Field content, instrument limits, and curved backgrounds

The geometry and a coarse covariance action do not identify a Hamiltonian or field algebra. The remaining dynamical obligations, corresponding to CONE's C8–C11, are:

C8: Specified invariant state and reversible sector.

A translation-invariant invariant state, declared number conservation, and a one-particle sector must be provided for a selected dynamics.

C9: Dispersion and polarization.

Low-momentum spectral branches and their eigenprojectors must converge to the intended relativistic data. For a Dirac-type limit, positive and negative branches must be handled; a positive square-root dispersion alone does not specify a local first-order Dirac generator.

C10: Generator comparison.

Self-adjoint generators need common Hilbert-space comparisons, domains, and a justified logarithm branch. Strong-resolvent convergence of actual self-adjoint operators can yield strong convergence of their unitary groups ReedSimon; an unspecified varying-space embedding is not that hypothesis.

C11: Field covariance.

The field-observable representation and its boost intertwiners must be constructed. Diamond norms belong to bounded channel interfaces, not automatically to unbounded field operators.

A one-particle limit is not by itself a many-body quantum field theory, a vacuum construction, or a dynamical local net. Specified QCA and quantum-walk models illustrate possible low-momentum constructions, but do not verify these obligations for the increasing-stencil family DArianoPerinotti,ArrighiFacchiniForets.

For the corrected multi-time fixed-wire class, Hypothesis T can be realized on a nested increasing-stencil family with vanishing complexity-independent coarse error.

There is a specified nontrivial subclass with common input/output encodings, reference-stable instrument convergence, compatible composition and local modifications, and a matching C6${}^{\mathrm{obs}}$ realization.

Supply a complete schedule with: (i) local control and branch-routing flags; (ii) a conflict-resolution invariant; (iii) quantitative waiting and departure-time bounds; (iv) paced spacetime route and dwell containment; (v) correctly timestamped acquisition events; and (vi) payload naturality for matched interventions. Then formalize those statements. Finite tests alone are not their uniform proof.

Characterize the laws realizable with a given compact support and tuple of acquisition events, retaining the difference between independent product preparations and correlated resources. One-port samplers make bare one-port support–law data rich; they do not characterize modification relations or multi-port laws.