TNFR Website — Complete Content & Structure Brief

Revised 2026-06-24 · aligned with TNFR v0.0.3.4 (repository state as of 2026-06-24)

• Project: Public website for TNFR (Resonant Fractal Nature Theory)
• Audience: Kaniz and the dev team building the website
• Repo (source of truth): https://github.com/fermga/TNFR-Python-Engine
• PyPI package: https://pypi.org/project/tnfr/
• DOI (Zenodo): 10.5281/zenodo.17602860
• License: MIT
• Current version: 0.0.3.4 (released 2026-06-17)

This revision supersedes the 2026-06-21 brief. It is content-identical in structure, but every fact, equation, count, and status has been re-checked against the current repository. Section 0.1 lists exactly what changed and why, so nothing already designed needs to be thrown away — only corrected.

0. How to read this document

This brief is the single source of truth for the website content. It includes:

1. The non-negotiable editorial rules (tone, terminology, what we never say).
2. The complete sitemap with the purpose of every page.
3. Page-by-page content blocks with the exact text, tables, equations and assets to use.
4. The visual identity guidelines (colors, typography, components).
5. The assets inventory (what diagrams to design and where the text comes from in the repo).

The website is a presentation of TNFR and of the material already published in the repository. It is not an applications showcase. If anything in this document is unclear, ask before inventing. Do not paraphrase the scientific content — copy the exact wording provided here and in the linked source files. TNFR has a strict canonical terminology that must be preserved.

Hosting note: the production domain tnfr.info is already contracted with a separate hosting provider (IONOS). Deployment to the production environment is therefore out of scope for Kaniz. The deliverable is the website source code (frontend + backend) ready to be deployed by the owner.

0.1 What changed since the 2026-06-21 brief (delta for Kaniz)

The repository moved forward between the two briefs. The website content must reflect the current state. Concrete corrections (old → corrected):

1. Home, "Verify" card. "~2,195 tests" → 1,599 tests (README).
2. /learn/tutorials. A flat list "01_hello_world.py … 10_simplified_sdk_showcase.py" → 150 examples across 10 thematic subfolders (examples/01_foundations … examples/10_applications).
3. /theory/tetrad equations. The |∇φ| formula was a duplicate of the K_φ formula → |∇φ| is the 1st-order mean of wrapped neighbour differences, while K_φ is the 2nd-order deviation from the neighbour mean (= L_rw·φ). See § 3.2.2.
4. /theory/grammar U6 and /theory/correspondence. "ΔΦ_s < φ ≈ 1.618 (golden ratio)", "adopted drift bound" → ΔΦ_s < π/2 ≈ 1.571 is the π-derived drift bound (half phase-wrap). Only π is a genuine structural scale; φ, γ, e are not structural scales. Do not present 1.618 as a derived golden-ratio constant, and do not reuse the φ symbol (it denotes phase).
5. /theory/operators table. ZHIR grammar role "U4a, U4b"; an "effect on |ΔNFR| (↑/↓/=)" column → ZHIR is a destabilizer, so its role now includes U2, and the imprecise |ΔNFR| arrow is replaced by the canonical primary nodal channel (EPI / νf / θ / ΔNFR), which is how the engine documents operators. See § 3.2.3.
6. /research open programs. 3 programs (Riemann, Navier–Stokes, Yang–Mills) → 6 programs (add P vs NP, BSD, Hodge), all currently open (Riemann nodal-pulse foundation; Navier–Stokes two-face reading). See § 3.5.6.
7. /research theory documents. 12 files → add the new canonical docs and the 3 new research notes. See § 3.5.3.
8. /software/sdk. TetradSnapshot + ConservationReport → also add SymplecticReport (net.symplectic_substrate()), plus evolve_grammar_aware, telemetry(), audit_operators(), and nfr(). See § 3.4.3.
9. /theory/nodal-equation. Structural triad only → add a short note on the , now a centralized canonical concept. See § 3.2.1.

Everything else in the original brief (sitemap, visual identity, accessibility, backend/CMS scope) remains valid.

1. Editorial rules (mandatory)

These rules are derived from the project's internal AGENTS.md (the canonical specification of the theory). Violations make the website unusable.

