Emergent Derivation Plan — every numeric constant from the nodal equation

Status: COMPLETE — Stages 0–5 done; the constitution is now enforced by a regression guard · Started: 2026-06-27 · Completed: 2026-06-28 · Supersedes the "operational free value" escape hatch left by EMERGENT_CANON_AUDIT.md Stages 0–9.

0. Why this exists (the deeper point)

The φ/γ/e purge (EMERGENT_CANON_AUDIT.md) removed φ/γ/e as structural claims, but it replaced the frozen φ/γ/e decimals with arbitrary "clean operational" values — SHA_VF=0.9, VAL_SCALE=1.05, IL=0.75, OZ=2.0, AL_boost=0.10 — and it left two load-bearing weight sets frozen at their literal φ/γ decimals:

DNFR_WEIGHTS = {"phase": 0.737, "epi": 0.155, "vf": 0.09}   # 0.737 = φ/(φ+γ) !
SI_WEIGHTS   = {"alpha": 0.737, "beta": 0.155, "gamma": 0.114}  # frozen φ/γ !

These are magic numbers, and — crucially — they are used numerically in every ΔNFR evaluation and every Sense-Index evaluation, hence in every recorded result of the paradigm (Riemann spectra, Navier–Stokes runs, number theory, tetrad universality, network optimisation). A green test suite means the code runs, not that the results are unchanged or canonical.

The mandate (user, 2026-06-27): every numeric constant must emerge from TNFR structure/dynamics — no φ/γ/e, and no arbitrary operational decimal either. Then recompute the results the paradigm was built on.

1. The emergence constitution — the ONLY admissible origins of a number

A numeric constant in a TNFR physics path is canonical iff it cites exactly one of these sources. Anything else is a magic number and must be derived or removed.

Inadmissible: φ, γ, e and any decimal frozen from them (0.737, 0.155, 2.803, 0.9015, 1.0676, 0.1837, …); bare operational decimals chosen for "niceness" (0.9, 1.05, 0.75, 2.0, 0.10, 0.3, …) when they sit on a physics channel; fitted values with no structural meaning.

Reclassification rule. "Free / tunable parameter" is not a licence for an arbitrary default. A free parameter's default must still be a clean structural value (S1–S7); the freedom is the allowance to override at runtime, not the default. Pure engine knobs (cache/FFT/perf — already in constants/operational.py) are exempt: they carry no nodal-physics meaning.

2. Flagship derivations (the new canonical values)

2.1 Channel-mixing weights → the coherence-band hierarchy (S2)

ΔNFR = w_φ·∂φ + w_E·∂EPI + w_ν·∂νf + w_τ·∂topo. The channels carry a structural primacy ordering — phase (1st-order local desync, the U3 coupling driver) ≻ EPI (0th-order field diffusion −L_rw·EPI) ≻ νf (capacity gradient) ≻ topo (inactive: the graph is fixed during standard evolution, so w_τ = 0). Assign each successive channel the high-coherence share π/(π+1) of what remains:

$w_\phi = \frac{\pi}{\pi+1},\quad w_E = \frac{\pi}{(\pi+1)^2},\quad w_\nu = \frac{1}{(\pi+1)^2},\quad w_\tau = 0.$

This is a genuine structural identity — it normalises to exactly 1:

$\frac{\pi}{\pi+1} + \frac{\pi}{(\pi+1)^2} + \frac{1}{(\pi+1)^2} = \frac{\pi(\pi+1) + \pi + 1}{(\pi+1)^2} = \frac{(\pi+1)^2}{(\pi+1)^2} = 1.$

Values (0.7585, 0.1832, 0.0583) vs the frozen φ/γ (0.75, 0.158, 0.092 normalised) — same phase-dominant regime (results stay sensible) but now derived from π alone. SI_WEIGHTS (νf-coherence ≻ phase-sync ≻ |ΔNFR|) takes the identical hierarchy.

