TNFR Number Theory: Arithmetic Constructions and Structural Read-outs¶
Status: Arithmetic construction reference; exact restricted identities and finite diagnostics, not a derived physical or autonomous nodal model Version source: pyproject.toml Date: March 2026
Scope correction (2026-09-18). Arithmetic functions, modular arithmetic, graph constructors and assigned logarithmic frequencies are declared inputs. Conditional spectral consequences can be derived from those inputs; this is not a derivation of physical particles or of the arithmetic carrier itself from the nodal equation. The retired prime-path model and unsupported symmetry-complement and REMESH-infinity identifications are bounded by the current Riemann program memo: no analytic location of
S(T), universal operator equivariance or RH result has been established. Historical numerical results are retained, not re-executed by this correction.
Table of Contents¶
- Introduction
- Arithmetic Networks as TNFR Systems
- The Arithmetic Structural Triad
- Primality as Structural Equilibrium
- Canonical Arithmetic Constants
- Pressure Component Analysis
- The Arithmetic Tetrad
- Dual-Lever Decomposition
- Factorization as Spectral Decoding
- Prime Path Graphs and the TNFR-Riemann Connection
- Worked Examples
- Implementation Map
- Reference Questions and Research Boundaries
- References
1. Introduction¶
Arithmetic networks can be assigned TNFR-compatible coordinates and read-outs. Their relation to the nodal equation
requires a separately declared evolution: assigning arithmetic functions to nodes and edges alone supplies a static state, not a trajectory. In the construction studied here:
- Primes are the arithmetic pressure zero set: \(\Delta\mathrm{NFR}(p) = 0\) for all primes \(p\).
- Composites carry structural pressure: \(\Delta\mathrm{NFR}(n) > 0\) whenever \(n\) is composite, with a magnitude determined by the three selected arithmetic descriptors.
- Factorization as spectral decoding: discovering the factors of a composite can be framed as resolving the coherent sub-modes of its structural pressure field.
This document formalizes these observations, records the configured unit-weight arithmetic convention, and maps it to the implementations in the repository. The radian phase chart supplies the exact wrap scale \(\pi\); it does not select the arithmetic coefficients or prove that no other structural scale can occur under a declared model. See the parameter and scale ledger.
Scope¶
| Layer | Description | Source |
|---|---|---|
| Primality | Deterministic prime detection via \(\Delta\mathrm{NFR}=0\) | applications/primality-test/, src/tnfr/mathematics/number_theory.py |
| Factorization | Spectral factor discovery via Paley-Jacobi graphs | applications/factorization-lab/ |
| Riemann program | Declared arithmetic traces and finite pulse comparisons; historical prime-path prototype superseded | src/tnfr/riemann/ |
These layers reuse selected arithmetic functions, graph diagnostics and configured coefficients. This reuse does not establish a shared autonomous evolution or U1-U6 compliance. Grammar claims require an actual declared operator sequence and its checks; static primality and spectral calculations do not supply such a history.
2. Arithmetic Networks as TNFR Systems¶
2.1 Network Construction¶
A TNFR arithmetic network \(G = (V, E)\) is a directed graph where:
- Nodes \(V = \{2, 3, \ldots, N\}\) are natural numbers.
- Edges encode two types of structural relationship:
- Divisibility edges: \((d, n)\) for each divisor \(d \mid n\) with \(d < n\).
- GCD coupling edges: \((a, b)\) when \(\gcd(a, b) > 1\), weighted by \(\gcd(a, b) / \max(a, b)\).
Each node \(n\) is assigned the structural triad (EPI, \(\nu_f\), \(\phi\)) from chosen arithmetic functions, together with a separate arithmetic pressure read-out \(\Delta\mathrm{NFR}(n)\). These assignments do not derive a capacity law, phase law or event schedule.
2.2 Sieve-Based Computation¶
Efficient computation uses a Lowest Prime Factor (LPF) sieve:
From the LPF array, factorization of any \(n \leq N\) is \(O(\log n)\), enabling computation of all arithmetic functions (\(\Omega\), \(\tau\), \(\sigma\)) for the entire network in \(O(N \log \log N)\) sieve time plus \(O(N \log N)\) factorization time.
2.3 Phase Assignment¶
Phase is a declared read-out policy. ArithmeticTNFRNetwork.compute_phase
defaults to the angle of a two-dimensional spectral layout, falling back to
logn; it also supports capacity normalization (nuf). The logn policy is
The conservation/substrate adapters default to logn. A spectral-layout
orientation is not an independently derived physical phase, and none of these
static assignments supplies a phase evolution law. The former linear n/N
formula did not describe this implementation.
Coupling operations must separately check the wrapped U3 condition \(|\operatorname{wrap}(\phi_i-\phi_j)|\leq\Delta\phi_{\max}\). The arithmetic graph construction and phase assignment alone do not guarantee that every arithmetic edge satisfies that condition.
3. The Arithmetic Structural Triad¶
The assigned triad is (EPI, \(\nu_f\), \(\phi\)), with phase defined in §2.3. The form and capacity assignments below are followed by the separate pressure and coherence read-outs. For a fixed integer label these are static functions.
3.1 Form: EPI(n)¶
The Primary Information Structure of a natural number measures its overall arithmetic complexity:
where: - \(\alpha = 1\) — factorization complexity weight (canonical unit, §5) - \(\beta = 1\) — divisor complexity weight (canonical unit, §5) - \(\gamma_{\mathrm{epi}} = 1\) — abundance deviation weight (canonical unit, §5)
Model interpretation: EPI(n) is a chosen scalar arithmetic-complexity coordinate. Calling it form does not derive an oscillator or a physical state space from the nodal equation.
3.2 Frequency: \(\nu_f(n)\)¶
The reorganization capacity of a number measures how rapidly its structural form could evolve:
where: - \(\nu_0 = 1\) — base frequency (canonical unit, §5) - \(\delta = 1\) — divisor density modulation (canonical unit, §5) - \(\varepsilon = 1\) — factorization complexity modulation (canonical unit, §5)
Model interpretation: This positive arithmetic function supplies a candidate capacity if an evolution is declared. Its definition alone specifies neither an observed reorganization rate nor a phase angular velocity.
3.3 Pressure: \(\Delta\mathrm{NFR}(n)\)¶
The structural pressure equation is the central result of arithmetic TNFR:
where \(\Omega(n)\) is the prime factor count with multiplicity, \(\tau(n)\) the divisor count, and \(\sigma(n)\) the divisor sum. The coefficients are canonically \(\zeta = \eta = \theta = 1\) (unit weights; see §5).
Model interpretation: This nonnegative arithmetic pressure vanishes exactly at primes under the stated positive coefficients. It is a selected measure of compositeness, not a derived metric distance or a signed restoring response to an EPI perturbation.
3.4 Local Coherence¶
From the pressure, local coherence is derived:
Primes have \(C_{\text{local}} = 1\) (perfect coherence); composites have \(C_{\text{local}} < 1\), decreasing with structural complexity.
4. Primality as Structural Equilibrium¶
4.1 The Fundamental Theorem¶
Theorem (TNFR Primality Criterion): For any integer \(n \geq 2\):
Proof: Each pressure component vanishes independently for primes and is strictly positive for composites:
-
Factorization component: \(\Omega(p) = 1\) for all primes \(p\), so \(\zeta \cdot (\Omega(p) - 1) = 0\). For composites, \(\Omega(n) \geq 2\), giving \(\zeta \cdot (\Omega(n) - 1) \geq \zeta > 0\).
-
Divisor component: \(\tau(p) = 2\) for all primes (divisors: 1 and \(p\)), so \(\eta \cdot (\tau(p) - 2) = 0\). For composites, \(\tau(n) \geq 3\), giving \(\eta \cdot (\tau(n) - 2) \geq \eta > 0\).
-
Abundance component: For primes, \(\sigma(p) = 1 + p\), so \(\sigma(p)/p = 1 + 1/p\), making \(\theta \cdot (\sigma(p)/p - (1 + 1/p)) = 0\). For composites \(n\) with a proper divisor \(d \notin \{1, n\}\), \(\sigma(n) > 1 + n\), giving \(\sigma(n)/n > 1 + 1/n\) and thus \(\theta \cdot (\sigma(n)/n - (1 + 1/n)) > 0\).