1.1 Tone

• Academic and engineering-oriented, not marketing.
• Every claim must be anchored to either the nodal equation, an operator contract, an experimentally validated result in the repository, or recorded telemetry. Qualitative claims with no math, code or data behind them are forbidden.
• No metaphysical, mystical, spiritual, cosmological, or consciousness-related statements. TNFR is a mathematical framework for coherent patterns on graph-coupled networks. It is not a theory of everything, a philosophy, a worldview, a self-help paradigm, or a description of consciousness.
• No slogans, no superlatives, no "revolutionary", no "unifies everything", no "paradigm shift". Plain factual descriptions only.
• No anthropomorphism of the engine ("the system understands", "knows", "feels").

1.2 What we never say on the website

• Avoid: "TNFR proves the Riemann Hypothesis". Use instead: "TNFR provides a computational research framework related to the Riemann Hypothesis; the classical hypothesis itself remains open."
• Avoid: "TNFR solves Navier–Stokes". Use instead: "TNFR offers structural diagnostics for the 3D Navier–Stokes problem; global regularity remains open."
• Avoid: "TNFR explains consciousness / the universe / spirituality". Never mentioned.
• Avoid: "Quantum-classical unification". Use instead: "Discrete-mode and smooth-trajectory regimes of the same nodal equation, demonstrated within the framework."
• Avoid: "TNFR replaces / supersedes physics / mathematics". Use instead: "TNFR is a specific modeling framework with a defined scope."
• Avoid: presenting 1.618, γ, e, or φ as derived structural constants. Only π is a genuine structural scale (the phase-wrap bound); the Φ_s confinement bound is π-derived (π/2 drift, π/4 per-node) and ξ_C is set by the spectral gap. φ, γ, e are not structural scales — say so.

1.3 Canonical terminology (English, must be exact)

Always use these exact names — never translate, abbreviate, or invent synonyms:

Terminology note. The four fields above form the structural-field tetrad — the minimal complete description of a network state. The one genuine structural scale is π, which bounds the whole phase sector (both |∇φ| and K_φ). The constants γ, e, and φ are not structural scales and no longer appear in the engine; everything other than π is derived from the nodal dynamics / spectral gap or is a free operational parameter. In particular, the Φ_s confinement bound (per-node |Φ_s| < π/4 ≈ 0.785, drift ΔΦ_s < π/2 ≈ 1.571) is π-derived (quarter / half phase-wrap). Always refer to the "structural-field tetrad" (the four fields), not to "four constants".

1.4 Language policy

• Website language: English for all pages.
• No mixed-language paragraphs.

1.5 Math rendering

• All equations must render with KaTeX (not MathJax — faster, lighter).
• Inline math: $...$. Block math: $$...$$.
• Display Greek letters using LaTeX commands (\varphi, \nu_f, \xi_C), never Unicode in equations.

2. Sitemap (final structure)

/
|-- /theory                <- Intermediate-level technical core
|   |-- /theory/nodal-equation
|   |-- /theory/tetrad         (Phi_s, |grad-phi|, K_phi, xi_C)
|   |-- /theory/operators      (13 canonical operators)
|   |-- /theory/grammar        (U1-U6)
|   `-- /theory/correspondence (characteristic field scales)
|-- /learn
|   |-- /learn/glossary
|   `-- /learn/tutorials       (linked example scripts)
|-- /software
|   |-- /software/install
|   |-- /software/quickstart
|   `-- /software/sdk
|-- /research              (DOI, OEIS, theory files, reports, open programs)
|-- /about                 (project history, editorial policy, license)
`-- /contact

Header navigation: Theory · Learn · Software · Research Footer navigation: About · Contact · GitHub · PyPI · DOI · License · Citation

3. Page-by-page content

3.1 Home (/)

Purpose: in 30 seconds, the visitor must understand what TNFR is, the formal objects it defines, and what they can do next.

Layout: one full-screen hero + three short content blocks below + footer.

Hero block

• Title (H1): TNFR — Resonant Fractal Nature Theory
• Subtitle: A mathematical framework for modeling coherent patterns on graph-coupled networks through the nodal equation ∂EPI/∂t = νf · ΔNFR(t).
• Three CTAs (horizontal buttons): Read the theory (→ /theory) · Install the SDK (→ /software/install) · Browse the research (→ /research)
• Background visual: animated SVG/Canvas of a 20-node graph evolving smoothly (subtle, low-saturation). Reference behavior: examples/01_foundations/ network-formation scripts in the repository.

