TNFR Structural Observability — the Symmetry-Sector Theorem (R1)

Status: diffusion-sector base case — DERIVED (representation theory) + MEASURED (finite implementation checks). Pointed operator/word covariance is tested on selected fixtures. A universal catalog equivariance theorem, a complete observability theorem and an analytic Riemann obstruction are not established here. Modules: src/tnfr/physics/symmetry_sectors.py, src/tnfr/physics/equivariance.py · Tests: tests/physics/test_symmetry_sectors.py · Origin: formalises example 123.

1. The question

What information can a TNFR dynamics generate or resolve when the graph, the initial condition, νf, the node selection and the operators all respect a symmetry group? The answer bounds every "wall" in the §9.5–§9.10 number-theory arc: some information is local, some lives in collective modes, some was injected from outside.

2. The diffusion-sector result (DERIVED)

Let Γ = Aut(G, W) act on ℝᴺ by the permutation representation σ ↦ P_σ. The structural-diffusion operator has rows L_rw = I − D⁻¹W at positive outgoing strength and zero rows at zero strength, including isolates (the EPI channel of the nodal equation, structural_diffusion.py). The automorphisms preserve weights and directed support when present, so this complete operator is equivariant:

\[P_\sigma\, L_{rw} = L_{rw}\, P_\sigma \qquad \forall\,\sigma\in\Gamma .\]

The Reynolds projector \(Q_\Gamma = \tfrac{1}{|\Gamma|}\sum_{\sigma}P_\sigma\) satisfies \(\operatorname{im}(Q_\Gamma)=\mathrm{Fix}(\Gamma)\) and \(\ker(Q_\Gamma)=\mathrm{Fix}(\Gamma)^\perp\), with

\[\dim\mathrm{Fix}(\Gamma) = \#\{\text{vertex orbits of }\Gamma\}.\]

By averaging the commutation identities over the finite group, an equivariant linear operator commutes with \(Q_\Gamma\), so it block-diagonalises:

\[\mathbb{R}^N = \mathrm{Fix}(\Gamma)\ \oplus\ \mathrm{Fix}(\Gamma)^\perp,\qquad [L_{rw}, Q_\Gamma] = 0 .\]

More generally, it preserves the isotypic components of this finite permutation representation; the displayed fixed/complement split is coarser.

Consequence for the flow. With an orbit-constant νf (so D_νf also commutes with P_σ), the overdamped nodal-equation flow \(\dot x = -D_{\nu f} L_{rw} x\) preserves both sectors. A symmetric seed stays in Fix(Γ); this fixed-support diffusion cannot manufacture per-node structure that separates two nodes in the same orbit. Any such separation must come from the seed or a departure from the stated equivariant evolution assumptions. This is not a claim that all possible TNFR evolutions, changing supports or effective descriptions satisfy those assumptions.

3. Certification (MEASURED)

verify_diffusion_equivariance measures, on each test graph, the equivariance residual \(\max_\sigma\lVert P_\sigma L - L P_\sigma\rVert_2\) and the sector-preservation residual \(\lVert L Q_\Gamma - Q_\Gamma L\rVert_2\), both against the derived tolerance \(\tau=\sqrt{\varepsilon}\, \lVert L\rVert_2\). Across cycle, complete, star, path and torus:

Graph	orbits = dim Fix(Γ)	equivariant	sectors preserved
cycle C₈ (vertex-transitive)	1	yes	yes
complete K₆	1	yes	yes
star K₁,₅	2 (center, leaves)	yes	yes
path P₆	3 (ends, near-ends, middle)	yes	yes
torus C₃×C₃	1	yes	yes

rank Q_Γ = #orbits in every case; the residuals are at machine precision. The star shows the result does not need a symmetric operator: L_rw there is non-symmetric yet still commutes with Aut = S₅. Results are relabel-invariant and weight/direction aware (a distinguished heavy edge lowers the symmetry and raises the orbit count; a directed cycle stays single-orbit under rotations).

4. Pointed graphs (declared symmetry breaking)

Selecting a single origin node o (a localized emission) is not equivariant. It changes the object of study from G to the pointed graph (G, o) and reduces the relevant group to the stabilizer Γ_o. This is legitimate provided the break is declared: the reduced symmetry is stated, not presented as spontaneous emergence. This distinction is the basis of the arithmetic-pulse programme (R2), which studies the pointed residue network (G_{p,k}, 0).

Formal structure (N07). The stabilizer Γ_o = {g ∈ Aut(G) : g(o) = o} is a subgroup of Aut(G) (pointed_symmetry.py), and three facts make the break exactly Aut(G) → Γ_o:

• Orbit–stabilizer (DERIVED group theory, MEASURED): |Aut(G)| = |Γ_o| · |orbit(o)| — verified 24 = 6·4 (star leaf) and 12 = 2·6 (cycle vertex).
• Residual sectors refine, never coarsen (DERIVED): Fix(Aut(G)) ⊆ Fix(Γ_o) and dim Fix(Γ_o) = #orbits(Γ_o) ≥ #orbits(Aut(G)). The origin splits its orbit: star 2 → 3 sectors ({0},{1,2,3,4} → {0},{1},{2,3,4}), cycle 1 → 4.
• Break localization (MEASURED): a pointed Emission maps Fix(Aut(G)) out of Fix(Aut(G)) (break_magnitude ≈ 0.07 > 0) yet into Fix(Γ_o) (stabilizer_residual = 0). The possible marked origins are parameterized by the coset set Aut(G)/Γ_o, in bijection with orbit(o); this discrete set is not a linear space of broken directions. A singleton-orbit origin (the star centre) breaks nothing (break_magnitude = 0).

No privileged origin (conjugation). Origins in one orbit are conjugate: Γ_{g(o)} = g Γ_o g⁻¹ and O@{g(o)} = g (O@o) g⁻¹ (MEASURED), so the pointed structures at o and g(o) are isomorphic — the choice of 0 in (G_{p,k}, 0) is a labelling convention, and the arithmetic-pulse observables are independent of it.

4b. Pointed per-operator covariance (finite isolated probes)

The probes compare actions at corresponding selected nodes: O_(σv)(P_σ x) = P_σ O_v(x), evaluated on a σ-invariant seed. This covariance of the family of pointed maps differs from equivariance of one map with a fixed selector. A single selected-node emission can break the seed symmetry while still obeying the pointed covariance identity (operator_equivariance.py, test_operator_equivariance.py).

Result (MEASURED). On the vertex-transitive cycle (uniform seed) and the two-orbit star (orbit-constant seed), all 13 canonical operators — AL, EN, IL, OZ, UM, RA, SHA, VAL, NUL, THOL, ZHIR, NAV, REMESH — are equivariant to the selected tolerance (residual \(<10^{-6}\), most exactly \(0\)) on these pointed comparisons. The helper reads four scalar channels on the original node set after refreshing pressure. It does not compare every history, created node, changed edge, selector or other state attribute; its boolean is_equivariant is a finite probe verdict, not an all-state theorem.

Root cause and fix (N01). The certification requires each operator to be measured from an independent, clean graph cache. TNFR content-keyed caches (_dnfr_prep_cache, _node_set_checksum_cache, the graph cache managers) live in G.graph and survive networkx's G.copy() by shared reference; keyed on a label-independent node-set checksum, they collide between the two isomorphic copies A and B = σ(A), so B reads A's cached ΔNFR prep. On reused seed graphs this leaked across operators and made a batch audit report spurious \(\sim 10^{-3}\) residuals (verified: Silence is \(0\) in isolation, \(\sim 10^{-3}\) only after other operators run). The fix (_isolate_graph_caches) drops those content caches on each copy, so every measurement is an independent experiment with a private cache (report R1-T01: prefer per-experiment local cache when a global content key cannot distinguish isomorphic siblings). With it the batch audit is exactly \(0\) for all 13 operators, independent of cache warmth, operator order, or repeated runs (test_equivariance_cache_isolation.py). The config keys (_dnfr_weights, _DNFR_META) are preserved — only the caches are isolated.

4c. Word composition closure (N06)

The following algebraic theorem assumes complete equivariance of each map on its stated domain. The finite pointed probes in §4b do not prove that premise.

