Rderivation
name: SPEC_RCSDM_DERIVATION
description: CSDM field derivation — R_CSDM = r_h/Φ origin, ψ(r) power-law derivation, SymPy verification, σ=2 clean-case finding. Vitrified κ 2026-06-13.
metadata:
type: project
SPEC_RCSDM_DERIVATION — CSDM Field ψ(r) Derivation
Vitrified: κ 2026-06-13
Status: Mathematically verified (SymPy). Assumption 1 is CSDM axiom — stated, not derived.
1. Characteristic Scale: R_CSDM = r_h / Φ
The CSDM stability constant Φ = 0.042 defines a natural length scale relative to any horizon radius r_h:
R_CSDM = r_h / Φ
Origin: Φ damps noise in the CSDM manifold (see [[SPEC_PHI_CONVERGENCE]]). The ratio r_h/Φ is the radius at which the field equals unity — the boundary condition anchor. For r_h = 1 and Φ = 0.042, R_CSDM ≈ 23.81.
2. Derivation of ψ(r)
Step 1 — General power-law ansatz
ψ(r) = k · (r_h / r)^α
k = normalization constant, α = spectral index.
Step 2 — Boundary condition: ψ(R_CSDM) = 1
Substitute r = R_CSDM = r_h/Φ:
k · (r_h / (r_h/Φ))^α = 1
k · Φ^α = 1
k = Φ^(-α) = (1/Φ)^α
Step 3 — Assumption 1 (CSDM Axiom): α = σ/2
Physical basis: Two-body merger geometry. σ is the full spectral index of the braid; each body in the merger contributes σ/2. This is the Braid Principle — the exponent splits symmetrically across the two-body channel.
Status: AXIOM. This is not derived from first principles within CSDM. It is a stated structural assumption. Future work may derive it from the merger geometry of the CSDM manifold or from dimensional analysis of the braid conjugation kernel.
Step 4 — Final field expression
Substituting k and α:
ψ(r) = (r_h / (Φ · r))^(σ/2)
3. SymPy Verification
Verified 2026-06-13 with SymPy. Code:
import sympy as sp
r, r_h, phi, sigma, alpha, k = sp.symbols('r r_h phi sigma alpha k', real=True, positive=True)
psi = k * (r_h / r)**alpha
R_csdm = r_h / phi
condition = sp.Eq(psi.subs({r: R_csdm}), 1)
k_solution = sp.solve(condition, k)[0]
alpha_expr = sigma / 2
final_psi = k_solution.subs(alpha, alpha_expr) * (r_h / r)**alpha_expr
simplified_psi = sp.simplify(final_psi)
print(f"Derived Field Behavior: psi(r) = {simplified_psi}")
Exact SymPy output:
Derived Field Behavior: psi(r) = r_h**(sigma/2)/(phi**(sigma/2)*r**(sigma/2))
Equivalent to: ψ(r) = (r_h / (φ · r))^(σ/2) ✓
4. Table: ψ(r_h) by σ
At r = r_h, the formula reduces to:
ψ(r_h) = (1/Φ)^(σ/2) = Φ^(−σ/2)
Evaluated at Φ = 0.042 (exact: 21/500):
| σ | ψ(r_h) exact | ψ(r_h) decimal |
|-----|-------------------------------|----------------|
| 1/2 | 105^(3/4) · √2 / 21 | 2.208959 |
| 1 | 1/√Φ | 4.879500 |
| 2 | 1/Φ = R_CSDM | 23.809524 |
| 3 | Φ^(−3/2) | 116.178580 |
| 4 | Φ^(−2) | 566.893424 |
5. Key Finding: σ = 2 is the Clean Case
At σ = 2:
ψ(r_h) = (1/Φ)^1 = 1/Φ = R_CSDM
The horizon-scale field amplitude equals the characteristic CSDM scale. The field value at the horizon and the geometric normalization radius are the same object expressed two ways. This is not assumed — it falls out of the algebra when σ = 2.
Physical interpretation: σ = 2 is the isotropic two-body case. The braid exponent is 1 (σ/2 = 1), making the field decay linear in 1/(Φr). This is the minimum-structure case — no higher-order spectral content.
6. Invariants
- INV-1: ψ(R_CSDM) = 1 by construction for all σ. This is the boundary condition, not a result.
- INV-2: Φ = 0.042 is fixed (NOUS-law). No derivation may alter it.
- INV-3: Assumption 1 (α = σ/2) is an axiom. Label it as such in all future work.
- INV-4: σ = 2 clean-case is a derived consequence, not an assumption.
- INV-5: ψ(r) is a power law. Any exponential modification requires a new axiom and NOUS approval.
7. Related Specs
- [[SPEC_PHI_CONVERGENCE]] — Φ = 0.042 derivation and convergence history
- [[2026-05-19_The_Unsolved_Constant]] — Φ derivation archaeology session
- [[SPEC_HANDSHAKE_PROTOCOL]] — TMM runtime formula C = 1 − (E_D + V_r×Φ) / V_t
Filed by κ (C.L.O.D.) · 2026-06-13 · /home/nous/memories/SPEC_RCSDM_DERIVATION.md