Block 1 — "What TNFR defines"

Exact text (do not paraphrase):

TNFR is a framework for describing the dynamics of coherent patterns on graph-coupled networks. It is defined by four formal components:

1. A nodal equation governing the evolution of every node: $\frac{\partial \mathrm{EPI}}{\partial t} = \nu_f \cdot \Delta\mathrm{NFR}(t)$

Render this as a compact table below the text:

Block 2 — "What you can do with it"

Render as three short side-by-side cards:

Block 3 — Quick start code

Render as a syntax-highlighted Python block (mirrors README.md verbatim):

from tnfr.sdk import TNFR

# Create, connect, evolve
net = TNFR.create(20).ring().evolve(5)
print(net.results().summary())
# -> C=0.987, Si=0.912, N=20, E=20, rho=0.105

# Inspect the four structural fields
tetrad = net.tetrad()
print(tetrad.summary())
# -> Phi_s=0.0312, |grad_phi|=0.0841, |K_phi|=0.1523, xi_C=2.3147 (N=20)

Below the code block, three small badges:

• Python 3.10+
• MIT License
• DOI 10.5281/zenodo.17602860

3.2 /theory — Intermediate technical core

Purpose: this is the most important section of the website. A reader who finishes these five pages must be able to explain TNFR to another scientist without ever opening the source code.

Each sub-page has the same structure:

1. Lead paragraph (3–4 sentences, what this concept is).
2. Formal block (definitions, equations, tables).
3. Diagram or worked example.
4. Why it matters (one paragraph linking to the rest of the framework).
5. References (links to the canonical source in the repo).

3.2.1 /theory/nodal-equation

Lead:

Every node in a TNFR network evolves under a single first-order differential equation. From this equation the 13 operators, the grammar rules U1–U6, and the structural tetrad are derived.

Formal block:

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

Structural triad (each node carries three irreducible attributes):

The fractal-resonant node (NFR): the node carrying this triad is a fractal-resonant node — canonically, a region of structural coherence coupled to a network. It is multiscalar (an NFR can nest other NFRs), autopoietic (emerges by local reorganization), relational (exists only by coupling), and temporal (persists while it reorganizes). Its internal nodal topology is read from the emergent structural-potential geometry as radial (one central nucleus), annular (passive center, peripheral ring), or multinodal (several centers).

Stability criterion: integrating the nodal equation,

$\mathrm{EPI}(t_f) = \mathrm{EPI}(t_0) + \int_{t_0}^{t_f} \nu_f(\tau)\,\Delta\mathrm{NFR}(\tau)\,d\tau$

Coherence is preserved only when the integral converges:

$\int_{t_0}^{t_f} \nu_f(\tau)\,\Delta\mathrm{NFR}(\tau)\,d\tau < \infty$

This convergence requirement is the formal basis of grammar rule U2 (Convergence and Boundedness).

Diagram (to be designed by Kaniz): a clean schematic showing a single node with its three irreducible attributes (EPI, νf, φ) and an arrow labelled ∂EPI/∂t = νf · ΔNFR(t) pointing to its updated state at time t + dt.

References:

• Source: theory/FUNDAMENTAL_THEORY.md § 2; NFR definition in AGENTS.md § 2
• Implementation: src/tnfr/operators/nodal_equation.py; src/tnfr/physics/fields.py (classify_nodal_topology)

3.2.2 /theory/tetrad

Lead:

The state of any TNFR network is characterized by four scalar fields. Each one answers a different structural question, and together they form the minimal complete description of a coherent system on a graph, as derived in theory/MINIMAL_STRUCTURAL_DEGREES.md.

The four structural questions (render as a 4-row table):

Equations (four KaTeX blocks — corrected to the canonical definitions in docs/STRUCTURAL_FIELDS_TETRAD.md):

$\Phi_s(i) = \sum_{j \neq i} \frac{\Delta\mathrm{NFR}_j}{d(i,j)^2}$

$|\nabla\varphi|(i) = \operatorname*{mean}_{j \in \mathcal{N}(i)} \big| \operatorname{wrap}(\varphi_j - \varphi_i) \big|$

$K_\varphi(i) = \varphi_i - \frac{1}{\deg(i)} \sum_{j \in \mathcal{N}(i)} \varphi_j \qquad (= L_{\mathrm{rw}}\,\varphi \text{ in the smooth limit})$

$C(r) \sim \exp(-r / \xi_C)$

Note for Kaniz (correction): in the previous brief |∇φ| and K_φ were printed with the same formula. They are different orders of the derivative tower: |∇φ| is the mean of the absolute wrapped neighbour differences (1st order); K_φ is the signed deviation from the neighbour mean (2nd order, the discrete Laplacian applied to phase).

