canonical-core

Canonical Notation Rules

This document establishes the official notation for Canonical Core. If you reference these papers, please use these symbols as defined.

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Core Symbols

Symbol	Name	Definition	Paper
⧉	Phase-Lift	Prefix operator for phase-structured branching	04
πₐ	Adaptive π	Adaptive π as a field (not constant 3.14159…)	02
π	Standard π	Mathematical constant (3.14159…)	—
CM	Curve Memory	Memory object encoding path + derivative history	03
CMA	Curve Memory Alphabet	Encoding system for CM	03
PROs	Phase-Resistant Objects	Objects that survive phase transitions	04
ARP	Adaptive Resistance Pattern	Resistance that adapts to strain	01
AIN	Adaptive Impedance Network	Network with adaptive impedance	01

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QPS-GR Mapping Symbols

Symbol	Name	Definition	Paper
τ_coh	Coherence time	Time scale for phase coherence	05
V_floor	Visibility floor	Minimum observable visibility	05
Δν	Frequency offset	Clock frequency difference	05
ε_strain	Strain parameter	Dimensionless strain	05
QPS	Quantum Phase Space	Phase space with quantum structure	05

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Notation Rules

1. ⧉ is a prefix operator

✅ Correct: ⧉(state, context)
❌ Wrong: state⧉ or ⧉state

2. π remains constant, πₐ is a field

• Use π for the mathematical constant (3.14159…)
• Use πₐ when π is adaptive (context-dependent field)
• Never mix them in the same expression
• πₐ defines the local phase wrap unit (not just a rescaling)

✅ Correct: A = π r² (circle area in Euclidean geometry)
✅ Correct: A = πₐ(context) · r² (adaptive geometry)
✅ Correct: θ = θ_R + 2πₐ(x,t) · w (phase with adaptive wrap)
❌ Wrong: A = π · πₐ · r²
❌ Wrong: θ = θ_R + 2π · πₐ · w (mixing fixed and adaptive)

Key distinction: In standard QM, phase wraps at fixed 2π. With πₐ, the wrap unit itself becomes a dynamic field:

• Standard: θ ≡ θ + 2πk (fixed period)
• Adaptive: θ = θ_R + 2πₐ(x,t) · w (dynamic period)

3. CM is a memory object

• CM is not a function; it’s a data structure
• Access CM contents via: CM.path, CM.derivatives, CM.encode()

✅ Correct: CM.path[t]
❌ Wrong: CM(t)

4. CMA is an encoding alphabet

• CMA defines the symbols used to encode CM
• Think of CMA as a “language” for memory

5. PROs are objects, not functions

• PROs carry properties across phase transitions
• They are instantiated, not called

✅ Correct: object = PRO(properties)
❌ Wrong: PRO(state) → result

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Typography

Subscripts and Superscripts

• Use subscripts for indices, labels, parameters: τ_coh, V_floor, πₐ
• Use superscripts for exponents, powers: r², e^{iθ}

Greek Letters

• π (pi) – Standard mathematical constant
• πₐ (pi-adaptive) – Adaptive π field
• τ (tau) – Time scales (τ_coh, τ_relax)
• ε (epsilon) – Small parameters (ε_strain)
• Δ (delta) – Differences (Δν, Δφ)

Special Operators

• ⧉ (Phase-Lift) – U+29C9 “Two Joined Squares”
• Use Unicode if possible; otherwise use \PhaseLift in LaTeX

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LaTeX Macros (Recommended)

If you’re writing a paper that references Canonical Core, use these macros:

\newcommand{\PhaseLift}{\ensuremath{\unicode{x29C9}}}
\newcommand{\pia}{\ensuremath{\pi_{\text{a}}}}
\newcommand{\CM}{\ensuremath{\text{CM}}}
\newcommand{\CMA}{\ensuremath{\text{CMA}}}
\newcommand{\PRO}{\ensuremath{\text{PRO}}}
\newcommand{\tcoh}{\ensuremath{\tau_{\text{coh}}}}
\newcommand{\Vfloor}{\ensuremath{V_{\text{floor}}}}

Then use:

\PhaseLift(state, context) generates a phase-structured branch with memory \CM.

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Special Topic: πₐ in Phase Equations

The adaptive-π field has specific notation conventions when used in phase equations:

Standard vs Adaptive Phase Wrap

Standard QM (fixed wrap):

θ ≡ θ + 2πk,  k ∈ ℤ

Phase wraps at fixed 2π intervals.

Adaptive-π (dynamic wrap):

θ = θ_R + 2πₐ(x,t) · w,  w ∈ ℤ

Where:

• θ_R = resolved (unwrapped) phase
• πₐ(x,t) = local phase-period field
• w = winding number

Key Properties

1. πₐ is a field, not a constant — It varies with position and time
2. πₐ defines the local wrap unit — Not just a rescaling factor
3. πₐ enables continuous phase transport — Phase doesn’t “jump” at branch cuts
4. πₐ → π recovers standard QM — Standard physics is a special case

When to Use πₐ

Use πₐ when:

• Tracking phase continuity along paths
• Computing winding numbers under deformation
• Implementing Phase-Lift (⧉) operations
• Working with parity tracking (ℤ₂)
• Dealing with topology changes that don’t change winding

Use π when:

• Working in standard Euclidean geometry
• Using fixed reference frames
• Computing with constant phase periods

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Citation Format

When referencing a specific symbol, cite the paper where it’s defined:

“The Phase-Lift operator ⧉ (Paper 04) creates phase-structured branches that carry Curve Memory CM (Paper 03) forward.”

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Future Extensions

As the framework evolves, new symbols may be added. This document is the canonical source for notation—check here first before inventing new symbols.

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Last Updated: 2026-01-09
Version: 0.1.0

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