ORDINATIVE SCIENCES EDUCATION · Repository ufficiale

Ordinative Algebras — Semantic Algebra (SA) and Proportional Algebra (PA)

Fabio Ghioni · Copia del 2026-09-18

Ordinative Algebras — Semantic Algebra (SA) and Proportional Algebra (PA)

English public mirror of two distinct formal frameworks developed within the Technology of Expressions / Ordinative Sciences research programme.

Two Frameworks, One Repository

This repository bundles two independent formal systems that share a structural relationship:

Framework Folder Focus
Semantic Algebra (SA) SA/ A formal method to extract universal structural invariants from natural language and re-project them into any target domain.
Proportional Algebra (PA) PA/ The formal space, metric, and operators that govern collapse from coherent content to expressed form across all domains — language, chemistry, music, emotion, medicine, AI, and more.

The relationship: PA Theorem 9.1 establishes that SA is mathematically a restriction of PA to the decoherent space D. This connects them formally without collapsing them into a single project. SA was developed first as a tool for AI training and cross-domain translation; PA was announced in the foundational TE text and provides the deeper theoretical structure.

For convenience, both are versioned and released together. They can be read independently.

Part of a Larger Ecosystem

This repository is one of four public repositories in the Ordinative Sciences programme:

Repository Purpose What you'll find there
ordinative_sciences_framework Theory The complete TE framework, core ontology, and operational modules.
te-ordinative-lora Practice Code, datasets, and scripts to fine-tune an LLM into a TE-compliant ordinative agent.
te-oct-framework-en Validation English mirror of the core framework, plus OCT (Ordinative Category Theory) datasets and benchmarks.
te-ordinative-algebras-en (this repo) Algebras The SA and PA formal frameworks — analytical operators and the proportional space they live in.

These repositories are designed to work together. Reading one in isolation can lead to incomplete understanding.

For a full map, see ECOSYSTEM.md.

First-Time Reader Shortcut

If this is your first visit, start here:

  1. START_HERE_FIRST_TIME.md
  2. SIMPLE_GLOSSARY.md
  3. SUPER_SIMPLE_FAQ.md

What Is in SA/

Semantic Algebra — an operator-based method that:

  1. Takes any natural-language expression and applies the Strip operator (S) to extract its structural content (invariant), removing field-specific vocabulary.
  2. Takes a known structural content and applies the Re-contextualization operator (π) to re-express it in any chosen target domain.
  3. Verifies that the round-trip S(π(I, D)) = I holds — a formal integrity test on the extraction.

Validated through:

  • A 7-text experiment across maximally distant domains (Lao Tzu, Shakespeare, Einstein, Rumi, Bhagavad Gita, Gödel, Ungaretti) producing an unprogrammed convergence.
  • A discrimination test on 4 expressions that simulate depth — 0 false positives.
  • A documented self-correction case (Ungaretti's M'illumino d'immenso), demonstrating the method's capacity to detect and correct projection errors.

Current invariant library: 10 validated invariants (ι₁ through ι₁₀).

What Is in PA/

Proportional Algebra — a formal grammar that:

  1. Defines the Proportional Space 𝒫 = (ℭ_h, 𝕀, 𝒟, ρ, ≤_𝓚) — the ground in which coherent content, identities, expressions, and their relations live.
  2. Defines three operators: Collapse (Φ), Strip (S), Resonance (⊗), and the Pulsation generator (τ) that produces time as an emergent quantity.
  3. Provides the Extended Round-Trip (ERT) — a four-step diagnostic that tests not only analytical fidelity (as SA does) but also genetic fidelity of the original collapse.
  4. States six explicit falsification criteria (F1–F6).

Demonstrated across five domains: chemistry, language, emotion, medicine, artificial intelligence.

A small Python reference engine implementing the PA operators is in PA/pa_engine/.

Important Disambiguation

  • "PA" in this repository means Proportional Algebra, not anything else (e.g., Pennsylvania, public address, etc.).
  • "Algebra" here refers to a structured grammar with operators and tests, not numerical algebra.
  • This is research material, not deployed software or a proven physical theory. Both frameworks include explicit falsification criteria and operational tests.

What You Will Find

  • Front-door files: START_HERE_FIRST_TIME.md, INDEX.md, SIMPLE_GLOSSARY.md, SUPER_SIMPLE_FAQ.md
  • Governance: LICENSE, CONTRIBUTING.md, CODE_OF_CONDUCT.md, SECURITY.md
  • Operational: CHANGELOG.md, RELEASE_CHECKLIST.md
  • Discovery: ECOSYSTEM.md, OBJECT_REGISTRY.md, object_registry.json, LOAD_PROFILES.md, PUBLICATION_SCOPE.md
  • SA/ — Semantic Algebra full corpus
  • PA/ — Proportional Algebra full corpus + reference engine
  • .github/ — issue and PR templates

What This Repository Is Not

  • Not a model-weight repository
  • Not a turnkey productized AI system
  • Not a claim of proven cross-domain unity without falsification evidence
  • Not LaTeX or print-ready material — the public mirror is Markdown-only by design (LaTeX projects exist privately and may be deposited separately on Zenodo)

Start Here (New Readers)

For SA-first reading:

  1. SA/README.md
  2. SA/START_HERE_SA.md
  3. SA/SA_BOOK/00_prologue.md
  4. SA/SA_BOOK/01_the_lossy_channel.md
  5. SA/SA_BOOK/06_the_strip_operator.md

For PA-first reading:

  1. PA/README.md
  2. PA/START_HERE_PA.md
  3. PA/PA_BOOK/01_why_the_sciences_cannot_speak.md
  4. PA/PA_BOOK/04_the_proportional_space.md
  5. PA/PA_BOOK/08_collapse_operator.md

Citation

To cite SA: Ghioni, F. Semantic Algebra: Foundations. Technology of Expressions — Ordinative Sciences. 2026.

To cite PA: Ghioni, F. Proportional Algebra: Foundations. Technology of Expressions — Ordinative Sciences. 2026.

Include the version tag (e.g., v1.0.0) and this repository URL.