TNFR–Yang–Mills Structural Gap Research Notes

Status: Pre-registered research programme; Y1–Y5 diagnostics implemented; closure classified as Branch B
Date: 2026-05-31
Scope: TNFR-internal structural gauge dynamics; not a proof of the Clay Yang–Mills and Mass Gap problem
Primary anchors: nodal equation ∂EPI/∂t = νf · ΔNFR(t), canonical operators, grammar U1–U6, structural field tetrad (Φ_s, |∇φ|, K_φ, ξ_C), complex geometric field Ψ = K_φ + i·J_φ

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0. Terminology Discipline

This programme must be formulated in TNFR language only.

TNFR does not introduce an independent entity called "quantum mechanics" or a separate microscopic ontology. The same nodal equation,

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

admits different coherence regimes:

• smooth-trajectory regimes, externally comparable to classical mechanics;
• discrete-mode / high-dissonance regimes, externally comparable to quantum-mechanical behaviour;
• gauge-structured regimes generated by the internal phase of Ψ = K_φ + i·J_φ.

Therefore, references to Yang–Mills and mass gap are treated as external comparison targets. The TNFR object is a nodal structural gauge dynamics problem: construct gauge-compatible EPI evolution, measure structural spectral separation, and test whether U1–U6 plus U6 confinement enforce a positive gap in the admissible nodal spectrum.

No claim in this document should be read as a solution of the Clay Millennium Problem. The Clay problem concerns rigorous four-dimensional non-Abelian Yang–Mills existence and a positive mass gap in the continuum. The current TNFR codebase contains a canonical U(1) gauge structure; any non-Abelian extension must be derived from the nodal equation and grammar before being called canonical.

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1. Existing Canonical Base in the Repository

The programme starts from already-shipped TNFR machinery:

Component	Existing source	Role
Complex geometric field	src/tnfr/physics/fields.py, src/tnfr/physics/unified.py	Ψ = K_φ + i·J_φ, geometric-transport sector
Gauge symmetry	src/tnfr/physics/gauge.py	local U(1) rotation Ψ(i) → e^{iα(i)}Ψ(i)
Gauge connection	compute_gauge_connection()	A_ij = arg(Ψ_j) − arg(Ψ_i) on edges
Gauge curvature	compute_gauge_curvature()	cycle holonomy F_C = Σ A_ij
Covariant derivative	compute_covariant_derivative()	D_ijΨ = Ψ(j) − e^{iA_ij}Ψ(i)
Yang–Mills-like action	compute_yang_mills_action()	S_YM = 1/2 Σ_C F_C^2
Field equations	compute_yang_mills_equations()	discrete gauge divergence vs matter current residuals
Conservation-gauge unification	src/tnfr/physics/conservation_gauge_unification.py	grammar → symmetry → conservation → gauge

This means the first TNFR–Yang–Mills step does not require a new canonical operator. It requires a spectral diagnostic built on top of the existing tetrad/gauge/conservation stack.

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2. TNFR-Native Reformulation of the Mass Gap Question

The external Clay statement asks for a rigorous four-dimensional Yang–Mills theory with compact simple gauge group and positive mass gap. TNFR reframes the first attack surface as follows.

Given a family of grammar-compliant nodal gauge graphs G(a, L) with lattice spacing a and size L, construct a self-adjoint structural gauge operator

$$H_{\mathrm{YM}}^{\mathrm{TNFR}}(a,L)$$

from canonical telemetry only:

$$(\Phi_s, |\nabla\phi|, K_\phi, \xi_C, J_\phi, J_{\Delta\mathrm{NFR}}, A_{ij}, F_C, D_{ij}\Psi).$$

Define the finite-graph structural gap

$$\Delta_{\mathrm{TNFR}}(a,L) = \lambda_1\!\left(H_{\mathrm{YM}}^{\mathrm{TNFR}}(a,L)\right) - \lambda_0\!\left(H_{\mathrm{YM}}^{\mathrm{TNFR}}(a,L)\right).$$

The first TNFR question is not the full Clay theorem, but the discrete structural precursor:

YMG-1: Under U1–U6, U6 confinement, and gauge-invariant construction from Ψ, does the finite TNFR gauge operator have a reproducible positive gap above the coherent vacuum mode?

The continuum-strength question is deferred:

YMG-5: Does liminf_{a→0, L→∞} Δ_TNFR(a,L) > 0 hold under a canonically specified scaling regime?

