Start Here (PA)

The easiest path for understanding Proportional Algebra in this repository.

Step-by-Step Path

Part I — The Problem of the Absent Grammar (motivation)

1. PA_BOOK/01_why_the_sciences_cannot_speak.md — the conference of the inaudible (forty-seven researchers, one structural law, no recognition).
2. PA_BOOK/02_the_principle_of_structural_isomorphism.md — the deduction from the Collapse Function.
3. PA_BOOK/03_what_is_needed.md — the architectural blueprint for the grammar.

Part II — The Proportional Space (formal foundations)

1. PA_BOOK/04_the_proportional_space.md — the space P = (C, R, D, ρ, ≤_𝓚).
2. PA_BOOK/05_the_resonance_metric.md — the 5-component metric ρ.
3. PA_BOOK/06_the_coherence_order.md — the partial order ≤_𝓚 on expressions.
4. PA_BOOK/07_semantic_vectors_and_remir.md — identity as algebraic structure.

Part III — The Operators (the working machinery)

1. PA_BOOK/08_collapse_operator.md — Φ as algebraic operation.
2. PA_BOOK/09_strip_and_recontextualisation.md — S and π as PA operators (Theorem 9.1: SA is PA restricted to D).
3. PA_BOOK/10_resonance_operator.md — ⊗ and the dialogic field.
4. PA_BOOK/11_extended_round_trip.md — the four-step ERT diagnostic.
5. PA_BOOK/12_pulsation_temporal_generator.md — τ generates time.

Part IV — Cross-Domain Applications

1. PA_BOOK/13_chemistry.md — bonds as proportional collapses.
2. PA_BOOK/14_language.md — syntax as vector geometry.
3. PA_BOOK/15_emotion.md — emotion as phase transition in P.
4. PA_BOOK/16_medicine.md — disease as ⟨𝓚⁵⟩ degradation.
5. PA_BOOK/17_artificial_intelligence.md — specification for a PA-aligned AI.

Part V — Completion

1. PA_BOOK/18_limits_and_open_questions.md — six limits + falsification criteria.
2. PA_BOOK/19_relation_to_oct_ogt.md — position within the 4-pillar Ordinative Sciences architecture.
3. PA_BOOK/20_epilogue.md — the grammar is one.

Appendices

• PA_BOOK/appendix_a_symbol_register.md — full symbol table.
• PA_BOOK/appendix_b_te_equations_in_pa.md — translation of the 12 TE equations into PA notation.
• PA_BOOK/appendix_c_sa_pa_equivalences.md — complete SA↔PA correspondence.
• PA_BOOK/appendix_d_bibliography.md — sources and intellectual debts.

If You Only Want the Bottom Line

1. PA defines a formal space P in which coherent content, identities, and expressions live.
2. Three operators act on P: Collapse (Φ), Strip (S), Resonance (⊗). Pulsation τ generates time.
3. The Extended Round-Trip tests both the analysis and the original collapse.
4. The grammar is demonstrated across five domains and falsifiable through six explicit criteria.
5. PA is the formalisation of what the TE Collapse Function E = Φ(C, I, K) already implied.

If You Want the Single-File Version

PA_FULL/Proportional_Algebra_Foundations_UNIFIED.md contains the full content (~3500 lines, 20 chapters + 4 appendices).

If You Want to Run Code

pa_engine/ is a Python reference engine for the PA operators. Start with simulation_orchestrator.py.

Cross-Reference to SA

PA generalises SA. The SA → PA reading direction is operational-first (concrete tools), then formal. The PA → SA reading direction is foundational-first (the space and operators), then specialisation.

For the cross-correspondence, see PA_BOOK/appendix_c_sa_pa_equivalences.md.