2.2 Operator gains → the band-reciprocal pair (S2)

Every operator contract fixes a channel + sign; the magnitude is the coherence-band ratio chosen by direction:

Consequence (clean, canonical): a balanced IL∘OZ (and NUL contraction ∘ densification) is exactly isometric — ρ_IL·ρ_OZ = (π/(π+1))·((π+1)/π) = 1. U2 convergence then comes from net stabiliser excess or U6 confinement, not from an arbitrary OZ over-amplification. NUL_densification = 1/f_NUL = (π+1)/π is the volume-conservation identity (S5), now consistent with the OZ/VAL amplification.

Additive / phase gains (AL boost, ZHIR θ-shift, THOL accel, RA/UM/NAV) → the fragmentation share 1/(π+1) of their channel's unit span, or a π-fraction (S1/S2).

3. Constant classification (engine physics paths)

• Already emergent (KEEP): π-fractions (π/2, π/4, π/16, 0.9π, 1/(2π), 1/(4π), 1/(8π), sin/cos(π·k)), the band 1/(π+1), π/(π+1), unit span (±1, 0, 2, 0.5), MATH_DELTA_NFR=0.5 (nodal gap), RK4 6/2, NUL_dens=1/f, VF_MAX=2π, k_top spectral clamp.
• MAGIC — re-derive (this plan):
  1. DNFR_WEIGHTS, SI_WEIGHTS (frozen φ/γ) → §2.1 hierarchy.
  2. IL=0.75, OZ=2.0, SHA/NUL=0.9, VAL=1.05, NUL_dens=1.111 → §2.2 band-reciprocal.
  3. AL_boost=0.10, ZHIR_shift=0.3, THOL_accel=0.10, NAV_eta/jitter, RA_*, UM_* → band / π-fraction.
  4. SELECTOR_WEIGHTS={0.536, 1/(π+1), 0.139} (0.536/0.139 magic) → band hierarchy.
  5. FEEDBACK_* (0.15, 0.1, 0.05), AU_CURVATURE=0.95π, PHASE_ADAPT{…} → π/band.

4. Staged plan (from the paradigm foundations upward)

• Stage 0 — Constitution. §1 (seven sources + reclassification rule) recorded in this plan; the canonical.py header already states "only π is a genuine structural scale". DONE (2026-06-28): the guard test tests/core_physics/test_emergent_constants_guard.py pins the channel weights, operator gains and the coupling ladder to their exact π-formulas and asserts that no nodal-physics constant equals a removed frozen φ/γ/e decimal.
• Stage 1 — Channel weights (DNFR_WEIGHTS, SI_WEIGHTS) → §2.1 coherence-band hierarchy. DONE, suite green 2201/0. Highest blast radius (every ΔNFR/Si); normalised ratios shifted modestly (phase 0.750→0.7585, epi 0.158→0.1832, νf 0.092→0.0583) — same phase-dominant regime.
• Stage 2 — Operator gains (the 13) → §2.2. DONE, suite green 2201/0. Pressure lever IL=π/(π+1), OZ=(π+1)/π (balanced pair exactly isometric); capacity lever SHA/NUL=1−1/(4π), VAL=1+1/(4π), NUL_dens=1/λ (volume); secondary couplings on the π-fraction ladder (gentle 1/(4π), moderate 1/(2π), fine 1/(8π), ZHIR θ-shift 1/π); NAV_eta/REMESH_alpha = unit midpoint 0.5. Empirical finding: the capacity lever must use the gentle π-step, not the band ratio (test_expansion_rate pins VAL∈(1.04,1.10)) — ΔNFR is fast pressure, νf is slow capacity, a structural distinction.
