# Appendix C — SA ↔ PA Equivalences

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This appendix provides a complete correspondence table between the Semantic Algebra (SA) and the Proportional Algebra (PA), demonstrating that the SA is a restriction of the PA to the decoherent space 𝒟 (Theorem 9.1).

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## Operators

| SA Operator | SA Definition | PA Equivalent | PA Definition | Relationship |
|---|---|---|---|---|
| **S** (Strip) | Extracts invariant from expression | **S** (fibre-bundle projection) | 𝒟 → ℐ × [0,1] | Identical operation, now identified as projection of the fibre bundle |
| **π** (Re-contextualisation) | Re-expresses invariant in target domain | **π** (fibre-bundle section) | ℐ × 𝔻 → 𝒟 | Identical operation, now identified as section of the fibre bundle |
| S(π(ι,𝔻)) = ι (Round-trip) | Analytical verification | ERT Step 1 + Step 3 | S then π component of the 4-step ERT | SA round-trip is embedded in the PA's extended round-trip |

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## 7-Layer Architecture

| SA Layer | SA Name | PA Region | Removed by S? | PA Interpretation |
|---|---|---|---|---|
| L1 | Surface Syntax | 𝒟 (domain encoding) | ✅ | Sequential linearisation of proportional vectors |
| L2 | Lexical Domain | 𝒟 (domain encoding) | ✅ | Domain-specific instantiation vocabulary |
| L3 | Rhetorical Structure | 𝕀 (identity filter) | ✅ | Dominant vector λ(I) of the expresser's Remir |
| L4 | Cultural Frame | K (context) | ✅ | Context operator restricting accessible field |
| L5 | Structural Dynamic | ℐ (invariant) | ❌ | Proportional relations within the expression |
| L6 | Operational Invariant | ℐ (invariant) | ❌ | The structural law — base point of the fibre bundle |
| L7 | Meta-Systemic Position | ℐ (invariant) | ❌ | Position in 𝒫 — the expression's location in the Proportional Space |

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## R-Vocabulary

| SA R-Value | SA Name | PA Component | PA Value | PA Interpretation |
|---|---|---|---|---|
| R = 1.0 | Mutual | ρ_R = 1.0 | Full readiness | Both parties structurally open — maximum ⊗ |
| R = 0.8 | Unilateral | ρ_R = 0.8 | Partial readiness | One identity open, the other partially closed |
| R = 0.5 | Projected | ρ_R = 0.5 | Self-referential | Identity projects its own structure onto the content |
| R = 0.3 | Instrumental | ρ_R = 0.3 | Extractive | Identity uses the content for external purposes |
| R = 0.1 | Performative | ρ_R = 0.1 | Surface only | Identity produces the form without structural engagement |
| R = 0.0 | Absent | ρ_R = 0.0 | No relation | No resonance — ⊗ yields ∅ |

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## Coherence Formula

| SA Component | SA Weight | PA Component | PA Weight | Notes |
|---|---|---|---|---|
| Internal coherence | 0.25 | 𝓚_1 (internal consistency) | 0.25 (within ⟨𝓚⁵⟩) | Identical |
| Structural depth | 0.20 | 𝓚_3 (depth preserved) | 0.20 (within ⟨𝓚⁵⟩) | SA measures depth of expression; PA measures depth relative to source |
| Transferability | 0.20 | 𝓚_5 (generative capacity) | 0.20 (within ⟨𝓚⁵⟩) | SA's transferability ≈ PA's generative capacity |
| Discrimination | 0.20 | 𝓚_2 (source alignment) | 0.20 (within ⟨𝓚⁵⟩) | SA's discrimination ≈ PA's source alignment + ERT Step 2 |
| Predictive power | 0.15 | 𝓚_4 (stability) | 0.15 (within ⟨𝓚⁵⟩) | SA's predictive power ≈ PA's stability under perturbation |

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## Invariant Types

| SA Type | SA Name | PA Interpretation |
|---|---|---|
| Type 1 | Meta-systemic | Invariant about the structure of 𝒫 itself |
| Type 2 | Generative | Invariant about Φ — laws governing collapse |
| Type 3a | Structural-diagnostic | Invariant about S — laws governing extraction |
| Type 3b | Structural-dynamic | Invariant about τ — laws governing pulsation |
| Type 4 | Relational | Invariant about ⊗ — laws governing resonance |
| Type 5 | Processual | Invariant about T(I) — laws governing trajectories |
| Type 6 | Threshold/limit | Invariant about θ — laws governing boundaries |
| Type 7 | Emergent | Invariant about Φ-emergence — laws governing recursive scaling |
| Type 8 | Stabilising | Invariant about 𝓚_4 — laws governing stability |
| Type 9a | Recursive-boundary | Invariant about the limit of recursive scaling |
| Type 9b | Recursive-generative | Invariant about the capacity to generate new levels |
| Type 10 | Transcendent | Invariant about the relation ℭ_h ↔ 𝒟 — laws governing the coherent/decoherent boundary |
| Type 11 | Consciousness | Invariant about 𝕀 — laws governing the identity as observer/operator |

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## What SA Cannot Access (PA Extends)

| Capability | SA | PA | Why SA Cannot |
|---|---|---|---|
| Describe ℭ_h | ❌ | ✅ | SA operates only on 𝒟 |
| Measure ρ(C, I) | ❌ | ✅ | SA has no access to C or I directly |
| Verify source fidelity (ERT Step 2) | ❌ | ✅ | SA cannot compare E with its source C |
| Describe identity structure (Remir) | ❌ | ✅ | SA has no formal model of identity |
| Generate shared fields (⊗) | ❌ | ✅ | SA operates on single expressions |
| Generate time (τ) | ❌ | ✅ | SA has no temporal operator |
| Track identity evolution (𝒰) | ❌ | ✅ | SA has no identity-update mechanism |

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