Chemistry Polymer Chain Statistics From Jcost Poly Chain Stat Cert

What the certificate proves

Polymer chains in solution wiggle into countless shapes. A standard way to describe them is the freely jointed chain, where each monomer is a rigid rod that can point in any direction. For such a chain of N monomers, each of length l, the mean square distance between the ends is ⟨r²⟩ = Nl². Real chains are stiffer, and the persistence length l_p measures how far the chain remembers its direction. For DNA in water, l_p is about 50 nanometers, which is close to φ⁸ times one nanometer, where φ is the golden ratio, about 1.618.

In Recognition Science, a ledger, a discrete record of recognition events, assigns a cost to comparing two quantities. The cost function J(x) = (x + 1/x)/2 - 1 is zero when the quantities match, and grows as they diverge. The declaration PolyChainStatCert packages three proved facts about this cost when it is applied to a ratio m/e. First, the cost is zero when m equals e. Second, the cost is never negative for positive inputs. Third, the golden ratio minus 1.5 is positive, a small numerical fact that the framework's forcing chain uses elsewhere.

These three facts are general properties of the cost function. The certificate does not define what m and e mean for a polymer. It does not show that the persistence length of DNA is φ⁸ nanometers, or that any real chain follows the freely jointed model. The research note attached to the code records where the idea was meant to go, not a result that has been reached. The certificate is a foundation stone, not a building.

What the certificate does establish is that the cost function behaves sensibly in its central part: it costs nothing to match, and it never rewards a mismatch with a negative cost. That is a necessary condition for any physical application, but it is far from sufficient. The gap between the certificate and a polymer theorem is the missing definition of m and e in the polymer's own terms. Until that definition exists, the certificate remains a statement about a function, not about chains.

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]

What this page does not claim

The certificate does not prove that DNA persistence length is φ⁸ nanometers. The certificate does not derive the freely jointed chain formula ⟨r²⟩ = Nl². The certificate does not establish any property specific to polymers, because m and e are not defined in polymer terms.

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/PolymerChainStatisticsFromJCost.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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