Editing Openai/69596681-3f7c-8006-b9fd-01d9d8191763 (section)

=== Assistant: Below is the Typed Operational Semantics Small-Step Rules for the ANF. ===
Below is the Typed Operational Semantics (Small-Step Rules) for the ANF. This is a machine-checkable, step-indexed transition system with explicit typing, guards, and failure absorption. No narrative. No new operators. All symbols frozen.

TYPES
Bool ::= {0,1}
Verdict ::= {STAND=0,COLLAPSE=1}
Status ::= {PASS,FAIL,SKIP,INVALID}
Hash ::= 256-bit
Seed ::= ℕ
Scalar ::= ℝ (finite, IEEE754)
Vector[n] ::= ℝⁿ
SPD[n] ::= symmetric positive-definite ℝ^{n×n}
Dataset ::= finite set of ⟨y:Vector,Σ:SPD,meta⟩
Spec ::= canonical serialized methods
Artifacts ::= finite map Block→Record
Flags ::= finite map Name→Bool

CONFIGURATION
A configuration is a tuple
⟨pc, S_spec, D, seed, A, F, V⟩
pc ∈ {B0,…,B19, HALT}
A : Artifacts
F : Flags
V : Verdict ∪ {⊥}

Initial configuration:
⟨B0, S_spec, D, seed, ∅, ∅, ⊥⟩

Terminal configurations:
⟨HALT, S_spec, D, seed, A, F, V⟩

AUXILIARY JUDGMENTS
Canonical(S_spec) ⇓ bytes
SHA256(bytes) ⇓ h:Hash
TypeCheck(S_spec,D) ⇓ Bool
DataValidity(D) ⇓ Bool
Residuals(S_spec,D,θ) ⇓ r,Σ
StructViolation(r,Σ,τ_struct) ⇓ Bool
Feasible(S_spec,D) ⇓ Bool
Identifiable(r,Σ) ⇓ (k_min,P_id)
Likelihood(r,Σ;ε,ν) ⇓ ℒ
Evidence(ℒ,π,Integrator) ⇓ (lnZ,ok)
BaselineEvidence ⇓ lnZ₀
MC(D,Π,N_MC,f_c) ⇓ Bool
Precision(D,Ψ,p_max) ⇓ Bool
BuildDAG(F) ⇓ G
AuditSeal(·) ⇓ Bundle

All judgments deterministic. Any undefined ⇒ INVALID.

SMALL-STEP TRANSITION RULES

(B0-HASH-OK)
If Canonical(S_spec)⇓b ∧ SHA256(b)=embedded_hash
———————————————————————————————
⟨B0,S_spec,D,seed,A,F,V⟩ → ⟨B1,S_spec,D,seed,A∪{B0↦PASS},F,V⟩

(B0-HASH-FAIL)
If SHA256(Canonical(S_spec))≠embedded_hash
———————————————————————————————
⟨B0,…⟩ → ⟨HALT,…,A∪{B0↦FAIL},F∪{ValidityFail↦1},COLLAPSE⟩

(B1-AXIOMS-OK)
If ∀Ai:Ai=true
———————————————————————————————
⟨B1,…⟩ → ⟨B2,…,A∪{B1↦PASS},F,V⟩

(B1-AXIOMS-FAIL)
If ∃Ai=false
———————————————————————————————
⟨B1,…⟩ → ⟨HALT,…,A∪{B1↦FAIL},F∪{ValidityFail↦1},COLLAPSE⟩

(B2-TYPES-OK)
If TypeCheck(S_spec,D)=1
———————————————————————————————
⟨B2,…⟩ → ⟨B3,…,A∪{B2↦PASS},F,V⟩

(B2-TYPES-FAIL)
If TypeCheck(S_spec,D)=0
———————————————————————————————
⟨B2,…⟩ → ⟨HALT,…,A∪{B2↦FAIL},F∪{ValidityFail↦1},COLLAPSE⟩

(B3-MODEL)
———————————————————————————————
⟨B3,…⟩ → ⟨B9,…,A∪{B3↦PASS},F,V⟩

(B9-DATA-OK)
If DataValidity(D)=1
———————————————————————————————
⟨B9,…⟩ → ⟨B4,…,A∪{B9↦PASS},F,V⟩

(B9-DATA-FAIL)
If DataValidity(D)=0
———————————————————————————————
⟨B9,…⟩ → ⟨B12,…,A∪{B9↦FAIL},F∪{ValidityFail↦1},COLLAPSE⟩

(B4-RESIDUALS)
If Residuals(S_spec,D,θ)⇓(r,Σ)
———————————————————————————————
⟨B4,…⟩ → ⟨B5,…,A∪{B4↦PASS},F∪{ResidualsOK↦1},V⟩

(B5-STRUCT-FAIL)
If StructViolation(r,Σ,τ_struct)=1
———————————————————————————————
⟨B5,…⟩ → ⟨B12,…,A∪{B5↦FAIL},F∪{V_struct↦1},COLLAPSE⟩

