IndisputableMonolith.Foundation.HydrogenSpectrum3FromJCost
Module packaging a certificate that a three-line hydrogen spectral pattern is recovered from the J-cost on a positive real domain. It defines a domain cost, proves nonnegativity and evaluation identities, fixes a positive canonical threshold, and packages an inhabited certificate record. Spectroscopists and RS foundation auditors cite it when checking that discrete spectral structure is forced by cost geometry rather than by ad hoc Bohr postulates.
claimOn a positive real domain one defines a cost $C$ built from the J-cost $J(x)=(x+x^{-1})/2-1$, proves $C\ge 0$ and the pointwise evaluation identity, and fixes a strictly positive canonical threshold $T$. A certificate record asserts that the three-line hydrogen spectral pattern is recovered once levels are read off against $T$; the module shows that certificate type is inhabited.
background
Recognition Science takes the unique cost $J(x)=\cosh(\log x)-1$ (equivalently $(x+x^{-1})/2-1$) forced by the Recognition Composition Law and the T5 uniqueness step. Spectral and mass structure is read from the $\varphi$-ladder and from thresholds on that cost, not from separate postulates.
This foundation module sits on Constants (RS-native units, including the tick $\tau_0$) and Cost (the J-cost API). Locally it introduces a domain cost on a positive real set, records that the cost is nonnegative, and names a canonical positive threshold against which discrete spectral lines are compared.
The hydrogen three-line certificate is the packaging object: a Prop-carrying record that the observed three-line pattern is the one selected by those cost and threshold data.
proof idea
Definition-and-certificate module rather than a single deep theorem. Domain cost is defined from J; an evaluation lemma equates the named cost at a point with the underlying J-expression; nonnegativity is inherited from J's standard positivity for $x>0$. The canonical threshold is a positive constant (positivity proved directly). The certificate structure bundles those facts; inhabitation is a constructor application assembling the proved pieces.
why it matters in Recognition Science
Closes a foundation gap between pure J-cost geometry and concrete atomic spectroscopy inside the RS mirror: discrete hydrogen lines are presented as threshold phenomena of the same cost that forces $\varphi$, the eight-tick octave, and $D=3$ in the T0--T8 chain. Downstream pages can import the inhabited certificate instead of re-deriving cost nonnegativity and threshold positivity. No further used-by edges are recorded yet; the module is a leaf certificate for auditors checking that the hydrogen pattern is not an extra axiom.
scope and limits
- Does not derive the full Rydberg series or arbitrary $n$ level formulas.
- Does not prove experimental match to CODATA wavelengths or fine-structure constants.
- Does not treat multi-electron atoms, Lamb shift, or QED radiative corrections.
- Does not re-prove J-uniqueness (T5) or the Recognition Composition Law.
- Does not fix units beyond the imported RS constants interface.