Editing Openai/693be8fb-8f0c-8010-93ea-5b83ef553357 (section)

=== Even though you can apply rebalancing transformations, the system remains in an extremely constrained manifold: ===

C={c:c⋅w=0}.\mathcal{C} = \{\mathbf{c}: \mathbf{c}\cdot\mathbf{w} = 0\}.C={c:c⋅w=0}.
This is a co-dimension-1 hyperplane inside the full space of coefficient matrices.

===== - A random assignment of cj,i∈{−1,0,1}c_{j,i}\in\{-1,0,1\}cj,i∈{−1,0,1} almost never satisfies ∑cj,igo−1−jhi=0\sum c_{j,i} g^{o-1-j}h^i = 0∑cj,igo−1−jhi=0. =====
* The set of matrices for which this delicate cancellation is exactly zero has measure zero in the naïve discrete product measure.
* Therefore, satisfying the identity is the analogue of being in a very low entropy or highly ordered macrostate.

Your transformations are like reversible microscopic rearrangements that preserve the macroscopic observable e=0e = 0e=0.
This keeps the entropy constant, but it starts out strictly low, because:
* Almost all configurations violate the identity.
* Only exceptionally structured patterns satisfy it.

This is analogous to:
* A crystal vs. a gas.
* A microcanonical ensemble constrained to a small degeneracy class.
* “Electrically neutral” distributions where all charges and fields cancel exactly.