Semiring Parsing the S-Matrix: Packed forests, recursion as dynamic programming, and typed pruning in perturbative field theory

Abstract

The sequel to the certified-attribute-grammar note above: that paper stopped at recognition (which sentences are legal); this one is about evaluation (what the parse is worth). The single load-bearing idea is imported from computational linguistics: once the grammar's derivations are organised as a packed parse forest — the polynomial-size chart whose nodes are shared subderivations — then legality, counting, and amplitude assembly are the same inside algorithm run over different semirings. The Boolean semiring is the legality front end (selection rules), the counting semiring counts Feynman skeletons, and the complex-weight version with couplings as production weights computes the amplitude skeleton — the Born rule is then the reading law converting inside scores to record probabilities. Meaning attaches to the forest, not to any single tree: interference is the difference between the coherent inside score and the incoherent per-tree sum. Three consequences. (1) The amplitudes community's recursion revolution — Berends–Giele currents and BCFW — is dynamic programming on the shared forest: an off-shell current is a memoized chart cell, and parsing theory predicts the polynomial-versus-factorial separation the field found empirically by ingenuity (Catalan-many trees share O(n²) spans evaluated in O(n³) work). (2) The planar/tree fragment is context-free-class, so 't Hooft's 1/N expansion becomes a complexity filtration — leading N selects exactly the maximally-sharing fragment, non-planar corrections are context-sensitive insertions with controlled sharing degradation, and full non-planar complexity is explicitly left open. (3) Typed pruning terminates illegal subderivations before expansion, where the path integral generates-and-cancels by interference — with the safe algorithmic claim kept strictly separate from the interpretation-fenced "nature prunes" reading. Unitarity enters as a constraint on admissible weightings, with cutting rules restated as forest splittings and crossing as one forest serving several sentences. Two examples run end to end: e⁺e⁻ → μ⁺μ⁻ as forest semantics with the γ/Z interference term exhibited explicitly (the forward–backward asymmetry at the Z pole is parse interference made observable), and proton decay p → e⁺π⁰ as a syntax error of class "new grammar" — no parse exists at any order because every production conserves B — whose repair is priced, not free: legalising it needs a leptoquark-class letter or dimension-six bridge whose anomaly-completeness partner content is forced, while superficially similar neutrinoless double-beta decay already parses through the ΔL = 2 Majorana portal, so it is a sensitivity class, not new grammar. A scope fence is stated twice: the semiring claim lives at the combinatorial skeleton layer — loop integrals are the weight-evaluation step, not the parse, and nothing here claims parsing complexity controls them. Every quantitative statement is asserted by a self-checking evaluator (chart work fitted at n³·⁰⁰, forest growth e¹·³² per leg against the Catalan rate ln 4 = 1.386, counts Catalan-exact to n = 16) reproduced on commodity hardware.

Keywords

How to cite

Elliman, D. (2026). Semiring Parsing the S-Matrix: Packed forests, recursion as dynamic programming, and typed pruning in perturbative field theory. Neuro-Symbolic Ltd technical report. https://doi.org/10.5281/zenodo.21204128

@techreport{elliman2026semiringparsingsmatrix,
  author      = {Elliman, David},
  title       = {Semiring Parsing the S-Matrix: Packed forests, recursion as dynamic programming, and typed pruning in perturbative field theory},
  institution = {Neuro-Symbolic Ltd},
  year        = {2026},
  doi         = {10.5281/zenodo.21204128},
  url         = {https://neusym.ai/papers/semiring_parsing_smatrix}
}