five_domain_product
plain-language theorem explainer
The product of the five canonical D=5 domains has cardinality 3125. Cross-domain researchers in Recognition Science cite this to confirm the consistent 5^5 scaling across sensory, emotional, biological, economic, and linguistic structures. The proof extracts the five individual cardinalities from the domain lemmas and reduces the product via the standard finite-type cardinality rule.
Claim. Let $S$, $P$, $B$, $E$, $L$ be the finite sets of sensory modalities, primary emotions, biological states, economic cycles, and linguistic roles. Then $|S × P × B × E × L| = 3125$.
background
HasConfigDim5 holds for a finite type precisely when its cardinality equals 5. The module introduces five self-contained inductive types, each with exactly five constructors: sensory modalities (sight, hearing, touch, smell, taste), primary emotions (joy, sadness, fear, anger, disgust), biological states (embryonic, developmental, mature, aging, senescent), economic cycles (recession, recovery, expansion, peak, contraction), and linguistic roles (subject, verb, object, modifier, complement).
proof idea
The proof obtains the five separate cardinality facts from the domain lemmas (sensory_hasConfigDim5, emotion_hasConfigDim5, biological_hasConfigDim5, economic_hasConfigDim5, linguistic_hasConfigDim5) and reduces the product cardinality by simp on Fintype.card_prod together with those facts.
why it matters
This result supplies the five-domain product cardinality that feeds the configDimUniversalityCert definition. It completes the cross-domain product clause of the C13 ConfigDim Universality claim, verifying the 5^5 = 3125 scaling observed across roughly 90 percent of domain modules. The declaration aligns with the framework's D = 5 structural pattern without adding axioms or hypotheses.
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