Information Information Qcap4 Deep Cert

The certificate

A certificate in a machine-checked library of formal theorems is a bundle of proofs that some object satisfies a list of required properties. The declaration QCap4DeepCert is such a bundle, and its name suggests a deep result about quantum capacity. The content, however, is far more modest. It proves three facts about a cost function Jcost, a measure of recognition cost that vanishes when two quantities are equal and grows as they diverge.

The three facts are these. First, when the two inputs to the cost function are equal and nonzero, the cost is zero. Second, for any two positive inputs, the cost is never negative. Third, a certain threshold value, defined as the golden ratio minus 1.5, is positive. Each statement is proved in the library from earlier theorems about the cost function. The certificate simply collects these three proofs into a single structure and shows that such a structure exists.

The name of the certificate is where the ambition lives. The research note attached to the module records an intended connection: quantum channel capacity, defined as the maximum over input states of the mutual information, might be related to the inverse of this cost function at a canonical coupling. That idea is a research note, not a result. The certificate itself contains no definition of a quantum channel, no mutual information, and no capacity formula. It proves nothing specific to information theory.

This pattern is deliberate. The same certificate body is shared verbatim across 2383 sibling modules, each with a different subject name. The shared content is stated once, universally quantified, in a template module. What would turn the certificate into a theorem about its subject is a definition of the two quantities in that subject's own terms. Until that definition exists, the certificate is a generic proof about a cost function, wearing a subject-specific name.

What the certificate does establish is useful groundwork. It shows that the cost function behaves sensibly at equality and on positive inputs, and that a canonical threshold is positive. These are the kind of facts a later, subject-specific proof would rely on. The reader should take the certificate for what it is: a small, verified foundation, not a deep claim about information.

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]

structure QCap4DeepCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

No claim that QCap4DeepCert proves anything about quantum channel capacity or information theory. No claim that the research note's proposed relation between Jcost and capacity has been derived. No claim that the certificate establishes the golden ratio threshold has any physical significance.

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/Information.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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