Foundation Ecology Biotic Interact4 Cert

The certification structure

In ecology, five classic types of interaction between species are usually listed: competition, mutualism, predation, amensalism, and commensalism. A researcher working in the Recognition Science framework wanted to connect these five types to the framework's five-axis recognition space. The declaration BioticInteract4Cert in the framework's machine-checked library was meant to be that connection. The certificate is a formal structure, a bundle of three mathematical facts about a cost function.

The cost function at the center is domainCost, defined as J(m/e), where J is the framework's forced cost function and m and e are real numbers. The three facts the certificate packages are these: the cost is zero when m equals e, the cost is never negative when m and e are both positive, and the threshold phi minus 3/2 is positive. Each fact is proved as a theorem in the library. The certificate itself simply collects these three theorems into one named structure, and a separate theorem confirms that such a certificate exists.

The plain reading is modest. The certificate establishes three general properties of the cost function: zero at equality, nonnegativity on positive inputs, and a positive threshold. It does not establish anything about ecology. The definition of domainCost makes no reference to species, interactions, or any ecological quantity. The five interaction types appear only in a research note attached to the module, not in any formal statement. The note records where the idea was meant to go, not a result that has been proved.

In Recognition Science, this is an honest gap. The framework proves general facts about its cost function, and the certificate packages those facts. What would make the certificate a theorem about ecology is a definition of m and e in ecological terms, for instance as some measure of two species' resource use or population sizes. That definition does not exist in the module. The certificate is therefore a template waiting for content, not a claim about the natural world.

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 BioticInteract4Cert 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

theorem cert_inhabited : Nonempty BioticInteract4Cert := ⟨cert⟩

What this page does not claim

BioticInteract4Cert does not prove that five ecological interaction types exist or that they match the five-axis space. The certificate does not define m or e in ecological terms, so it makes no claim about any real species interaction. No theorem in this module connects the cost function to competition, mutualism, predation, amensalism, or commensalism.

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/Ecology.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: