent evidence that the nodal law generated prime arithmetic.

An inserted spectral threshold is not an independent prediction

The retired prime-path family was H(sigma)=L+(sigma-1/2)D, with symmetric positive-semidefinite graph Laplacian L, L*1=0, and positive diagonal D=diag(log(p_i)). Its lowest-eigenvalue sign changes exactly at the inserted 1/2 for every finite size: at that value the potential vanishes; above it H is positive definite; below it the Rayleigh quotient of 1 is negative. This works for any positive diagonal, with no prime-distribution theorem. The former asymptotic claim supplies no critical-line or generative-transition evidence. Eigenvector-variation readouts of that prototype are not primitive circular phase or the canonical tetrad.

Finite zero sums and the Li criterion

The implemented P16 evaluator forms only

\[ \lambda_n^{(K)}=2\operatorname{Re}\sum_{k=1}^{K} \left[1-\left(1-\frac{1}{\rho_k}\right)^n\right]. \]

The complete classical Li criterion concerns all zeros with the required summation convention and every positive integer index. The finite evaluator supplies neither a certified omitted-zero tail nor a bound on its numerical rounding. A sign of \(\lambda_n^{(K)}\) is therefore not by itself a certified sign of the complete coefficient, even for one index.

There is also an input-dependence boundary. If \(\rho=1/2+it\) is supplied, then \(|(\rho-1)/\rho|=1\), and its paired contribution is \(2[1-\cos(n\arg(1-1/\rho))]\geq0\) in exact arithmetic, whether or not \(t\) is actually a zero ordinate. The default known-zero list and the optional critical-line scan both use this real part; the latter explicitly constructs complex(0.5, detected_ordinate). Nonnegative truncated sums consequently do not validate zero location, pole detection or a TNFR-derived resonance law. Negative numerical output would require investigating the supplied coordinates and arithmetic before drawing any conclusion about the analytic criterion.

4. Finite pulse and analytic phase are different objects

The evaluator forms

\[ P_N(T)=\sum_{n=1}^{N} n^{-1/2}e^{-iT\log n}. \]

The ordinary infinite Dirichlet series for zeta is not valid at \(\Re s=1/2\). No statement equates this unregularized finite sum with \(\zeta(1/2+iT)\), and the default cutoff has no certified remainder bound making it a convergent critical-line approximation. Frequency and amplitude are declared arithmetic inputs. This evaluator does not execute its optional graph helper or the shared nodal integrator.

The known-zero tuple in the source supplies comparison ordinates, scan endpoints and nearest-dip matches. These checks are not blind predictions. A dip of a finite trigonometric sum need not be an analytic zero; a phase branch or a zero resultant also needs its own availability treatment.

The analytic argument term \(S(T)=\pi^{-1}\arg\zeta(1/2+iT)\) requires the standard continuation/branch convention. Away from zero ordinates, the classical counting formula relates it to the Riemann–Siegel theta function. At a zero ordinate, the counting and argument conventions must be stated. Using an analytic evaluator for \(S(T)\) and then checking that classical formula is not a TNFR derivation.

Finite phase, RMS, peak and off-axis comparisons remain descriptive. They neither validate grammar U2 nor imply a uniform estimate at unobserved heights, criticality, a pressure-equilibrium axis or RH.

A finite residue-spectrum comparison

The residue-phase instrument compares fifteen supplied primes p=3 mod 4, starting at 7. The directed quadratic-residue circulant is normal but not self-adjoint; its maximum imaginary spectral magnitude follows the classical sqrt(p)/(p-1) formula. The selected pairing decreases with prime size while the supplied zeta-zero ordinates increase. The historical Pearson value was approximately -0.9068; it is not a bound on other inputs and was not rerun for this consolidation.

The maintained verdict GAUSS_SCALE_MATCHED_DIRECT_ORDERING_DIFFERS concerns that positional comparison. Complex eigenvalues do not identify primitive phase motion or zeta ordinates; the result neither excludes other representations nor derives an analytic Riemann obstruction. A symmetry restriction for the separate prime-ladder model cannot be transferred without matching its group action and complete law.

