Materials Trained Neural Network From Jcost

Overparameterization and the cost floor

In machine learning, overparameterization means giving a model more parameters than the minimum needed to fit the training data. A common practical rule is that a neural network needs roughly 5 to 10 times more parameters than data points to generalize well. The question in Recognition Science is whether that ratio can be derived from a deeper principle rather than observed as a heuristic.

The framework's starting point is a cost function, a measure of how expensive a recognition event is, written J(x) = (x + 1/x)/2 - 1. This function has three proved properties in the machine-checked library of formal theorems: it equals zero when its input is 1, it never goes negative for positive inputs, and the golden-ratio constant phi minus 3/2 is positive. The module TrainedNeuralNetworkFromJCost defines a domain cost as J(m/e), where m is parameters and e is data, and proves those three general facts about it.

What the module does not do is connect m and e to any specific neural-network subject. The definition of domainCost is just J(m/e) with no further structure. The three theorems hold for any positive numbers m and e, so they say nothing about what m and e should be for a trained network. The research note in the source states the intended target: an optimal ratio of about 8.47 times the minimum parameters for the recognition budget. That number comes from inverting J at the golden ratio, but it is a stated goal, not a proved result.

In Recognition Science, the framework models the ratio as a recognition budget problem. The claim would be that the cost floor forces a specific overparameterization ratio, just as the framework forces other constants. But the current module only certifies the cost function's basic analytic properties. The bridge from those properties to a neural-network-specific statement is missing: there is no definition of m and e in terms of network architecture, data distribution, or training dynamics.

The practical lesson is a caution about provenance. A machine-checked theorem about a generic function is not a machine-checked theorem about neural networks. The three proved facts are real, but they are universal truths about J, not discoveries about overparameterization. The neural-network claim remains a hypothesis with a named falsifier: if a trained network generalizes well at a ratio substantially below 8.47 times the minimum, the proposed bound fails.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The optimal overparameterization ratio of about 8.47 is not proved, only stated as a research note. The three proved theorems apply to any positive numbers m and e, not specifically to neural networks. No claim is made that the framework derives the 5-10x heuristic from first principles.

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

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