Syntactic Structure, Quantum Weights

TL;DR

The paper argues that universal features of physics—local additive actions and exponential Euclidean weights—originate from minimal syntactic constraints on finite histories encoded as prefix-free, sequentially decodable programs. By showing that such syntax forces any continuous local descriptive cost into an Euler–Lagrange equivalence class and that exponential redundancy yields a Euclidean weight P[x] ∝ exp[−S_E[x]/ħ_eff], it reframes quantum weights as a structural consequence of description, not dynamics. The author then demonstrates that requiring the weight to be real and normalizable selects a real Euclidean representative, with OS positivity enabling a Lorentzian reconstruction when applicable. A cosmological calibration using holographic information content fixes ħ_eff to be of order ħ, linking microscopic quantum scales to the Universe’s global descriptive capacity. Overall, the work separates a universal syntactic layer (EL class and Euclidean weight) from system-specific semantic input (microscopic actions) and suggests a path toward understanding the Planck scale as a calibration outcome of cosmic information bounds.

Abstract

Why do local actions and exponential Euclidean weights arise so universally in classical, statistical, and quantum theories? We offer a structural explanation from minimal constraints on finite descriptions of admissible histories. Assume that histories admit finite, self-delimiting (prefix-free) generative codes that can be decoded sequentially in a single forward pass. These purely syntactic requirements define a minimal descriptive cost, interpretable as a smoothed minimal program length, that is additive over local segments. First, any continuous local additive cost whose stationary sector coincides with the empirically identified classical variational sector is forced into a unique Euler--Lagrange equivalence class. Hence the universal form of an action is fixed by descriptional structure alone, while the specific microscopic Lagrangian and couplings remain system-dependent semantic input. Second, independently of microscopic stochasticity, finite prefix-free languages exhibit exponential redundancy: many distinct programs encode the same coarse history, and this redundancy induces a universal exponential multiplicity weight on histories. Requiring this weight to be real and bounded below selects a real Euclidean representative for stable local bosonic systems, yielding the standard Euclidean path-integral form. When Osterwalder--Schrader reflection positivity holds, the Euclidean measure reconstructs a unitary Lorentzian amplitude.

Figures (1)

• Figure 1: Schematic overview of the syntactic framework and the structure of the paper. Finite-resolution, prefix-free generative programs assign each trajectory a minimal program length $|p_x|:=\min_{p\to x}|p|$, and a continuous local descriptive cost $\ell[x]$ obtained by smoothing $|p_x|$ at fixed descriptive resolution. Section \ref{['sec:local']} shows that syntactic locality together with variational/classical correspondence forces any continuous additive local cost into the Euler--Lagrange equivalence class of a local action, $\ell[x]=\alpha S[x]+B$. Section \ref{['sec:redundancy']} counts redundant programs encoding the same coarse trajectory; exponential redundancy with rate $\Lambda$ yields a Euclidean weight $P[x]\propto \exp(-S_E[x]/\hbar_{\rm eff})$ with $\hbar_{\rm eff}^{-1}=\alpha\Lambda$. When OS reflection positivity holds, OS reconstruction promotes this Euclidean theory to a unitary Lorentzian/Minkowskian amplitude $\exp(+iS_M[x]/\hbar_{\rm eff})$. Section \ref{['sec:cosmo']} compares the emergent $\hbar_{\rm eff}$ with the cosmological value of $\hbar$, providing a global consistency check. The present work closes within the real Euclidean sector; Lorentzian physics enters only conditionally through OS reconstruction.

Theorems & Definitions (4)

• Lemma 1: EL-locality
• Lemma 2: Prefix-free necessity
• proof
• Lemma 3: EL--locality lemma