TAQUERIA: Traced Autocatalytic Quantum Units of Energy Realizing Iterated Actions

In the spirit of the Particle Apothecary, TAQUERIA models spacetime as a series of transformation-based events— discrete, flavored packets of geometry and energy, assembled via local rules and global constraints. Each particle path is a trail of transformations. Each decay adds garnish to the emerging topology.

More formally, this experiment builds on the observation that particle transformations in an autocatalytic network (as formalized in the Particle Apothecary Model) can define both an emergent notion of spacetime and a dual structure encoding energy-momentum transitions. By tracing actualized transformation paths and measuring how energy decays across them, we aim to construct a toy-scale analog of the observed cosmic ray spectrum.

The broader hypothesis motivating this work is that quantum gravity may emerge from a discrete process of action-constrained transformation actualization. Here, the network of actualized events defines spacetime geometry, while its dual graph reflects momentum-space connectivity—together forming a self-organizing, background-free computational universe.

Objective

Use the dual structure of actualized transformations from the Particle Apothecary Model (PAM) to construct an emergent flux vs energy spectrum. This is a conceptual approximation ("cartoon") of the observed cosmic ray spectrum, where:

\[\Phi(E) = \text{Number of particle paths terminating at energy } E\]

Method

1. Particle Initialization

Each transformation (e.g., \(2\gamma \rightarrow \mu^+ \mu^-\)) creates new particles.
Assign initial energy \(E_0\) based on transformation type or from a distribution (e.g., exponential or power-law).

2. Path Tracing

As particles undergo subsequent transformations, track their path-history and update their energy by subtracting an action cost per transformation.
Each particle’s life becomes a discrete trajectory through the dual graph.

3. Action Cost Table

Assign an action cost to each transformation (e.g., photon conversion = 1.0, weak interaction = 3.0).
Store in a lookup table: action-cost["reaction-id"] = value.
Apply per-step decay: \(E_n = E_{n-1} - \text{action-cost}\).

4. Decay Conditions

Particles stop propagating when:

\(E \leq 0\)
No allowed transformations remain
Max path length reached

5. Flux Extraction

For each terminated path:

Record final energy \(E_f\)
Bin and count across energy ranges
Plot histogram \(\Phi(E)\)

Note on Broader Context

While we do not expect this toy-scale model to quantitatively reproduce the full cosmic ray spectrum (e.g., \(E^{-2.7}\)), it offers a concrete mechanism by which:

Spacetime emerges from discrete transformation events,
Energy and momentum are encoded in the dual graph of face-sharing transitions,
Action-based path selection gives rise to flux-energy distributions reminiscent of high-energy particle behavior.

This framework contributes to a growing class of models where causality, energy, and geometry are not fundamental fields, but emergent properties of a transformation network that evolves according to local rules and global least-action principles.