Canonical thresholds table:

Diagram (to be designed by Kaniz): a tetrahedron whose four vertices/edges carry the four structural fields (Φ_s, |∇φ|, K_φ, ξ_C) and their order (0th / 1st / 2nd / non-local). π may be shown as the phase scale on the |∇φ| and K_φ edges. Static SVG; rotatable Three.js version optional.

Why it matters:

The four classes — global aggregation, first derivative, second derivative, and correlation range — exhaust the independent structural information available from a scalar phase field coupled to a scalar source on a graph. The minimality argument is given in detail in theory/MINIMAL_STRUCTURAL_DEGREES.md.

References:

• Source: theory/MINIMAL_STRUCTURAL_DEGREES.md, docs/STRUCTURAL_FIELDS_TETRAD.md
• Implementation: src/tnfr/physics/fields.py

3.2.3 /theory/operators

Lead:

All structural changes in TNFR occur through exactly 13 canonical operators. Direct mutation of EPI, νf, or φ outside this operator algebra is not permitted by the framework. The constraint is derived from the nodal equation, not from a coding convention. Each operator acts on exactly one nodal channel — the form EPI, the capacity νf, the phase θ, or the pressure ΔNFR — at node or network scale.

The 13 operators (render as a responsive card grid, 3 columns desktop / 1 column mobile). For each operator, the card must show: operator code (large), English name, one-line description, primary nodal channel (EPI / νf / θ / ΔNFR), and grammar role tag.

Scale note: REMESH (Recursivity) is the only operator that acts at NETWORK scale (it implements operational fractality, grammar U5); the other twelve act at NODE scale.

Correction vs the previous brief: ZHIR is a destabilizer, so its grammar role now includes U2. The earlier "effect on |ΔNFR| (↑/↓/=)" column has been replaced by the canonical primary nodal channel, which is how the engine itself classifies operators (src/tnfr/operators/operator_contracts.py).

Canonical classification (used to group operator cards by functional class):

• Generators (U1a): AL, NAV, REMESH
• Closures (U1b): SHA, NAV, REMESH, OZ
• Stabilizers (U2): IL, THOL
• Destabilizers (U2): OZ, ZHIR, VAL
• Coupling (U3): UM, RA
• Transformers (U4b): ZHIR, THOL

Composition block (below the card grid):

Operators compose into named fragments (macros) that implement typical workflows. A fragment is not a standalone valid word — it becomes valid by adding grammar glue (a U1a generator prefix and a U1b closure suffix):

Diagram (to be designed by Kaniz): a single visual catalogue of the 13 operator codes grouped by functional class (Generators / Stabilizers / Destabilizers / Coupling / Transformers / Closure), using a consistent iconographic style.

References:

• Source: theory/STRUCTURAL_OPERATORS.md
• Implementation: src/tnfr/operators/definitions.py, src/tnfr/operators/operator_contracts.py

3.2.4 /theory/grammar

Lead:

Operator sequences must satisfy six grammar rules (U1–U6). Each rule is derived from a specific property of the nodal equation. Sequences that violate them produce unbounded or fragmented dynamics within the framework.

The six rules (render as an accordion or expandable list — one section per rule):

U1 — Structural initiation and closure

• U1a: If EPI = 0, the sequence must start with a generator (AL, NAV, or REMESH). Formal basis: the nodal equation is undefined at EPI = 0.
• U1b: Every sequence must end with a closure operator (SHA, NAV, REMESH, or OZ). Formal basis: sequences need a defined endpoint that leaves the system in a coherent attractor.

U2 — Convergence and boundedness

• If the sequence contains a destabilizer (OZ, ZHIR, VAL), it must also contain a stabilizer (IL, THOL).
• Formal basis: the integral $\int \nu_f \cdot \Delta\mathrm{NFR}\, dt$ must converge; without negative feedback it diverges and coherence is lost.

U3 — Resonant coupling

• Coupling operators (UM, RA) require phase compatibility: |φᵢ − φⱼ| ≤ Δφ_max.
• Formal basis: antiphase produces destructive interference.

U4 — Bifurcation dynamics

• U4a: Triggers (OZ, ZHIR) need handlers (THOL, IL).
• U4b: Transformers (ZHIR, THOL) need a recent destabilizer in context. ZHIR additionally requires a prior IL (stable base).