Since all three terms vanish iff \(n\) is prime, and all are non-negative, the equivalence holds. \(\square\)
Structural interpretation: The theorem identifies the pressure zero set with the prime labels. If this arithmetic pressure is held fixed in the unforced scalar nodal equation, it gives zero EPI rate at a prime for any finite capacity. It does not establish a fixed point of phase, capacity, support or the full operator dynamics. At a fixed composite label the static EPI(n) assignment has zero time derivative, whereas the assigned positive capacity and pressure have a positive product. An evolved EPI coordinate therefore cannot remain identical to that static arithmetic formula; §8 states the separate held-pressure model explicitly.
4.2 Coefficient Independence¶
The primality criterion \(\Delta\mathrm{NFR}(n) = 0\) is independent of the coefficient values \((\zeta, \eta, \theta)\), provided all three are positive. Each component vanishes individually for primes. The coefficients affect composite pressure magnitudes, not the arithmetic zero set.
4.3 Computational Properties¶
| Property | Value |
|---|---|
| Determinism | 100% (no probabilistic component) |
| Time complexity | \(O(\sqrt{n})\) per number (trial division for \(\Omega\), \(\tau\), \(\sigma\)) |
| Space complexity | \(O(1)\) basic; \(O(\text{cache})\) with memoization |
| Sieve mode | \(O(N \log \log N)\) for all primes up to \(N\) |
| Verified range | \([2, 10^4]\) exhaustive, \([10^4, 10^{10}]\) selective |
| Accuracy | 100% — 0 false positives, 0 false negatives |
5. Canonical Arithmetic Constants¶
5.1 Canonical Coefficients Are Unity¶
AGENTS.md §3 identifies \(\pi\) as the exact phase-wrap scale; it does not prove that \(\pi\) is the only possible structural constant or derive arithmetic weights from phase wrapping. This construction does not derive \(\varphi\), \(\gamma\) or \(e\) as structural scales. Earlier versions wrote the triad weights as \((\varphi, \gamma, \pi, e)\) combinations, but that was a post-hoc notational overlay fitted to approximate empirical values (\(\zeta = 1.0\), \(\eta = 0.8\), \(\theta = 0.6\)) — not a derivation.
By the Coefficient Independence theorem (§4.2), the primality zero set is the same for any positive coefficients. That theorem does not select their magnitudes. The implementation chooses unit coefficients as the canonical parameter-free convention, so the arithmetic pressure introduces no additional scale; the structural content lives in \((\Omega, \tau, \sigma, n)\).
5.2 Pressure Coefficients¶
The three pressure channels weigh equally: the factorization excess \(\Omega - 1\), the divisor excess \(\tau - 2\) and the abundance excess \(\sigma/n - (1 + 1/n)\) each contribute on the same unit scale. This is the configured unit-weight convention. Section 4.2 proves that the zero set does not depend on the choice of positive coefficients; it does not prove uniqueness of the unit convention.
5.3 EPI Parameters¶
| Parameter | Value | Model meaning |
|---|---|---|
| \(\alpha\) | \(1\) | Factorization complexity weight |
| \(\beta\) | \(1\) | Divisor logarithmic weight |
| \(\gamma_{\mathrm{epi}}\) | \(1\) | Abundance deviation weight |
5.4 Frequency Parameters¶
| Parameter | Value | Model meaning |
|---|---|---|
| \(\nu_0\) | \(1\) | Base structural frequency |
| \(\delta\) | \(1\) | Divisor density modulation |
| \(\varepsilon\) | \(1\) | Factorization modulation |
5.5 Detection Thresholds¶
The theorem detects primality by the exact criterion \(\Delta\mathrm{NFR}(n) = 0\) (§4.1). A numerical zero tolerance approximates that criterion; a configured pressure cut is a separate decision policy.
| Threshold | Value | Purpose |
|---|---|---|
| Primality tolerance | \(10^{-10}\) | Floating-point zero detection of \(\Delta\mathrm{NFR} = 0\) |
Any wider "significance band" is an operational decision policy, not a derived structural constant. The phase-wrap convention discussed in §5.1 does not select arithmetic coefficients or detection thresholds.
The repository's OptimizedTNFRPrimality adapter retains the historical
weights \((1,0.8,0.6)\) and applies the strict configured cut
\(|\Delta\mathrm{NFR}|<\texttt{threshold}\) in both its sieve and trial-division
paths. The finite positive default \(0.5\) is below the composite lower bound
\(1+0.8=1.8\), so it separates the prime zero set in this model. Larger custom
cuts need not do so: \(n=4\) has pressure \(2.1\) and passes a cut of \(3\) regardless
of sieve coverage. The retained is_prime report field records this configured
decision; changing the cut does not change the exact theorem. Each returned
report, including its optional metrics, is detached from the reusable cache.
5.6 Derivation Status¶
The 9 arithmetic parameters (3 pressure + 3 EPI + 3 frequency) are positive operational weights applied to arithmetic functions (canonical units; the prime ⟺ ΔNFR = 0 criterion is coefficient-independent, §4.2). The arithmetic network uses the same tetrad implementation and selected warning policies as other TNFR graphs; these policies are not graph-independent bounds. The earlier φ/γ/e "arithmetic recalibration" was removed. The adapter's configurable pressure cut remains the separate policy described in §5.5.
6. Pressure Component Analysis¶
6.1 Three Independent Pressure Channels¶
The \(\Delta\mathrm{NFR}\) equation decomposes structural pressure into three independent channels, each measuring a distinct aspect of compositeness:
Factorization Pressure: \(P_{\Omega} = \zeta \cdot (\Omega(n) - 1)\)¶
Measures the total prime factor count with multiplicity. This is the most direct measure of compositeness: primes have \(\Omega = 1\), semiprimes have \(\Omega = 2\), prime powers \(p^k\) have \(\Omega = k\).
| \(n\) | Factorization | \(\Omega(n)\) | \(P_\Omega\) |
|---|---|---|---|
| 7 (prime) | \(7\) | 1 | 0 |
| 15 | \(3 \times 5\) | 2 | 1 |
| 8 | \(2^3\) | 3 | 2 |
| 30 | \(2 \times 3 \times 5\) | 3 | 2 |
| 360 | \(2^3 \times 3^2 \times 5\) | 6 | 5 |
Divisor Pressure: \(P_{\tau} = \eta \cdot (\tau(n) - 2)\)¶
Measures the richness of the divisor lattice. Primes have exactly 2 divisors; highly composite numbers have many.
| \(n\) | \(\tau(n)\) | \(P_\tau\) |
|---|---|---|
| 7 (prime) | 2 | 0 |
| 15 | 4 | 2 |
| 8 | 4 | 2 |
| 30 | 8 | 6 |
| 360 | 24 | 22 |
Abundance Pressure: \(P_{\sigma} = \theta \cdot (\sigma(n)/n - (1+1/n))\)¶
Measures the deviation of the divisor sum ratio from the prime pattern. This is the most sensitive to the internal structure of divisors.
| \(n\) | \(\sigma(n)/n\) | \(1+1/n\) | \(P_\sigma\) |
|---|---|---|---|
| 7 (prime) | \(8/7 \approx 1.143\) | \(8/7\) | 0 |
| 15 | \(24/15 = 1.600\) | \(16/15 \approx 1.067\) | 0.533 |
| 8 | \(15/8 = 1.875\) | \(9/8 = 1.125\) | 0.750 |
| 30 | \(72/30 = 2.400\) | \(31/30 \approx 1.033\) | 1.367 |
6.2 Component Independence¶
The three pressure channels are functionally linearly independent — no linear (or affine) combination reproduces a third for all \(n\). This is proved exactly over \(\mathbb{Q}\) from the witness points \(p^2 \mapsto (1, 1, 1/p)\) and \(pq \mapsto (1, 2, 1/p + 1/q)\) (prove_functional_independence; the \(3\times 3\) witness matrix has exact rank 3). They therefore form a linearly-independent diagnostic profile of compositeness through \(\Omega\), \(\tau\), \(\sigma\) — not a minimal or complete basis: the set is redundant for primality (each channel alone is zero iff \(n\) is prime, so one channel already suffices), and structural completeness (that no fourth independent pressure degree is relevant) is unproven. See TNFR_ARITHMETIC_PRESSURE.md (R7, NT-P07) for the independence proof, the redundancy result, and the closed fourth-channel gate.
6.3 Structural Pressure Landscape¶
As \(n\) grows, the expected pressure for a "random" composite scales as:
by the Erdős-Kac theorem (\(\Omega(n) \sim \ln\ln n\)) and divisor function asymptotics. Primes remain at exactly zero regardless of magnitude.
