Information Compression Ratio Rs

Compression ratio

Information compression ratio rs is a proposed measure, within the Recognition Science framework, of how much a piece of structured data can be losslessly compressed. The idea is that data with a certain kind of internal complexity, called phi-rung complexity, can be reduced to about 8.47 times smaller. Random data, by contrast, is incompressible, so its ratio is 1. This ratio is meant to be a practical counterpart to the theoretical Kolmogorov limit, which says there is a maximum lossless compression for any string.

The framework's central object is the cost of recognition, a number that measures how hard it is to reconcile two descriptions of the same thing. The framework defines a domain-specific cost as the universal cost function J applied to the ratio of two quantities, m and e. The formal library of machine-checked theorems proves three general facts about this cost: it is zero when the two quantities are equal, it is never negative for positive inputs, and a certain threshold value involving the golden ratio is positive.

In Recognition Science, the universal cost function is J(x) = (x + 1/x)/2 - 1. The domain-specific cost is simply J(m/e). The three proved facts are exactly what one would want from a cost function: it costs nothing to compare a thing with itself, it never gives a negative cost, and the golden-ratio threshold is a real, positive number. These are the basic sanity checks that any such measure must pass.

What the framework does not do is prove that 8.47 is the correct compression ratio. The docstring states that as a research note, a description of where the idea was meant to go, not as a result. The formal theorems prove properties of the cost function, but they do not define what m and e are in terms of actual data. Without that definition, the framework is a template, not a theorem about compression.

The practical upshot is a clear distinction. The framework has a proved, general cost function with desirable properties. The specific claim about 8.47x compression for structured data remains a hypothesis, a target for future work, not an established fact. A reader should treat the number as an aspiration, and the three formal properties as the solid ground.

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]

What this page does not claim

The 8.47x compression ratio is a proved theorem. The framework defines what m and e are for actual data. The framework proves anything about the compression of specific data types.

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

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