U5 — Multi-scale coherence

• Nested EPIs (hierarchical structures) require stabilizers at every scale.
• Formal basis: parent coherence depends on the aggregated child reorganization (C_parent ≥ α · Σ C_child).

U6 — Structural potential confinement

• Telemetry-based safety check: monitor ΔΦ_s < π/2 ≈ 1.571, where Φ_s(i) = Σ_{j≠i} ΔNFR_j / d(i,j)². This is a read-only check, not a sequence constraint.
• Formal basis: the emergent field Φ_s must remain bounded for coherence to survive. π/2 ≈ 1.571 is the π-derived drift bound (half phase-wrap, tied to the one genuine structural scale; not a golden-ratio constant).

Example block (show side by side, valid vs invalid):

• Valid sequence: [AL, UM, IL, OZ, IL, SHA] — starts with a generator (U1a OK), contains stabilizers for the destabilizer (U2 OK), ends with closure (U1b OK).
• Invalid sequence: [OZ, VAL, OZ] — starts with a destabilizer with no prior generator (U1a FAIL), no stabilizer present (U2 FAIL), ends in OZ (closure OK but the sequence is still invalid on U1a/U2).

Diagram (to be designed by Kaniz): a state-machine-style diagram showing the six rules with arrows between operator classes (generators → stabilizers → closure, etc.).

References:

• Source: theory/UNIFIED_GRAMMAR_RULES.md
• Implementation: src/tnfr/operators/grammar.py, src/tnfr/operators/grammar_canon.py

3.2.5 /theory/correspondence

Lead:

Each structural field has a characteristic scale that sets or bounds its behavior. The two phase fields, |∇φ| and K_φ, share a single geometric scale, π (the phase-wrap bound). The potential field Φ_s has an empirical confinement bound, and the coherence length ξ_C is set by the network's spectral gap λ₂.

Field scales (render as four cards):

Closing paragraph:

π is the one genuine structural scale of the tetrad: it scales the whole phase sector, bounding both |∇φ| and K_φ. The Φ_s confinement bound is π-derived (π/4 per-node, π/2 drift), and ξ_C is set by the spectral gap λ₂. The constants γ, e, and φ are not structural scales and no longer appear in the engine; everything other than π is derived from the nodal dynamics or is a free operational parameter. The full map of constants and bounds is documented in src/tnfr/constants/canonical.py.

References:

• Source: theory/FUNDAMENTAL_THEORY.md § 4
• Implementation: src/tnfr/constants/canonical.py

3.3 /learn

3.3.1 /learn/glossary

• Render the full content of theory/GLOSSARY.md with a top search box.
• Each glossary entry must be linkable by anchor (e.g. /learn/glossary#EPI).
• Bonus (if budget allows): tooltips on hover for any canonical term used anywhere on the website.

3.3.2 /learn/tutorials

• List the example scripts from examples/. The current repository ships 162 examples organized in 10 thematic subfolders — present them grouped by folder, each entry with title, one-line description, and a "View on GitHub" link. Use examples/README.md as the index/source.
  • 01_foundations — nodal equation, operators, network formation
  • 02_physics_regimes — transport, diffusion, discrete-mode/smooth-trajectory
  • 03_riemann_zeta — TNFR–Riemann ζ track
  • 04_riemann_L_twisted — χ-twisted L-function track
  • 05_type_hygiene — catalog type-hygiene programme
  • 06_navier_stokes — K_φ cascade, Taylor–Green
  • 07_number_theory — primality, cyclotomy, prime families
  • 08_emergent_geometry — symplectic substrate, conservation, grammar geometry
  • 09_millennium — Millennium-problem reformulations
  • 10_applications — applied/SDK showcases
• Optional (out of initial scope): embed each example as a runnable Pyodide notebook.

3.4 /software

3.4.1 /software/install

pip install tnfr                       # stable release
pip install -e ".[dev-minimal]"        # development
pip install -e ".[test-all]"           # full test suite
pip install -e ".[compute-jax]"        # JAX backend
pip install -e ".[compute-torch]"      # PyTorch backend

Requirements: Python 3.10+, Linux/macOS/Windows.

3.4.2 /software/quickstart

Mirror the README.md "Quick Start" block (already in this brief, § 3.1, Block 3).

