IndisputableMonolith.Physics.MagnonDispersion2
Defines the second magnon-dispersion certificate for Recognition Science: a nonnegative domain cost on the Brillouin zone together with a strictly positive canonical energy threshold. Condensed-matter theorists working in RS-native units cite it when matching spin-wave spectra to the J-cost ladder. The module is mostly definitional, with short positivity and evaluation lemmas and an inhabited certificate record.
claimThe module introduces a domain cost $C_{\mathrm{dom}}(k)$ on magnon wave-vectors, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at a reference point, and fixes a canonical threshold $E_{\ast}>0$. These data are packaged as an inhabited certificate $\mathrm{MagnonDisp2Cert}$.
background
Recognition Science measures energetic cost by the unique J-functional forced at T5, $J(x)=(x+x^{-1})/2-1$. In condensed-matter settings the same cost is evaluated on dimensionless ratios built from lattice momenta and the RS time quantum $\tau_0=1$ tick (from Constants). The Cost import supplies the ambient nonnegative cost infrastructure.
Magnon dispersion concerns the low-lying spin-wave branch $\omega(k)$ on a magnetic lattice. The present module does not derive $\omega(k)$ from a microscopic Hamiltonian; it records the RS-side bookkeeping: a domain cost that stays nonnegative across the Brillouin zone and a strictly positive canonical threshold against which the branch may later be compared.
Sibling definitions therefore split into the cost family (domainCost, its evaluation lemma, nonnegativity) and the threshold family (canonicalThreshold, positivity), then assemble them into the certificate record MagnonDisp2Cert with an inhabitation witness.
proof idea
Definition module with thin lemma layer. domainCost is introduced as a concrete nonnegative expression; domainCost_nonneg is a direct nonnegativity argument, and domainCost_at_eq is an evaluation identity at a fixed reference momentum. canonicalThreshold is a positive constant built from RS units; canonicalThreshold_pos is immediate positivity. The certificate record simply packages these fields, and cert_inhabited supplies a canonical inhabitant. No deep tactic scripts or external analytic estimates appear.
why it matters in Recognition Science
Places magnon kinematics inside the same cost calculus used for particle masses and the eight-tick octave. Downstream physics developments that need a certified nonnegative domain cost or a positive energy floor for spin waves can import the inhabited MagnonDisp2Cert rather than re-proving positivity. The module sits in the Physics domain and currently has no recorded reverse dependents, so it functions as a leaf certificate ready for later spectral-matching theorems. It does not itself invoke T5–T8 forcing, but inherits the J-cost and $\tau_0$ conventions those steps fix.
scope and limits
- Does not derive a microscopic Heisenberg or Hubbard Hamiltonian.
- Does not prove that experimental magnon branches equal the RS domain cost.
- Does not fix lattice dimension, spin value, or exchange constants.
- Does not connect the threshold to the Berry creation scale $\varphi^{-1}$.
- Does not supply numerical fits or material-specific dispersion curves.