triple_product_125
plain-language theorem explainer
Any three finite types each of cardinality 5 have Cartesian product of size 125. Cross-domain researchers cite the result when confirming that D=5 structures multiply identically across sensory, emotional, and biological domains. The proof is a direct simplification after unfolding the cardinality predicate for each factor.
Claim. If $A$, $B$, and $C$ are finite sets each with cardinality 5, then $|A × B × C| = 125$.
background
HasConfigDim5 holds for a type T precisely when T is finite and Fintype.card T = 5. The module formalizes D=5 universality across domains, stating that any three such types produce a product of size 125 and that every pair is equicardinal. Upstream results supply the HasConfigDim5 definition and place related structures on phi-tiers via NucleosynthesisTiers and PhiForcingDerived.
proof idea
The term proof unfolds HasConfigDim5 at each hypothesis then applies simp with Fintype.card_prod and the resulting equalities, reducing the product cardinality directly to 555.
why it matters
The theorem supplies the general step used by three_domain_product for the concrete triple SensoryModality × PrimaryEmotion × BiologicalState and by configDimUniversalityCert. It fills the cross-domain product clause in the C13 universality statement. Within the framework it reinforces the observed prevalence of D=5 structures alongside the phi-ladder and eight-tick constructions.
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