Foundation Rs Ad Scft Rs Rsad Scftrs

The declaration's reach

AdS/CFT duality is a conjecture in theoretical physics that connects a gravitational theory in a curved space of one dimension to a quantum field theory living on the boundary of that space, with one fewer dimension. The declaration RSAdSCFTRS, from the machine-checked library of formal theorems in the Recognition Science framework, does not prove that duality. Its name records where the idea was meant to go, not a result. The formal content is narrower and entirely general.

The declaration assembles three facts about a cost function, a measure of recognition effort defined as J(m/e), where J is the framework's forced cost function. First, when the two inputs are equal and nonzero, the cost is zero. Second, for positive inputs, the cost is never negative. Third, the number phi minus 3/2 is positive, where phi is the golden ratio. These are the three fields of the structure, and a certificate proves they hold.

The theorem cert_inhabited shows that such a structure exists. The proof of each fact is short: the zero-at-equality fact follows from the unit property of J, the nonnegativity from the positivity of the ratio, and the threshold fact from the known bound that phi exceeds 1.5. The declaration proves nothing about anti-de Sitter space, conformal field theory, or their duality, because the quantities m and e are never defined in those terms.

In Recognition Science, the framework models the cost of recognition as forced by five conditions, and its library proves that any such cost must equal J. The declaration RSAdSCFTRS is a template: it states what would need to be true for a subject-specific cost to be well-behaved. To become a theorem about AdS/CFT, one would need a definition of m and e within that subject's own terms. The current file lacks that definition, so the duality claim remains a research note, not a proved result.

What the declaration does establish is that the cost function is a sensible measure whenever m and e are positive: it is zero exactly when the inputs match, and it never dips below zero. That is a portable lesson. The same three facts hold for any subject whose quantities can be cast as a ratio, which is why the framework reuses this body across many sibling modules. The declaration is a well-formed container, not a filled one.

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 AdS/CFT duality itself is not proved by this declaration. The declaration does not establish that the golden ratio threshold has any physical meaning for AdS/CFT. The certificate does not prove that any actual physical system realizes the cost function.

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/RS_AdSCFT_RS.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: