TNFR–P vs NP Structural Synthesis Research Notes

Status: Pre-registered research programme; PNP-1 diagnostic implemented; obstruction classified as Branch B (open) Date: 2026-06-13 Scope: TNFR-internal structural synthesis-vs-verification dynamics; not a proof or disproof of the Clay P vs NP problem Primary anchors: nodal equation ∂EPI/∂t = νf · ΔNFR(t) as gradient flow ΔNFR = −∂V/∂EPI, canonical operators, grammar U1–U6, structural field tetrad (Φ_s, |∇φ|, K_φ, ξ_C)

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0. Terminology Discipline

This programme is formulated in TNFR language only. TNFR does not introduce a separate "computation" ontology. The same nodal equation,

$$\frac{\partial \mathrm{EPI}}{\partial t} = \nu_f \cdot \Delta\mathrm{NFR}(t), \qquad \Delta\mathrm{NFR} = -\frac{\partial V}{\partial \mathrm{EPI}},$$

is a gradient flow on the structural potential V (established in src/tnfr/physics/variational.py). Coherence relaxation descends V.

References to P vs NP are treated as an external comparison target. The TNFR object is a nodal structural question: is synthesising a globally coherent configuration fundamentally harder than verifying one? No claim in this document should be read as a solution of the Clay Millennium Problem. The Clay problem concerns worst-case separation of the complexity classes P and NP for general decision problems; nothing here establishes or refutes that separation.

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1. Existing Canonical Base in the Repository

The programme starts from already-shipped TNFR machinery; it requires no new canonical operator.

Component	Existing source	Role
Nodal equation as gradient flow	src/tnfr/physics/variational.py	ΔNFR = −∂V/∂EPI; relaxation descends V
Structural potential	src/tnfr/physics/variational.py	`V = ½[Φ_s² +
Phase channel (Kuramoto coupling)	src/tnfr/dynamics/dnfr.py, src/tnfr/physics/structural_diffusion.py	circular neighbour coupling; sign sets align/antialign
Dissonance operator OZ	src/tnfr/operators/dissonance.py	controlled instability; basin-escape move (PNP-2)
Coherence C(t), Sense Index Si	src/tnfr/metrics/	polynomial-cost verification telemetry

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2. TNFR-Native Reformulation

P vs NP, read through the nodal gradient flow, is the asymmetry between two structural tasks on a graph-coupled network:

• Verification. Given a configuration, evaluate its coherence (e.g. the cut value / frustration energy, or C(t)). Cost O(|E|) — polynomial. This is the TNFR analogue of checking an NP witness.
• Synthesis. Find a globally coherent configuration by nodal relaxation. On a frustrated topology the potential V has many local optima = dissonance (OZ) basins; gradient flow descends to the nearest basin.

PNP-1: On a family of frustrated instances of growing size, does bare coherence relaxation (gradient flow) trap in local optima — hit rate of the global optimum dropping, required restarts growing — while verification stays O(|E|)?

PNP-2 (open): Does the full canonical operator catalog (OZ-controlled dissonance, ZHIR mutation, THOL re-organization, REMESH cross-scale echo) collapse the trapping to polynomial-cost synthesis, or do the dissonance basins remain exponentially many?

PNP-2 is the TNFR-native P-vs-NP boundary and is not assumed.

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3. Encoding (MAX-CUT as TNFR antiphase coupling)

Each node carries a phase θ. Every edge demands antiphase (a cut). The relaxation

$$\frac{d\theta_i}{dt} = \sum_{j \sim i} \sin(\theta_i - \theta_j)$$

is the canonical TNFR phase channel with the anti-aligning sign — an all-edge dissonance (OZ) demand. The global minimum of the frustration energy E = Σ_{(i,j)} cos(θ_i − θ_j) over θ ∈ {0, π}^n is exactly the MAX-CUT of the graph (an NP-hard objective). Frustration arises on odd cycles, which cannot satisfy all antiphase demands simultaneously — the structural origin of the local-optima (dissonance-basin) landscape.

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4. PNP-1 Result (DONE)

Measured on random 3-regular graphs (a standard frustrated family), 3 instances per size, R = 200 random initial conditions each; exact MAX-CUT by enumeration; relaxation 400 steps, dt = 0.1. Reproducible in examples/09_millennium/109_p_vs_np_coherence_synthesis.py.

n	|E|	global reachable	hit rate	restarts ≈ 1/hr
8	12	yes	0.737	1.36
10	15	yes	0.648	1.54
12	18	yes	0.642	1.56
14	21	yes	0.558	1.79
16	24	yes	0.422	2.37
18	27	yes	0.412	2.43

• Hit-rate trend slope d(hit_rate)/dn = −0.0341 per node; monotone decreasing across all sizes.
• "global reachable = yes" (best over all restarts reaches the exact MAX-CUT at every size): the low hit rate is genuine trapping in local optima, not an encoding failure.

PNP-1 verdict: coherence synthesis by bare gradient flow is trapped by dissonance basins, with trapping increasing in size, while verification stays O(|E|). The synthesis-vs-verification asymmetry is the TNFR-native reflection of the P-vs-NP asymmetry.

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5. Honest Obstruction Classification

Using the same A/B trichotomy as the other TNFR Millennium programs:

• Branch A (closure inside the existing catalog) — not established. PNP-1 only measures bare gradient flow, which is one descent strategy.
• Branch B (open; current classification) — the trapping is real, but whether the full canonical catalog (OZ/ZHIR/THOL/REMESH escape moves) synthesizes in polynomial time, or whether the dissonance basins are exponentially many, is open (PNP-2). The honest expectation — frustrated landscapes have exponentially many local optima — reflects P ≠ NP but is unproven here.
• Branch B3 (no TNFR closure) — not decidable from PNP-1.

This obstruction is structurally analogous to:

• the Riemann residual S(T) (oscillatory half of the admissible rescaling, RH-equivalent);
• the Navier–Stokes production residual (cascade at scale → 0, Clay-open);
• the Yang–Mills continuum-limit gap (YMG-5, Clay-open).

In each case TNFR reformulates the problem and localises the obstruction precisely, without closing it.

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6. Milestone Roadmap

PNP	Title	Status
PNP-1	Synthesis-vs-verification trapping on frustrated MAX-CUT	DONE (examples/109)
PNP-2	Full-catalog escape (OZ/ZHIR/THOL/REMESH): does trapping collapse to polynomial?	open
PNP-3	Basin-count scaling: are local optima exponentially many under U1–U6?	open
PNP-4	Encoding generality: SAT / graph-colouring beyond MAX-CUT	open
PNP-5	Worst-case separation (Clay-hard boundary)	open, not assumed

PNP-5 is the Clay-strength statement and is not claimed.

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7. What This Program Does and Does Not Do

Does: provide a TNFR-native reformulation of P vs NP as coherence synthesis vs verification; supply a reproducible diagnostic (PNP-1) showing the trapping signature; classify the obstruction honestly (Branch B, open).

Does not: prove or disprove P = NP; claim that TNFR relaxation is an efficient general solver; assume the full operator catalog escapes the traps. The program follows the disciplined pattern of the Riemann, Navier–Stokes, and Yang–Mills programs: reformulate, measure, localise the obstruction, and remain honest about the open boundary.