Canonical Core
Pinned, non-ranked anchor equations with rubric scores.
Scoring Rubric (0-100)
- Tractability (0-20)
- Physical plausibility (0-20)
- Validation (0-20)
- Artifact completeness (0-10)
- Total normalized from a 70-point base
- Novelty is shown as a dated tag only
CORE
Phase Ambiguity (multi-valuedness axiom)
Canonical equation
$$z=re^{i\theta},\quad \theta\equiv\theta+2\pi k,\ k\in\mathbb{Z}$$
Reference: Equation Sheet v1.1 §A (Eq.1)
Description
Multi-valuedness comes from the (2π)-quotient on phase. Starting axiom motivating the Phase-Lift framework.
Rubric
T 20/20, P 20/20, V 20/20, A 0/10, normalized to 86/100
Novelty tag
-
Repository
Animation
planned
Image/Diagram
planned
CORE
Phase-Lift operator definition (⧉)
Canonical equation
$$(\,\,\,\,\,\u29c9 f\,\,\,)(z;\theta_{\rm ref}) := f(z)\ \text{computed using}\ \theta_R=\mathrm{unwrap}(\arg z;\theta_{\rm ref})$$
Reference: canonical-core paper 04 / Eq.Sheet §A (Eq.2)
Description
Defines Phase-Lift as explicit phase resolution (branch selection) via unwrapping.
Rubric
T 16/20, P 16/20, V 19/20, A 2/10, normalized to 76/100
Novelty tag
-
Repository
Animation
Image/Diagram
planned
CORE
Deterministic Unwrapping Rule
Canonical equation
$$\mathrm{unwrap}(\theta;\theta_{\rm ref})=\theta+2\pi\,\mathrm{round}\!\Big(\frac{\theta_{\rm ref}-\theta}{2\pi}\Big)$$
Reference: Equation Sheet v1.1 §A (Eq.3)
Description
Pick the representative closest to the reference. The concrete algorithm behind Phase-Lift: a single deterministic formula for branch selection.
Rubric
T 19/20, P 19/20, V 20/20, A 0/10, normalized to 83/100
Novelty tag
-
Repository
Animation
planned
Image/Diagram
planned
CORE
Path Continuity (Stateful Phase-Lift)
Canonical equation
$$\theta_{R,k}=\mathrm{unwrap}(\arg z_k;\,\theta_{R,k-1})$$
Reference: Equation Sheet v1.1 §A (Eq.4)
Description
Turns phase into a continuous real trajectory by chaining the unwrap rule sample-to-sample (except at true singularities like z=0).
Rubric
T 18/20, P 18/20, V 15/20, A 0/10, normalized to 73/100
Novelty tag
-
Repository
Animation
planned
Image/Diagram
planned
CORE
PR-Root as a special case of Phase-Lift (⧉√)
Canonical equation
$$\u29c9\sqrt{z;\theta_{ref}}:=\sqrt{r}\,e^{i\theta_R/2}\ "where"\ z=re^{i\theta}\ "and"\ \theta_R=\mathrm{unwrap}(\arg z;\theta_{ref})$$
Reference: canonical-core paper 04 / Eq.Sheet §A (Eq.5)
Description
Phase-resolved square root: deterministic branch choice encoded by the resolved phase.
Rubric
T 16/20, P 16/20, V 13/20, A 2/10, normalized to 67/100
Novelty tag
-
Repository
Animation
Image/Diagram
planned
CORE
Winding Number + ℤ₂ Parity Invariants
Canonical equation
$$w=\frac{\Delta\theta_R}{2\pi}\in\mathbb{Z},\qquad \nu:=w\bmod 2\in\{0,1\},\quad b=(-1)^w\in\{\pm1\}$$
Reference: canonical-core papers 02/04 / Eq.Sheet §B (Eq.6–7)
Description
Counts how many times the lifted phase winds (w); ℤ₂ parity b predicts whether PR-Root returns to the same or opposite sheet.
Rubric
T 18/20, P 17/20, V 20/20, A 2/10, normalized to 81/100
Novelty tag
-
Canonical source
Repository
Animation
Image/Diagram
planned
CORE
Adaptive-π Conformal Scale Factor and Metric
Canonical equation
$$\Omega(x,t):=\frac{\pi_a(x,t)}{\pi},\qquad g_{ij}(x,t)=\Omega(x,t)^2\,\delta_{ij}$$
Reference: Equation Sheet v1.1 §C (Eq.8)
Description
πₐ/π acts like a local ruler scaling. Defines the conformal metric induced by the adaptive-π field on the underlying flat geometry.
