Astrophysics Cmb Lensing3 From Jcost

A lensing note, not a lensing theorem

Cosmic microwave background (CMB) lensing is the bending of the oldest light in the universe by the gravity of intervening matter. Astronomers map that bending through the lensing potential, whose power spectrum shows a peak at an angular scale around l = 60 to 100. In the Planck satellite's lensing maps, that peak sits near l = 70 to 80. The file CMB_Lensing3_FromJCost in the Recognition Science library takes that empirical peak as a target: it notes that phi^9, where phi is the golden ratio, equals about 76, which lands inside the observed range.

That is where the subject-specific story ends. The file's own header says so in plain terms: the Lean code proves three general facts about a cost function, and it proves nothing specific to CMB lensing, because the cost function is defined without any reference to lensing. The paragraph about the peak is a research note recording where the idea was meant to go, not a result. The same note is shared verbatim with 2383 sibling modules, each named for a different subject, and the universal content lives in one template file.

What the file actually proves, in the machine-checked library of formal theorems, are three facts about the cost function Jcost(m/e). First, the cost vanishes when the two inputs are equal: Jcost(r/r) = 0 for any nonzero r. Second, the cost is never negative for positive inputs. Third, the constant phi minus 3/2 is positive, which is just the statement that the golden ratio exceeds 1.5. These are collected into a certificate structure and shown to be inhabited, meaning the three facts are consistent and provable together.

In Recognition Science, the cost function is the central object: a measure of the forced expense of a recognition event, with the uniqueness theorem J(x) = (x + 1/x)/2 - 1. The framework models many physical scales as phi-power ladders, so the suggestion that the lensing peak sits at phi^9 is a natural move within that account. But the move is a hypothesis with a named falsifier: if future CMB lensing measurements placed the peak clearly outside the range l = 70 to 80, the identification would fail. The module itself does not even go that far; it only records the numerical coincidence.

What a reader can take away is a clean distinction between a theorem and a research note. The three proved facts are generic properties of any cost function of this form, true for any positive inputs, with no lensing content. The lensing peak claim is a separate, unformalized suggestion. The library is honest about which is which, and the honesty is the useful part: it shows how a formal framework can carry a speculative application without pretending the speculation is proved.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

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 module does not prove that the CMB lensing peak is at l = 76. The module does not derive the lensing power spectrum from the cost function. The phi^9 identification is not a proved theorem about the universe, only a recorded numerical suggestion.

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/Astrophysics/CMB_Lensing3_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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