Information Error Correction3 Deep From Jcost

Error correction capacity

Error-correcting codes let a message survive noise. A sender adds redundancy, a receiver uses it to detect and fix flipped bits. The Hamming bound is a classical limit on this: a code of length n that corrects t errors needs at least a certain number of redundant patterns, expressed by the inequality 2^(n-k) >= V_H(n,t). It is a counting argument, and it has been known since the 1950s.

Recognition Science (RS) is a framework that starts from a ledger, a discrete record of recognition events, and derives structure from the forced cost of recognition. Its central object is the cost function J(x) = (x + 1/x)/2 - 1, which a proved theorem in the framework's machine-checked library of formal theorems forces from five plain conditions. The module ErrorCorrection3DeepFromJCost attempts to connect this cost to error correction.

The module defines a domain cost as J(m/e), the cost of a ratio between two real numbers m and e. It proves three facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and the number phi - 3/2 is positive (where phi is the golden ratio). These are general properties of the cost function itself, and the proofs are machine-checked. The module packages them into a certificate structure, a formal object that bundles these three properties together.

What the module does not do is establish anything specific about error correction. The definition of domainCost as J(m/e) makes no reference to codes, messages, or noise. The docstring records the intended direction: at a rate R = J(phi) = 0.118, the code would support an error fraction t/n = J(phi)/2 = 0.059, using the Hamming bound as a structural guide. But that paragraph is explicitly a research note, not a result. The module proves no theorem about codes; it proves properties of J applied to a ratio.

In plain language: the module shows that a particular cost function has three basic mathematical properties, and it packages those properties into a certificate. The leap from that cost to actual error-correcting codes, with a defined m and e in terms of code parameters, is not made. That step remains open. The certificate is a tool waiting for a subject, not a theorem about one.

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

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]

theorem cert_inhabited : Nonempty ErrCorr3DeepCert := ⟨cert⟩

What this page does not claim

This module does not prove any bound on error-correcting codes. The rate 0.118 and error fraction 0.059 are research notes, not derived theorems. The certificate structure does not by itself constitute an error-correcting code.

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/ErrorCorrection3DeepFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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