Foundation Hydrogen Spectrum3 From Jcost

A template, not a spectrum

The hydrogen Balmer series is the set of visible spectral lines emitted when an electron in a hydrogen atom drops from a higher energy level to the second one. The first three lines have wavelengths 656 nm (H-alpha), 486 nm (H-beta), and 434 nm (H-gamma), measured in the 1880s by Johann Balmer, who found they fit a simple formula. The ratio of the first two wavelengths is 656/486 ≈ 1.35, which is close to the golden ratio φ ≈ 1.618 raised to the power 0.8.

In Recognition Science, the framework models recognition events as a ledger, a discrete record of events, and assigns each comparison a cost, a number measuring how expensive the recognition is. The framework's core theorem forces the cost function to be J(x) = (x + 1/x)/2 - 1. The module named HydrogenSpectrum3FromJCost defines a cost for a ratio m/e as J(m/e), and then proves three facts: the cost is zero when m equals e, it is never negative for positive inputs, and the quantity φ - 3/2 is positive.

Those three facts are true, but they are true for any positive numbers m and e. The module never defines what m and e mean for hydrogen, so it proves nothing about the Balmer series. The docstring notes the intended connection: adjacent Balmer lines differ by a factor of about φ^0.8 in wavelength, but that observation is a research note, not a theorem. The file itself says plainly that it proves nothing specific to hydrogen, because the cost is defined without reference to the subject.

What the module does establish is a reusable template. The same three generic facts about J are stated once in a shared file and repeated in 2383 sibling modules, each with a different subject name attached. To turn this template into a theorem about hydrogen, someone would need to define the electron mass and charge in hydrogen's own terms. Until then, the module is a placeholder: a certificate that the cost function behaves well, waiting for the physics that would make it specific.

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 any fact specific to hydrogen, because m and e are never defined in hydrogen's terms. The φ^0.8 wavelength ratio is a research note, not a theorem. The module does not derive the Balmer series or any spectral line wavelength.

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/HydrogenSpectrum3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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