Foundation Ecology

The ecological ledger

Ecology classically recognizes five types of interaction between two species: competition, mutualism, predation, amensalism, and commensalism. These are the standard categories taught in every introductory biology course. The question Recognition Science asks is whether this particular list of five, rather than four or six, is an accident of natural history or a structural necessity.

The framework's answer begins with a ledger, a discrete record of events, and a cost, the forced price of recording a recognition event. The central proved result in the framework is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This is a theorem checked by a machine-checked library of formal theorems, with no hidden assumptions. The cost is zero when the two sides are equal, and it grows as they diverge.

Foundation ecology applies this cost to the ratio of two species' ecological values, m and e. The module defines domainCost(m, e) = J(m / e). Three facts follow immediately and are proved in the library: the cost vanishes when m equals e, the cost is never negative for positive inputs, and the golden ratio threshold phi - 3/2 is positive. These are general properties of the cost function, not yet claims about any particular ecosystem.

The five interaction types are then identified with the five axes of a recognition space. This identification is a research note, not a proved result. What the module proves is only the general cost facts. The step from those facts to the five biotic interactions requires defining m and e in ecological terms, which the module does not yet do.

In plain language: the framework has shown that any recognition system pays a specific, unavoidable cost. Ecology's five interaction types may be the shape that cost takes when the recognition is between species. The number five comes from the framework's five-axis recognition space, but the bridge from cost to competition and mutualism remains to be built.

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

What this page does not claim

The module proves nothing specific to ecology; it proves general cost facts. The identification of five interaction types with five axes is a research note, not a proved result. No claim is made that competition or mutualism follows from the cost function without defining m and e.

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: