gap45
gap45 is the natural number 45 that sets the base regulatory ceiling in the cross-domain model for 8-state recognition systems. Researchers applying Recognition Science to gene regulation would cite it to verify that the peak of 70 mutually consistent configurations fits under the doubled bound of 90. The declaration is a direct constant assignment with no reduction steps.
claimDefine the gap-45 ceiling as the natural number $45$.
background
The module treats the regulatory ceiling on the 8-element set Q3, where the central binomial coefficient reaches its maximum value of C(8,4)=70. This peak is shown to lie strictly below twice the gap-45 value. The local setting is the C9 structural claim that an 8-state regulatory module can sustain at most 70 half-activated configurations before decoherence.
proof idea
Direct definition that assigns the constant 45 to gap45.
why it matters in Recognition Science
The definition supplies the numerical bound used to confirm that the binomial peak of 70 fits inside the doubled ceiling of 90. It completes the C9 regulatory ceiling step that links the eight-tick octave (T7) to concrete regulatory predictions in Recognition Science. No open scaffolding remains in the module.
scope and limits
- Does not derive 45 from the phi-ladder or any upstream RS constant.
- Does not prove that C(8,4) equals 70.
- Does not extend the bound to sets larger than 8 elements.
formal statement (Lean)
29def gap45 : ℕ := 45