Rubric
T 16/20, P 16/20, V 19/20, A 0/10, normalized to 73/100
Novelty tag
-
Repository
Animation
planned
Image/Diagram
planned
CORE
Adaptive Arc Length
Canonical equation
$$L_g(\gamma)=\int_0^1 \Omega(\gamma(t),t)\,|\dot\gamma(t)|\,dt$$
Reference: Equation Sheet v1.1 §C (Eq.9)
Description
Distances and penalties are measured in the adaptive geometry: path lengths scale with the local πₐ field.
Rubric
T 16/20, P 15/20, V 9/20, A 0/10, normalized to 57/100
Novelty tag
-
Repository
Animation
planned
Image/Diagram
planned
CORE
Adaptive-π field definition + limit (πₐ → π)
Canonical equation
$$\theta=\theta_R+2\pi_a(x,t)\,w,\ "with"\ w\in\mathbb{Z}\ "and"\ \pi_a\to\pi$$
Reference: canonical-core paper 02 + paper 04
Description
Generalizes the phase wrap unit; reduces to standard 2π wrapping when πₐ becomes constant π.
Rubric
T 16/20, P 15/20, V 19/20, A 2/10, normalized to 74/100
Novelty tag
-
Repository
Animation
Image/Diagram
planned
CORE
Reinforce/Decay Dynamics for πₐ
Canonical equation
$$\frac{d\pi_a}{dt}=\alpha_\pi\,s(x,t)-\mu_\pi(\pi_a-\pi_0)$$
Reference: Equation Sheet v1.1 §C (Eq.10)
Description
πₐ increases under stimulus (events/errors/curvature) and relaxes toward π₀ (often π). The core adaptive mechanism of the geometry layer.
Rubric
T 17/20, P 16/20, V 20/20, A 0/10, normalized to 76/100
Novelty tag
-
Repository
Animation
planned
Image/Diagram
planned
CORE
ARP Core Law (canonical ODE)
Canonical equation
$$\frac{dG_{ij}}{dt}=\alpha_G\,|I_{ij}(t)|-\mu_G\,G_{ij}(t)$$
Reference: canonical-core paper 01 / Eq.Sheet §D (Eq.11)
Description
Canonical Adaptive Resistance Principle update rule for edge conductance. Edges that carry activity reinforce; unused edges decay (self-organizing backbones).
Rubric
T 19/20, P 19/20, V 20/20, A 2/10, normalized to 86/100
Novelty tag
-
Canonical source
Repository
Animation
Image/Diagram
planned
CORE
Curvature as Salience (Curve-Memory Primitive)
Canonical equation
$$\kappa(s)=|\gamma''(s)|$$
Reference: Equation Sheet v1.1 §E (Eq.12)
Description
Salience equals curvature: sharp bends in a trajectory mark events and transitions. Defines the geometric primitive used by curve-memory.
Rubric
T 20/20, P 20/20, V 20/20, A 0/10, normalized to 86/100
Novelty tag
-
Repository
Animation
planned
Image/Diagram
planned
CORE
Reinforce/Decay Memory Law (generic)
Canonical equation
$$\frac{dM}{dt}=\alpha\,S(t)-\mu\,M(t)$$
Reference: Equation Sheet v1.1 §E (Eq.13)
Description
A generic time-constant memory law: M grows under stimulus S and decays at rate μ. Can drive s(x,t) for πₐ or any curve-memory trace.
Rubric
T 18/20, P 17/20, V 20/20, A 0/10, normalized to 79/100
Novelty tag
-
Canonical source
Repository
Animation
planned
Image/Diagram
planned
CORE
Phase-Lifted Stokes Quantization (Adaptive-π)
Canonical equation
$$\theta_R[\gamma]:=\mathrm{unwrap}\!\left(\oint_{\gamma} A\,;\theta_{\rm ref}\right)=\int_{S:\,\partial S=\gamma}F+2\pi_a\,w,\quad w\in\mathbb{Z},\quad b(\gamma)=(-1)^w$$
Reference: canonical-core papers 04 (Phase-Lift) + 02 (Adaptive-π)
Description
Turns the usual Stokes/holonomy statement ‘mod 2π’ into an exact equality by Phase-Lifting the phase (real-valued branch) and explicitly tracking the integer sector w; adaptive-π generalizes the period to 2πₐ.
Rubric
T 14/20, P 15/20, V 19/20, A 2/10, normalized to 71/100
Novelty tag
-
Repository
Animation
Image/Diagram
planned