PA — Proportional Algebra

The formal space, metric, and operators that govern the collapse from coherent content to expressed form across all domains.

Scope

This folder contains the Proportional Algebra corpus in English for:

• GitHub public repository content
• Zenodo / OSF deposits
• Hugging Face documentation/dataset cards
• Reference engine (pa_engine/) for experimental verification of PA operators

What PA Is

Proportional Algebra is announced in the foundational TE text and is here published for the first time. It provides:

1. The Proportional Space P = (C, R, D, ρ, ≤_𝓚) — the formal ground in which coherent content (C), identities (R), expressions (D), and their relations live.
2. Three operators:
   • Φ (Collapse) — the central operation that generates expressions from coherent content via an identity in a context.
   • S (Strip) — the partial inverse of Φ; identical to the SA operator and identified in PA as the projection of a fibre bundle from D onto invariant space I.
   • ⊗ (Resonance) — generates shared coherent fields between two identities.
3. The Pulsation generator τ — produces time as an emergent quantity from the cycle of collapse and return, rather than treating time as an external parameter.
4. The Extended Round-Trip (ERT) — a four-step diagnostic that tests both analytical fidelity (as SA does) and genetic fidelity of the original collapse.
5. Six explicit falsification criteria (F1–F6).

What PA Is Not

• Not a metaphysics — every operator has procedural definitions and falsifiable properties.
• Not a numerical algebra — "algebra" here means a structured grammar with operations and tests.
• Not a "theory of everything" — PA explicitly states its limits (Chapter 18) and identifies the questions that belong to OCT/OGT.

Cross-Domain Demonstrations

PA is demonstrated across five domains in Part IV (chapters 13–17):

• Chemistry — bonds as proportional collapses; H₂O as worked example with full ERT.
• Language — syntax as geometry of proportional vectors; ambiguity as superposition.
• Emotion — emotional dynamics as phase transitions in P.
• Medicine — disease as ⟨𝓚⁵⟩ degradation; therapy as re-coherence.
• Artificial Intelligence — specification for a PA-aligned AI; alignment over scale.

Each demonstration applies the full operator set and reports concrete coherence/resonance values.

Relationship to Semantic Algebra

PA Theorem 9.1 establishes that SA is mathematically a restriction of PA to the decoherent space D. The 10 SA invariants become base points of the fibre bundle in PA. The 7-layer SA architecture maps onto PA regions: layers 1–4 are the fibre (removed by S); layers 5–7 are the base (preserved by S).

This connects them formally without collapsing them into a single project. SA was developed first and has independent applications. See PA_BOOK/appendix_c_sa_pa_equivalences.md for the complete correspondence table.

Reference Engine

pa_engine/ is a small Python package (~700 lines, 7 files) implementing the PA operators and metric for experimentation:

• remir.py — the Remir structure (semantic vectors + resonance matrix)
• metric.py — the resonance metric ρ with its 5 components
• operators.py — the Collapse, Strip, and Resonance operators
• dynamics.py — temporal evolution and pulsation
• ert_diagnostic.py — the Extended Round-Trip test
• test_cases.py — worked examples
• simulation_orchestrator.py — entry point

The engine is for experimental verification of the framework. It is not production software.

Start Here

1. START_HERE_PA.md — orientation and reading paths
2. PA_BOOK/01_why_the_sciences_cannot_speak.md — motivation
3. PA_BOOK/04_the_proportional_space.md — the formal space
4. PA_BOOK/08_collapse_operator.md — the central operation
5. PA_BOOK/11_extended_round_trip.md — the integrity test

Falsification

Six explicit falsification criteria are listed in PA_BOOK/03_what_is_needed.md §3.6 and PA_BOOK/18_limits_and_open_questions.md:

• F1 — Same (C, I, K) produces structurally different E
• F2 — Two expressions classified as structurally isomorphic carry different content
• F3 — ρ assigns wrong compatibility values
• F4 — S yields the same invariant for all expressions
• F5 — ≤_𝓚 reverses independent coherence judgements
• F6 — I₁ ⊗ I₂ ≠ I₂ ⊗ I₁ without contextual cause

None of these have been observed. All are testable.

Editorial Note

The single-file manuscript PA_FULL/Proportional_Algebra_Foundations_UNIFIED.md is the unified compilation. The chapter-level files in PA_BOOK/ are the canonical source of truth. The Python engine in pa_engine/ is the runnable companion.