Field arena

In plain terms

Here is the problem. Everything so far talks about finitely many outcomes, and the world's working theory is quantum field theory -- infinitely many modes, infinitely many quanta. A reconstruction that cannot even gesture at fields is a reconstruction of a toy.

The field arena is the gesture made precise, deliberately at finite resolution. Keep finitely many modes; let each hold at most N quanta. On the projective space of those configurations, the field-theory canon is then re-proved as exact finite theorems: free propagators oscillating at the right frequency, interaction effects priced linearly in the coupling, a genuine light cone with an explicit velocity bound, thermal states satisfying the exact KMS condition.

This is breadth, and it is labelled as such: the image of standard physics inside the programme's arena, not the frontier of the reconstruction.

In CSD

The field arena is the CV track's stage: FieldArena K N, the projective space over configurations of K modes at cutoff N. Its role in the programme is honestly bounded -- it is the isolated-piece image, showing that effective field-theory structure survives at finite cutoff on exactly the kind of arena the reconstruction runs on, with no continuum limit and no RG flow claimed.

The bridge theorems tie it back to the record layer: arena observations are density pairings, dynamics act by kicks, disjointly supported observables commute exactly, and the Lieb-Robinson cone bounds how fast a record cell in the fibre can be steered from afar.

Mathematically

Mode-local subalgebras are unital star-subalgebras supported on their blocks; free and interacting drives are exact matrix exponentials; the two-point and four-point functions close into Wick's pairing sum at explicit thresholds; the commutator bound decays factorially in graph distance with velocity constant 2e^2 times the coupling norm; and composite arenas compose by mode concatenation, with no-signalling exact for all states, entangled included.