IndisputableMonolith.Physics.RecognitionHamiltonianSpectrum
Defines the spectral sector decomposition of the Recognition Hamiltonian: vacuum (J=0), excited levels, and a lattice spacing gap witness. Supplies a certificate bundle packing those facts for downstream physics lemmas. Anyone citing discrete spectrum structure or the vacuum cost floor in RS units lands here. Content is definitional plus short algebraic witnesses, not a deep existence proof.
claimThe Recognition Hamiltonian spectrum is partitioned into spectral sectors. The vacuum sector has cost $J=0$; excited sectors carry positive $J$-cost. A lattice spacing gap $\Delta>0$ separates the vacuum from the first excited level, and a certificate packages these identities for use in RS physics arguments.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law. In RS-native units the vacuum configuration sits at $J=0$; any nontrivial excitation raises the cost.
This module sits in the Physics layer and imports only Constants (the tick $\tau_0$) and Cost (the $J$ functional). It introduces a finite enumeration of spectral sectors, named vacuum and excited $J$-costs, and a positive lattice spacing gap separating them. The gap is the discrete analogue of a mass gap for the Recognition Hamiltonian on the $\phi$-ladder.
The certificate type bundles the sector count, the vacuum identity $J_{\mathrm{vac}}=0$, the excited cost formula, and the gap witness into one Prop-carrying structure that downstream lemmas can assume or discharge.
proof idea
Definition module with short algebraic witnesses. Spectral sectors and the sector count are plain inductive or numeric definitions. Vacuum cost is the identity $J=0$ on the neutral configuration. Excited cost and the lattice gap are closed-form expressions in $J$ and the rung spacing; the gap witness is a one-line positivity check. The certificate constructor assembles those four facts; no heavy tactic proof is required.
why it matters in Recognition Science
Gives the Physics layer a named spectrum object so mass-ladder and hopping arguments can cite a vacuum floor and a strict gap rather than re-deriving $J\ge 0$ ad hoc. Aligns with the forcing chain: T5 fixes $J$, T6 fixes $\phi$, and the eight-tick octave (T7) supplies the discrete time base on which the Hamiltonian acts. No downstream consumers are wired yet in the graph; the certificate is the intended hook for later spectrum or stability theorems. Closes the definitional gap between raw Cost and any claim that the Recognition Hamiltonian has a discrete gapped spectrum.
scope and limits
- Does not prove uniqueness of the Hamiltonian operator itself.
- Does not derive continuum eigenvalues or scattering states.
- Does not connect the gap to measured particle masses.
- Does not discharge dynamics or time evolution on the spectrum.
- Does not claim the sector count is forced by T0–T8 alone.