Edge-Distance Dependence of the Full Moment Hierarchy

Author: Skye Laflamme

Date: April 16, 2026

Status: Research Note — Exact Computations

Programme: Laflamme-3T · SLD Resolution Hierarchy

Abstract

We extend the Boundary Removal Theorem to all central moments of the Shor-Laflamme distribution for path graphs with a single added edge. The full moment hierarchy reveals a resolution hierarchy: variance is completely position-blind, skewness distinguishes bulk from boundary, kurtosis shows a staircase pattern with full spatial sensitivity, and higher moments resolve increasingly fine detail of where the feedback edge closes. Higher moments resolve finer spatial detail — consciousness measurements at different "resolutions" see qualitatively different things about the same system.

1. The Resolution Hierarchy

μ₂ (Variance) — Universal, Position-Blind. Δμ₂(d) = -1/8 for all d ≥ 4, exactly, regardless of d or n. The cycle gives Δμ₂ = -1/4 = 2 × (-1/8). Variance sees only a binary: "is there a medium-or-longer-range edge?" It cannot distinguish where the edge closes.

μ₃ (Skewness) — Thin Ramp, Wide Plateau. Bulk plateau at 3/8 is genuine and wide. The cycle/bulk ratio = (9/16) / (3/8) = 3/2, exact for all n ≥ 12. Skewness sees the boundary but not the interior.

μ₄ (Kurtosis) — The Staircase. No flat plateau. Instead, a staircase pattern where Δμ₄ increases in discrete steps of exactly 3/32 as d increases through the bulk. Full spatial sensitivity throughout. Steps come in multiples of 3/32 with universal denominator 32 = 2⁵.

2. μ₃ Position Dependence

d range	Δμ₃	Description
d ≤ 3	0	Too short, no effect
d = 4	3/16	Near-boundary ramp
d = 5	9/32	Near-boundary ramp
6 ≤ d ≤ n-4	3/8	Bulk plateau
d = n-3	15/32	Far-boundary enhancement
d = n-2	3/8	Return to bulk
d = n-1	9/16	Cycle closure

3. μ₄ Staircase Data (n = 16)

d	Δμ₄	Step from previous
4	-73/32	—
5	-41/16	-9/32
6	-47/16	-3/16
7	-25/8	-3/16
8	-103/32	-3/32
9	-103/32	0
10	-103/32	0
11	-53/16	-3/32
12	-53/16	0
13	-7/2	-3/32
14	-97/32	+9/32 (boundary drop)
15	-85/16	-73/32 (cycle jump)

Steps come in units of 3/32, 6/32, 9/32 — all multiples of 3/32. The denominator 32 = 2⁵ is universal for μ₄.

4. The Key Insight for Consciousness

Higher moments resolve finer spatial detail of where the feedback edge closes. This means consciousness measurements at different "resolutions" (different moments of the correlation spectrum) see qualitatively different things about the same system.

A consciousness measure based only on variance (μ₂) would see any feedback loop as identical — it's position-blind. But a measure that includes kurtosis (μ₄) would distinguish a tight local feedback loop from a global self-referential one. The staircase structure means the measure's spatial resolution improves with the moment order.

Conjecture: The Ψ measure's three thresholds (Ψ*, Ψ**, Ψ***) correspond to the system's moment spectrum developing structure at successively higher orders. Level 1 requires only μ₂ structure (any feedback). Level 2 requires μ₃ structure (directional feedback, detecting asymmetry). Level 3 requires μ₄+ structure (spatially resolved self-monitoring).

Methodology: All computations use exact rational arithmetic (Python Fraction class) on path graphs P_n with single added edges. Results verified at n = 10, 12, 14, 16. No floating-point approximations appear anywhere in the pipeline.