Description

Abstract (English)

DESCRIPTION

Paper 17 defines the high-density breakdown regime of the composite sector.

1. Operational Block Scale

An operational correlator-matching block scale b is defined as the minimal coarse-graining length such that:

• Directional anisotropy of the composite correlator falls below tolerance.
• The correlator tail is well-described by a single dominant decay mode.

This scale provides the physical volume V_b = b^3 used for density definitions.

1. Composite Density

Composite density is defined operationally as:

rho = N_comp / V_b.

This density is not defined at the primitive scale but relative to the correlator-matching block.

1. Overlap Threshold

Let xi_core denote the composite core size extracted from correlator decay.

Mean separation d scales as rho^(-1/3).

Isolation fails when d is less than or comparable to xi_core.

Define the overlap threshold:

rho_star = xi_core^(-3).

When rho exceeds rho_star, single-composite states cease to be approximately orthogonal to multi-composite states, invalidating the dilute truncation used in Paper 15.

1. Gauge-Invariant Curvature Loading

Let W_p denote plaquette holonomy.

Define gauge-invariant curvature-loading observables:

chi = 1 - Re <W_p>
nu = < |1 - W_p|^2 >

Because W_p belongs to a compact group, phase cancellation can suppress Re <W_p>.  Therefore nu is taken as the primary curvature-loading order parameter.

Interpretation:

• nu much less than 1 indicates weak curvature loading.
• nu of order 1 indicates nonperturbative curvature loading.

1. Curvature Threshold

Define the linear response coefficient:

kappa = partial chi / partial rho evaluated at rho approaching 0.

Expressed dimensionlessly in terms of rho b^3.

Define a curvature transition density rho_c by:

nu(rho_c) = nu_star,

for fixed threshold nu_star between 0 and 1.

The ordering between rho_star and rho_c is model-dependent.

1. Breakdown of the Effective Expansion

Paper 16 relies on:

• rho b^3 much less than 1.
• nu much less than 1.
• Small gradient expansion parameters.

When rho approaches rho_star or rho_c:

• Holonomy fluctuations become order 1.
• The derivative expansion loses convergence.
• The quasiparticle description fails.

Thus Paper 17 specifies the boundary of validity for Papers 13–16.

1. Recurrence Density and Curvature Bias

Define a grading operator acting on the doubled-sector composite field.

Define a recurrence density s as the block-averaged expectation value of this grading operator.

In the high-density regime:

nu becomes a functional of s,
chi becomes a functional of s.

Composite excitations carry both holonomy flux and recurrence content.  At high density, the ensemble distribution of recurrence states biases holonomy statistics.

This curvature-bias functional is the structural input for Paper 18.

Programme Flow;-

Paper 16 - Low-density observable deviations.
Paper 17 - Density thresholds and curvature loading.
Paper 18 - Curvature response functional and emergent geometry.
Paper 19 - Parameter reduction and quantum gravity definition.
Paper 20 - Consolidated falsifiability and experimental closure.