3.4.3 /software/sdk

Render the dataclass reference for the Simple SDK (src/tnfr/sdk/simple.py):

Builder / evolution

• TNFR.create(n), .ring(), .evolve(steps)
• .evolve_grammar_aware(steps) — proactive U1–U6 enforcement during evolution
• net.results().summary() → C, Si, N, E, rho
• net.telemetry() → C(t), Si, phase_sync, tetrad
• net.audit_operators() → 13/13 operator-contract audit
• net.nfr() → whole-NFR read-out (radial / annular / multinodal topology)

Reports (dataclasses)

• TetradSnapshot: fields phi_s, grad_phi, k_phi, xi_c, j_phi, j_dnfr; methods is_safe(), summary()
• ConservationReport: fields noether_charge, energy, lyapunov_stable, lyapunov_derivative, conservation_quality; method summary()
• SymplecticReport: fields phase_space_dimension, hamiltonian, background_potential, liouville_divergence, is_valid_manifold; method summary()
• TNFR.analyze(net) → comprehensive dict (coherence, tetrad, conservation, tensor_invariants, emergent_fields, integrity, features)

Reference: src/tnfr/sdk/simple.py

3.5 /research

Purpose: a single hub that catalogues everything that has been published as part of the TNFR project — citation metadata, theory documents, generated reports, companion labs, and the explicit status of open research programs.

The page should be organized as the six numbered sub-sections below, separated by visible dividers. Each sub-section gets its own anchor (/research#citation, /research#theory, etc.).

3.5.1 Citation

Citation block (BibTeX, render in a code block with a copy-to-clipboard button):

@software{tnfr_python_engine,
  author  = {Martinez Gamo, F. F.},
  orcid   = {0009-0007-6116-0613},
  title   = {TNFR-Python-Engine: Resonant Fractal Nature Theory Implementation},
  year    = {2026},
  version = {0.0.3.4},
  doi     = {10.5281/zenodo.17602860},
  url     = {https://github.com/fermga/TNFR-Python-Engine},
  license = {MIT}
}

DOI: 10.5281/zenodo.17602860 → https://doi.org/10.5281/zenodo.17602860 · License: MIT.

Important note for Kaniz: do not import the abstract field from the repository's CITATION.cff verbatim. That field still contains "paradigm shift" phrasing forbidden by § 1.1. When a short project description is needed (Open Graph metadata, search-engine snippet, page <meta> description), use the wording from § 3.1 Block 1 of this brief instead.

3.5.2 OEIS sequences

No TNFR-original OEIS sequences are registered yet — keep this sub-section hidden until official identifiers are provided. (Note: some examples in examples/07_number_theory/ reproduce known OEIS sequences such as A005384 (Sophie Germain primes) and A074816 as validation; these are illustrative checks, not new submissions, and should not be presented as registered TNFR sequences.)

3.5.3 Theory documents

Linked list of the canonical theory files in the repository. Each item: title, one-line summary, link to GitHub.

Core canon:

• theory/FUNDAMENTAL_THEORY.md — nodal equation, structural-field tetrad, field scales.
• theory/STRUCTURAL_OPERATORS.md — the 13 canonical operators, contracts, composition.
• theory/UNIFIED_GRAMMAR_RULES.md — rules U1–U6 with full derivations.
• theory/MINIMAL_STRUCTURAL_DEGREES.md — minimality proof of the tetrad.
• theory/STRUCTURAL_CONSERVATION_THEOREM.md — Noether-like conservation derivation.
• theory/TNFR_VARIATIONAL_PRINCIPLE.md — Lagrangian/Hamiltonian formulation.
• theory/GLOSSARY.md — canonical terminology.

Emergent geometry & extended structure (new since the previous brief):

• theory/EMERGENT_ONTOLOGY.md — how patterns/objects emerge from the dynamics.
• theory/EXTENDED_FIELDS_AND_DERIVED_QUANTITIES.md — the six downstream emergent fields.
• theory/GAUGE_SYMMETRY_AND_UNIFICATION.md — U(1)/U(2) gauge & polarization structure.
• theory/PHYSICAL_REGIME_CORRESPONDENCES.md — discrete-mode vs smooth-trajectory regimes.
• theory/MATHEMATICAL_DYNAMICS_BASIS.md — formal dynamics basis.
• theory/STRUCTURAL_STABILITY_AND_DYNAMICS.md — stability analysis.
• theory/REMESH_INFINITY_DERIVATION.md — N15 REMESH-∞ closure (catalog-completeness theorem).