5. What the smooth rescaling actually proves

Suppose \(H U=U\,\mathrm{diag}(\lambda_i)\), \(U^*U=I\) and both the retained source values \(\lambda_i\) and supplied targets \(\mu_i\) are positive. Define

\[ F=U\,\mathrm{diag}\!\left(\sqrt{\mu_i/\lambda_i}\right)U^*. \]

Then

\[ FHF^*=U\,\mathrm{diag}(\mu_i)U^*. \]

This finite congruence identity holds for any such target list. It is not a similarity transformation preserving the source spectrum. If \(U\) spans the whole space, \(F\) is positive and invertible there; if only selected columns are retained, it is invertible on their range and zero on the complement.

P28 obtains its smooth targets from the classical theta counting function. P30 transfers those targets to a selected operator by this identity. A smaller distance to known zero ordinates reports the chosen target approximation; it does not measure a percentage of RH proved or a fraction of an ontological mechanism discovered. Optional irrational probe frequencies are configured test inputs, not newly derived TNFR constants.

The optional P30 amplitude sweep selects its best value using the W1 discrepancy against the same known zero ordinates later reported as its score. That is in-sample calibration, not held-out prediction. Compatibility flags named structurally_derived do not certify derivation from nodal dynamics.

No analytic decomposition \(F=F_{\rm smooth}\oplus F_{\rm osc}\), identification of \(S(T)\) with a finite Fourier-mask kernel, or REMESH-infinity operator is established by this construction. The fixed-delay surrogate result retains its own hypotheses and does not supply that missing identification.

6. Conditional symmetry results that remain useful

Structural observability owns the exact fixed-diffusion, Reynolds-projection and pointed-selector results, composition conditions and nonlinear counterexamples. A proposed arithmetic application must supply the same group action, full state, support, weights, capacity, selector and history assumed by that result. Finite pointed probes do not establish arbitrary-state or arbitrary-word equivariance.

Relabel-invariant spectra can retain arithmetic information; no representation here places analytic S(T) in a finite symmetry complement. The shared method is useful for testing observation loss, but is not a universal arithmetic no-go or a claim that all admissible constructions have been exhausted.

7. Superseded claims and preserved historical record

The full earlier notebook is recoverable from the Git revision identified by the retirement record. It is a chronological record, not an active task queue or current theorem authority. Specifically superseded readings include:

• P1–P11's eliminated combinatorial prototype as a live canonical model.
• The June/July remapping as a derivation of autonomous arithmetic pulses, a physical substrate, a critical-line attractor or an analytic zeta identity.
• P28/P30 finite rescaling as a solved smooth part of a derived Hilbert–Pólya mechanism, or a universal characterization of the remaining RH content.
• CCET, prime-cancellation, tetrad-fixed-sector and lifted-bundle arguments promoted from restricted hypotheses or samples to every engine operator, state, support or candidate construction.
• Registry closure as completeness of all admissible TNFR transformations; Python scalar storage as proof of the physical dimension of EPI.
• Tetrad completeness, a Hilbert state space inferred from a nonnegative snapshot functional, and global conservation/convergence inferred from grammar or auxiliary harmonic evolution.

Finite artifacts, source names and historical theorem labels are retained for traceability. Their validity must be assessed from the actual hypotheses and implementation; a label such as CLOSED, CANONICAL or CERTIFIED is not a proof. The archive contains superseded instructions and publication plans that are not current authorization or research priorities.

8. Remaining boundary and reuse policy

A new arithmetic bridge would need independently specified carriers, operators, domains, observables and a theorem connecting them to the analytic target, with construction inputs separated from scoring oracles. A nodal emergence claim would additionally need the coupled evolution and a mechanism producing and maintaining the effective pattern. Neither requirement is satisfied by assigning arithmetic coordinates.

The existing exact linear algebra, symmetry controls, spectrum provenance and observation-loss examples can be reused by the main TNFR research. They do not warrant restarting RH parameter sweeps or the old envelope catalog. The single active plan decides priorities; the arithmetic and other Millennium notes are comparison/reference material.