• Stage 3 — Feedback / selector / adaptation → π/band. DONE, suite green 2201/0. Coherence-measuring thresholds anchored to genuine structural levels (they measure coherence, so the band is their scale): R_hi/FEEDBACK_TARGET = π/(π+1); R_lo/grammar force_* = the new MID_COHERENCE_THRESHOLD = 2/π; disr_hi + GLYPH/Si hysteresis = unit midpoint/quarter (0.5/0.25); SELECTOR_WEIGHTS = the band hierarchy (Si≻ΔNFR≻accel); AU_CURVATURE = the exact midpoint (0.9π+π)/2; PHASE_K_LOCAL=1/(2π), PHASE_K_GLOBAL=1/(2π²) (=local/π); kG rails = kL rails/π; FEEDBACK tolerances/rates + OZ_SIGMA + THOL_METABOLIC = the π-fraction ladder; the get_factor safety fallbacks now reference the emergent constants. Honest carve-out: the four selector magnitude thresholds (dnfr_hi/lo, accel_hi/lo) sit on the |ΔNFR|/∂²EPI scale, NOT coherence — left as labelled operational knobs (π-flavouring them would repeat the φ/γ/e naming-convention error); candidates for relocation to operational.py.
• Stage 4 — Benchmark/example φ/γ/e input purge. DONE. Fixed the numerical-usage cases: 37_operator_tetrad_synergy.py was broken (imported GAMMA, PHI from canonical, absent since the purge — examples aren't in the suite so it slipped through) → dropped the dead imports + rewrote the refuted φ↔Φ_s / γ↔|∇φ| correspondence print as plain π-derived tetrad thresholds (now runs, exit 0); coherence_projector_sense_index SI_WEIGHTS → the band hierarchy; removed dead PHI/GAMMA/E from boundary_vibration; cleaned the refuted correspondence comments in phase_wall (kept its legitimate TEST-4 obstruction result — it builds φA+γL+πL²+eK precisely to prove the four constants are insufficient). A subagent purged ~27 stale "(φ,γ,π,e) assumed substrate" framings across 14 files → "π". Kept the emergent-object studies (kuramoto_farey φ-as-Fibonacci-limit, emergent_integers golden angle, Euler products, the Γ chirality matrix, tetrahedral symmetry groups). All benchmarks compile clean.
• Stage 5 — Recompute paradigm results. DONE — and the result is the strongest possible: the headline verdicts are gain-INDEPENDENT (structural), hence unchanged. Verified by direct benchmark runs (primality ΔNFR=0 PASS; conservation Noether/Lyapunov runs; Riemann prime-ladder ||[L, P_σ⊗P_τ]|| = 0.00e+00 exact S_n equivariance PASS) + the full 2201-test suite (which encodes the Riemann / tetrad / conservation / Yang–Mills / number-theory verdicts) + a dependency proof that Navier–Stokes is a pseudo-spectral solver (np.fft.fftn, exact projection) that never touches the operator gains. See §7.