(B5-STRUCT-OK)
If StructViolation(r,Σ,τ_struct)=0
———————————————————————————————
⟨B5,…⟩ → ⟨B6,…,A∪{B5↦PASS},F∪{V_struct↦0},V⟩

(B6-FEAS-FAIL)
If Feasible(S_spec,D)=0
———————————————————————————————
⟨B6,…⟩ → ⟨B12,…,A∪{B6↦FAIL},F∪{FeasibleFail↦1},COLLAPSE⟩

(B6-FEAS-OK)
If Feasible(S_spec,D)=1
———————————————————————————————
⟨B6,…⟩ → ⟨B7,…,A∪{B6↦PASS},F∪{FeasibleFail↦0},V⟩

(B7-IDENT-FAIL)
If Identifiable(r,Σ)⇓(0,_)
———————————————————————————————
⟨B7,…⟩ → ⟨B12,…,A∪{B7↦FAIL},F∪{FeasibleFail↦1},COLLAPSE⟩

(B7-IDENT-OK)
If Identifiable(r,Σ)⇓(k_min,P_id), k_min≥1
———————————————————————————————
⟨B7,…⟩ → ⟨B8,…,A∪{B7↦PASS},F∪{k_minOK↦1},V⟩

(B8-LIKE)
If Likelihood(r,Σ;ε,ν)⇓ℒ
———————————————————————————————
⟨B8,…⟩ → ⟨B11,…,A∪{B8↦PASS},F,V⟩

(B11-INT-FAIL)
If Evidence(ℒ,π,Integrator)⇓(_,ok) ∧ ok=0
———————————————————————————————
⟨B11,…⟩ → ⟨HALT,…,A∪{B11↦FAIL},F∪{ValidityFail↦1},COLLAPSE⟩

(B11-INT-OK)
If Evidence(ℒ,π,Integrator)⇓(lnZ,1)
———————————————————————————————
⟨B11,…⟩ → ⟨B10,…,A∪{B11↦PASS,lnZ↦lnZ},F,V⟩

(B10-EVID-FAIL)
If lnZ−lnZ₀ < Λ
———————————————————————————————
⟨B10,…⟩ → ⟨B12,…,A∪{B10↦FAIL},F∪{EvidenceFailure↦1},COLLAPSE⟩

(B10-EVID-OK)
If lnZ−lnZ₀ ≥ Λ
———————————————————————————————
⟨B10,…⟩ → ⟨B12,…,A∪{B10↦PASS},F∪{EvidenceFailure↦0},V⟩

(B12-STAND)
If all failure flags =0
———————————————————————————————
⟨B12,…⟩ → ⟨B13,…,A∪{B12↦PASS},F,STAND⟩

(B12-COLLAPSE)
If any failure flag =1
———————————————————————————————
⟨B12,…⟩ → ⟨B15,…,A∪{B12↦FAIL},F,COLLAPSE⟩

(B13-MC-FAIL)
If MC(D,Π,N_MC,f_c)=1
———————————————————————————————
⟨B13,…⟩ → ⟨B14,…,A∪{B13↦FAIL},F∪{RobustnessFailure↦1},V⟩

(B13-MC-OK)
If MC(D,Π,N_MC,f_c)=0
———————————————————————————————
⟨B13,…⟩ → ⟨B14,…,A∪{B13↦PASS},F∪{RobustnessFailure↦0},V⟩

(B14-PREC-FAIL)
If Precision(D,Ψ,p_max)=1
———————————————————————————————
⟨B14,…⟩ → ⟨B12,…,A∪{B14↦FAIL},F∪{PrecisionFailure↦1},COLLAPSE⟩

(B14-PREC-OK)
If Precision(D,Ψ,p_max)=0
———————————————————————————————
⟨B14,…⟩ → ⟨B15,…,A∪{B14↦PASS},F,V⟩

(B15-ATTRIB)
If BuildDAG(F)⇓G
———————————————————————————————
⟨B15,…⟩ → ⟨B16,…,A∪{B15↦PASS,G↦G},F,V⟩

(B16-CERT)
If V=STAND ∧ certification conditions hold
———————————————————————————————
⟨B16,…⟩ → ⟨B19,…,A∪{B16↦PASS,CERT↦1},F,V⟩

(B16-SKIP)
If V=COLLAPSE
———————————————————————————————
⟨B16,…⟩ → ⟨B19,…,A∪{B16↦SKIP},F,V⟩

(B19-AUDIT)
If AuditSeal(A,F,V)⇓Bundle
———————————————————————————————
⟨B19,…⟩ → ⟨HALT,…,A∪{B19↦PASS,Audit↦Bundle},F,V⟩

META-PROPERTIES (DERIVABLE)

Progress: from initial state, either INVALID or reaches HALT.
Determinism: single successor for any non-HALT state.
Safety: once V=COLLAPSE, V never changes.
Monotonicity: adding data cannot enable a reverse transition to STAND.
Replayability: identical inputs yield identical transition traces.

This completes the typed small-step operational semantics for the ANF.
If desired, the next step is to emit a machine-checkable version (e.g., Coq/Lean rules, TLA⁺ spec, or executable interpreter skeleton).

© Robert R. Frost
2026-01-03