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def definition def or abbrev high

entrainmentConfidence

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The definition assigns to each natural number k the real value 1/phi^k, which gives the relative entrainment confidence at rung k on the phi-ladder. Device engineers specifying transcranial neuromodulation hardware cite it when stating the pulse schedule for 5 phi Hz cortical resonance. It is introduced as a direct one-line abbreviation with no further computation.

claimFor each rung $k$ in the natural numbers, the entrainment confidence is defined by $c(k) = 1/phi^k$, where $phi$ denotes the golden ratio fixed point.

background

The Cortical Neuromodulation Device module derives the engineering specification for a transcranial neuromodulation device (RS_PAT_025) that targets cortical-column resonance at carrier frequency 5 phi Hz, lying inside the interval (8.05, 8.10) Hz, together with optimal pulse spacing tau approximately 124 ms. The phi-ladder is the discrete scaling sequence generated by successive powers of phi, the self-similar fixed point forced by the Recognition Composition Law and the J-uniqueness property in the upstream forcing chain. This definition supplies the explicit scaling factor for confidence levels measured relative to the baseline value 1 at rung zero.

proof idea

One-line definition that directly encodes the phi-rung scaling as 1/phi^k.

why it matters in Recognition Science

The definition supplies the numerical values required by the CorticalNeuromodulationDeviceCert structure and by the neuromod_one_statement theorem, which packages the carrier band, spacing band, and the strict anti-monotonicity property of the ladder. It thereby completes the engineering derivation track that begins from the phi fixed point (T6) and the eight-tick octave. No open scaffolding remains inside the module.

scope and limits

formal statement (Lean)

  62def entrainmentConfidence (k : ℕ) : ℝ := 1 / phi ^ k

proof body

Definition body.

  63

used by (6)

From the project-wide theorem graph. These declarations reference this one in their body.