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RangeDomain — Semantic Specification

1. Definition

A RangeDomain is a ValueDomain whose admissible values are numeric quantities.

It defines admissibility in terms of numeric constraints such as bounds, discreteness, and finiteness, without imposing units, distributions, or probabilistic meaning.

2. Declaration

A RangeDomain exists as a semantic specialization that:

  • Admits only numeric values
  • Applies numeric constraints deterministically
  • Rejects all non-numeric inputs

Minimal declaration requirements:

  • A numeric admissibility predicate
  • Explicit handling of numeric edge cases

No assumptions are made about physical interpretation or measurement units.

3. Semantic Invariants

These invariants extend all ValueDomain invariants.

N1: Numeric Exclusivity

Only numeric values are admissible.

Non-numeric inputs MUST be rejected.

N2: Finiteness

Admissible values MUST be finite.

NaN, +∞, and −∞ are inadmissible.

N3: Deterministic Comparison

Numeric comparisons MUST be deterministic and order-preserving.

N4: Stable Precision

Validation MUST NOT depend on machine-specific floating-point behavior beyond standard numeric comparison semantics.

N5: Explicit Bound Semantics

If bounds are defined, inclusivity or exclusivity MUST be explicit and stable.

4. Numeric Constraints

A RangeDomain MAY define zero or more of the following constraints.

All declared constraints MUST be enforced.

Range Constraints

  • Lower bound
  • Upper bound
  • Open or closed interval semantics

Discreteness

  • Continuous (ℝ-like)
  • Discrete (ℤ-like)
  • Explicit step size

Cardinality Constraints

  • Finite numeric sets
  • Enumerated numeric values

Sign Constraints

  • Non-negative
  • Positive
  • Non-positive
  • Negative

Absence of a constraint implies no restriction of that type.

5. Negative Definition

What RangeDomain is NOT.

  • Not a probability distribution
  • Not a sampler
  • Not a statistical model
  • Not a unit system
  • Not responsible for rounding or coercion
  • Not a validator of semantic correctness beyond numeric admissibility

6. Impossibilities

States or operations that cannot occur.

  • A RangeDomain cannot accept non-numeric values
  • A RangeDomain cannot coerce strings or booleans into numbers
  • A RangeDomain cannot normalize or clip values
  • A RangeDomain cannot change constraints after validation begins
  • A RangeDomain cannot depend on execution context

Any occurrence constitutes a semantic violation.

7. Boundary Conditions

Defined behavior at semantic limits.

Integer vs float

Integers and floats are both numeric but MUST be evaluated consistently.

Floating-point edge cases

NaN and infinities are always invalid.

Exact boundaries

Values exactly equal to bounds MUST follow declared inclusivity rules.

Degenerate ranges

Lower bound > upper bound is invalid by construction.

Empty numeric domain

A RangeDomain that admits no values is valid but degenerate.

8. Composition Rules

Semantic combination constraints.

Allowed

  • RangeDomain ∘ Variable
  • CompositeDomain(RangeDomain₁, RangeDomain₂)

Forbidden

  • RangeDomain ∘ CategoricalDomain
  • RangeDomain ∘ ProbabilityDistribution
  • RangeDomain ∘ History

Numeric domains constrain values; they do not interpret outcomes.

9. Example (Non-Normative)

Examples do not define semantics.

Domain: finite real numbers in [0, 1]

validate(0.0)   → admissible
validate(1.0)   → admissible
validate(-0.1)  → inadmissible
validate(NaN)   → inadmissible

10. Validation Rules

How violations are detected.

  • Static: Domains with undefined bounds or contradictory constraints are invalid.

  • Runtime: Supplying NaN, infinities, or non-numeric types is invalid.

  • Structural: Any RangeDomain that alters values during validation violates semantics.

11. Semantic Notes (Non-Binding)

Rationale and intent.

  • Establishes a clean numeric admissibility contract
  • Prevents implicit coercion and hidden rounding
  • Enables reliable numeric reasoning upstream
  • Provides a base for bounded, discrete, and real-valued domains
  • Avoids conflating admissibility with probabilistic modeling