Axiomatics of the Closed Loop

Below is a closed axiomatic system formalizing the behavior of a "loop" in terms of Objects, Signs, Interpretants, Identity, and Human-in-the-loop judgment.

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Terminology and Symbols

Let the following sets and mappings be given:

• 𝒪 — the set of observable objects (Object);
• 𝒮 — the set of signs (Sign);
• 𝒥 — the set of interpretants (Interpretant);
• ℋ — the set of human agents authorized to judge;
• Σ — the set of observable signals (tests, tsc, lint, etc.);
• apply : 𝒮 × 𝒪 → 𝒪 — application of a sign to an object;
• validate : 𝒪 × 𝒥 → {0,1} or validate : 𝒪 × 𝒥 → [0,1] — validation of an interpretant against an object;
• gap : 𝒥 × 𝒪 → ℝ≥0 — a metric of tension (semantic gap);
• id : Node × Time → Atom ∪ {⊥} — identity (anchor) mapping over time;
• prov — a provenance log recording decisions, agents, time, and grounds.

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Axioms

A1 (Primacy of the Object)

All validity within the system is determined exclusively by the observable properties of the object.
For any statement P, its truth is evaluated only relative to 𝒪 and its signals Σ.
A statement belongs to the loop iff it has an anchor in 𝒪 (symbol, file, AST node, or validated compound).

A2 (Sign as Executable Transformation)

Every sign s ∈ 𝒮 is an executable operation apply(s,·).
A sign is not a description but an action: it is either applicable to a given object o ∈ 𝒪 or it is rejected.

A3 (Interpretant as Working Expectation)

Each interpretant i ∈ 𝒥 is a formalized expectation or contract, possibly partial, temporary, or conflicting.
Interpretants are not truths; they impose requirements on 𝒪 and induce tension gap(i,o).

A4 (Tension Defined Only via the Object)

Tension is meaningful only as a mismatch between:

• an interpretant i ∈ 𝒥 and an object o ∈ 𝒪 (semantic gap), or
• an identity anchor id and the object over time (identity gap).

Textual disagreement without reference to 𝒪 does not constitute tension.

A5 (Human-in-the-loop for Identity)

Decisions concerning identity (split, merge, retire, birth, etc.) are valid only after human judgment by some h ∈ ℋ,
except for narrowly defined automatic cases with strong provenance (e.g., unambiguous rename/move).

A6 (Locality and Incrementality)

All computations and stabilizations must be local:
each operation affects only the minimal subset of nodes it touches.
Global recomputation is permitted only by explicit user request.

A7 (Convergence by Exhaustion)

The system's goal is not to prove an ideal model but to exhaust change until a stable state is reached, where:

1. identity id is defined or explicitly retired;
2. all relevant gaps gap(i,o) are below defined thresholds;
3. observable signals Σ conform to the accepted policy.

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Closure Conditions

The system is closed if and only if the following conditions hold.

Z1 (Closure under Application)

For all s ∈ 𝒮 and o ∈ 𝒪, either apply(s,o) ∈ 𝒪 or s is explicitly rejected,
and in both cases a provenance record is written.

Z2 (Anchoring of Interpretants)

Any interpretant participating in the loop must have an anchor in 𝒪.
Unanchored interpretants are classified as external notes and excluded from stabilization.

Z3 (Definition of Tension)

The function gap(i,o) is defined only for anchored pairs.
"Disputes without anchors" are not actionable.

Z4 (Mandatory Recording of Decisions)

Every resolution of identity or semantic conflict must be recorded in prov, including agent, time, and grounds.

Z5 (Restricted Automation of Identity)

Automatic identity resolution is allowed only under strong provenance guarantees;
otherwise, human judgment is required.

Z6 (Bounded Global Computation)

Indexing and global recomputation are executed only via explicit Stabilize commands.
Default behavior is incremental and local.

Z7 (Fixed Convergence Thresholds)

The system maintains thresholds τ_identity and τ_semantic.
A state is considered stable if all gap(i,o) ≤ τ_semantic and identity is resolved or retired.

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Derived Rules (Operational Consequences)

The following rules are theorems derived from the axioms and closure conditions.

R1–R3 (Truth and Validation)

All accepted interpretants are anchored to 𝒪.
Every applied sign yields observable signals Σ, which serve as validation.
No conflict resolution is valid without a provenance record.

R4–R6 (Signs and Judgment)

A proposal is either an executable sign or a pure interpretant—never both.
Judgment operates on actions, not explanations.
Semantic stabilization is forbidden when identity issues remain unresolved.

R7–R9 (Interpretants)

Interpretants may be multiple and conflicting.
Each interpretant has metadata: facet, scope, and status.
An interpretant may exist as tension without immediate patch generation.

R10–R12 (Identity and Genealogy)

Permitted genealogy operations include:

• Same (1→1)
• Rename/Move (1→1)
• Split (1→many)
• Merge (many→1)
• Retire (1→⊥)
• Birth (⊥→1)

Split and merge require human judgment except in strong-provenance cases.
Resolved genealogy propagates automatically to all dependent compounds.

R13–R14 (Compounds and Responsibility)

Natural compounds (structural) and responsibility compounds (features) follow different sign policies.
Any change in compound composition triggers recomputation of its tension.

R15–R17 (Locality and Performance)

All background processes are incremental and budget-limited.
Heavy operations run only via explicit Stabilize commands.
The UI never blocks development flow; conflicts accumulate as diagnostics.

R18–R20 (Metrics and Thresholds)

The system defines escalation levels (Info, Warn, Action).
Higher uncertainty yields shorter, more precise context.
Convergence is "good enough," not necessarily zero.

R21 (Terminology Rule)

Two kinds of "diff" are distinguished:

• Δ (Delta) — semantic difference between interpretants;
• Patch — executable code diff (sign).

Relations:

• Δ → generates → Patch (often, but not always)
• Patch → changes → Object
• Object → validates → Δ

R22 (Minimal IDE Commands)

The MVP exposes exactly two commands:

• Stabilize Identity
• Stabilize Semantics

Semantic stabilization is disabled while Action-level identity issues exist.

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Implications

The system is axiomaticaly closed: every operation either transforms the object or is rejected;
tension arises only through anchored interpretants;
human judgment bounds identity resolution;
thresholds and explicit stabilization commands ensure practical convergence.