Foundation Bitkernel Families3

A kernel family's scope

The module foundation bitkernel families3 is a file in the Recognition Science framework's machine-checked library of formal theorems. Its subject line names a family of kernel functions, but its actual content is deliberately narrower. The file defines a cost function, domainCost, as the framework's central cost J applied to the ratio of two real numbers, m divided by e. It then proves three general facts about this function: the cost is zero when m equals e, the cost is never negative when both inputs are positive, and a certain threshold value, phi minus 3/2, is greater than zero.

These three facts are genuine theorems, checked by the machine. The first, domainCost_at_eq, says that comparing a thing to itself costs nothing. The second, domainCost_nonneg, says that comparing two positive things never yields a negative cost. The third, canonicalThreshold_pos, establishes that the golden ratio phi, approximately 1.618, is more than 1.5 by a specific margin. The module packages these three theorems into a single structure, a certificate, and proves that this certificate exists.

What the module does not do is what its title suggests. The definition of domainCost refers only to the ratio m over e, with no connection to any specific physical subject. The file's own documentation is explicit on this point: it proves nothing specific to its named subject, because the subject's defining quantities, m and e, are not defined in the subject's own terms. The paragraph above the formal code is a research note recording where the idea was meant to go, not a result.

In Recognition Science, this pattern is intentional. The framework's library contains many modules that share this same body of three theorems, each with a different subject name attached. The shared content is stated once, universally, in a template module, and each family module inherits it. What would turn this particular module into a theorem about its named subject is a definition of m and e in that subject's own terms. Until that definition exists, the module is a general-purpose tool, proven and ready, waiting for its subject to be specified.

The practical consequence is a clear division of labor. The framework can certify that a cost function behaves well: zero at equality, nonnegative for positive inputs, and with a positive threshold. These are the properties any recognition cost must have. But the framework does not pretend that a general cost function, applied to an unspecified ratio, constitutes a physical theory. The module is scaffolding, not a building. It is the proven foundation on which a specific model, once m and e are given meaning, would rest.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not establish any physical theory about dark energy or a specific kernel family. The module does not prove that the golden ratio phi has any special significance beyond being greater than 1.5. The module does not define what m and e represent in any physical context.

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/Foundation/BITKernelFamilies3.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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