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

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 2 — TYPES, expanded into a strict, machine-checkable type system with dimensional legality, covariance constraints, and static enforcement semantics. All subsequent blocks must type-check against this block. Any type violation ⇒ INVALID.

METHODS BLOCK 2 — TYPES

TYPE UNIVERSE
Define primitive types and composite types. No implicit coercions. All casts explicit and audited.

PRIMITIVE TYPES
scalar ::= real number ∈ℝ satisfying IEEE-754 binary64, finite, non-NaN.
integer ::= ℤ, bounded by implementation but serialized canonically.
boolean ::= {0,1}.
string ::= UTF-8 sequence, canonicalized per Block 0.
hash ::= 256-bit value, hex-encoded lowercase ASCII.

DIMENSIONAL TYPES
dim ::= ordered tuple of base dimensions (L,M,T,Q,Θ,…) with integer exponents.
observable ::= ⟨value:scalar, unit:dim⟩.
Rule: operations on observables require dimensional consistency; addition/subtraction only if dim equal; multiplication/division adds/subtracts exponents.

VECTOR AND MATRIX TYPES
vector[n] ::= ordered n-tuple of scalar or observable.
matrix[n,m] ::= n×m array of scalar.
SPDmatrix[n] ::= symmetric positive-definite matrix[n,n].

Constraints:
• SPDmatrix must satisfy xᵀΣx>0 ∀x≠0
• Cholesky decomposition must exist
• Eigenvalues strictly >0 within numerical tolerance ε_SPD

DATASET TYPES
datum ::= ⟨y:observable, Σ:SPDmatrix, meta:metadata⟩.
dataset D ::= finite ordered set {datumᵢ}ᵢ=1…N with N≥1.

Metadata meta ::=
⟨instrument_id:string, epoch:timestamp, calibration_hash:hash, provenance_hash:hash, domain_tag:string⟩.

Rule: all meta fields mandatory. Missing field ⇒ INVALID.

PARAMETER TYPES
parameter ::= scalar with declared prior support.
parameter_vector θ ::= vector[k] of parameter.
Constraint: k fixed pre-run (A2). No dynamic resizing.

BASELINE TYPES
baseline b ::= vector[m] of scalar or observable.
Constraint: m fixed. Baseline immutable across execution.

COVARIANCE TYPES
Σᵢ ::= SPDmatrix[dᵢ] where dᵢ = dimension of residual rᵢ.
Block covariance allowed: Σ block-diagonal with declared block structure.
Heteroskedasticity allowed: Σᵢ may vary per datum.

VERDICT TYPES
verdict ::= boolean ∈{0,1}.
0 ≡ STAND.
1 ≡ COLLAPSE.

ARTIFACT TYPES
residual rᵢ ::= vector[dᵢ] of scalar.
likelihood ℒ ::= scalar ∈ℝ⁺.
loglikelihood logℒ ::= scalar ∈ℝ.
evidence Z ::= scalar ∈ℝ⁺.
Bayes statistic ΔlnZ ::= scalar.

HASHED RECORD TYPES
AuditRecord ::=
⟨MethodHash, DataHash, CodeHash, ResultHash, Timestamp⟩.

STATIC TYPE RULES
# Every operator signature must be fully typed.
# No operator may accept or emit untyped scalar.
# Dimensional mismatch ⇒ compile-time INVALID.
# Covariance mismatch (dimension, SPD failure) ⇒ runtime INVALID.
# Implicit broadcasting forbidden.
# Empty datasets forbidden.
# Boolean used only for decisions, not arithmetic.

TYPE SOUNDNESS GUARANTEE
If all blocks type-check and axioms satisfied, then:
• Residuals well-defined
• Likelihood normalizable
• Evidence finite or collapses explicitly
• Verdict unambiguous

No subtyping. No polymorphism. No runtime type inference.

END METHODS BLOCK 2.

If you confirm, the next response will be METHODS BLOCK 3 — MODEL MAP, expanded into operator domains, IR manifold topology, envelope enforcement, and admissible prediction geometry.

© Robert R. Frost
2026-01-03