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IndisputableMonolith.CrossDomain.ProductRecognitionLattice

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This module supplies concrete arithmetic lemmas on powers of five, centered on the bound 5^6 < 2^14, to certify that cognitive-oncology joint states fit inside a 14-bit encoding inside the product recognition lattice. Cross-domain researchers cite these facts when sizing state spaces or verifying bit budgets. The module consists of direct computational lemmas discharged by Mathlib arithmetic tactics.

claim$5^6 < 2^{14}$ together with the auxiliary facts $5^n = 5 imes 5^{n-1}$ for $n=2, o,8$ and the companion inequalities $5^5 < 2^{13}$, $5^4 < 2^{10}$, $5^3 < 2^7$, $5^2 < 2^5$.

background

The module lives in the CrossDomain.ProductRecognitionLattice namespace and imports only Mathlib. It introduces the family of definitions five_pow_n (n = 2 to 8) that compute successive powers of 5 and the named inequalities that bound those powers against powers of 2. The supplied module doc-comment states the intended use: the bound 5^6 < 2^14 shows that a cognitive-oncology joint state fits in 14 bits.

proof idea

This is a lemma module whose proofs are one-line wrappers that apply norm_num or decide. Each five_pow_n is introduced by a direct computation or recursive unfolding; each inequality is discharged by evaluating both sides numerically inside Mathlib.

why it matters in Recognition Science

The module supplies the numerical anchors required by the product recognition lattice construction. It directly discharges the bit-width claim stated in the module doc-comment and thereby feeds any downstream cross-domain theorem that must reason about state encoding size.

scope and limits

declarations in this module (24)