Information Ldpccode Rate From Jcost Ldpccert

The LDPC certificate

Low-density parity-check (LDPC) codes are error-correcting codes that approach the Shannon limit, the theoretical maximum rate for reliable communication over a noisy channel. Practical LDPC codes typically operate within 5 to 15 percent of that limit, a gap that shrinks as block length grows. A machine-checked library of formal theorems contains a structure named LDPCCert that packages three claims about a particular cost function, but the certificate itself says nothing specific about LDPC codes. The name is a research aspiration, not a description of what the formal proof establishes.

The cost function at the center is J(x) = (x + 1/x)/2 - 1, which the framework treats as a universal measure of mismatch between a measured value and an expected value. The certificate packages three facts about this function. First, when the measured value equals the expected value, the cost is zero. Second, for positive inputs, the cost is never negative. Third, the constant phi - 3/2 is positive, where phi is the golden ratio, approximately 1.618. Each of these is proved in the library as a general theorem about the cost function, and the certificate simply bundles them into a single structure.

The certificate's cost function is defined as J(measured / expected), a ratio of two real numbers. Nothing in the definition says what "measured" and "expected" mean for an LDPC code. The certificate never defines a code rate, a block length, a channel model, or a capacity. It proves no relationship between the cost function and any property of LDPC codes. The research note attached to the module records an intended direction: that the gap between designed rate and Shannon capacity for practical LDPC codes might equal J(phi), about 0.118, an 11.8 percent capacity penalty. That is a hypothesis recorded in a comment, not a theorem proved in the library.

What the certificate does establish, in plain terms, is that the cost function has three basic properties: it vanishes at agreement, it never reports a negative mismatch for positive inputs, and a particular golden-ratio constant is positive. These are true for any pair of positive real numbers, no LDPC context required. The certificate is inhabited, meaning the three properties are consistent and can be instantiated together. That is the full extent of the formal content.

In the framework's own terms, the certificate is a model of what a cost function should look like, not a theorem about coding theory. The gap between the certificate's name and its content is a useful example of why provenance matters: the machine-checked part is narrow and solid, while the intended application remains a research note. A reader who wants an LDPC-specific result should look for a definition of measured and expected in coding-theoretic terms, which this certificate does not provide.

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

theorem domainCost_at_equilibrium (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 (measured expected : ℝ) : ℝ := Jcost (measured / expected)

What this page does not claim

The certificate does not prove any property specific to LDPC codes. The 11.8 percent capacity penalty is a research note, not a proved result. The certificate does not relate the cost function to Shannon capacity or any coding-theoretic quantity.

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/LDPCCodeRateFromJCost.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: