Information Local Cache Jcost Pos Of Ne One

The cost of imbalance

The Recognition Science framework starts from a simple bookkeeping idea: reality keeps a ledger, a discrete record of events, and every act of recognition carries a forced cost. The central object is the cost function J(x) = (x + 1/x)/2 - 1, which measures how expensive it is to recognize a ratio x between two quantities. The theorem Jcost_pos_of_ne_one states a sharp fact about this function: for any positive x that is not exactly 1, the cost J(x) is strictly greater than zero. In plainer terms, any deviation from perfect balance, whether the ratio is 2 or 1/2, costs something; only the perfectly balanced ratio x = 1 costs nothing.

The proof is short and rests on a rewriting of the cost function. For positive x, J(x) equals (x-1)²/(2x), a square divided by twice the input. Since the numerator is a square, it is never negative, and it is zero only when x = 1. The division by 2x keeps the expression positive for all positive x away from 1. The theorem is machine-checked in the framework's library of formal theorems, meaning the chain of reasoning has been verified by a computer, not merely asserted in prose.

This positivity fact carries a concrete interpretation in the framework's model of neural firing. The framework defines synapse cost as J(f_u/f_v), where f_u and f_v are firing rates of two neurons. The theorem then says: when two neurons fire at different rates, the synapse carries positive cost; when they fire at the same rate, the cost is zero. The framework reads this as the mathematical content of Hebbian learning, the classical rule that neurons which fire together wire together. Correlated firing, where rates match, minimizes cost; uncorrelated firing, where rates differ, incurs a penalty that learning can reduce.

What the theorem does not claim is just as important. It proves positivity of J away from 1, but it does not by itself prove that J is the unique cost function satisfying the framework's axioms; that is a separate result. It does not claim that any particular neural system actually minimizes this cost, only that the cost function has this mathematical shape. And it does not assert that the golden ratio or any other constant emerges from this theorem alone; those conclusions come from additional assumptions about cache hierarchies and partition recurrences, which are separate results in the same library.

The practical upshot is a clean mathematical fact with a memorable shape: in this framework, balance is free and imbalance is not. The theorem gives a precise sense in which a system that recognizes ratios pays nothing only at the point of perfect symmetry, and pays a positive price everywhere else. That single fact anchors the framework's account of why correlated activity is favored and why deviation carries a cost that learning must overcome.

/-- J-cost is strictly positive when r ≠ 1. -/
theorem Jcost_pos_of_ne_one (r : ℝ) (hr : 0 < r) (hr_ne : r ≠ 1) :
    0 < Jcost r := by
  have h := Jcost_eq_sq (ne_of_gt hr)
  rw [h]
  apply div_pos
  · have : r - 1 ≠ 0 := sub_ne_zero.mpr hr_ne
    positivity
  · positivity

/-- J-cost is minimized at r = 1 (balanced firing rates). -/
theorem Jcost_min_at_one : Jcost 1 = 0 := Jcost_unit0

/-- J-cost is strictly positive when r ≠ 1. -/
theorem Jcost_pos_of_ne_one (r : ℝ) (hr : 0 < r) (hr_ne : r ≠ 1) :
    0 < Jcost r := by
  have h := Jcost_eq_sq (ne_of_gt hr)
  rw [h]
  apply div_pos
  · have : r - 1 ≠ 0 := sub_ne_zero.mpr hr_ne
    positivity
  · positivity

What this page does not claim

The theorem does not prove that J is the unique cost function satisfying the framework's axioms. The theorem does not claim that any actual neural system minimizes this cost function. The theorem does not by itself derive the golden ratio or any other constant from the cost function alone.

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/Information/LocalCache.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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