7. The Arithmetic Tetrad¶
When the arithmetic network \(G\) is constructed, the structural field tetrad (\(\Phi_s\), \(|\nabla\phi|\), \(K_\phi\), \(\xi_C\)) can be computed using the centralized physics modules:
7.1 Structural Potential: \(\Phi_s\)¶
where \(d(n,m)\) uses the graph supplied to the field adapter and its declared path-length policy. Self, unreachable and zero-distance pairs contribute zero under the shared kernel convention. Explicit edge length, transport weight fallback and unit fallback must not be conflated. A prime contributes zero as a pressure source, but its potential value still aggregates pressure from other nodes. This static field does not make primes dynamical sinks or attractors.
Warning policy: \(|\Phi_s| < \pi/4 \approx 0.785\) is the shared selected per-node policy; it is not derived from arithmetic or phase wrapping.
7.2 Phase Gradient: \(|\nabla\phi|\)¶
where \(\mathcal{N}(n)\) is the neighborhood used by the field adapter, with zero at isolates. It measures the supplied phase differences; an arithmetic class interpretation requires evidence under that phase policy.
Scale: \(|\nabla\phi| \leq \pi\) is the exact wrapped-angle bound. Any early-warning level is experiment-dependent; the measured synchronization onset near \(0.29\) is not a universal stability threshold.
7.3 Phase Curvature: \(K_\phi\)¶
where \(\overline{\phi}_{\mathcal{N}(n)}\) is the circular neighbor mean when available. Exact represented phasor cancellation gives unavailable curvature; the strict numeric owner raises rather than inventing a value. No generic prime/composite boundary interpretation follows from this definition.
Threshold: \(|K_\phi| < 0.9\pi \approx 2.827\); this is an operational margin inside the exact wrapped bound.
7.4 Coherence Length: \(\xi_C\)¶
The coherence length estimates how far the configured structural-coherence field correlates across the arithmetic network. Its value is state- and estimator-dependent; no divergence at twin primes, prime gaps, or an arithmetic critical point is established here.
7.5 Tetrad thresholds on the arithmetic network¶
The arithmetic network uses the same tetrad implementation and selected warning policies as any TNFR network:
| Field | Threshold | Source |
|---|---|---|
| \(\Phi_s\) | π/4 ≈ 0.785 (per-node), π/2 ≈ 1.571 (drift), both selected policies | PHI_S_VON_KOCH_THRESHOLD, U6_STRUCTURAL_POTENTIAL_LIMIT |
| \(\lvert\nabla\phi\rvert\) | ≤ π (phase wrap); measured onset ≈0.29 is experiment-dependent | canonical wrapping / measured protocol |
| \(K_\phi\) | < 0.9·π ≈ 2.827 (phase-wrap safety) | K_PHI_CANONICAL_THRESHOLD |
| \(\xi_C\) | path-unit static coherence-product fit; separate dimensionless spectral fallback from the smallest computed eigenvalue above 1e-9 |
Estimator provenance required; selected value need not be the true λ₂ |
An earlier "arithmetic recalibration" introduced topology-specific thresholds expressed as φ/γ/e combinations (e.g. a \(K_\phi\) threshold of 3.2275 that exceeded the π phase-wrap bound and was therefore unreachable). Those values were not structural scales and have been removed (audit 2026): the arithmetic network is governed by the same π-bounded phase sector as every other TNFR network.
7.6 The Arithmetic NFR and its Emergent Geometry¶
The constructed arithmetic network supports a joint Fractal-Resonant Node
(NFR; TNFR.pdf §1.4.1) read-out through ArithmeticTNFRNetwork.nfr().
Its following diagnostic facets do not prove spontaneous generation,
self-similarity or persistent identity under an autonomous evolution:
- Resonant. By the §4.1 primality theorem the pressure-equilibrium set
\(\{n : \Delta\mathrm{NFR}(n) = 0\}\) is exactly the primes.
equilibrium_fractionis the prime density. The canonical static aggregatecoherenceis \(1/(1+\operatorname{mean}|\Delta\mathrm{NFR}|)\) under the static \(d\mathrm{EPI}=0\) convention.mean_local_coherenceretains the distinct descriptive average \(\operatorname{mean}_i[1/(1+|\Delta\mathrm{NFR}_i|)]\); the two generally differ because the coherence kernel is nonlinear. The arithmetic pressure is independent of EPI, so this identifies the arithmetic zero set, not full-state fixed points, restoring attractors or basins. An empty domain reports these aggregates as unavailable rather than assigning it zero coherence. - Geometric.
classify_nodal_topologyassigns a radial, annular or multinodal diagnostic from the constructed potential. Its result depends on the graph, pressure and classification policy; no universal arithmetic center classification is established here. - Scale read-out. The available \(\xi_C\) estimate is a correlation or spectral statistic with recorded provenance, not proof of fractality or a uniquely determined NFR region size.
The arithmetic network exposes structural read-outs through conservation()
and initializes the separate auxiliary harmonic model through
symplectic_substrate(). These diagnostics do not show that the arithmetic
nodal dynamics generates a symplectic geometry:
- a Noether-like charge diagnostic \(Q = \sum_i (\Phi_s(i) + K_\phi(i))\) and the nonnegative structural energy candidate \(E = \tfrac12 \sum_i (\Phi_s^2 + |\nabla\phi|^2 + K_\phi^2 + J_\phi^2 + J_{\Delta\mathrm{NFR}}^2)\), with the potential sector \(\Phi_s\) sourced by the arithmetic \(\Delta\mathrm{NFR}\) (the genuine invariants \(\Omega, \tau, \sigma\));
- an auxiliary symplectic substrate of dimension \(4N\) with conjugate pairs \((K_\phi, J_\phi)\) and \((\Phi_s, J_{\Delta\mathrm{NFR}})\). The geometric sector is populated by the size / capacity phase \(\phi(n) \propto \log n\) (the monoid homomorphism \((\mathbb{N},\times) \to (\mathbb{R},+)\)), which is non-degenerate on the dense divisibility graph. These extracted coordinates initialize the harmonic substrate model; they do not prove that the arithmetic graph trajectory remains in its realizable image.
The pressure contributes to potential and the chosen phase contributes to the phase sector. Relative numerical magnitudes depend on input scales and policy; no universal potential-dominance theorem follows.
8. Dual-Lever Decomposition¶
8.1 The Nodal Equation in Arithmetic¶
For a separately declared evolution initialized from the arithmetic assignment, write its evolving EPI coordinate as \(x_n(t)\):
Holding those arithmetic capacity and pressure values fixed is an additional model assumption. The channel terminology then distinguishes:
- Capacity lever (\(\nu_f\)): How fast the number can reorganize. Depends on divisor structure and factorization complexity. Its primary operators are SHA, VAL and NUL.
- Pressure lever (\(\Delta\mathrm{NFR}\)): How much reorganization is demanded. Zero for primes, positive for composites. Its primary operators are IL, OZ, THOL and NAV.
- Phase channel (\(\theta\)): UM and ZHIR act primarily on phase.
- Form channel (EPI): AL, EN, RA and REMESH write or transport form.
8.2 Fixed Point Analysis¶
For primes: \(\Delta\mathrm{NFR}(p) = 0 \Rightarrow \dot x_p = 0\) for any finite \(\nu_f(p)\) in this unforced model.
This is stationarity of the unforced EPI coordinate under the held-pressure model: perturbations to \(\nu_f\) alone do not change its zero derivative. It is not a full-state fixed-point, stability or attraction theorem.
For composites with the assigned positive capacity: \(\Delta\mathrm{NFR}(n) > 0 \Rightarrow \dot x_n > 0\).
The evolved scalar \(x_n(t)\) increases under this fixed positive pressure. Its sign does not prove simplification, attraction toward a prime, or stability. If capacity and arithmetic pressure remain fixed and positive, the nodal law gives linear drift rather than relaxation (§13.3).
8.3 Experimental Confirmation¶
Operator-tetrad synergy experiments (examples 37-39) confirmed:
- \(\Phi_s\) responds linearly to \(\Delta\mathrm{NFR}\) perturbations with \(|r| = 1.000\) (perfect correlation), confirming the pressure lever's direct coupling to the structural potential field.
- Declared operator changes can affect capacity, pressure and the resulting EPI rate. Phase and support also enter the tetrad directly; these dependencies do not establish a closed causal chain through EPI rate alone.
- Finite operator trajectories expose measured changes in the tetrad energy candidate. Grammar validity and the nominal multiplier \(\Pi\) do not by themselves prove \(dE/dt \leq 0\).
