Description

Abstract (English)

DESCRIPTION 

Paper 19 performs parameter reduction for the holonomy–occupation curvature response derived in Paper 18.

1. Primitive Scales and Closure Axiom

Finite Reversible Closure assumes:

• Primitive spatial and temporal steps (lp, tp).

• Invariant causal slope c = lp / tp.

• Finite-dimensional local Hilbert space.

• Finite-depth local unitary update per tick U.

Primitive Closure Axiom:

The relational sector of U contains a non-zero minimal eigenphase that cannot be removed by any local gauge transformation.

Define:

phi_min as the smallest non-zero relational eigenphase.

Define minimal action-per-tick:

S0 = phi_min times hbar (or treated directly as primitive action scale).

Define energy per primitive relational excitation:

E0 = S0 / tp.

1. Occupation Density and Energy Mapping

From Paper 15, the composite sector has a gapped infrared dispersion.

Define coarse occupation density:

epsilon(x) = block-averaged Psi dagger Psi.

Energy density maps as:

rho_E(x) = E0 times epsilon(x).

Thus energy density derives directly from relational update structure.

1. Response Coefficients from Update Dynamics

From Paper 18, curvature-loading observable nu depends on occupation and recurrence density.

Define linear response coefficients as time-integrated connected correlators of the microscopic update dynamics.

These coefficients are determined entirely by U.

Dimensional scaling shows:

alpha_1 = b^3 times alpha_tilde_1,

with alpha_tilde_1 dimensionless and fixed by update spectrum.

1. Emergent Gravitational Coupling

From the curvature response relation:

K(x) proportional to (alpha_1 / b^2) epsilon(x).

Using rho_E = E0 epsilon, we obtain:

K proportional to (alpha_1 / (b^2 E0)) rho_E.

Comparing dimensions with weak-field curvature scaling yields:

G_eff proportional to c^4 times alpha_1 divided by (b^2 E0).

Substituting primitive relations gives:

G_eff proportional to (lp^4 b) divided by (tp^3 S0), times alpha_tilde_1.

Thus the effective gravitational coupling reduces entirely to:

• Primitive scales (lp, tp),

• Minimal relational action scale S0,

• Dimensionless response coefficients from U.

No independent gravitational constant is inserted.

1. Coarse-Scale Stability and Boundedness

In the dilute infrared regime:

• The block scale b is fixed by correlator matching.

• Response coefficients are finite.

• Holonomy-loading variance is bounded due to finite Hilbert space.

Collapse corresponds to breakdown of infrared compression mapping, not divergence of microscopic holonomy statistics.

1. Operational Definition of Quantum Gravity

Quantum Gravity in the FRC framework is defined as;

The parameter-closed regime of finite reversible closure dynamics in which curvature-loading observables derived from gauge holonomy statistics are determined entirely by occupation and recurrence content through update-generated response functionals, without assuming prior manifold geometry.

All dimensional constants reduce to primitive scales and update-spectrum data.

1. Failure Conditions

The framework fails if;-

• No non-zero minimal relational eigenphase exists.

• Response coefficients diverge.

• Weak-field limit disagrees with observation.

• Correlator matching fails to produce stable infrared regime.

• Holonomy statistics are unbounded.

Programme Flow;-

Paper 18 - Curvature response functional.
Paper 19 - Parameter reduction and operational quantum gravity definition.
Paper 20 - Consolidated empirical tests and falsifiability.