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Objective-Sector Exchange: Same Macro Behavior, Different Internal Allocation

A loss can care about aggregate sector statistics while being indifferent to which sector carries them. Then different internal allocations become equivalent at the objective level.

This is not usually an exact symmetry of the microscopic dynamics. It is a degeneracy of what the objective pays attention to.

Theorem Reference

Lean anchors. LossSectorExchangeInvariant, objective_sector_exchange_of_invariant_statistic, attractor_multiplicity_of_sector_statistics

Math statement.

\mathrm{stat}(\pi \!\cdot\! x)=\mathrm{stat}(x) \;\Longrightarrow\; f(\pi \!\cdot\! x)=f(x) \quad \text{for every observable } f \text{ factoring through } \mathrm{stat}.

In English. If the objective only sees a sector statistic and that statistic is unchanged by relabeling sectors, then the whole objective is blind to which sector index carries the burden.

Physical intuition. Different runs can land in different internal sector assignments while preserving the same coarse task-level objective.

physics: degenerate macrostate training: multiple attractor allocations engineering: do not over-interpret sector labels

Two allocations, one macro count

The theorem is about objective blindness to relabeling, not about the microscopic states being literally equal.

Physical Use

RLHF intuition

Different training regimes may settle on different balances of mechanism types while matching task behavior.

Attractors

The same objective landscape can support multiple internally re-labeled basins.

Caution

If you narrate sectors too literally, you may mistake allocation choices for new capabilities.

Feynman reading: thermodynamics often cares about macro variables, not particle labels. This theorem family is the same kind of move, but for training objectives.