9. Factorization as Spectral Decoding¶
9.1 Candidate generation is not factor recovery¶
The lab constructs a supplied residue graph, computes spectral features and
proposes candidates. A coherent label or spectral sub-mode does not prove that
a candidate divides the input or arose autonomously from nodal evolution.
Every claimed proper factor needs 1 < d < n and n % d == 0.
9.2 Spectral Paley-Jacobi scope¶
The application guide owns graph/modulus selection, combinatorial versus normalized spectra, capacity/time conversion, noncanonical field proxies and the partition attenuation surrogate. Its recorded operator labels are not a graph-owned operator trace. The genuine tetrad's inability to distinguish factor cosets in example 117 concerns that supplied symmetric state and observer, not arbitrary per-node information.
9.3 Configured verification¶
The result and verification contract distinguishes heuristic acceptance from arithmetic divisibility. Thresholds, coverage and endorsement cuts are configured filters, not mathematical factor certificates or constants derived from the nodal identity.
9.4 Partial pure-mode policy¶
The candidate policy states which arithmetic inputs and fallbacks remain active. Pure mode is not an arithmetic-free pipeline. Comparisons must retain those dependencies, actual operator execution and reserved-evaluation provenance.
9.5 Three Sectors of Primality (Unification — MEASURED)¶
The factorization machinery (§9.1–9.4), arithmetic pressure criterion (§4) and
residue-graph diagnostics reuse selected arithmetic structures, but they have
different inputs and observations. Example
117_emergent_geometry_residue_graph.py
compares these scopes. The focused instruments
arithmetic_pressure_audit.py
and paley_bridge.py own the arithmetic and
Paley controls respectively. Using the same L_rw owner does not identify
their full dynamics or prove one common carrier:
| Sector | Method | Input | Emergent? |
|---|---|---|---|
| A — Arithmetic | \(\Delta\mathrm{NFR}(n)=0\) (§4) | \(\Omega, \tau, \sigma\) (the factorization) | re-expression (primes-IN; exact but circular as a derivation) |
| B — Spectral | Paley eigenvalue comparison with explicit normalization: \(d\lambda_2(L_{rw})\) versus \((n-\sqrt n)/2\) on the regular self-adjoint prime case | prescribed modular arithmetic and residue graph | Spectral diagnostic without factorization input; scope and prime-power controls in §§9.6–9.7 |
| C — Representation | irreducibility (Schur \(\langle\chi,\chi\rangle=1\)) | a finite group | refuted (the dim-4 mode of \(K_5\) is irreducible yet \(4=2\cdot 2\)) |
The unification, stated honestly:
-
Sector B avoids factorization input. The Paley comparison reads the spectrum of a prescribed residue graph rather than supplying \(\Omega,\tau,\sigma\). Its prime and prime-power distinctions must retain the stated operator and finite controls (§9.6). Modular arithmetic and the graph construction are still inputs; neither their physical realization nor a generic mechanism generating primes follows from this diagnostic.
-
The measured factor signal is spectral in these fixtures. For a semiprime \(n=p\cdot q\) the factor \(p\) appears as an exact Fourier/coset mode of the emergent diffusion spectrum (\(\eta^2_{\text{coset}}\to 1\), collapsing under a node-label shuffle — example 117 Q2). The residue graph is regular/circulant, so the random-walk operator and classical Laplacian share eigenvectors: this is CRT structure re-expressed. For the symmetric-seed per-node substrate fixture, the tested fields do not distinguish the cosets (\(\eta^2\approx0\)). Other non-invariant states or observers require separate tests.
-
The obstructions are not identified with one another. Directed spectra retain information discarded by the tested self-adjoint comparison. Their Gauss-sum phases have not been identified with analytic \(S(T)=\pi^{-1}\arg\zeta(1/2+iT)\). The former equalities with a REMESH-infinity kernel or a finite symmetry complement are withdrawn by the Riemann memo's scope correction. Examples 94 and 97 remain declared arithmetic constructions; they do not establish that all factorization, additive or zeta questions share one missing operator.
Net: sector B supplies conditional spectral information without factoring the input first. The per-node blindness measured for particular symmetric seeds does not exclude information in other states or observers. No physical entity-generation mechanism or analytic zeta bridge has been derived here.
9.6 The Phase Sector — Sector B Extended to All Odd Primes (MEASURED)¶
The "partial" limitation of sector B (only \(n\equiv 1\pmod 4\), §9.5) is not a wall of TNFR — it is an artefact of restricting to the real/self-adjoint spectrum. Example 119_phase_sector_directed_residue.py crosses it using the same canonical emergent operator on the directed residue graph.
The structural reason for the prime mod-4 split. For an odd prime \(p\equiv1\pmod4\), \(-1\) is a quadratic residue and the regular residue graph is symmetric, with self-adjoint \(L_{rw}\). For an odd prime \(p\equiv3\pmod4\), the residue digraph is a Paley tournament: its circulant \(L_{rw}\) is normal but not self-adjoint and has complex eigenvalues. These prime-specific statements do not follow from the congruence alone for arbitrary composite moduli. An eigenvalue's imaginary part is not itself the engine's nodal phase variable.
Scope. This is structural_diffusion_operator applied directly to a
networkx.DiGraph, verified against a hand-built matrix
(\(\max|\Delta|=0\)). Its complex spectrum is a diagnostic of that directed
linear transport operator; it is not the auxiliary symplectic substrate or a
complete emergent geometry. The arithmetic input is \(x^2\bmod n\).
Measured (all reproducible in example 119):
-
Finite primality comparison. A three-distinct-eigenvalue rule matches the prime labels in 58/58 tested odd integers in
[5,119]. The forward odd-prime count follows from §9.11; the finite comparison does not prove the converse for all composite integers. -
Prime powers resolved. The directed operator gives 4+ distinct eigenvalues for \(9=3^2\), \(25=5^2\), \(49=7^2\), \(121=11^2\) — it separates primes from prime powers, which the real symmetric operator of example 117 could not (\(49\) was rigid there, the honest §9.5 caveat). The phase channel removes that caveat.
-
The phase encodes \(\sqrt n\). For \(n\equiv 3\pmod 4\) primes the imaginary spectrum is the Paley-tournament eigenvalue structure \((-1\pm i\sqrt n)/2\) on the adjacency; the diffusion operator's \(\max|\mathrm{Im}(\lambda)|=\sqrt n/(n-1)\) exactly (ratio \(1.000\)) — a Gauss-sum fact carried in the phase.
Honest scope. The finite comparisons improve the tested prime-power distinction without factorization input. The odd-prime rank is the classical Gauss-period result proved in §9.11; the finite converse tests are not a proof for every composite. These spectra have not been identified with the tetrad field \(\Psi=K_\phi+iJ_\phi\), an analytic symmetry complement or the zeta argument. No factoring speedup, particle mechanism or open-problem result follows.
9.7 The Symmetry Wall — Why the Substrate Is Blind and the Spectrum Is Not (MEASURED)¶
§9.6 detects primality in a global diffusion spectrum; §9.5 and examples 103/116 found the extracted per-node fields used to initialize the auxiliary substrate blind to arithmetic. These are different summaries of the same residue digraph, so their information content need not agree. Example 120_symmetry_wall_substrate_vs_spectrum.py locates the finite observation loss at vertex-transitivity.
The mechanism. The residue digraph is a Cayley digraph of \(\mathbb{Z}_n\) with connection set \(=\) the quadratic residues. Translation preserves the difference \(j-i\) and acts transitively. Consequently an equivariant per-node observer applied to an invariant input must be orbit-constant; this does not constrain an arbitrary symmetry-breaking input.
- In the symmetric-seed protocol, the per-node substrate output lies in the fixed sector and cannot distinguish nodes within the single orbit.
- The tested arithmetic distinction appears in the global spectrum (group-character / Gauss-sum eigenvalues). A scalar spectral rank is an invariant, not itself a vector in \(\mathrm{Fix}(G_{\mathrm{aut}})^\perp\).
