Appendix A — Symbol Register

Spaces

Symbol	Name	Definition	Chapter
𝒫	Proportional Space	(ℭ_h, 𝕀, 𝒟, ρ, ≤_𝓚)	4
ℭ_h	Coherent Field	Space of un-collapsed structured potential	4.2
𝕀	Identity Space	Space of Remirs	4.2
𝒟	Decoherent Space	Space of expressed realities	4.2
ℐ	Invariant Space	Space of structural invariants (base of the fibre bundle)	9.2
ℭ_shared	Shared Coherent Field	Region of ℭ_h accessible to two identities via ⊗	10.2
N(I)	Accessible Field	{C ∈ ℭ_h : ρ(C, I) ≥ θ} — neighbourhood of I in ℭ_h	5.5.4

Objects

Symbol	Name	Definition	Chapter
C	Coherent Content	Element of ℭ_h — un-collapsed structured potential	2.2
I	Identity	Active functional vector that performs the collapse	2.2
K	Context	Constraint on ℭ_h that restricts accessible content: K(ℭ_h) → ℭ_K	4.5
E	Expression	Element of 𝒟 — collapsed, observable output	2.2
ℛ(I)	Remir	(V_I, B_I) — semantic vectors + resonance matrix	7.2
V_I	Semantic Vectors	{v₁, v₂, ..., vₙ} — directed intensities of the identity	7.2
B_I	Resonance Matrix	V_I × V_I → [-1, 1] — internal proportional structure	7.2
λ(I)	Dominant Vector	Eigenvector of B_I with largest eigenvalue	7.3
T(I)	Trajectory	Ordered sequence of an identity's collapses	3.2
ι_k	Invariant k	Structural law extracted by S — base point of fibre bundle	9.5

Operations

Symbol	Name	Type Signature	Chapter
Φ	Collapse	ℭ_h × 𝕀 × K → 𝒟	8.2
S	Strip	𝒟 → ℐ × [0, 1]	9.2
π	Re-contextualisation	ℐ × 𝔻_target → 𝒟	9.3
⊗	Resonance	𝕀 × 𝕀 → ℭ_shared	10.2
𝒰	Identity Update	𝕀 × 𝒟 → 𝕀	8.6
τ	Pulsation	ℭ_h × 𝒟 × 𝕀 → ℝ⁺	12.2
μ	Isomorphism Map	𝔻₁ → 𝔻₂ (structure-preserving)	2.3

Metrics and Measures

Symbol	Name	Range	Chapter
ρ	Resonance Metric	[0, 1]	5.1
ρ_v	Vectorial Alignment	[0, 1]	5.2
ρ_d	Proportional Depth	[0, 1]	5.2
ρ_K	Contextual Compatibility	[0, 1]	5.2
ρ_τ	Temporal Phase	[0, 1]	5.2
ρ_R	Relational Readiness	{1.0, 0.8, 0.5, 0.3, 0.1, 0.0}	5.2
𝓚⁵	Coherence Vector	(𝓚_1, 𝓚_2, 𝓚_3, 𝓚_4, 𝓚_5) — 5-dimensional extension of OST's intra-set coherence vector 𝓚	6.2
⟨𝓚⁵⟩	Coherence Aggregate	[0, 1] — weighted mean of the 𝓚⁵ components (weights per §6.2)	6.2
𝓚_1	Internal Consistency	[0, 1]	6.2
𝓚_2	Source Alignment	[0, 1]	6.2
𝓚_3	Depth Preserved	[0, 1]	6.2
𝓚_4	Stability	[0, 1]	6.2
𝓚_5	Generative Capacity	[0, 1]	6.2
θ	Collapse Threshold	[0, 1]	5.4
δ	Fidelity	[0, 1]	11.2
b̄(I)	Average Internal Coherence	[-1, 1]	7.2
Ξ	Terminal Compatibility	[0, 1]	16.5

Relations

Symbol	Name	Type	Chapter
≤_𝓚	Coherence Order	Partial order on 𝒟	6.3
≡_S	Structural Equivalence	Equivalence relation on 𝒟	3.4
∼_ρ	Compatibility	Tolerance relation on ℭ_h × 𝕀	3.4

Collapse Types

Type	Condition	Description	Chapter
A	⟨𝓚⁵⟩ ≥ 0.7, ρ ≥ θ	Coherent — faithful carrier	8.5
B	0.3 ≤ ⟨𝓚⁵⟩ < 0.7	Partial — shallow but accurate	8.5
C	⟨𝓚⁵⟩ < 0.3, S(E) ≠ ∅	Distorted — content altered	8.5
D	S(E) = ∅ or ρ < θ	Failed — no structural content	8.5

Falsification Criteria

Code	Would Falsify	Chapter
F1	Φ as well-defined operation	3.6
F2	The isomorphism map μ	3.6
F3	The resonance metric ρ	3.6
F4	S as extraction (not projection)	3.6
F5	The coherence order ≤_𝓚	3.6
F6	The symmetry of ⊗	3.6

Canon Alignment Note (Symbol Canon v1.0 — ratified 2026-06-11)

This register follows the Symbol Canon for the Ordinative Sciences Programme. Renames applied in v2.0 relative to v1.0.1 (notation only — no semantic changes):

v1.0.1	v2.0	Object
𝒞, 𝒞_shared, 𝒞_K	ℭ_h, ℭ_shared, ℭ_K	Coherent Field and derived regions (frees 𝒞 = TE Collective Field)
ℛ (space)	𝕀	Identity Space — the Remir ℛ(I) is unchanged
κ, κ_1…κ_5, ≤_κ	⟨𝓚⁵⟩, 𝓚_1…𝓚_5, ≤_𝓚	Coherence vector 𝓚⁵ and aggregate — extends OST's intra-set 𝓚 (3-vector); TE's pairwise κ(I_a, I_b) is a distinct object coexisting by operand distinction
I_k	ι_k	Invariants (ι-family programme-wide)
Α / Β / Γ / Δ	A / B / C / D	Collapse types — state taxonomy shared with OCT admissibility states (A admissible · B sterile/partial · C degenerative/distorted · D invalid/failed); PA instantiates it over collapse quality, OCT over morphism admissibility
D₁, D₂, D_target	𝔻₁, 𝔻₂, 𝔻_target	Domains (𝔻 namespace; 𝒟 remains the Decoherent Space, declared equivalent to TE's plain D)

Unchanged by design: Φ (Collapse, TE-aligned), S (Strip), π (Re-contextualisation), ⊗ (Resonance), τ (Pulsation), ρ and its five components (Resonance family), θ, 𝒰, μ, Ξ, ℐ (Invariant Space), V_I, B_I, λ(I).