YMG-5 is the Clay-hard boundary and is not assumed.

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3. Structural Interpretation of the Gap

TNFR does not treat "particle mass" as primitive. A mass gap is interpreted as spectral isolation of the first non-trivial stable nodal reorganisation mode.

External term	TNFR structural object
Vacuum	grammar-compliant coherent attractor minimising structural gauge energy
Excitation	admissible non-zero EPI reorganisation mode with gauge curvature or covariant-gradient content
Mass gap	positive separation between the coherent attractor and first admissible non-trivial structural mode
Confinement	U6-bounded structural potential plus non-zero gauge curvature preventing arbitrarily cheap free modes
Gauge field	internal geometric-transport phase structure of Ψ = K_φ + i·J_φ

The working hypothesis is:

$$\Delta_{\mathrm{TNFR}} > 0 \quad\Longleftrightarrow\quad \text{U6 confinement + gauge curvature + grammar-compliant bounded evolution prevent zero-cost non-trivial modes.}$$

This is a TNFR statement about nodal dynamics, not an ontological statement about a separate quantum layer.

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4. Candidate Operator Surface

The initial finite-graph operator should be assembled from already canonical pieces. A minimal candidate family is:

$$H_{\mathrm{YM}}^{\mathrm{TNFR}} = L_A + V_{F} + V_{\mathrm{U6}},$$

where:

• L_A is a gauge-covariant graph Laplacian derived from D_ijΨ;
• V_F is a curvature potential derived from cycle holonomies F_C;
• V_U6 is a confinement barrier derived from the structural potential channel Φ_s and the U6 threshold.

No term may depend on external labels, empirical tuning, or non-canonical per-node parameters. Any coefficient must be traceable to the canonical structural scale π, the nodal dynamics, or to graph-level normalisation.

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5. Gap Ledger

Gap	Question	Status
YMG-0	TNFR-native terminology and scope discipline	CLOSED by pre-registration
YMG-1	Finite-graph TNFR gauge gap diagnostic	IMPLEMENTED by tnfr.yang_mills.compute_structural_gauge_gap(); finite graph only
YMG-2	Gauge invariance / Ward identity compatibility for the gap operator	PARTIALLY SUPPORTED by existing gauge.py, conservation_gauge_unification.py, and Y1/Y2 spectral-invariance checks
YMG-3	U6 confinement lower-bound argument for finite graphs	EMPIRICAL SURFACE CREATED by tnfr.yang_mills.run_u6_confinement_sweep(); proof remains open
YMG-4	Non-Abelian derivability audit from the nodal equation	AUDITED: OPEN_DERIVABILITY_GAP; current canonical gauge implementation remains U(1)
YMG-5	Continuum + thermodynamic scaling liminf Δ > 0	FINITE SCALING DIAGNOSTIC IMPLEMENTED by Y4; continuum limit remains OPEN / Clay-hard
YMG-6	Closure / obstruction classification	CLASSIFIED: BRANCH_B_OBSTRUCTION_CLASSIFIED by Y5

The key honesty constraint is YMG-4: classical Yang–Mills mass gap is non-Abelian. A multi-channel or non-Abelian TNFR gauge sector cannot be assumed merely because external Yang–Mills uses it. It must be derived as a structural consequence of the nodal equation, tetrad, operators, and U1–U6.

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6. Pre-Registered Milestones

Y1 — Finite Structural Gauge Gap Diagnostic

Implementation status (2026-05-31): DIAGNOSTIC_SURFACE_CREATED.

Implemented in:

• src/tnfr/yang_mills/__init__.py
• src/tnfr/yang_mills/structural_gap.py
• tests/physics/test_yang_mills_structural_gap.py

The implemented operator is:

$$H_{\mathrm{YM}}^{\mathrm{TNFR}} = L_A + V_F + V_{\mathrm{U6}},$$

where L_A is the gauge-covariant graph Laplacian assembled from A_ij, V_F is the cycle-curvature potential normalised by π², and V_U6 is the structural-potential confinement term normalised by (π/2)² (the U6 drift bound U6_STRUCTURAL_POTENTIAL_LIMIT = π/2). The diagnostic reports (λ0, λ1, Δ), self-adjointness, seeded local-U(1) spectral invariance, U6 metadata, Yang–Mills action, gauge coupling, and grammar-rule counts. It is read-only with respect to EPI and phase attributes.