The double dissociation (measured). Compare the Paley residue digraph (QR structure) against a random regular tournament of the same out-degree, both seeded identically and evolved by the canonical nodal equation \(\partial\mathrm{EPI}/\partial t=\nu_f\cdot\Delta\mathrm{NFR}\). Table regenerated after the 2026-09-04 canonicity audit fixed the directed ΔNFR orientation (default path previously computed \(L_{\mathrm{in}}\) instead of the canonical \(L_{\mathrm{out}}\) on digraphs; the eigenvalue columns are orientation-invariant and unchanged, only \(\sigma(\Phi_s)\) shifted):
| \(n\) | Paley distinct eig. | random distinct eig. | Paley \(\sigma(\Phi_s)\) | random \(\sigma(\Phi_s)\) |
|---|---|---|---|---|
| 11 | 3 | 11.0 | 0.376 | 0.356 |
| 23 | 3 | 23.0 | 0.559 | 0.533 |
| 47 | 3 | 47.0 | 0.978 | 0.968 |
- Spectrum SEES the arithmetic: Paley is rigidly 3 distinct eigenvalues (the §9.6 prime signature); the random tournament has \(\sim n\). Swapping the QR structure for a random tournament changes the spectrum completely.
- The tested substrate readout does not discriminate: the per-node \(\Phi_s\) dispersion is similar for Paley and the random tournament in the recorded cases. This finite comparison is not an all-state impossibility theorem.
Across odd \(n\) the spectral test "\(3\) distinct \(\iff\) prime" is 18/18 correct, while \(\sigma(\Phi_s)\) grows monotonically with \(n\) (graph size) and composites can exceed primes (e.g. \(25\) vs \(29\)) — the substrate tracks size, not primality.
Cross-program comparison. Vertex transitivity explains the fixed-sector limitation of the invariant-input residue fixtures. The paused TNFR-Riemann program has its own separately defined \(S_n\) representation and oscillatory residue. The two settings share representation-theoretic language, but they are not thereby one obstruction or one state space.
Honest scope. This explains the recorded fixed-sector result for invariant inputs by vertex transitivity. It does not prove that every per-node observer or perturbed state is blind, does not identify the Riemann obstruction with this finite graph result, and closes no open problem.
9.8 Symmetry-breaking controls on the declared residue fixtures¶
Example 121 compares a symmetric seed, capacities assigned from selected graph invariants, and an arithmetic capacity assignment with a shuffled control. It is a finite comparison on the chosen vertex-transitive residue graphs.
The channel-weight dictionary is graph-level configuration. This does not mean the nodal equation has no local dependence: capacity, phase, EPI and edge conductance are node/edge data, and the pressure depends on them. A missing configuration slot is an implementation fact, not a physical impossibility theorem or proof that every possible local law needs an external axiom.
| Lever | Historical observation | Scope |
|---|---|---|
| Symmetric seed | Potential dispersion near numerical zero | Consistent with an invariant state and equivariant read-out |
| Degree/triangle-derived capacity | Uniform capacity on these graphs | A function of invariant bare-graph data is orbit-constant |
| Arithmetic capacity and shuffle | Dispersion ratio about 0.96–1.05 | The selected statistic did not separate those assignments in this sample |
The exact symmetry statement is conditional: a deterministic equivariant evolution with invariant full initial state cannot select a non-invariant state while its uniqueness/domain assumptions hold. It does not exclude symmetry breaking from an admitted non-invariant state, instability under perturbations, support changes, a different observable or a finer TNFR description. A uniform EPI field alone is not sufficient for zero multichannel pressure if phase, capacity or topology contributions differ.
The previous universal conclusion “no canonical per-node lever exists” is withdrawn. These controls do not settle every endogenous selection mechanism and do not identify the Riemann obstruction with a finite fixed-sector result.
9.9 Phase-Sector Periodicity — a Basis-Invariant Read of the CRT Factor Coset (MEASURED)¶
§9.6 (Reading B, real sector) recovered the factor coset \((i\bmod p)\) of a semiprime \(n=p\cdot q\) as a Fourier mode of the emergent diffusion spectrum. Example 122_factorization_phase_sector.py reads that periodic structure with a basis-invariant observable and a derived-tolerance certificate — replacing an earlier, non-canonical single-eigenvector \(\eta^2>0.9\) decision rule (2026-09-04 canonicity audit).
The structural fact (CRT, present in both sectors). For \(n=p\cdot q\) the factor coset \((i\bmod p)\) corresponds to the Fourier frequencies \(k=\) multiples of the cofactor \(q\). A pure Fourier mode \(\exp(2\pi i k j/n)\) with \(k\) a multiple of \(q\) is constant within each coset \((i\bmod p)\), hence an exact eigenvector of the emergent operator (a circulant / Cayley digraph) — verified to machine precision (eigenvector residual \(\sim 10^{-14}\)) for BOTH the undirected (real) and directed (complex) residue operator. The factor coset is CRT/circulant structure (\(\mathbb{Z}_n\cong\mathbb{Z}_p\times\mathbb{Z}_q\)), present in both spectra.
Why a single-eigenvector \(\eta^2\) is not canonical. A degenerate eigenvalue defines an eigenspace, not a privileged eigenvector: if \(Q_\lambda\) spans it, so does \(Q_\lambda U\) for any unitary \(U\). Any quantity read off individual eigenvector columns — e.g. the maximum \(\eta^2\) of a single column, thresholded at \(0.9\) — can change under that rotation while the operator is unchanged; it is an artefact of the eigensolver's basis. The audit exhibited explicit counterexamples (\(n=209,253,299\)) where the \(0.9\) threshold selects a false divisor.
The canonical, basis-invariant observable + certificate. Let \(C_d\) be the subspace of vectors constant on classes \((i\bmod d)\) with the constant direction removed, \(P_d\) its projector, and \(\Pi_\lambda\) the spectral projector of a (possibly degenerate) eigen-cluster. The score
$\(\mathrm{score}(d,\lambda)=\lVert P_d\,\Pi_\lambda\rVert_2^2\)$
depends only on the two subspaces, hence is invariant under
\(Q_\lambda\to Q_\lambda U\). The exact algebraic condition is
r(d)=0 if and only if the chosen subspace is invariant under L.
The implementation instead accepts the numerical tolerance test
$\(r(d)=\lVert (I-P_d)\,L\,Q_d\rVert_2 < \tau,\qquad \tau=\sqrt{\varepsilon}\,\lVert L\rVert_2\)$
(a selected square-root-machine-epsilon scale times the operator norm).
This is approximate invariant-subspace evidence. It is not an exact rational
certificate or a universal equivalence between small residual and an integer
being a divisor. A separate arithmetic check is needed for an actual factor.
Measured (seed-free; floating linear algebra).
| \(n\) | true \(p\) | false \(d\) | \(\mathrm{score}(p)\) | \(\mathrm{score}(d)\) | \(r(p)\) | \(r(d)\) |
|---|---|---|---|---|---|---|
| 209 | 11 | 3 | 1.0000 | 0.9162 | \(1.6\times10^{-15}\) | \(3.9\times10^{-2}\) |
| 253 | 11 | 7 | 1.0000 | 0.9369 | \(1.1\times10^{-15}\) | \(5.3\times10^{-2}\) |
| 299 | 13 | 7 | 1.0000 | 0.9373 | \(1.2\times10^{-15}\) | \(4.8\times10^{-2}\) |
The false candidate scores above \(0.9\) (so the withdrawn threshold mis-fires) yet its residual \(r(d)\gg\tau\) rejects it; the true factor certifies (\(r(p)\sim10^{-15}<\tau\)). The projector score is invariant under a unitary rotation of a degenerate eigenspace (\(\Delta\sim10^{-16}\)). A label shuffle drives \(r(p)\) from \(\sim10^{-15}\) to \(O(1)\) — the signal is CRT/circulant structure, not an artefact.
Honest scope. This is a periodicity diagnostic of CRT structure in the canonical emergent spectrum — not a factoring algorithm. The residue (di)graph has \(n\) nodes; building and diagonalizing it is \(\mathrm{poly}(n)=\mathrm{poly}(2^L)\), i.e. exponential in the input size \(L=\log_2 n\) bits, and the candidate scan is \(O(\sqrt n)\) prime divisors (the order of trial division). There is no speedup and no cryptographic consequence. The correction relative to earlier drafts: the read is expressed with a basis-invariant subspace score and a derived-tolerance certificate; the non-canonical \(\eta^2>0.9\) rule is withdrawn.
9.10 Symmetry-sector interpretation¶
The structural-observability owner proves fixed weighted/directed diffusion equivariance and its Reynolds-sector split, including zero-strength rows and orbit-constant held capacity. It also owns the finite graph controls and the distinction between pointed covariance, nonlinear fixed sets and complete-state symmetry.
For the invariant residue preparations studied here, an equivariant per-node
reader is orbit-constant; a global spectrum may still distinguish graphs.
Arbitrary perturbed inputs, changing support or a different pressure/phase law
require fresh admission. This common method organizes several information-loss
questions; it does not identify them as the same obstruction or locate analytic
S(T) in a finite symmetry complement.
