hamiltonianSpectrumCert
plain-language theorem explainer
The definition constructs an explicit certificate for the spectrum of the Recognition Hamiltonian by assembling witnesses for five spectral sectors, vanishing vacuum J-cost, positive excitation costs, and a positive gap on any lattice spacing. Researchers addressing the Yang-Mills mass gap via the J-cost functional would cite it to confirm the discrete spectrum structure. The construction is a direct record that invokes the four supporting results on sector count, vacuum energy, excitation positivity, and lattice gap.
Claim. The Recognition Hamiltonian spectrum admits a certificate in which the number of spectral sectors equals five, the J-cost vanishes at scale one, the J-cost is strictly positive for every positive scale different from one, and the lattice spacing gap is positive for every positive spacing.
background
The Recognition Hamiltonian is defined on the space H_RS whose spectrum is governed by the J-cost function. Ground state energy is zero at the unit scale while excited states carry positive J-cost; the spectral gap on a discretized lattice with spacing a is the minimum positive J-cost away from unity. The module situates this construction inside the Recognition Science treatment of the Yang-Mills mass gap, where five canonical sectors arise from the configuration dimension D = 5.
proof idea
The definition is a direct structure constructor that supplies the four fields of the certificate by invoking spectralSectorCount for the sector cardinality, vacuum_jcost for the ground-state energy, excited_jcost for the positivity of excitations, and lattice_gap_witness for the gap on the lattice.
why it matters
This definition supplies the concrete certificate required to certify the spectrum structure that underpins the mass-gap claim for Yang-Mills within the Recognition Science framework. It connects to the five-sector decomposition that follows from the configuration dimension and provides the interface for further spectral analysis in the T5-T8 forcing chain. No downstream theorems currently depend on it.
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