Cost Ndim Neutrality

Ledger neutrality surface

Cost ndim neutrality is a surface in the space of recognition states. In Recognition Science, a recognition state is a vector of positive numbers, and a cost function assigns a nonnegative cost to each state. The aggregate cost, written as a product of powers, equals one exactly on this surface. The module establishes a plain equivalence: the aggregate equals one if and only if the weighted sum of the logarithms of the state entries is zero. That weighted log sum is the ledger's bookkeeping condition for a neutral state.

The same surface is the zero-cost set of the n-dimensional cost function. Two theorems in the module state this directly: zero cost holds if and only if the weighted log sum vanishes, and zero cost holds if and only if the aggregate equals one. The two conditions are the same condition seen from two sides. On the neutrality surface, recognition costs nothing, and the ledger balances.

This is a structural fact, not a numerical accident. The theorems are machine-checked with no special axioms beyond the standard kernel postulates. The neutrality surface is the first step toward understanding how higher-dimensional recognition states behave, and it gives a precise target for what a neutral state must satisfy.

THEOREM aggregate_eq_one_iff · IndisputableMonolith/Cost/Ndim/Neutrality.lean

THEOREM zero_cost_iff_dot_zero · IndisputableMonolith/Cost/Ndim/Neutrality.lean

THEOREM zero_cost_iff_aggregate_one · IndisputableMonolith/Cost/Ndim/Neutrality.lean

What this page does not claim

This answer does not claim that the neutrality surface is derived from physical principles. This answer does not claim that all recognition states with zero cost are physically realizable.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/Ndim/Neutrality.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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