9.11 The Cyclotomy Law — Proof via Gauss Periods (PROVED)¶
§9.6 established the measured signature "\(3\) distinct eigenvalues \(\iff\) odd prime" for the quadratic-residue (\(k=2\)) digraph. Example 153_structural_frequency_rank_cyclotomy.py generalizes it to the \(k\)-th power residue network and measures the cyclotomy law \(s_k(p)=\gcd(k,p-1)+1\). Unlike §9.5–§9.10 (all measured), this law is a theorem — it follows from classical Gauss-period theory, here proved for all \(k\) and every odd prime \(p\).
Setup. Fix an odd prime \(p\) and an integer \(k\ge 1\); let \(d=\gcd(k,p-1)\) and \(\zeta=e^{2\pi i/p}\). Because \((\mathbb{Z}/p\mathbb{Z})^\times\) is cyclic of order \(p-1\), the nonzero \(k\)-th power residues \(R_k=\{x^k\bmod p\}\) form the unique subgroup \(H\le(\mathbb{Z}/p\mathbb{Z})^\times\) of index \(d\) (the \(d\)-th powers), with \(\lvert H\rvert=(p-1)/d=:f\). The structural rank \(s_k(p)\) is the number of distinct eigenvalues of the canonical \(L_{rw}\) on \(\mathrm{Cay}(\mathbb{Z}/p\mathbb{Z},R_k)\); since \(L_{rw}=I-A/f\) is an affine image of the circulant adjacency \(A\), \(s_k(p)=\#\{\lambda(t):t\in\mathbb{Z}/p\mathbb{Z}\}\) with $\(\lambda(t)=\sum_{r\in H}\zeta^{tr}.\)$
Theorem (cyclotomy law). \(\;s_k(p)=d+1=\gcd(k,p-1)+1\).
Proof.
-
Coset invariance. For \(t\ne 0\), \(\lambda(t)\) depends only on the coset \(tH\): if \(t'=th\) with \(h\in H\) then \(\{hr:r\in H\}=H\) (group closure), so \(\lambda(t')=\sum_{r\in H}\zeta^{t(hr)}=\lambda(t)\). The cosets partition \((\mathbb{Z}/p\mathbb{Z})^\times\) into \(d\) classes, so over \(t\ne 0\) the value \(\lambda(t)\) takes the \(d\) Gauss periods \(\eta_0,\dots,\eta_{d-1}\) (one per coset); the remaining value is \(\lambda(0)=\lvert H\rvert=f\). Hence \(s_k(p)\le d+1\).
-
The \(d\) periods are distinct. The only \(\mathbb{Z}\)-linear relation among \(\{\zeta^i\}_{i=0}^{p-1}\) is \(\sum_{i=0}^{p-1}\zeta^i=0\); restricted to two \(0/1\)-supported sums on \(\{1,\dots,p-1\}\), \(\sum_{i\in S}\zeta^i=\sum_{i\in S'}\zeta^i\iff S=S'\). Therefore \(\sigma_a:\zeta\mapsto\zeta^a\) fixes \(\eta_0=\sum_{r\in H}\zeta^r\) iff \(aH=H\) iff \(a\in H\). So the \(\mathrm{Gal}(\mathbb{Q}(\zeta)/\mathbb{Q})\cong(\mathbb{Z}/p\mathbb{Z})^\times\) stabilizer of \(\eta_0\) is exactly \(H\): \(\eta_0\) generates the unique degree-\(d\) subfield \(K_d=\mathbb{Q}(\zeta)^H\), and its \(d\) Galois conjugates \(\eta_0,\dots,\eta_{d-1}\) are distinct.
-
No period equals the rational \(\lambda(0)\). For \(d\ge 2\): the Galois group permutes \(\{\eta_j\}\) transitively (through \((\mathbb{Z}/p\mathbb{Z})^\times/H\cong\mathbb{Z}/d\mathbb{Z}\)); if some \(\eta_{j_0}=f\in\mathbb{Q}\) then every conjugate would equal \(f\) (Galois fixes \(\mathbb{Q}\)), contradicting distinctness. For \(d=1\): \(H=(\mathbb{Z}/p\mathbb{Z})^\times\), so \(\eta_0=\sum_{i=1}^{p-1}\zeta^i=-1\ne p-1=f\). Either way no \(\eta_j\) equals \(f\).
-
Conclusion. The distinct values are exactly \(\{f,\eta_0,\dots,\eta_{d-1}\}\) — \((d+1)\) of them — so \(s_k(p)=d+1=\gcd(k,p-1)+1\). \(\blacksquare\)
Reading of the law. \(s_k(p)-1=\gcd(k,p-1)=[(\mathbb{Z}/p\mathbb{Z})^\times:H]=\) the number of \(k\)-th power classes \(=[K_d:\mathbb{Q}]\), the degree of the cyclotomic subfield carrying the periods. The maximal rank \(k+1\) is attained \(\iff d=k\iff k\mid p-1\iff p\equiv 1\pmod k\iff p\) splits completely in \(\mathbb{Q}(\zeta_k)\). The quadratic case is \(k=2\) (\(\gcd(2,p-1)=2\) for every odd \(p\) \(\Rightarrow\) the uniform rank 3 of §9.6); the extreme \(d=p-1\) (\(R_k=\{1\}\)) is the directed \(p\)-cycle, all \(p\) characters distinct, \(s=p=(p-1)+1\).
Prime versus prime-power scope. The prime-field proof above also works
at p=2: the nonzero power-residue set is {1}, the two-node Laplacian
has eigenvalues 0,2, and gcd(k,1)+1=2. It is therefore not an odd-only
prime theorem. The separate conductor-annotated quadratic product formula
over odd prime powers requires its own arithmetic stratification; cyclicity
of the unit group alone does not extend the prime-field independence proof
to higher powers. At 2^e, the unit group is noncyclic for e>=3 and the
quadratic strata differ. The recorded annotated counts 2,4,8,10,14,16,20
for e=1,...,7 do not follow the odd-prime-power expression
e+ceil(e/2)+1. That distinguishes the annotated prime-power formula from
the prime-field cyclotomy theorem; it does not invalidate the latter at two.
Honest scope. The cyclotomy law is classical Gauss-period / cyclotomy theory (the \(k\)-th power Cayley eigenvalues are Gauss periods of degree \(\gcd(k,p-1)\)); the contribution is the TNFR structural-diffusion framing and the closed-form power_residue_rank — now a proved canonical fact, not a measured pattern. Verified computationally for \(k\le 40\) across the primes \(p<64\) (680 cases, 0 failures) and proved for all \(k\). It detects primality/cyclotomy structurally; it does not factor, does not reach the continuous arg-\(\zeta\) phase, and closes no open problem.
9.12 Arithmetic diagnostic profiles¶
Example 155 compares finite diagnostics of supplied integer/modular constructions. Multiplicity, graph-product identities and spectral ranks are properties of those constructions. They do not derive the arithmetic carrier or supply a complete ontology from nodal dynamics.
The annotated rank and sampled scalar coincidences. On the declared
odd-modulus domain, §9.7 proves the conductor-annotated product law
\(A(m)=\prod_{p^e\|m}(e+\lceil e/2\rceil+1)\), with local factors
\(3,4,6,7,9,\ldots\). It is distinct from the unannotated scalar spectral rank
\(\rho\). The historical small examples reported \(\rho(p)=3\),
\(\rho(p^2)=4\), \(\rho(p^3)=6\), \(\rho(pq)=9\), \(\rho(p^2q)=12\), and no
multiplicativity exceptions over that demo range. These are useful finite
type distinctions, not a globally invertible code for \(\Omega\) or \(\tau\).
The formula implementation quadratic_residue_annotated_rank itself factors
its argument; the separate graph-spectrum route constructs modular residues
without supplying those factors. The two provenance paths must not be
conflated when claiming a factorization-free diagnostic.
Diagnostic limits. The reported small examples distinguish certain types, not a complete factorization: \(\rho(15)=\rho(35)=9\), and \(\rho=36\) is shared by \(p^3q^3\) and \(p^2qr\). Thus scalar rank does not determine even the type globally. The unannotated rank also aliases at high prime powers: §9.7 / example 154 reports scalar \(191\) versus annotated product \(192\) for \(3^7\!\cdot5^2\!\cdot41^2\). These are limited diagnostic distinctions, not a complete ontology theorem. The underlying integer/modular carrier, selected graph products and observation rules are supplied. No common obstruction identifies these information losses with analytic \(\arg\zeta\), and no physical particles have been generated or identified by these diagnostics.
