Chemistry Gel Point From Jcost Gelation Crit Exp Cert

The certification structure

In the study of gels, a critical point is the concentration or conversion at which a liquid suddenly turns into a network that spans the whole sample. Near that point, the viscosity does not simply grow; it diverges, and the sharpness of that divergence is described by a critical exponent. For percolation gels, the standard empirical range for that exponent is roughly 1.3 to 1.5, a value measured in many experiments and simulations.

The Recognition Science framework contributes a candidate value for this exponent, derived from its cost function. The cost function, written J(x), measures the price of a recognition event, and the framework's central theorem forces it to have the form J(x) = (x + 1/x)/2 - 1. The framework's library of machine-checked theorems then computes a predicted exponent of about 1.133, which falls short of the empirical 1.3 to 1.5 range. That partial match is a research note, not a proved result.

What the Lean declaration GelationCritExpCert actually establishes is much narrower. It is a structure, a bundle of three general facts about the cost function applied to a ratio of two quantities, here called m and e. First, when m equals e, the cost is zero. Second, for positive m and e, the cost is never negative. Third, the number phi - 3/2, where phi is the golden ratio, is positive. These three statements are proved in the machine-checked library, and the certificate simply packages them together.

The certificate does not define what m and e mean in the context of gelation. In the source file, the cost is defined as J(m/e) without any reference to percolation, viscosity, or a critical point. The certificate proves facts about the cost function for any positive inputs, but nothing ties those inputs to the physical quantities of a gel. The text above the code records the intended research direction, but the formal content stops at the three general properties.

In plain terms, the certificate is a small, verified package of three harmless properties of a cost function. It is not a theorem about gelation, not a derivation of the critical exponent, and not a match to experiment. The gap between the certificate and the gelation claim is exactly the missing definition of m and e in the subject's own terms. Until that definition exists, the certificate remains what it is: a true but empty shell.

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 certificate does not prove that the gelation critical exponent is 1.133. The certificate does not match the empirical exponent range, and the partial match is a research note, not a result. The certificate does not define what m and e mean for a gel.

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/Chemistry/GelPoint_FromJCost.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.

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