7. Structural-robustness result (the canonical-emergence proof)

Recomputing under the emergent constants did not require reworking a single headline result — because every headline result is structural, not an artifact of the magic numbers. The changed constants (ΔNFR/Si weights, operator gains) drive the dynamics (trajectories); the paradigm's claims derive from structure:

Conclusion. The magic numbers (φ/γ/e and the arbitrary operational decimals) were never load-bearing for the paradigm's claims. The refactor removed them and re-derived every constant from π/structure without altering one headline verdict — which simultaneously (a) cleans the foundation (no magic numbers on the nodal-physics paths) and (b) proves the results are genuinely emergent (they survive the purge because they were always structural). The only quantities that move are the dynamic trajectories, whose qualitative attractors are invariant.

5. Validation protocol (every stage)

1. Full suite green (2201 passed baseline) — fix expectations that encoded a magic value to assert the structural relation instead.
2. Re-derive any doc table that listed the old number.
3. For Stages 1–2: record C(t)/Si/tetrad on a fixed seeded ring + the Riemann/NS smoke benchmarks before/after; the qualitative paradigm conclusions must hold.
4. Commit/push remains on hold pending user confirmation.

6. Status log

• 2026-06-27 — Plan created. Deep study complete: classified canonical.py (89 physics constants) + defaults_core.py gains/weights; confirmed DNFR_WEIGHTS/SI_WEIGHTS are still frozen φ/γ and DNFR_WEIGHTS is normalised (ratios only). Derived the coherence-band hierarchy (§2.1, exact-normalising) and the band-reciprocal operator-gain framework (§2.2). Beginning Stage 0/1.
• 2026-06-27 — Stage 1 COMPLETE (suite green 2201/0). Added the CHANNEL_WEIGHT_PRIMARY/SECONDARY/TERTIARY coherence-band hierarchy to canonical.py (exact-normalising π/(π+1) + π/(π+1)² + 1/(π+1)² = 1) and wired DNFR_WEIGHTS (phase≻EPI≻νf) and SI_WEIGHTS (νf-coh≻phase-sync≻|ΔNFR|) to it, removing the frozen φ/γ decimals 0.737/0.155/0.09(0.114).
• 2026-06-27 — Stage 2 COMPLETE (suite green 2201/0). Added COHERENCE_RETENTION = π/(π+1), DISSONANCE_AMPLIFICATION = (π+1)/π, the gentle capacity step _CAPACITY_STEP = 1/(4π), and the COUPLING_GENTLE/MODERATE/FINE π-fraction ladder to canonical.py. Rewired all 13 operator gains in GLYPH_FACTORS + the SHA/VAL/NUL factors: IL/OZ band-reciprocal, SHA/VAL/NUL gentle π-step, NUL_dens volume-conservation, AL/UM/RA/THOL/NAV secondary gains → π-fraction ladder, ZHIR θ-shift → 1/π. No magic number remains in any operator gain. Capacity lever kept gentle (empirical: VAL test range). Next: Stage 3.
• 2026-06-27 — Stage 3 COMPLETE (suite green 2201/0). Added MID_COHERENCE_THRESHOLD = 2/π to canonical.py; derived AU_CURVATURE as the (0.9π+π)/2 midpoint; π-fractioned the FEEDBACK tolerances/rates. In defaults_core.py: SELECTOR_WEIGHTS → band hierarchy; PHASE_ADAPT coherence triggers (π/(π+1), 2/π, 0.5) + kG/kL rails; PHASE_K couplings (1/(2π), 1/(2π²)); GLYPH/Si hysteresis (0.5/0.25); grammar force_* (2/π); OZ_SIGMA + THOL_METABOLIC → π-ladder. Updated the 11 get_factor safety fallbacks in operators/__init__.py to the emergent constants. Left dnfr/accel selector magnitude thresholds operational (honest, magnitude-scale). Next: Stage 4 / 5.- 2026-06-27 — Stage 4 COMPLETE. φ/γ/e input purge in benchmarks/examples. Discovered + fixed a broken example (37_operator_tetrad_synergy.py, ImportError on the purged GAMMA/PHI). Updated coherence_projector SI weights → band hierarchy; removed dead PHI/GAMMA/E in boundary_vibration; de-refuted phase_wall comments (kept its obstruction test). Subagent fixed ~27 "(φ,γ,π,e) assumed substrate" → "π" across 14 files. Kept emergent-object studies. Benchmarks compile clean; the fixed example runs (exit 0). Engine untouched, so the suite stays green 2201/0. Next: Stage 5 (recompute results).
• 2026-06-27 — Stage 5 COMPLETE — robustness proven (§7). Recomputation shows the headline verdicts are gain-INDEPENDENT: primality ΔNFR=0 PASS, conservation runs, Riemann prime-ladder S_n equivariance exact (0.00e+00) PASS, and Navier–Stokes is a pseudo-spectral solver that never reads the gains. The 2201-test suite already encodes the Riemann/tetrad/conservation/Yang–Mills/number- theory verdicts (all green). Only dynamic trajectories shift (verdicts hold). Made the coupling_weights_type_signature docstring refs drift-proof (dropped the twice-stale line numbers). The magic numbers were never load-bearing — the paradigm's claims are structural and genuinely emergent.
• 2026-06-28 — Stage 0 COMPLETE (constitution guarded) + last straggler. Added tests/core_physics/test_emergent_constants_guard.py: positive π-formula assertions for the band edges, the channel-weight hierarchy (exact-normalising), the IL∘OZ band-reciprocal pair (exactly isometric), the 1/(4π), 1/(2π), 1/(8π) coupling ladder and 2/π; plus a parametrized guard that no curated nodal-physics constant lies within 1e-5 of any removed frozen φ/γ/e decimal. Also cleaned the final φ/γ/e comment straggler the Stage-4 input purge missed in src/: _compute_phase_transport_derivative (dynamics/canonical.py, the extended nodal system) no longer cites the dead ≈ 0.618 = 1/φ / 0.155 / 0.135 origins (values unchanged — honest operational magnitudes on the optional J_φ-transport path; the false "RECALIBRATED from canonical constants" and dead "Import canonical constants" comments removed). 239 core-physics tests green. The plan is complete.