Implement a finite-graph diagnostic that:

1. constructs a grammar-compliant gauge graph;
2. computes Ψ, A_ij, F_C, D_ijΨ, S_YM;
3. assembles a self-adjoint finite matrix H_YM_TNFR;
4. reports (λ0, λ1, Δ) with seed, graph, and threshold metadata;
5. verifies gauge-invariant diagnostics before and after local U(1) rotations.

First verdict: DIAGNOSTIC_SURFACE_CREATED, not closure. The gap reported by Y1 is a finite-graph structural spectral gap only; it does not address non-Abelian derivability or the continuum / thermodynamic limit.

Y2 — U6 Confinement Sweep

Implementation status (2026-05-31): EMPIRICAL_FINITE_GRAPH_ONLY.

Implemented in:

• src/tnfr/yang_mills/u6_sweep.py
• tests/physics/test_yang_mills_u6_sweep.py

The Y2 sweep wraps the Y1 operator across finite graph families and target ratios

$$\rho_{\mathrm{U6}} = \frac{\max_i |\Phi_s(i)|}{\pi/2},$$

where ρ_U6 < 1 is U6-confined and ρ_U6 ≥ 1 intentionally probes unconfined structural-potential regimes. The sweep records gap statistics, U6 confinement status, Yang–Mills equation residuals, curvature activity, grammar-rule counts, self-adjointness, and seeded local-U(1) spectral invariance. The ratios are sampling targets only; the canonical threshold remains ρ_U6 = 1.

Sweep graph families and U6-safe / U6-unsafe regimes. Test whether positive gap correlates with:

• bounded Φ_s;
• non-zero but controlled F_C;
• low Yang–Mills equation residual;
• grammar-compliant operator histories.

First verdict: EMPIRICAL_FINITE_GRAPH_ONLY. This creates the finite empirical surface needed to study YMG-3, but it does not prove a U6 lower bound and does not address YMG-4/YMG-5.

Y3 — Non-Abelian Derivability Audit

Implementation status (2026-05-31): OPEN_DERIVABILITY_GAP.

Implemented in:

• src/tnfr/yang_mills/derivability.py
• tests/physics/test_yang_mills_derivability.py

The Y3 audit evaluates whether any candidate route supplies all of the following without external input: a TNFR-native multiplet, a canonical connection mixing multiplet components, non-commuting generators derived from nodal dynamics, and U1–U6 compatibility. The implemented candidate routes are:

Route	Result	Obstruction
u5_nested_epi_multiplet	OPEN_MULTIPLET_WITHOUT_CANONICAL_CONNECTION if nested EPI data are present; otherwise FAILED_NO_TNFR_MULTIPLET	Nested EPI can provide components, but no canonical component-mixing connection or non-commuting generator algebra is derived
thol_remesh_internal_space	OPEN_HISTORY_WITHOUT_CANONICAL_GENERATORS if operator history exists; otherwise FAILED_NO_OPERATOR_INTERNAL_SPACE	THOL/REMESH history does not expose a derived non-commuting generator algebra
cycle_basis_bundle	FAILED_BASIS_DEPENDENT_EXTERNAL_SELECTION when enough cycles exist	Cycle-basis generator selection depends on non-canonical basis/orientation choices

Net verdict: the repository still has a canonical local U(1) gauge sector only. No non-Abelian TNFR gauge sector is promoted by Y3.

Attempt to derive multi-component gauge structure from TNFR-internal data only. Candidate routes:

• multi-channel Ψ multiplets from nested EPI levels (U5);
• operator-induced internal state spaces from THOL/REMESH;
• graph-cycle basis bundles from canonical topology.

Acceptance requires a nodal-equation derivation. If the construction needs external group labels or hand-selected generators, it is non-canonical.

First verdict: OPEN_DERIVABILITY_GAP.

Y4 — Scaling Study

Implementation status (2026-05-31): FINITE_SCALING_EVIDENCE or GAP_COLLAPSE_OBSERVED, depending on the sampled finite family.

Implemented in:

• src/tnfr/yang_mills/scaling.py
• tests/physics/test_yang_mills_scaling.py

The Y4 diagnostic evaluates graph-size surrogates using the node count n and fixed U6 target ratios. For each (topology, ρ_U6) family it records mean gap by size and a finite log-log slope of gap versus n. The slope is a diagnostic of sampled finite behaviour only; it is not a continuum exponent and does not define a thermodynamic limit.

