Information Moore Law Rs

The phi growth law

Moore's Law is the observation that the number of transistors on an integrated circuit doubles roughly every 1.5 to 2 years. It has held since the 1960s as an engineering target and a self-fulfilling prophecy for the semiconductor industry. In Recognition Science (RS), the same observation is recast as a consequence of a deeper cost structure, and the doubling period becomes a specific number derived from the golden ratio phi, approximately 1.618.

In RS, the cost of recognition is forced by five plain conditions and must equal J(x) = (x + 1/x)/2 - 1. From this cost function, the golden ratio emerges as the unique self-similar scaling. The RS Moore Law module applies this to transistor growth: if the cost of recognizing a transistor count is measured against an expected count, the cost vanishes when the two match, and the framework's scaling law predicts growth as phi^(2t) where t is in years. That is an annual growth factor of phi^2 = 2.618, compared to Moore's 2x. The doubling time becomes 1/phi = 0.618 phi-years, or about 0.618 times the golden-section year.

The formal content is deliberately modest. It defines a domain cost function cost, a measure of recognition effort, as J(m/e) where m is the measured count and e is the expected count. It proves three general facts: the cost is zero when m equals e, the cost is never negative for positive inputs, and the threshold phi - 3/2 is positive. These are properties of the cost function itself, not of transistors. The module assembles these into a certificate structure, a formal package showing the cost function satisfies its basic sanity conditions.

What the module does not do is connect this cost function to any specific definition of m and e for semiconductors. The docstring is explicit: the phi-law prediction is a research note recording where the idea was meant to go, not a result. The formal theorems prove facts about J(m/e) universally, and they would apply to any domain where m and e are positive real numbers. The step from the general cost function to a statement about actual transistor counts requires a definition of m and e in the semiconductor's own terms, and that definition is absent.

In plain language: the framework has a proved cost function, and it has a proved scaling law that gives phi as the natural growth factor. The Moore Law module shows the cost function behaves well, and it records the intended application to transistor growth. The application itself, the claim that real transistors follow phi^(2t), remains a hypothesis waiting for a definition that ties the abstract cost to actual chip data. The formal library proves the cost function's properties; it does not prove Moore's Law.

The consequence for the reader is a clear separation: the mathematics of the cost function is settled, and the empirical claim about semiconductors is open. The module is honest about this gap, and that honesty is the point. It shows what a machine-checked theorem can and cannot do when a domain-specific definition is missing.

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 module does not prove that real transistor counts grow at phi^(2t); that is a stated research note, not a theorem. The framework's cost function J is proved, but its application to semiconductors is not a derived result. No empirical comparison against semiconductor industry data is made in this module.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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