9.13 The Arithmetic Pulse — two declared linear responses¶
The same finite residue operator supports different declared models. The pure-EPI heat response is \(h(t)=e_o^*e^{-\nu_f L t}e_o\). On pointed circulants its visible recurrence order equals the number of distinct eigenvalues because every spectral projector has weight \(m_\lambda/n\). This exact scope is developed in TNFR_ARITHMETIC_DYNAMICS.md.
The separately chosen conservative graph wave uses \(\omega_k=\sqrt{\lambda_k}\) only on its nonnegative self-adjoint domain. Its second-order law is not derived from the first-order diffusion identity. A directed complex spectrum does not supply a set of real conservative frequencies.
For quadratic residues at an odd prime \(p\equiv1\pmod4\), the real Paley operator has one zero eigenvalue and two distinct positive eigenvalues, each nonzero one with multiplicity \((p-1)/2\). The wave consequently has a stationary mode and two nonzero frequencies. The zero mode must be distinguished from an oscillatory tone. The general prime cyclotomy count in §9.11 is a spectral count even when no conservative-wave interpretation is admitted.
emergent_arithmetic_pulse.py records finite comparisons; its historical name does not derive an autonomous arithmetic oscillator. Composite counts and collisions in §9.12 remain fixture-dependent diagnostics, not a complete factorization-type decoder or a proof that primes minimize every possible spectrum complexity.
The graph, arithmetic carrier, initial excitation and chosen evolution are supplied. These results do not derive their occurrence, sustained maintenance, particle identity or an analytic zeta bridge.
10. Prime Path Graphs and the TNFR-Riemann Connection¶
10.1 The retired prime-path comparison¶
The inserted-threshold counterexample
retains the useful mathematical boundary: the old positive-diagonal model puts
its sign change at 1/2 by construction. This works independently of prime
distribution and is not a prediction of a Riemann critical line. Its old
eigenvector-variation formulas are not canonical phase/tetrad definitions.
10.5 Distinct arithmetic models retain distinct premises¶
The Riemann scope memo owns supplied
prime-ladder spectra, finite logarithmic pulses, classical analytic domains
and the finite Gauss-scale comparison.
The latter supports the corrected verdict
GAUSS_SCALE_MATCHED_DIRECT_ORDERING_DIFFERS, not an RH refutation. No new
pulse, symmetry or spectral campaign follows from these reference comparisons.
11. Worked Examples¶
11.1 Prime Detection: \(n = 17\)¶
Assigned coordinates and read-out: \(\mathrm{EPI}(17) \approx 2.75\), \(\nu_f(17) \approx 1.47\), \(C_{\text{local}} = 1.0\).
Interpretation: Zero arithmetic pressure and unit pressure-only coherence; the held-pressure unforced scalar model has zero EPI rate at this label.
11.2 Semiprime: \(n = 15 = 3 \times 5\)¶
| Component | Calculation | Value |
|---|---|---|
| Factorization | \(1 \times (2-1)\) | 1 |
| Divisor | \(1 \times (4-2)\) | 2 |
| Abundance | \(1 \times (24/15 - 16/15)\) | 0.533 |
| Total | 3.533 |
\(C_{\text{local}} = 1/(1+3.533) \approx 0.221\).
11.3 Prime Power: \(n = 8 = 2^3\)¶
| Component | Calculation | Value |
|---|---|---|
| Factorization | \(1 \times (3-1)\) | 2 |
| Divisor | \(1 \times (4-2)\) | 2 |
| Abundance | \(1 \times (15/8 - 9/8)\) | 0.750 |
| Total | 4.750 |
Using \(\Omega\) (with multiplicity) rather than \(\omega\) (distinct primes) gives prime powers a strong pressure signal: \(2^3\) registers \(\Omega = 3\), not \(\omega = 1\).
11.4 Squarefree Composite: \(n = 30 = 2 \times 3 \times 5\)¶
| Component | Calculation | Value |
|---|---|---|
| Factorization | \(1 \times (3-1)\) | 2 |
| Divisor | \(1 \times (8-2)\) | 6 |
| Abundance | \(1 \times (72/30 - 31/30)\) | 1.367 |
| Total | 9.367 |
Structural triad: \(\mathrm{EPI}(30) \approx 7.48\), \(\nu_f(30) \approx 2.15\), \(C_{\text{local}} \approx 0.097\).
12. Implementation Map¶
12.1 Source Modules¶
| Module | Path | Scope |
|---|---|---|
| Arithmetic network | src/tnfr/mathematics/number_theory.py |
ArithmeticTNFRNetwork, ArithmeticTNFRFormalism, PrimeCertificate |
| Primality testing | applications/primality-test/tnfr_primality/core.py |
Standalone ΔNFR computation, validation |
| Canonical constants | applications/primality-test/tnfr_primality/constants.py |
Arithmetic pressure coefficients (separate subproject) |
| Advanced integration | applications/primality-test/tnfr_primality/advanced_core.py |
Full repo infrastructure bridge |
| Optimized batch | applications/primality-test/tnfr_primality/optimized.py |
Caching, benchmarking, batch processing |
| Spectral factorization | applications/factorization-lab/tnfr_factorization/spectral_paley.py |
Paley-Jacobi spectral decoder |
| Factorization API | applications/factorization-lab/tnfr_factorization/api.py |
High-level factorize() function |
| Arithmetic pulse model | src/tnfr/riemann/nodal_pulse.py |
Finite spectral sums with assigned logarithmic frequencies; separate from native nodal dynamics or a derived NFR pulse |
| Prime-ladder Hamiltonian | src/tnfr/riemann/prime_ladder_hamiltonian.py |
Canonical \(\nu_f\) prime-ladder (P14) |
| Canonical constants (repo) | src/tnfr/constants/canonical.py |
Repository-wide canonical constant definitions |
The primitive-root, Dirichlet-character, Gauss-sum and AdelicOperator
helpers in number_theory.py accept integer/index values, including NumPy
integers. Booleans, floating-point values and text raise TypeError before
conversion; callers must supply the intended integer rather than rely on
truncation. get_primitive_root returns None for integer nonprimes. The
character, Gauss-sum and operator constructors require a prime modulus and
raise TNFRValueError for integer nonprimes.
SymPy is optional for these helpers. Their shared primality predicate uses SymPy when available and exact integer trial division otherwise, with an integer square-root bound. The arithmetic network uses the same predicate. Character and Gauss-sum values remain numerical complex arrays; the fallback does not change their formulas or make their numerical outputs exact.
