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Exact support cost and structured table families

Abstract

The quaternary coding gives an exact capacity count. A smaller structured generator is a genuine model restriction unless the omitted tables already have zero mass or are excluded by justified cross-stratum structure.

Definition 1.1 (Cover every positive-mass table).

Lean statement: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.SupportsLaw

Formalization. D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.SupportsLaw (✓ std3).

Source. Repository-derived.

Commentary.

A deterministic latent-state generator is exact for a law only when every positive-mass atom is produced by some latent state.

Theorem 1.2 (Universal table generation needs 4^k states).

Lean statement: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.surjective_response_table_generator_requires_four_pow

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.surjective_response_table_generator_requires_four_pow (✓ std3). ∎

Source. Repository-derived.

Commentary.

Surjectivity onto all k-row Boolean response tables forces at least 4^k deterministic latent states by finite cardinality.

Theorem 1.3 (Every smaller family omits a table).

Lean statement: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.small_generator_not_universal

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.small_generator_not_universal (✓ std3). ∎

Source. Repository-derived.

Commentary.

Any latent family below the radix capacity fails to cover the unrestricted table carrier. This is a support statement, not a DFAO-state lower bound.

Theorem 1.4 (Twenty-one latent atoms cannot cover three unrestricted rows).

Lean statement: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.twenty_one_latent_states_not_universal_three_rows

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.twenty_one_latent_states_not_universal_three_rows (✓ std3). ∎

Source. Repository-derived.

Commentary.

Three quaternary rows already give 64 possible tables, so a 21-element latent carrier is not universal. The number 21 mirrors the current golden powers-only draft scale solely to make the complexity distinction explicit; no automaton-minimality conclusion follows.

Theorem 1.5 (Positive row kernels fill the full table space).

Lean statement: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.independentResponseTable_full_support

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.independentResponseTable_full_support (✓ std3). ∎

Source. Repository-derived.

Commentary.

Strict positivity of all four row-response masses makes every complete table have positive product mass.

Theorem 1.6 (Full-support probability laws inherit the capacity lower bound).

Lean statement: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.positive_independent_table_law_generator_lower_bound

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.positive_independent_table_law_generator_lower_bound (✓ std3). ∎

Source. Repository-derived.

Commentary.

An exact deterministic latent generator of the positive independent-row law must cover every table and therefore needs at least 4^k states.

Theorem 1.7 (Compression requires a restricted model family).

Lean statement: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.positive_independent_table_law_not_supported_by_small_generator

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.positive_independent_table_law_not_supported_by_small_generator (✓ std3). ∎

Source. Repository-derived.

Commentary.

A smaller latent carrier cannot reproduce a full-support independent-row law exactly. Automata can still give short algorithmic descriptions of structured tables because that is a different complexity notion.

References

  • Truth anchor: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.SupportsLaw
  • Truth anchor: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.independentResponseTable_full_support
  • Truth anchor: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.positive_independent_table_law_generator_lower_bound
  • Truth anchor: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.positive_independent_table_law_not_supported_by_small_generator
  • Truth anchor: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.small_generator_not_universal
  • Truth anchor: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.surjective_response_table_generator_requires_four_pow
  • Truth anchor: D5/S3/ConceptDynamics/CausalMoments/StructuredResponseTableSupport.twenty_one_latent_states_not_universal_three_rows
  • Dependency: D5/S3/ConceptDynamics/CausalMoments/QuaternaryResponseTableCoding
  • Dependency: D5/S3/ConceptDynamics/PartialIdentification/FiniteIndependentSourceGrouping