Information Bandwidth Phi Rs Bandwidth Phi Cert

The bandwidth certificate

Shannon's channel capacity theorem says a communication channel of bandwidth B hertz can carry at most C = B·log₂(1+SNR) bits per second, where SNR is the signal-to-noise ratio. The Recognition Science framework defines a cost function, a measure of how far one quantity is from another, and its machine-checked library of formal theorems contains a declaration named BandwidthPhiCert. That name suggests a result about bandwidth, but the certificate itself proves something narrower and more general.

The certificate packages three facts about the cost function J(x) = (x + 1/x)/2 − 1. First, when the two quantities are equal, the cost is zero. Second, for positive inputs, the cost is never negative. Third, the number φ − 3/2 is positive, where φ is the golden ratio. These are the three fields of the certificate structure, and a proof that they hold is provided. The three facts are general properties of the cost function; they mention no channel, no bandwidth, and no signal-to-noise ratio.

The docstring above the declaration records an intended application: at a phi-resonance, the signal-to-noise ratio would be φ³ ≈ 4.24, or about 6.27 dB, giving a channel capacity of B·log₂(φ³+1) ≈ B·2.39 bits/Hz. That paragraph is a research note, not a proved result. The Lean code proves nothing specific to bandwidth, because the cost function is defined as J(m/e) without any definition of what m and e mean in this context. The declaration is shared verbatim with 2383 sibling modules, each with a different subject name but the same three general facts.

What the certificate does establish is that the cost function has the three listed properties, and that these properties are consistent, since the certificate type is inhabited. This matters because the cost function is the foundation of the framework's forcing chain: if the cost function failed to be nonnegative or failed to vanish at equality, the framework's later results would be suspect. The certificate is a small but real check on that foundation. It does not, however, turn the bandwidth note into a theorem.

structure BandwidthPhiCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

No theorem about channel capacity or bandwidth is proved in this module. The value B·log₂(φ³+1) ≈ B·2.39 bits/Hz is not derived from the certificate. The three facts are not specific to information theory; they hold for any positive m and e.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/Bandwidth_Phi_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: