Deepening 2 of 6

Sector Attractors: Many Basins, One Macro Objective

The basic sector-exchange page said the objective may ignore which sector carries the burden. This page sharpens that into an attractor picture.

The mathematical point is not that the microscopic states are equal. It is that they lie in the same observable equivalence class for the loss or sector statistic you chose.

Theorem Reference

Lean anchors. attractor_multiplicity_of_sector_exchange, attractor_multiplicity_of_sector_statistics

Math statement.

L(\pi \!\cdot\! x)=L(x) \;\Longrightarrow\; \pi \!\cdot\! x \sim_{\mathcal O_L} x, \qquad \sigma(\pi \!\cdot\! x)=\sigma(x) \;\Longrightarrow\; \pi \!\cdot\! x \sim_{\sigma^\ast \mathcal O} x.

In English. A relabeled sector allocation and the original allocation become indistinguishable once your observables factor only through the loss or through an exchange-invariant sector statistic.

Physical intuition. Different training runs can occupy different internal basins while sharing the same coarse macro description. The attractors are distinct microscopically but degenerate at the level the objective cares about.

physics: degenerate free-energy basin training: multiple endpoint allocations analysis: compare macro variables, not labels

Two basins, same macro readout

The formal equivalence lives at the level of observables, not at the level of raw parameter identity.

Use for Model Analysis

Base / SFT / DPO

Different regimes may rebalance which sectors do the work without introducing a wholly new macro capability.

Run-to-run drift

Different seeds can end in different allocations that still score similarly on the same coarse objective.

Interpretation warning

Do not treat a changed sector label as a changed mechanism unless the observables actually resolve it.

Feynman reading: many microscopic arrangements can realize the same thermodynamic macrostate. This is the training-objective version of that idea.