Cosmology Primordial Gw3 From Jcost

A template, not a result

Primordial gravitational waves are ripples in spacetime predicted to have been generated during cosmic inflation, the brief exponential expansion of the universe in its first instants. Their imprint on the polarization of the cosmic microwave background is a major target for observational cosmology. The most recent combined limits from the Planck satellite and the BICEP experiment place the tensor-to-scalar ratio r, a measure of their amplitude, below 0.036.

The module cosmology ledger, a discrete record of recognition events, contains a file named for this topic. Its formal content is modest. The file defines a cost function as J(m/e), where J is the forced cost function from the framework's core, and m and e are real numbers. It proves three general facts: the cost vanishes when m equals e, it is nonnegative for positive inputs, and the constant phi minus 3/2 is positive. These are properties of the cost function itself, not of gravitational waves. The file's own documentation states this plainly: it proves nothing specific to cosmology, because the ratio m/e is defined without reference to any physical quantity.

The research note attached to the file records the intended direction. It computes a predicted value of r = 2/(45*phi^2) ≈ 0.0170, which is consistent with the observational bound r < 0.036. This is a numerical agreement, not a theorem. The formal library proves the general cost facts; the cosmological application remains a proposal. The module's structure, a certificate bundling the three proven facts, is a template shared verbatim with 2383 sibling modules. What would turn this template into a theorem about its subject is a definition of m and e in the subject's own terms, for instance as a ratio of energy scales tied to inflation.

For a reader, the distinction matters. The framework's machine-checked library establishes the cost function's properties with certainty. The step from those properties to a statement about the early universe is not yet formalized. The module is an honest placeholder: it shows where the argument would go, not that it has arrived.

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)

def canonicalThreshold : ℝ := phi - 3 / 2

What this page does not claim

This module proves any statement about primordial gravitational waves. The numerical agreement for r is a derived theorem rather than a research note. The value phi minus 3/2 has any physical interpretation in this module.

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)

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Derived articles

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