Axiom-0

Collapse is generative.
Only what returns is real.

Collapsus generativus est; solum quod redit, reale est.

Explore the Kernel ↓ Try the Calculator →

The Framework

What is GCD?

Generative Collapse Dynamics is a mathematical framework that measures what survives collapse and what returns. It begins from a single axiom — Axiom-0 — and derives a complete measurement protocol that works identically across 23 scientific domains, from particle physics to consciousness studies.

Within GCD, UMCP (Universal Measurement Contract Protocol) is the measurement functional system — alongside RCFT (which explores) and ULRC (which preserves language). Its frozen parameters are seam-derived — discovered by the mathematics, not chosen by convention.

At a Glance

20,540

Tests

Domains

Identities

Lemmas

246

Closures

Languages

Six Invariants, Three DOF

The Kernel

The kernel function K: [0,1]ⁿ × Δⁿ → (F, ω, S, C, κ, IC) maps any trace vector to six invariants — but only 3 effective degrees of freedom.

Click any invariant to learn what it measures, why it exists, and how it connects to the others.

Primitive — computed directly from the trace vector

Derived — follows algebraically from a primitive

How the Six Invariants Relate

4 are primitive (computed directly), 2 are derived (follow algebraically). Together, they have only 3 effective degrees of freedom because S is asymptotically determined by F and C.

Every Claim Follows This Path

The Spine

Every claim, validation, and discourse follows a fixed five-stop spine. Click each stop to see what happens there.

Derived from Gates, Never Asserted

Three Regimes

Regime classification is derived from gates on (ω, F, S, C) — never asserted. Click to explore each regime.

Fisher Space Partition

12.5%

24.4%

63.1%

Stability is rare — 87.5% of the manifold lies outside it. Return from collapse to stability is what the axiom measures.

Minimal Prose Vocabulary

The Five Words

The minimal prose vocabulary for telling the story of any collapse-return cycle. Click a word to see its ledger role and operational definition.

Separation of Powers

Two-Axis Architecture

GCD generates two orthogonal axes. The authority axis governs what may be changed; the functional axis governs what is being done. No system may both perform the work and rewrite the law under which the work is judged.

The Constitution — object before field, field before theory

Reditus · object → Structura Reditus · field → GCD · theory

Authority Axis — what may be changed

Every symbol, function, and artifact belongs to exactly one tier. Click to understand each layer.

One-Way Dependency (No Back-Edges)

Tier-1 → Tier-0 → Tier-2

No feedback from Tier-2 to Tier-1 or Tier-0 within a frozen run.

Functional Axis — what is being done

Three functional systems operate across all three tiers. A functional system is never a tier.

UMCP

measures

Universal Measurement Contract Protocol — the contract-first measurement system.

“Do not judge first. Freeze first.” → make return evaluable.

RCFT

explores

Recursive Collapse Field Theory — the discovery engine. It develops Tier-2 closures but is not a tier.

“Explore freely. Freeze carefully.” → make discovery traceable.

ULRC

preserves language

Unified Language of Recursive Collapse — the cross-tier language system.

“Translate without drift.” → make meaning return.

The two axes are orthogonal

Tiers govern authority · systems govern operation · the systems operate across the tiers, the tiers constrain what each system may alter

Compounding Parametrized Sweeps

20,540 Tests

Not 20,540 separate assertions — compounding parametrized sweeps across 23 domains. ~800 base test functions expand 18× through domain × entity × configuration parametrization.

How Test Compounding Works

~800

Base test functions

Written once per pattern

~18×

Parametrization factor

Domain × entity × config

20,540

Collected test items

Every invariant, every domain

Example: One Tier-1 identity test compounds across 23 domains

# One parametrized test function:

@pytest.mark.parametrize("domain", ALL_20_DOMAINS)

def test_duality_identity(domain):

traces = domain.get_all_entities() # 12–118 entities each

for trace in traces:

F, ω = kernel(trace)

assert abs(F + ω - 1) == 0.0 # Exact to 0.0e+00

→ 1 function × 23 domains × 12–118 entities = hundreds of assertions from one test

Test Distribution by Domain

Click a tier to see its test domains.

233 test files, numbered test_000 through test_338

Proto

Core

Domains

Verified to Machine Precision

44 Structural Identities

All derived from Axiom-0 and verified to machine precision. They fall into four series. Click a series to see its identities and what they prove.

6 Connection Clusters

The 44 identities form a connected network. These clusters reveal the deep structure.

Explore all 44 identities in detail →

Same Formulas, Three Implementations

Three Languages, One Kernel

The same kernel function K: [0,1]ⁿ × Δⁿ → (F, ω, S, C, κ, IC) is implemented in three languages — each serving a distinct architectural role. Click to explore each layer.

Three-Layer Sandwich Architecture

Python Domain closures · CLI · Dashboard · Fleet

C++17 Types · Validation · pybind11 zero-copy

C99 Raw math · Kernel · Seam · Pipeline

Same formulas, same frozen parameters — three layers of the same truth

Formal Foundation

47 Lemmas

The formal foundation beneath the kernel. Each lemma proves a property used by the implementation — tagged as OPT-* in the code. Click a category to explore.