Phase Geometry (Basis5)

This page documents the exact discrete geometry used by LEE 3.0.

Phases

Phase	Angle	Unit Vector
ALIVE	0°	(1, 0)
JAM	90°	(0, 1)
MEM	180°	(-1, 0)

• Angles live in ℤ/360ℤ; we only use multiples of 90°.
• Vectors are integer unit vectors on ℤ². No floats, no amplitudes.

Rotations

• Allowed deltas: {0, 90, 180, 270} degrees.
• Composition is modular addition: Δ(a→b) = angle(b) - angle(a) mod 360.

from src.core.basis5 import project_phase, rotation_delta_deg, transition_basis

print(project_phase("ALIVE"))   # {'phase':'ALIVE','angle_deg':0,'vec':{'x':1,'y':0}}
print(rotation_delta_deg("ALIVE","JAM"))  # 90
print(transition_basis("ALIVE","MEM"))
# {'before':{...},'after':{...},'delta_deg':180}

Winding (per run)

build_winding(phases) accumulates the rotation across a phase trace and summarizes it.

from src.core.basis5 import build_winding
w = build_winding(["ALIVE","JAM","MEM"])
print(w["summary"])
# {'total_winding_deg': 180, 'avg_vector': {'x': {'num':0,'den':3}, 'y': {'num':1,'den':3}}, 'unique_phases':['ALIVE','JAM','MEM']}

Witness projection

witness_basis(expr) projects integer flags from syntax; e.g. implication (1 -> 0) or local refutation (p & ~p).

from src.core.basis5 import witness_basis
print(witness_basis("1 -> 0"))     # {'jam':1,'detach':1,'mp':1}
print(witness_basis("p & ~p"))     # {'jam':1,'refute':1}
print(witness_basis("1 -> 1"))     # {'jam':0}

Invariants

• Discrete rotation conservation: closed loops sum to multiples of 360°.
• Detachment preservation: implication witnesses survive into MEM and are recorded in enrichment/provenance.

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StressIndex

The StressIndex measures accumulated logical strain in a run. It is a function of: - Total winding degrees (cumulative angular displacement across phases) - Transition density (phase changes per unit time/steps) - Jam persistence (duration spent in JAM without resolution)

A simple baseline metric:

StressIndex = (total_winding_deg / 360) * jam_ratio
# where jam_ratio = jam_time / run_time

Interpretation: - Low (<0.2) – stable reasoning - Moderate (0.2–0.5) – contradictions present, resolvable - High (>0.5) – potential deadlock or over-constraint

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Resistance

Resistance is a measure of opposition to phase change. It increases with: - Mean absolute rotation delta per transition - Number of return cycles to a prior phase state - Prolonged stalls near JAM without resolution

Use Resistance alongside StressIndex to diagnose stability under load.

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Figure