Information Ldpccode Rate From Jcost

The LDPC rate gap

Low-density parity-check (LDPC) codes are error-correcting codes used in modern communication systems, from satellite links to 5G networks. Their practical importance comes from a theorem: LDPC codes can achieve the Shannon capacity, the theoretical maximum rate at which information can be transmitted reliably over a noisy channel. In practice, however, finite block lengths impose a penalty, and well-designed LDPC codes typically operate within 5 to 15 percent of the Shannon limit. The Recognition Science framework offers a formal expression for this gap.

In the framework's ledger, a discrete record of events, the cost of recognition is forced to follow a specific function. The cost function J(x) = (x + 1/x)/2 - 1 measures the price of a mismatch between an expected and an observed quantity. The framework evaluates the penalty at the golden ratio φ ≈ 1.618, which is the unique self-similar scaling of the framework. The value J(φ) ≈ 0.118 gives an 11.8 percent capacity penalty, a number that sits inside the empirically observed 5 to 15 percent band for practical LDPC codes.

What the machine-checked library of formal theorems actually proves is narrower and precise. It defines a domain cost as J(m/e), the cost function applied to the ratio of a measured value to an expected value. Three general facts are proved: the cost vanishes when the measured value equals the expected value, the cost is nonnegative for positive inputs, and the threshold φ - 3/2 is positive. These are properties of the cost function itself, not of LDPC codes.

The module does not define what m and e mean for LDPC codes. It proves nothing specific to coding theory, because the definition of domainCost is generic. The connection to the 11.8 percent gap is a research note, not a theorem. The formal content is a template: if one could define the measured rate and the expected rate in coding-theoretic terms, then the framework's cost function would apply. That definition is the missing step that would turn the template into a theorem about LDPC codes.

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 framework does not prove that LDPC codes achieve Shannon capacity; that is a classical result in information theory. The module does not prove that the 11.8 percent figure is the actual capacity penalty for any specific LDPC code. The golden ratio appears as the evaluation point because it is the framework's self-similar scaling, not because of a coding-theoretic derivation.

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)

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