TNFR Arithmetic Operators — Transformation-to-Operator Certification (R8)

Status: two candidate contract comparisons and four rejected identifications. The positive records do not execute AL/RA or certify a full engine transition. No fourteenth operator is invented; the general "which arithmetic maps are operators" question is OPEN (NT-P08). Module: src/tnfr/mathematics/operator_certificates.py · Tests: tests/mathematics/test_arithmetic_operator_certificates.py · Benchmark: benchmarks/arithmetic_operator_certification.py · Depends on: C2 (U3 hard invariant), C5 (claim manifest); contracts from operator_contracts.py.

1. The certificate

Whether an arithmetic transformation is a TNFR operator is a contract question. ArithmeticOperatorCertificate records, for each candidate, the transformation, the canonical operator (or None), the nodal state_channel and scale, the pre/postconditions, the grammar word, and the measured residuals. The current certified property only checks that canonical_operator is not None; it is not a sealed executor certificate. grammar_valid separately checks the listed word through validate_sequence. These records reuse contract_for metadata and selected synthetic residuals. They do not validate every operator pre/postcondition or run a graph-owned history, and the U3 phase premise of the propagation record is recorded as text rather than measured.

2. Positive candidate records (synthetic comparisons)

transformation	operator	channel / scale	residual
localized emission at zero (e_0 seed)	Emission (AL)	EPI / NODE	epi_direction_violation = 0 (ΔEPI = +1)
propagation over residue edges (additive Cayley transport)	Resonance (RA)	EPI / NODE	conservation_defect = 0 (identity preserved)

The first function assigns synthetic scalars epi_before=0, epi_after=1. It tests the sign of that increment and records an AL contract name plus the word [emission, coherence, silence]; it neither invokes AL nor derives creation from vacuum. Zero EPI is not absence of a nodal substrate.

The second multiplies a localized vector by a declared Cayley transition matrix. Total preservation follows from that matrix's column sums, not from executing RA. It records the word [emission, resonance, coupling, silence] and a U3 premise without constructing phases. EPI-sum preservation alone is not preservation of a complete pattern identity or a proof of RA equivalence.

3. Negative certificates — the boundary result (MEASURED)

A negative record documents why the tested identification was not certified. It does not exclude every future canonical realization of an arithmetic map.

transformation	verdict	evidence
CRT projection σ	rejected — relabeling	bijection, channel_modification = 0
p-adic lift ℤ/p^eℤ → ℤ/p^{e+1}ℤ	rejected — REMESH unverified	commutation = 0 but remesh_conditions_unmet = 4
affine x ↦ ax + b	rejected — automorphism	bijection, channel_modification = 0
power map x ↦ x^k	rejected — endomorphism	many_to_one_factor = gcd(k, ord)

• CRT projection (R3) is a pure coordinate change; it modifies no nodal channel, so it realizes U5 structure without being an operator.
• p-adic lift (R4) is transport-consistent, but the REMESH contract (EPI recursion, NETWORK scale, preserved identity, U5) is unverified — naming it REMESH is forbidden until remesh_contract_audit().realizes_remesh is True.
• affine maps with invertible multiplier are permutations. For a Cayley graph, they are automorphisms only when the multiplier preserves its connection set. The current helper checks bijectivity, not edge preservation.
• power maps are group endomorphisms; their "contraction" reduces the state space to a subgroup, not a node's νf, so certifying them as Contraction would be decorative.

4. Structural morphisms of the declared linear EPI flow (N08)

The companion helper classifies selected finite linear maps by shape, rank and intertwining residual (structural_morphism.py).

Genus (DERIVED). For the EPI channel with a fixed common ν_f>0 the nodal equation is dEPI/dt = −ν_f L · EPI with flow EPI(t) = e^{−ν_f t L} EPI(0). A linear map M : (V_src, L_src) → (V_tgt, L_tgt) carries every source solution to a target solution, M e^{−s L_src} = e^{−s L_tgt} M for all s, iff it intertwines the generators, M L_src = L_tgt M (differentiate at s = 0 for ⇒; both sides solve the same ODE d/ds(·) = −L_tgt(·) with equal initial data for ⇐). So a linear flow morphism is exactly an intertwiner for this fixed EPI model. intertwining_residual measures the generator defect. The distinct nodal_flow_preservation_residual samples a trajectory from one initial state; its numerical magnitude need not equal the generator defect, and a zero sampled result does not prove all-state transport. This uses the additional constitutive choice pressure=-L*EPI, fixed support and common capacity; it is not derived from the nodal identity alone.

Classification vocabulary. Dimension change, rank type and additional matrix predicates supply labels for the tested maps. These labels do not uniquely determine a map or classify all intertwiners:

helper label	dimension	rank	checked construction	finite intertwining result
AUTOMORPHISM ⊆ RELABELING	preserve	iso (permutation)	R1 symmetry / relabel-invariance (C1)	yes
INTERTWINER	preserve	iso (non-permutation)	change of coordinates	yes
PROJECTION ⊇ COARSE_GRAINING	preserve / reduce	idempotent / quotient	R1 sector projector Q_Γ / U5 fiber quotient (R4)	yes
LIFT	increase	embedding	U5 prolongation (R4)	yes
ENDOMORPHISM	preserve	rank-deficient fold	— (does not intertwine)	no

Measured (audit_structural_morphisms): six supplied constructions have nodal_flow_residual ≈ 0; the selected folding power map x ↦ x² (mod p) has ≈ 1.9, so it fails that generator test. This is not a classification of every folding map. Notably the Reynolds sector projector Q_Γ (R1) is an idempotent intertwiner: a rank test alone would misfile it as an endomorphism, but it is the helper's PROJECTION onto Fix(Γ).

Operator vs linear morphism. A relabeling, an automorphism, a fiber quotient and a lift are classified as morphisms (is_operator = False) in this helper. The distinction is not a universal "within versus between" dichotomy: canonical THOL can change support, and a projector can act on one state space. A joint morphism must also transport the actual phase, capacity, support and output laws.

5. Claim ledger

Claim	Basis	Status
synthetic emission compared with AL	assigned EPI increment + contract metadata	CANDIDATE; AL not executed
transition step compared with RA	EPI sum + declared U3 premise	CANDIDATE; RA/U3 not executed
CRT / affine are operators	relabeling changes no channel	NEGATIVE
p-adic lift is REMESH	REMESH contract unverified	NEGATIVE (R4)
power map is Contraction	endomorphism, not νf	NEGATIVE
a fourteenth operator is needed	—	not established (NT-P08 OPEN)
intertwiner transports every solution of the fixed linear EPI law	differentiate semigroup / ODE uniqueness	DERIVED, with stated hypotheses
Reynolds Q_Γ is a sector PROJECTION (not an endomorphism)	idempotent intertwiner (N08)	MEASURED
the selected folding map fails to intertwine	nodal_flow_residual ≈ 1.9 (N08)	MEASURED (fixture boundary)

The current records support selected arithmetic-to-contract comparisons and linear-flow identities. Full AL/RA transition equivalence, joint-state morphisms and a general classification remain open. Source API names such as certified retain compatibility and must be read within this scope.