Theorems — Generative Collapse Dynamics

Every theorem derives from Axiom-0. Classical results emerge as degenerate limits.

31 theorems across 5 modules

Catalog

ID	Name	Module
`T-CT-1`	Two types of collapse produce separable (Δ, C) signatures.	`collapse_taxonomy`
`T-CT-2`	IC/F measures integration quality, independent of F magnitude.	`collapse_taxonomy`
`T-CT-3`	Symmetry breaking = selective channel death, universally.	`collapse_taxonomy`
`T-CT-4`	Protected systems cluster at F ∈ [c*, 1] with IC/F > 0.95.	`collapse_taxonomy`
`T-CT-5`	Return is structurally possible iff collapse is selective.	`collapse_taxonomy`
`T-CT-6`	The triple (IC/F, Δ, C) is a complete diagnostic gauge.	`collapse_taxonomy`
`T-SI-1`	The kernel is a stateless function; time enters only via the budget.	`emergent_structural_insights`
`T-SI-2`	The equator c = 1/2 is a saddle point of S + κ.	`emergent_structural_insights`
`T-SI-3`	Channel death cascades exponentially: IC ∝ exp(−β · n_dead / n).	`emergent_structural_insights`
`T-SI-4`	The Fisher-weighted entropy integral equals 2ζ(2) = π²/3.	`emergent_structural_insights`
`T-SI-5`	F + ω = 1 is the only exact symmetry; all others break by design.	`emergent_structural_insights`
`T-SI-6`	The Fisher metric is flat (g_F = 1); all complexity is extrinsic.	`emergent_structural_insights`
`T-KS-1`	Dimensionality Fragility Law.	`kernel_structural_theorems`
`T-KS-2`	Positional Democracy of Slaughter.	`kernel_structural_theorems`
`T-KS-3`	Weight-Induced Fragility Hierarchy.	`kernel_structural_theorems`
`T-KS-4`	Monitoring Paradox (Quantified).	`kernel_structural_theorems`
`T-KS-5`	Approximation Boundary.	`kernel_structural_theorems`
`T-KS-6`	U-Curve of Degradation.	`kernel_structural_theorems`
`T-KS-7`	p=3 Unification Web.	`kernel_structural_theorems`
`T-RF-1`	Architectural Integrity Bound.	`reflexive_closure`
`T-RF-2`	Concentration Fragility (Geometric Slaughter of Architectures).	`reflexive_closure`
`T-RF-3`	Substrate Invariance of Epistemic Evaluation.	`reflexive_closure`
`T-RF-4`	Epistemic Channel Death.	`reflexive_closure`
`T-RF-5`	Reflexive Fixed Point.	`reflexive_closure`
`T-URC-1`	Tier-1 Kernel Identities.	`universal_regime_calibration`
`T-URC-2`	Universal Regime Concordance.	`universal_regime_calibration`
`T-URC-3`	Super-Exponential Repair.	`universal_regime_calibration`
`T-URC-4`	IC–Channel Vulnerability.	`universal_regime_calibration`
`T-URC-5`	Cross-Domain F Separation.	`universal_regime_calibration`
`T-URC-6`	Curvature–Regime Entailment.	`universal_regime_calibration`
`T-URC-7`	Entropy–Regime Correlation.	`universal_regime_calibration`

Collapse Taxonomy

Source: closures/gcd/collapse_taxonomy.py

T-CT-1: Two types of collapse produce separable (Δ, C) signatures.

Function: theorem_TCT1_collapse_typing()
Module: collapse_taxonomy

T-CT-2: IC/F measures integration quality, independent of F magnitude.

Function: theorem_TCT2_integration_magnitude()
Module: collapse_taxonomy

T-CT-3: Symmetry breaking = selective channel death, universally.

Function: theorem_TCT3_selective_death_universality()
Module: collapse_taxonomy

T-CT-4: Protected systems cluster at F ∈ [c*, 1] with IC/F > 0.95.

Function: theorem_TCT4_protected_basin()
Module: collapse_taxonomy

T-CT-5: Return is structurally possible iff collapse is selective.

Function: theorem_TCT5_return_predicate()
Module: collapse_taxonomy

T-CT-6: The triple (IC/F, Δ, C) is a complete diagnostic gauge.

