Rotor antiparticles and inverse topplings: a common quotient but different full dynamics

Definitions and mathematical statement

Jim Propp's original question asks whether Andrea Sportiello's inverse-toppling model [12] is the same as a particle creation/deletion model. Neither model nor "the same" is defined in the extracted question. The attached aid identifies two plausible readings: comparison of Caracciolo--Paoletti--Sportiello (CPS) addition/removal with rotor-routing particles/antiparticles, or comparison with the Karmakar--Manna particle--hole protocol. It says CPS explicitly identify a chain with the latter protocol, but that does not answer the rotor-routing comparison.

For the first reading, work on a finite undirected graph with a sink reachable from every vertex. Let $L$ be its reduced Laplacian and $J={\mathbb{Z}}^{V\setminus\{t\}}/L{\mathbb{Z}}^{V\setminus\{t\}}$. In a zero-based convention, stable CPS heights satisfy $0\le\eta(v)<\deg(v)$; addition stabilizes $\eta+e_v$, and removal applies the corresponding inverse-toppling stabilization to $\eta-e_v$. The aid states that their images modulo $L{\mathbb{Z}}^{V\setminus\{t\}}$ are translations by $+[e_v]$ and $-[e_v]$. With chosen cyclic rotor orders, particle and antiparticle addition induce those same signed translations on rotor-equivalence classes, a $J$-torsor. Identifying that torsor with $J$ requires a base class. Agreement of this quotient action does not assert agreement of the full states or trajectories.

The aid's explicit full-state obstruction uses nonsinks $c,l,r$, sink $t$, edges $ct,cl,cr,lt,rt$, and rotor orders \[ c:(t,l,r),\qquad l:(t,c),\qquad r:(t,c). \] There are $3\cdot2\cdot2=12$ stable-height states and $12$ unrestricted rotor states. Under particle routing with advance-then-move convention, the center addition maps are reported to have fiber-size multisets, including zeros, \[ \{0,0,0,1,1,1,1,1,1,2,2,2\}\quad\text{(CPS)}, \] \[ \{0,0,0,0,1,1,1,1,1,2,2,3\}\quad\text{(rotor particle)}. \] Their image sizes are respectively $9$ and $8$, so no bijection of the full state sets conjugates these labeled center maps. Nevertheless their common critical group has order $8$, and both quotient maps translate by $[e_c]$. This disproves that specific full-dynamics conjugacy, while preserving the signed critical-group comparison. It does not rule out other equivalence notions or replace the second protocol reading with a rotor claim.

Short English statement

The inverse-toppling comparison has two plausible intended particle models. For the rotor interpretation, the supplied aid finds the same signed critical-group quotient action but an explicit full-state conjugacy obstruction. For the particle--hole reading it reports an existing protocol identification.

Sources

• Local UnsolvedMath registry, permanent ID $20000997$, source code catalogue entry; canonical aim-combinatorics-notes.json, notes[121].
• American Institute of Mathematics, Problems from the AIM Chip-Firing Workshop, problem 32, attributed to Jim Propp: https://aimath.org/pastworkshops/chipfiringproblems.pdf. The source's Sportiello reference [12] is not fully supplied. No PDF inspection is claimed here.
• Attached research attempt 1: partial result, run and registry status partially solved, rigor proved, novelty confidence low, classification PARTIAL-PROGRESS. It names Caracciolo--Paoletti--Sportiello and Karmakar--Manna and invokes an official workshop-summary interpretation without full bibliographic details. The record is reconstructed and unverified; its machine-generated aid is not peer reviewed and requires independent verification before citation.