Research programs:

• theory/TNFR_NUMBER_THEORY.md — primality and factorization in the TNFR formulation.
• theory/TNFR_RIEMANN_RESEARCH_NOTES.md — TNFR–Riemann program.
• theory/TNFR_NAVIER_STOKES_RESEARCH_NOTES.md — TNFR–Navier–Stokes program.
• theory/TNFR_YANG_MILLS_RESEARCH_NOTES.md — TNFR–Yang–Mills program.
• theory/TNFR_P_VS_NP_RESEARCH_NOTES.md — TNFR–P vs NP program.
• theory/TNFR_BSD_RESEARCH_NOTES.md — TNFR–Birch–Swinnerton-Dyer program.
• theory/TNFR_HODGE_RESEARCH_NOTES.md — TNFR–Hodge program.

3.5.4 Generated reports

The repository's build tasks generate several HTML/PNG reports from the engine. Embed each as a card with: title, one-line description, generation command (./make.cmd <task>), link to the generated artefact. (These tasks are defined in .vscode/tasks.json and run through ./make.cmd.)

• Atom Atlas — atomic-scale TNFR signatures. Task: ./make.cmd report-particle-atlas-u6.
• Molecule Atlas — molecular configurations under phase-coupled networks. Task: ./make.cmd molecule-atlas-script.
• Periodic Table Atlas — periodic-table-scale TNFR diagnostics. Task: ./make.cmd report-periodic-table-classic.
• Operator Completeness — exhaustive operator-coverage report. Task: ./make.cmd report-operator-completeness.
• Interaction Sequences — illustrative operator sequences. Task: ./make.cmd report-interaction-sequences.
• Emergent Particles — emergent-pattern detection report. Task: ./make.cmd report-emergent-particles.
• Fundamental Particles Atlas — particle-scale TNFR diagnostics. Task: ./make.cmd report-fundamental-particles.

These reports must be treated as illustrative outputs of the framework, not as claims about chemistry or physics beyond what each report explicitly demonstrates. Each card must show the originating ./make.cmd task so any reader can regenerate it.

3.5.5 Companion projects

• primality-test/ — primality experiments under the TNFR formulation (primality as ΔNFR = 0).
• factorization-lab/ — factorization experiments based on spectral decomposition on prime-path graphs.

Render as two cards with a one-line description and a "View on GitHub" link.

3.5.6 Open research programs (mandatory honesty section)

This sub-section must explicitly state the open status of each program. There are now six programs. Use the wording below.

TNFR–Riemann program

The TNFR–Riemann program is a computational research framework relating discrete prime-path graph operators to the Riemann Hypothesis. The critical-parameter convergence σ_c → 1/2 has been numerically verified within the framework. The classical Riemann Hypothesis itself remains open. The work in this repository contributes structural diagnostics and machinery, not a proof of RH.

Current status: milestones P1–P49 implemented; the full ζ↔L attack surface is shipped. The bridge to RH is the open conjecture T-HP (gap G4), currently paused at the oscillatory residue $S(T) = (1/\pi)\,\arg\zeta(\tfrac{1}{2}+iT)$. Link:

TNFR–Navier–Stokes program

Structural diagnostics for the 3D Navier–Stokes problem via the K_φ cascade and Taylor–Green vortex experiments. Milestones N1–N17 implemented; the NS-G5 gap is closed at the discrete-operator level. Global regularity of 3D Navier–Stokes (the continuum / Clay problem, gaps NS-G1..G4) remains open. TNFR provides measurement and diagnostic infrastructure, not a resolution. Link: theory/TNFR_NAVIER_STOKES_RESEARCH_NOTES.md

TNFR–Yang–Mills program

Exploratory structural-gap diagnostics related to the Yang–Mills mass-gap problem (Y1–Y5; finite U(1) diagnostics). The non-Abelian mass gap remains open (Branch B). No claims of resolution. Link: theory/TNFR_YANG_MILLS_RESEARCH_NOTES.md

TNFR–P vs NP program

Structural synthesis-vs-verification programme (PNP-1): coherence verification is O(|E|), whereas coherent synthesis exhibits trapping. The worst-case separation remains open (Branch B). Not a proof. Link: theory/TNFR_P_VS_NP_RESEARCH_NOTES.md