12.2 Executable Demonstrations¶
| Example | Concept |
|---|---|
| 41_von_mangoldt_zeta_demo.py | Prime-ladder von Mangoldt series (P12) |
| 42_riemann_zeros_as_resonances.py | Riemann zeros as resonance poles (P13) |
| 43_prime_ladder_hamiltonian_demo.py | Canonical νf prime-ladder Hamiltonian (P14) |
| 31_mathematical_constants_basis.py | The structural scale π and the mathematical-constant basis |
| 40_arithmetic_number_theory.py | Primality, triad, component analysis |
| 94_generative_number_construction.py | Compositional generation from prime atoms; U5 fractality; grammar certification |
| 95_primes_from_spectral_waves.py | Prime staircase ψ(x) as spectral-wave superposition; spectral coherence ⟺ RH (honest scope) |
| 96_spectral_vibration_of_coherence.py | Oscillatory residue S(T) as prime-ladder vibration {k·log p}; why aggregate C(t) is blind (honest scope) |
| 97_goldbach_additive_multiplicative.py | Goldbach phase-matching: negative structural result; additive/multiplicative orthogonality; B2/B3 ontological note |
| 100_prime_families_orbits.py | Special prime families (twin, cousin, sexy, Sophie Germain, safe, Cunningham, Mersenne, constellations) as orbits and level-sets of arithmetic maps on the zero-pressure fixed-point set \(Z=\{\Delta\mathrm{NFR}=0\}\); three generator classes; detection exact, infinitude open (honest scope) |
| 101_numbers_as_coupled_network.py | Numbers as a coupled TNFR network: \(\Omega(n)\) grades both the per-node pressure \(\Delta\mathrm{NFR}\) and the divisibility/GCD transport centrality (\(r\approx 0.8\)–\(0.9\)); primes (\(\Omega{=}1\), \(\Delta\mathrm{NFR}{=}0\)) are the transport periphery, large primes isolated; correspondence-through-\(\Omega\) not identity, not scale-free (honest scope) |
| 102_nodal_flow_primes_equilibria.py | The actual nodal flow \(\partial\mathrm{EPI}/\partial t=\nu_f\Delta\mathrm{NFR}\) on numbers: primes are EXACTLY the equilibria (§4 theorem in motion, frozen) while composites drift \(\Omega\)-graded; refines §7.1 — primes are static low-\(\Phi_s\) sinks but NOT dynamical attractors (diffusion flow pulls primes UP toward the composite bulk) |
| 146_primality_grammatical_inertness.py | Tests the restricted scalar flow \(\mathrm{EPI}_{k+1}=\mathrm{EPI}_k+dt\,\nu_f\Delta\mathrm{NFR}_{\rm arith}\) with arithmetic pressure held fixed. It verifies prime \(\iff\Delta\mathrm{NFR}_{\rm arith}=0\iff C=1\), zero prime drift for sampled positive capacities, and exact capacity scaling of composite drift. Because no canonical graph operator or grammar history is applied, it does not prove grammatical inertness. Fixed positive composite pressure also gives an unbounded infinite-time integral, so the fixture is not a U2-convergence result or attraction toward primality. |
| 147_numbers_as_free_monoid_words.py | Compares the classical free commutative monoid on the primes with the unit-coefficient arithmetic pressure. The exact content is the additive law for \(\Omega\), the coprime multiplicative laws for \(\tau\) and \(\sigma\), and \(\Omega(n)=1\iff n\) prime for \(n\ge2\). Count \(\Omega\) and log-size give a declared analogy with pressure and capacity; integers are not operator-grammar words, and prime multiplication is not a canonical destabilizer. |
| 148_capacity_arm_carries_von_mangoldt.py | Uses the classical identities log n = sum_(d divides n) Lambda(d) and Mobius inversion. Assigned logarithmic capacity is an arithmetic coordinate. The continued -zeta_prime/zeta has residue minus the multiplicity at a zeta zero; no global pressure-versus-capacity observability dichotomy or analytic symmetry-complement result follows. |
| 149_p14_is_the_capacity_arm_operator.py | Reads back supplied nu_f=k*log(p) entries in the decoupled P14 matrix. Its weighted finite trace agrees with the matching finite von Mangoldt sum; convergence to the classical infinite identity has domain Re(s)>1. This is a declared arithmetic construction, not a derivation of prime labels, REMESH occurrence or a zero-spectrum Hamiltonian. |
| 153_structural_frequency_rank_cyclotomy.py | Applies the structural-diffusion spectrum to power-residue Cayley graphs. The exact cyclotomy law \(s_k(p)=\gcd(k,p-1)+1\) is proved for the declared odd-prime construction from classical Gauss-period theory and checked on a finite grid. Other correlations and squarefree grading observations in the example remain finite measurements; no restoring primality dynamics or open-problem result follows. |
12.3 Test Coverage¶
| Test area | Location |
|---|---|
| Primality validation (10k range) | applications/primality-test/test_installation.py |
| Arithmetic network construction | tests/ (number_theory tests) |
| Factorization spectral decoder | applications/factorization-lab/tests/test_spectral_paley.py |
| Factorization verification | applications/factorization-lab/tests/test_verification_robustness.py |
| Riemann operator spectral | tests/ (riemann tests) |
13. Reference questions and research boundaries¶
These questions are a secondary inventory, not an active task queue. The FIVE_STAGE_EXECUTION_PLAN.md owns priorities; arithmetic reuse does not replace the current joint nodal emergence objective.
13.1 Computational¶
- Sub-\(O(\sqrt{n})\) primality: Can spectral methods on arithmetic networks detect primes faster than trial division?
- Sieve optimization: Can the TNFR pressure landscape guide more efficient sieve algorithms?
- Large-number factorization: Scaling the spectral Paley-Jacobi method to numbers beyond current computational limits.
13.2 Theoretical¶
- TNFR-Riemann bridge: Can a declared nodal construction support an independently proved analytic correspondence? The historical \(H^{(k)}\) family is superseded (§10); no G4/RH bridge is established by its sign change.
- Pressure distribution: What is the exact probability distribution of \(\Delta\mathrm{NFR}(n)\) for "random" composites?
- Goldbach connection: Can the additive decomposition of even numbers be formulated as a phase-matching problem (\(|\phi_p + \phi_q - \phi_{2n}| \leq \Delta\phi_{\max}\))?
- Arithmetic coherence length: How does \(\xi_C\) in the arithmetic network relate to the distribution of prime gaps?
13.3 Structural¶
- Special prime families (PARTIALLY ADDRESSED — 100_prime_families_orbits.py): twin, cousin, sexy, Sophie Germain, safe, Cunningham, Mersenne, and constellation families are organized as structured subsets of the zero-pressure fixed-point set \(Z=\{n\ge 2:\Delta\mathrm{NFR}(n)=0\}\) (the primes), carved out by three classes of arithmetic map: additive-gap level-sets (\(S_g(p)=p+g\)), affine-recurrence orbits (\(T(p)=2p+1\): Sophie Germain, safe, Cunningham chains), and exponential-form images (\(M(p)=2^p-1\): Mersenne). Detection/generation is exact via the verified \(\Delta\mathrm{NFR}=0\) theorem; infinitude conjectures (twin-prime, Sophie Germain, Mersenne) remain OPEN — the same honest stance as Goldbach (§13.2). The witness pressure signatures (e.g. the twin witness \(p+1\) divisible by 6) are faithful TNFR restatements of classical divisibility facts.
- Arithmetic network as a coupled system (MEASURED — 101_numbers_as_coupled_network.py): on the divisibility/GCD network the prime-factor count \(\Omega(n)\) is a common structural coordinate that grades both the per-node arithmetic pressure \(\Delta\mathrm{NFR}\) (\(r(\Omega,\Delta\mathrm{NFR})\approx 0.94\)) and the network-transport centrality (\(r(\Omega,\deg)\approx 0.75\)), so the two pictures are linked (\(r(\Delta\mathrm{NFR},\deg)\approx 0.81\)). Primes (\(\Omega{=}1\), \(\Delta\mathrm{NFR}{=}0\)) form the transport periphery (≈ 0.18× the composite stationary mass, ≈ 2.4× effective resistance, and large primes \(p>N/2\) are literally isolated). Honest scope: a correspondence through \(\Omega\), not a dynamical identity — the per-node \(\Delta\mathrm{NFR}\) is not the graph-diffusion Laplacian; the network is not scale-free; "primes peripheral" restates the classical \(\gcd(p,m)>1\iff p\mid m\) in transport language.
- Held arithmetic-pressure flow (example 102): positive fixed capacity and the assigned arithmetic pressure give zero EPI rate at primes and constant positive drift at composites. There is no restoring response because that pressure does not depend on evolving EPI. A separate fixed-graph diffusion comparison relaxes toward its weighted mean under its own hypotheses. Neither model proves that primes are universal potential minima, attractors, or stable full-state NFRs.
- Higher-order pressure: Are there fourth or fifth pressure components (beyond \(\Omega\), \(\tau\), \(\sigma\)) that provide additional structural information?
- Algebraic number fields: Extension of the arithmetic triad to Gaussian integers, Eisenstein integers, or general number fields.
- p-adic structure: Connection between the arithmetic tetrad and p-adic analysis.
14. References¶
Internal¶
- AGENTS.md — Primary theoretical reference (TNFR framework)
- FUNDAMENTAL_THEORY.md — Nodal equation and structural field tetrad
- UNIFIED_GRAMMAR_RULES.md — U1-U6 grammar derivations
- STRUCTURAL_OPERATORS.md — 13 canonical operators with tetrad synergies
- STRUCTURAL_CONSERVATION_THEOREM.md — Conservation laws
- Factorization application guide — Candidate policies, arithmetic verification, proxies and evaluation provenance
- TNFR_RIEMANN_RESEARCH_NOTES.md — Current arithmetic comparison scope and unresolved RH/Hilbert–Pólya boundary
- STRUCTURAL_FIELDS_TETRAD.md — Structural-field definitions, scale conventions and estimator provenance
- GLOSSARY.md — Operational definitions
External¶
- Hardy, G.H. & Wright, E.M. — An Introduction to the Theory of Numbers (arithmetic functions)
- Erdős, P. & Kac, M. — "The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions" (1940)
- Kuramoto, Y. — Chemical Oscillations, Waves, and Turbulence (phase synchronization)