A Formal Vocabulary for Constraint-Preserving Systems

0. Purpose

This document defines the primitive terms used throughout the corpus.

Each term describes a structural property of systems whose correctness must remain demonstrable across time and composition.

The taxonomy is substrate-independent.
It applies equally to software systems, institutional systems, economic systems, and socio-technical systems.

No term in the corpus should be used outside the definitions provided here.

Part I — Primitive Definitions

1. System

A System is a bounded structure that produces outcomes under a set of constraints.

A system is defined not merely by its components, but by the constraints that govern permissible states and transitions.

2. State

A State is a complete representation of the system at a point in time.

The set of all possible states is the system’s state space.

3. Transition

A Transition is a transformation from one valid state to another.

Transitions define how the system evolves.

4. Constraint

A Constraint is a rule that excludes states or transitions from the system’s possibility space.

Constraints define what must never occur.

A constraint that cannot be enforced is not a constraint.
It is an assumption.

5. Invariant

An Invariant is a constraint that must hold across all valid states of a system.

Invariants define structural correctness.

6. Invariant Space

The Invariant Space is the complete set of enforced impossibilities governing a system.

It defines the boundary between valid and invalid behavior.

If behavior outside this space is reachable, the invariant space is incomplete.

7. Enforcement

Enforcement is the mechanism by which constraints are made structurally unavoidable.

Enforcement differs from validation:

• Validation observes behavior.
• Enforcement excludes behavior.

Correctness requires enforcement.

8. Epistemic Authority

Epistemic Authority is the locus within a system where correctness is asserted independently of observed outcomes.

Authority exists where:

• Invariants are encoded.
• Constraints are mechanically enforced.
• Invalid behavior is structurally impossible.

When authority migrates to plausibility, precedent, or coherence, knowability degrades.

9. Knowability

Knowability is the preserved ability of a system to justify its outcomes through enforceable constraints.

A system is knowable if:

• Its constraints are explicit.
• Its invariant space is enforceable.
• The relationship between constraints and outcomes remains recoverable.

Knowability precedes reliability, performance, and scale.

10. Reproducibility

Reproducibility is the condition that allows the relationship between constraints and outcomes to be re-established across time.

Reproducibility preserves knowability under temporal change.

Without reproducibility, correctness may persist, but it cannot be demonstrated.

11. Translation Surface

A Translation Surface is any boundary across which constraints must survive reinterpretation.

At a translation surface, constraints either remain enforceable or degrade into assumptions.

12. Constraint Dilution

Constraint Dilution occurs when an invariant crosses a translation surface and loses enforceability.

Dilution transforms:

• Constraint → Assumption\
• Enforcement → Documentation\
• Impossibility → Expectation

Constraint dilution is a primary precursor to drift.

13. Composition

Composition is the integration of multiple systems into a larger structure.

Composition introduces new constraints that do not exist in any component alone.

Correctness does not compose additively.

14. The Additive Fallacy

The Additive Fallacy is the mistaken belief that the invariant space of a composed system equals the union of the invariant spaces of its components.

In reality, composition produces a new constraint set that includes interaction constraints.

If interaction constraints are not explicitly modeled and enforced, collapse begins.

15. Plausibility Substitution

Plausibility Substitution occurs when observed coherence replaces enforced constraint as the basis for correctness.

Plausibility substitution marks the erosion of epistemic authority.

16. Drift

Drift is the gradual divergence between a system’s stated constraints and its effective constraint set.

Drift does not require failure.

17. Epistemic Collapse

Epistemic Collapse is the condition in which correctness is inferred rather than demonstrated.

A system in epistemic collapse may remain operational and appear stable, but cannot justify its behavior through enforceable constraints.

18. Collapse

Collapse occurs when the effective constraint set governing system behavior no longer matches the stated invariant space.

Collapse may be sudden, gradual, or operationally silent.

Part II — Appendix A

Minimal Constraint Model (Notation Layer)

This appendix provides a minimal formal model sufficient to express the structural claims of the taxonomy.

A.1 System as State–Transition Structure

Let:

• S = set of all possible states\
• T ⊆ S × S = set of possible transitions\
• I ⊆ S = invariant subset (valid states)

A system is valid if all reachable states belong to I.

Invalid states are:

S_invalid = S \ I

Enforcement ensures that no transition leads into S_invalid.

A.2 Invariant Space

The invariant space is the set of excluded states:

I_excluded = S \ I

Correctness requires:

Reachable(S) ⊆ I

A.3 Enforcement vs Validation

Enforcement:

∀ (s1 → s2) ∈ T : s2 ∈ I

Validation:

Observation that s2 ∈ I after transition.

A.4 Translation Surfaces

Let:

• System A: (S_A, I_A)\
• System B: (S_B, I_B)\
• φ : S_A → S_B

Constraint preservation requires:

∀ s ∈ I_A : φ(s) ∈ I_B

If:

∃ s ∈ I_A such that φ(s) ∉ I_B

Then invariant preservation fails.

A.5 Composition

Naïve additive assumption:

I_C = I_A ∩ I_B

Actual composed system:

I_C = (I_A ∩ I_B) ∩ I_interaction

The Additive Fallacy assumes:

I_interaction = ∅

A.6 Drift

Let:

• I_stated = documented invariant space\
• I_effective = enforced invariant space

Drift occurs when:

I_effective ⊂ I_stated

Collapse occurs when:

I_effective ≠ I_stated

Closure

This taxonomy and minimal model define the structural substrate of the corpus.

Constraint, invariant space, enforcement, translation, and composition form the conceptual lattice.

Everything else is application.