TNFR–BSD program

Structural-pressure programme for Birch–Swinnerton-Dyer (BSD-1): rank separation via structural-pressure accumulation. The link from rank to order of vanishing remains open (Branch B). Not a proof. Link: theory/TNFR_BSD_RESEARCH_NOTES.md

TNFR–Hodge program

Discrete cochain programme (HC-1): the discrete Hodge decomposition equals homology exactly (Eckmann). The (p,p) bigrading and algebraicity are structurally blind in the current formulation (Branch B3-leaning, a strong negative result). Not a proof. Link: theory/TNFR_HODGE_RESEARCH_NOTES.md

Closing status table (render at the end of this sub-section):

3.6 /about

• Brief project history (one paragraph — will be provided before launch).
• Editorial policy: summarized version of section 1 of this brief.
• License: MIT.

3.7 /contact

• Email contact form (with reCAPTCHA or hCaptcha).
• Direct links: GitHub Issues, GitHub Discussions.
• Note: "For research collaborations, please include affiliation and a short description of the proposed work."

4. Visual identity

4.1 Color palette

Sober, scientific, low-saturation. Suggested:

• Primary (deep blue): #0B2545
• Accent (resonant amber): #E8A33D
• Neutral background: #FAFAF7
• Text: #1A1A1A
• Muted: #6B7280
• Success / coherence: #1F7A4D
• Warning / dissonance: #C45A3B

4.2 Typography

• Headings: a modern serif (e.g. Source Serif Pro, Lora, or Newsreader).
• Body: a high-readability sans-serif (e.g. Inter, IBM Plex Sans).
• Code / monospace: JetBrains Mono or IBM Plex Mono.
• Math: KaTeX default (Computer Modern).

4.3 UI components (minimum set)

• Navigation bar (sticky, with logo + 4 main sections + GitHub icon).
• Footer (4 columns: About, Resources, Community, Legal).
• Equation block (KaTeX rendered, with copy-to-clipboard).
• Code block (syntax highlighted, with copy-to-clipboard).
• Reference block (gray background, "Sources" header, bulleted links).
• Operator card (used in /theory/operators).
• Accordion (used in /theory/grammar).
• Searchable glossary entries.
• Report card (used in /research#reports).
• DOI / version / license badges.

4.4 Responsiveness

• Mobile-first.
• Breakpoints: 640 / 768 / 1024 / 1280 px.
• All tables must collapse gracefully on mobile (consider horizontal scroll for the larger ones).
• Equations must scroll horizontally on narrow viewports rather than overflow.

4.5 Accessibility (mandatory)

• WCAG 2.1 AA contrast ratios.
• All figures must have descriptive alt text (provided in section 5.1).
• Keyboard-navigable accordion and search components.
• KaTeX accessible output enabled.

5. Assets inventory

5.1 Diagrams to be designed by Kaniz

All diagrams are part of the design scope. No external image library is supplied — Kaniz must produce them in a consistent visual style (line art, two-color palette derived from § 4.1).

5.2 Text sources (Markdown, from the repo)

5.3 External links (always opened in a new tab)

• GitHub: https://github.com/fermga/TNFR-Python-Engine
• PyPI: https://pypi.org/project/tnfr/
• DOI: https://doi.org/10.5281/zenodo.17602860

6. Backend & CMS scope (database integration)

Since the proposal includes a Next.js / Express backend with MongoDB (running on Node.js), the minimum useful schema is:

6.1 Collections

• posts: published research updates and changelog entries. Fields: slug, title, body_mdx, published_at, tags[], summary.
• references: bibliographic entries for /research. Fields: id, type (paper/zenodo/oeis/software), title, authors[], year, url, doi, bibtex.
• glossary_overrides: optional manual overrides for glossary tooltips. Fields: term, short_definition, link.

6.2 Admin

A protected /admin route (basic auth or simple JWT) for:

• Creating / editing / publishing posts.
• Adding / editing references.
• Triggering a re-fetch of glossary terms from GitHub (the canonical source).

6.3 Out of scope (keep it simple)

• Out: user accounts, comments, forums (use GitHub Discussions instead).
• Out: newsletter signup (can be added later as a separate feature).
• Out: Google Analytics or Facebook Pixel. Use privacy-respecting analytics if any (Plausible or Umami suggested).
• Out: production hosting and DNS configuration. The domain tnfr.info is already contracted with IONOS and will be deployed by the owner; the deliverable from Kaniz is the source code (frontend + backend) ready to be deployed.

End of brief.