Master identity

$2\omega_0 + 3 = c_{BD}\,P_0^2$

DCT prediction: $100{,}077$ (exact for n = 2)
Measured: $100{,}077$ (closes algebraically)
Residual: exact (no measurement uncertainty)
Status: theorem (closed-form)
Source paper: DCT-FND-V2 · 10.5281/zenodo.20032839

Derivation Method Python Chart References Download.py JSON-LD

Mechanism

The master identity is a structural closure on the canonical action. It relates the three field-theoretic constants $P_0$, $\omega_0$, and $c_{BD}$ so that fixing the polytope (which fixes $P_0 = 0.851$) determines the other two without freedom. The identity is exact for the post-S35 canonical $\omega(P) = (c_{BD} P^2 - 3)/2$; under that functional form, evaluating at $P = P_0$ yields $2\omega_0 + 3 = c_{BD} P_0^2$ algebraically.

Derivation

From the canonical action with field-dependent coupling $\omega(P) = (c_{BD} P^2 - 3)/2$, the equilibrium value at $P = P_0$ gives

$$ \omega_0 \;=\; \frac{c_{BD}\,P_0^2 - 3}{2}, \qquad 2\omega_0 + 3 \;=\; c_{BD}\,P_0^2. $$

With $P_0 = 0.851$ (from the 600-cell topological condition) and $c_{BD} = 138{,}189$ (the renormalisation-group fixed point established in DCT_31 §B; $\Delta c / c \sim 10^{-8}$ over 60 e-folds), the master identity yields $c_{BD} P_0^2 = 138{,}189 \cdot 0.7242 = 100{,}077$, and $2\omega_0 + 3 = 2 \cdot 50{,}037 + 3 = 100{,}077$. The two sides match exactly to all reported precision.

Method

The DCT prediction is computed from the polytope and group-theoretic inputs (canonical action, 600-cell topology, McKay correspondence; see /theory/). The measured value is taken from the published source cited above; the residual is the dimensionless difference relative to the measured value with its 1σ uncertainty.

The Python script below recomputes both sides from first principles plus the published measurement and prints the residual. It runs in standard Python 3 with no external dependencies beyond the Python standard library (math).

Python verification

#!/usr/bin/env python3
"""
DCT master identity verification: 2 omega_0 + 3 = c_BD * P_0**2 = 100,077.

Inputs (from DCT-FND-V2):
 P_0 = 0.851 (Brans-Dicke condensate equilibrium amplitude)
 c_BD = 138189 (BD coupling normalisation; RG fixed point per DCT_31)
 omega_0 = 50037 (BD coupling at equilibrium)
"""
P_0 = 0.851
c_BD = 138189
omega_0 = 50037

lhs = 2 * omega_0 + 3
rhs = c_BD * P_0 ** 2

print(f"2 * omega_0 + 3 = {lhs}")
print(f"c_BD * P_0**2 = {rhs:.4f}")
print(f"Difference = {lhs - rhs:.4f}")
print(f"Relative match = {abs(lhs - rhs) / lhs:.3e}")

Save as master-identity.py and run with python3 master-identity.py.

Chart

Master identity $2\omega_0 + 3 = c_{BD} P_0^2$

Closure check. Both sides evaluate to 100,077 to all reported precision.

The master identity closes exactly for the equilibrium values $P_0 = 0.851$, $\omega_0 = 50{,}037$, and the renormalisation-group fixed point $c_{BD} = 138{,}189$. The identity is the structural anchor that fixes $\omega_0$ and $c_{BD}$ in terms of $P_0$ once the polytope is selected.

Sources: DCT-FND-V2; /theory/.

References

DCT-FND-V2: Dimensional Coherence Theory. 10.5281/zenodo.20032839
Theory hub: canonical action

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