Causality and Graph Dynamics

Lieb-Robinson Sector Diagnostics and the Independence Theorem

Executive Summary (Plain English)

Investigates causal propagation when interaction graphs are controlled or superposed. Proves the Causal Nonseparability vs. Environmental Objectivity Independence Theorem (neither implies the other) and establishes sector-family Lieb-Robinson diagnostics (C_inf, C_1, C_bar).

Evidence Class: Analytic Theorem·Scope: Layer 3 causal propagation and Layer 1 causal order independence

Last Audited: 2026-09-18

Claim Boundary & Exclusions Audit

Scope: Layer 3 causal propagation and Layer 1 causal order independence

What Is Claimed (Within Scope)

•Independence Theorem: Environmental objectivity and indefinite causal order (causal nonseparability) are operationally independent resources.
•Topology here means interaction graph connectivity, not physical continuum topology or operator topology.
•Lieb-Robinson diagnostic taxonomy: C_inf (uniform essential supremum), C_1 (weighted mean), C_bar (averaged commutator norm under common embedding).
•Lower-front sector obstruction: essential unboundedness of sector velocities precludes uniform exponential envelopes.

What Is NOT Claimed (Prohibited Inferences)

•Lieb-Robinson velocity is a bound parameter within a fixed graph model, not a universal relativistic speed of light c.
•Does not derive spacetime light cones or continuous causal structure.

Technical Exposition & Formulation

FA-CB-02 uses direct-integral operator algebras to analyze propagation across ensembles of interaction topologies. FA-QD-PRXQ-01 proves that quantum-controlled ordering (e.g. quantum switch) does not automatically generate redundant classical environmental records.

Theorems & Mathematical Propositions

Environmental Objectivity / Causal Nonseparability Independence Theorem

[EXA]Exact result

“Within the composite class containing fixed-order circuits, the quantum switch, and record-augmented switch processes, environmental objectivity is neither necessary nor sufficient for causal nonseparability.”

Hypotheses & Assumptions:

Finite-dimensional Hilbert spaces.
Access decomposition Lambda (pointer), T (control), F (fragment).

Piecewise Uniform Sector-Envelope Criterion (Prop. 1)

[EXA]Exact result

“If sector Lieb-Robinson data satisfy C_x <= C, xi_x <= xi, v_x <= v_ess, and d_x >= d almost everywhere, then the uniform decomposable norm obeys C_inf(t) <= C ||A||_inf ||B||_inf exp(-(d - v_ess t)/xi) for d - v_ess t > 0.”

Hypotheses & Assumptions: