HamiltonianSpectrumCert
plain-language theorem explainer
HamiltonianSpectrumCert packages the spectral properties of the Recognition Hamiltonian into a single structure: exactly five sectors, zero J-cost at the vacuum, strictly positive J-cost for all other positive r, and a positive gap on any discretized lattice. Recognition Science researchers formalizing the Yang-Mills mass gap would cite this certificate when verifying the discrete spectrum. The declaration is a bare structure definition whose four fields are later populated by explicit witnesses.
Claim. A structure asserting that the set of spectral sectors has cardinality 5, that the J-cost function satisfies $J(1)=0$, that $J(r)>0$ for all $r>0$ with $r≠1$, and that for every lattice spacing $a>0$ there exists a gap $Δ>0$ with $Δ<J(1+a)$. Here $J$ is the J-cost function and the five sectors are vacuum, Goldstone, massive scalar, massive vector, and massive tensor.
background
The Recognition Hamiltonian Ĥ_RS acts on the space H_RS with J-cost measuring recognition cost of states. The five sectors are enumerated by the local inductive type SpectralSector with constructors vacuum, goldstone, massiveScalar, massiveVector, massiveTensor. The predicate latticeSpacingGap a ha is defined as the existence of Δ>0 such that Δ < Jcost(1+a), ensuring the continuous infimum becomes a strict minimum once the lattice is introduced. The module imports Constants and Cost and sits downstream of SpectralEmergence, where a related four-sector decomposition appears for the gauge group layers.
proof idea
This is a structure definition with an empty proof body. It simply declares four fields: the Fintype cardinality on SpectralSector, the vacuum J-cost equation, the universal positivity statement for excited states, and the lattice gap predicate. The structure is instantiated downstream by hamiltonianSpectrumCert, which supplies the concrete witnesses spectralSectorCount, vacuum_jcost, excited_jcost, and lattice_gap_witness.
why it matters
The certificate supplies the structural backbone for the discrete spectrum of the Recognition Hamiltonian and directly supports the mass-gap claim in the Yang-Mills unification step. It feeds the definition hamiltonianSpectrumCert that assembles the witnesses. In the framework it closes the passage from continuous J-cost positivity to the lattice-regularized gap required for a positive mass gap, consistent with the emergence of D=3 and the eight-tick octave. An open question remains the explicit numerical evaluation of the gap in RS-native units.
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