Information Compression3 Deep From Jcost

The compression overhead

Information theory sets a floor on how short a message can be made. For a source that emits symbols with known probabilities, the average number of bits per symbol cannot go below the entropy H, a quantity Claude Shannon defined in 1948. A code that reaches that floor is called optimal. The floor is a number, not a promise; real codes always sit at or above it.

In Recognition Science, the framework's cost function J adds a fixed overhead to that floor. The claim is that an RS-optimal code uses H + J(phi) bits per symbol on average, where phi is the golden ratio, about 1.618. The overhead J(phi) is about 0.118 bits per symbol. At H = 1 bit per symbol, the RS code length would be 1.118 bits per symbol on average, a number that appears in the research note as a target, not as a proved result.

The Compression3DeepFromJCost module itself does not prove that overhead claim. Its machine-checked theorems are more modest: they establish three general facts about the cost function J applied to a ratio m/e. First, when m equals e, the cost is exactly zero. Second, for positive m and e, the cost is never negative. Third, the quantity phi minus 3/2, about 0.118, is positive. These are properties any overhead term must have: zero at the no-overhead point, nonnegative everywhere, and a positive threshold.

What the module does not do is tie m and e to any particular subject. Its definition of domainCost is just J(m/e), with no reference to a source, a message, or a compression problem. The same three theorems are shared verbatim across 2383 sibling modules, each one a template waiting for a subject-specific definition of m and e. The research note records where the idea was meant to go, not where it has arrived.

In plain language, the module is a foundation stone, not a building. It proves the cost function behaves correctly as a measure of overhead, and it leaves the next step open: define m and e in the terms of a real compression problem, and the overhead claim becomes a theorem about that problem. Until then, the 1.118 figure is a research note, a direction, not a result.

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

The 1.118 bits per symbol figure is not a proved result, only a research note. The module does not prove that any real compression problem achieves the RS overhead. The three theorems are general facts about J, not about compression specifically.

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/Compression3DeepFromJCost.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

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