Theorem (composition closure). If each factor O_i is Γ-equivariant (O_i ρ(g) = ρ(g) O_i for every automorphism g), then the grammar word W = O_k ∘ … ∘ O_1 is Γ-equivariant. Proof (induction on length). Length 1 is the per-operator base case. For W' = O ∘ W with W, O equivariant,

\[W' \rho(g) = O\,(W \rho(g)) = O\,(\rho(g)\,W) = (O \rho(g))\,W = \rho(g)\,(O W) = \rho(g)\,W'. \qquad\blacksquare\]

Corollary (Fix(Γ) preservation). An equivariant W maps Fix(Γ) into Fix(Γ): if ρ(g)x = x for all g then ρ(g) W(x) = W(ρ(g)x) = W(x). A symmetric configuration therefore has no new component in Fix(Γ)^⊥ under such a word. Grammar admission alone does not imply the premise, and no representation placing analytic S(T) in this finite complement is supplied.

Status. The conditional composition theorem is exact. Small residuals on a finite base-case sample do not establish its hypothesis on later inputs, or control error accumulation under arbitrary composition. The independent probes in word_equivariance.py report that the five selected words (Bootstrap, Bootstrap+close, Stabilize, Propagate, Explore) have exactly zero residual on both test cases, every prefix stays within tolerance (composition_closure_holds), and a Γ-symmetric sweep keeps the seed orbit-constant (word_preserves_fix) (test_word_equivariance.py). This checks the recorded words and prefixes. A sequential sweep also needs its ordering respected by the group action; merely visiting every node does not prove equivariance. The pointed-selector result remains distinct (§4).

5. Honest scope

The reusable results are fixed-graph diffusion equivariance, the conditional composition theorem, and the group theory of pointed selectors. The finite operator/word residuals supplement those proofs without extending their quantifiers. An invariant input and equivariant observer explain nodewise constancy on vertex-transitive residue fixtures; arbitrary perturbed states fall outside that conclusion.

A global spectral invariant is not a vector in the nontrivial sector merely because it distinguishes graphs. No map identifies analytic S(T) with a finite symmetry complement or REMESH kernel. The R1 decomposition is useful but not a complete joint-state observability theory, an autonomous symmetry selection mechanism, or a solution to an external open problem. Current cross-scale dynamic and geometric output obligations are maintained in TNFR_SCALE_GEOMETRY_AND_BRIDGE.md.

6. Limits beyond linear symmetry

The same action and complete state must be used at every step. In addition to support, this includes capacities, phases, form, histories, selectors and ordering. Bare-graph symmetry or uniform configuration gains cannot substitute for symmetry of those inputs. The following limits apply to any proposed coarse-state or observation argument:

• Nonlinear fixed sets and complements differ. Equivariance preserves the fixed set, but not generally its orthogonal complement. Under coordinate exchange, (x_1,x_2) -> (x_1^2,x_2^2) is equivariant yet sends (1,-1) to (1,1). The linear two-sector decomposition cannot be transferred to an arbitrary nonlinear operator or tensor construction.
• An invariant observation can retain information. A matrix spectrum is unchanged by simultaneous relabeling yet can distinguish graphs or unordered frequency data. Swapping values while holding other data fixed is a different intervention. Relabeling covariance is not arithmetic or analytic blindness.
• An equivariant field reader needs invariant input to produce an invariant output. Arbitrary phase/pressure states need not give orbit-constant tetrad fields. Undefined curvature cannot be replaced with an invariant zero.
• One failed connection does not exhaust possible connections. A selected invariant edge construction can lose an observation without excluding every other state-derived construction. Non-invariance under bare-graph symmetry does not by itself mean that a connection was supplied externally.

For a specified linear generator, [L,P]=0 implies [f(L),P]=0 for polynomials and functional calculi defined on that generator. Equivariant factors similarly give equivariant Kronecker products and compatible Kronecker sums under the product action. Induced line-graph actions must also preserve the selected edge weights and observations. These conditional identities do not place analytic S(T) in a finite symmetry complement or establish a catalog-wide obstruction.

The linear flow-morphism owner proves the intertwining criterion; CRT and p-adic transport retain their product/reduction laws. The Riemann scope owns the external-target limits. A finite INDETERMINATE_DEGENERATE_CONSTRUCTION verdict establishes only failure of the selected statistic on its stated fixture and tolerance, not impossibility for an entire family or all future constructions.