Function: theorem_TCT6_gauge_completeness()
Module: collapse_taxonomy

Emergent Structural Insights

Source: closures/gcd/emergent_structural_insights.py

T-SI-1: The kernel is a stateless function; time enters only via the budget.

Function: theorem_TSI1_stateless_mirror()
Module: emergent_structural_insights

T-SI-2: The equator c = 1/2 is a saddle point of S + κ.

Function: theorem_TSI2_equator_saddle()
Module: emergent_structural_insights

T-SI-3: Channel death cascades exponentially: IC ∝ exp(−β · n_dead / n).

Function: theorem_TSI3_cascading_channel_death()
Module: emergent_structural_insights

T-SI-4: The Fisher-weighted entropy integral equals 2ζ(2) = π²/3.

Function: theorem_TSI4_fisher_entropy_zeta_bridge()
Module: emergent_structural_insights

T-SI-5: F + ω = 1 is the only exact symmetry; all others break by design.

Function: theorem_TSI5_symmetry_hierarchy()
Module: emergent_structural_insights

T-SI-6: The Fisher metric is flat (g_F = 1); all complexity is extrinsic.

Function: theorem_TSI6_flat_base_extrinsic()
Module: emergent_structural_insights

Kernel Structural Theorems

Source: closures/gcd/kernel_structural_theorems.py

T-KS-1: Dimensionality Fragility Law.

Function: theorem_TKS1_dimensionality_fragility()
Module: kernel_structural_theorems

T-KS-2: Positional Democracy of Slaughter.

Function: theorem_TKS2_positional_democracy()
Module: kernel_structural_theorems

T-KS-3: Weight-Induced Fragility Hierarchy.

Function: theorem_TKS3_weight_fragility()
Module: kernel_structural_theorems

T-KS-4: Monitoring Paradox (Quantified).

Function: theorem_TKS4_monitoring_paradox()
Module: kernel_structural_theorems

T-KS-5: Approximation Boundary.

Function: theorem_TKS5_approximation_boundary()
Module: kernel_structural_theorems

T-KS-6: U-Curve of Degradation.

Function: theorem_TKS6_u_curve()
Module: kernel_structural_theorems

T-KS-7: p=3 Unification Web.

Function: theorem_TKS7_p3_unification()
Module: kernel_structural_theorems

Reflexive Closure

Source: closures/gcd/reflexive_closure.py

T-RF-1: Architectural Integrity Bound.

Function: theorem_TRF1_architectural_integrity_bound()
Module: reflexive_closure

T-RF-2: Concentration Fragility (Geometric Slaughter of Architectures).

Function: theorem_TRF2_concentration_fragility()
Module: reflexive_closure

T-RF-3: Substrate Invariance of Epistemic Evaluation.

Function: theorem_TRF3_substrate_invariance()
Module: reflexive_closure

T-RF-4: Epistemic Channel Death.

Function: theorem_TRF4_epistemic_channel_death()
Module: reflexive_closure

T-RF-5: Reflexive Fixed Point.

Function: theorem_TRF5_reflexive_fixed_point()
Module: reflexive_closure

Universal Regime Calibration

Source: closures/gcd/universal_regime_calibration.py

T-URC-1: Tier-1 Kernel Identities.

Function: theorem_T_URC_1_tier1()
Module: universal_regime_calibration

T-URC-2: Universal Regime Concordance.

Function: theorem_T_URC_2_regime_concordance()
Module: universal_regime_calibration

T-URC-3: Super-Exponential Repair.

Function: theorem_T_URC_3_super_exponential_repair()
Module: universal_regime_calibration

T-URC-4: IC–Channel Vulnerability.

Function: theorem_T_URC_4_channel_vulnerability()
Module: universal_regime_calibration

T-URC-5: Cross-Domain F Separation.

Function: theorem_T_URC_5_F_separation()
Module: universal_regime_calibration

T-URC-6: Curvature–Regime Entailment.

Function: theorem_T_URC_6_curvature_entailment()
Module: universal_regime_calibration

T-URC-7: Entropy–Regime Correlation.

Function: theorem_T_URC_7_entropy_correlation()
Module: universal_regime_calibration

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