The report classifies finite samples as:

Verdict	Meaning
FINITE_SCALING_EVIDENCE	all sampled finite points are self-adjoint, gauge-invariant, and have positive gap above tolerance
GAP_COLLAPSE_OBSERVED	at least one sampled finite point has gap at or below tolerance
SCALING_FAILED_NON_SELF_ADJOINT	a sampled operator violates Hermiticity
SCALING_FAILED_GAUGE_VARIANCE	a sampled spectrum is not invariant under seeded local U(1) rotation

If Y1–Y3 provide a stable finite diagnostic surface while keeping YMG-4 explicitly open, evaluate Δ(a,L) across graph spacing / size surrogates. This is still not the Clay theorem; it is a TNFR scaling diagnostic.

First verdict: FINITE_SCALING_EVIDENCE for stable finite samples, or GAP_COLLAPSE_OBSERVED for sampled collapse. Both verdicts remain finite-diagnostic only.

Y5 — Closure / Obstruction Theorem

Implementation status (2026-05-31): BRANCH_B_OBSTRUCTION_CLASSIFIED.

Implemented in:

• src/tnfr/yang_mills/closure.py
• tests/physics/test_yang_mills_closure.py

Y5 separates two logically different questions:

1. Finite TNFR-internal result — Y1–Y4 support a finite, self-adjoint, seeded-gauge-invariant U(1) structural gauge diagnostic surface built from Ψ, A_ij, F_C, and Φ_s.
2. Clay-strength result — not closed, because Y3 leaves non-Abelian derivability open and Y4 does not prove a continuum / thermodynamic lower-bound theorem.

Current classification:

Layer	Verdict	Meaning
Finite TNFR layer	A_FINITE_U1_DIAGNOSTIC_SURFACE	existing catalog supports a finite U(1) structural gap diagnostic surface
Clay-strength layer	B_REQUIRES_NEW_CANONICAL_NONABELIAN_DERIVATION	a Clay-strength route requires a new TNFR-native non-Abelian derivation plus a continuum lower-bound theorem
Programme verdict	BRANCH_B_OBSTRUCTION_CLASSIFIED	the obstruction is now localized, not removed

Y5 therefore does not claim the Yang–Mills Millennium Problem is solved. It closes the first TNFR programme pass by identifying the exact boundary: finite U(1) structural diagnostics are available; non-Abelian derivability and continuum lower bounds remain the open requirements.

Classify the programme into one of three branches:

Branch	Meaning
A	finite TNFR structural gap is derivable from existing catalog and U6
B	finite gap requires a new canonical derivation, likely non-Abelian/multiplet
C	no TNFR-internal mass-gap analogue survives canonical constraints

Current Y5 result: Branch B at Clay-strength scope, with Branch A only at the finite U(1) diagnostic layer.

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7. Acceptance Criteria

Any claimed TNFR–Yang–Mills result must satisfy:

1. Nodal derivability — every term traces to ∂EPI/∂t = νf · ΔNFR(t) or canonical tetrad telemetry.
2. Operator discipline — no EPI mutation outside the 13 canonical operators.
3. Grammar compliance — U1–U6 constraints are checked or explicitly scoped.
4. Gauge invariance — reported gap must be invariant under local Ψ → e^{iα}Ψ transformations, or any gauge dependence must be classified as a diagnostic failure.
5. No empirical constants — parameters derive from the nodal dynamics, the structural scale π, or graph-level normalisation.
6. Reproducibility — graph family, seed, spectra, thresholds, and residuals are recorded.
7. Scope honesty — finite TNFR gap results are not called a Clay proof unless non-Abelian existence and continuum scaling are addressed rigorously.

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8. Immediate Next Step

The next research target is Y6 / Branch-B derivation search: attempt to derive a TNFR-native non-Abelian connection and non-commuting generator algebra from the nodal equation, nested EPI structure, and canonical operator histories. If such a derivation cannot be found without external group labels, the Yang–Mills programme should remain paused at Branch B rather than extending finite diagnostics indefinitely.

The programme begins from TNFR's own structural dynamics. External Yang–Mills terminology is used only to name the comparison problem and to define the mass-gap target surface.