Cosmology Primordial Gw3 From Jcost Prim Gw3 Cert

PrimGW3Cert

PrimGW3Cert is a formal structure in the Recognition Science framework's machine-checked library of formal theorems. It bundles three general properties of a cost function, a mathematical rule that assigns a nonnegative penalty to a comparison between two quantities. The first property states that when the two quantities are equal, the cost is exactly zero. The second states that for any two positive quantities, the cost is never negative. The third states that a particular constant, the golden ratio minus 1.5, is greater than zero.

The certificate is built from a specific cost function, Jcost, evaluated on the ratio of two real numbers. The library proves that this function vanishes at equality, is nonnegative for positive inputs, and that the golden ratio exceeds 1.5. These are the three facts that PrimGW3Cert packages together. The structure is inhabited, meaning the certificate can be constructed, and the library demonstrates this by providing an explicit instance.

The name suggests a connection to primordial gravitational waves, and a research note attached to the module records an intended application: a predicted tensor-to-scalar ratio of 0.0170, consistent with the Planck+BICEP bound of r < 0.036. The formal theorems themselves, however, do not reference gravitational waves, cosmology, or any physical quantity. The cost function is defined on arbitrary real numbers, with no definition linking m and e to masses, energies, or any cosmological observable. The note is a statement of where the idea was meant to go, not a result.

In Recognition Science, this certificate is a step in a larger program. It establishes the abstract mathematical facts that a physical theory built on this cost function would use. What it does not do is claim that those facts apply to the early universe. That connection, defining m and e in cosmological terms, remains an open target. The certificate is a proof about a function, not a measurement of the sky.

structure PrimGW3Cert 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

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 any fact about gravitational waves, cosmology, or the early universe. The tensor-to-scalar ratio of 0.0170 is not a theorem; it is a research note recording an intended application. The cost function's properties hold for arbitrary real numbers, not for any specific physical system.

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/Cosmology/PrimordialGW3_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.

Derived articles

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