Physics Sociology

The shared template

Physics sociology is a name in the Recognition Science framework for a research direction that never got its own content. The idea was to take the framework's universal cost function, a measure of how far a ratio of two quantities is from balance, and apply it to social measurements such as city size. The module Physics/Sociology.lean defines a cost for the ratio of two real numbers m and e, but it does not define what m and e mean for a city or a society. The docstring says plainly that the module proves nothing specific to the subject.

What the module does prove are three general facts about the cost function J(x) = (x + 1/x)/2 − 1. First, when the two inputs are equal, the cost is zero. Second, for positive inputs, the cost is never negative. Third, the constant φ − 3/2, called the canonical threshold, is positive. These are all properties of the cost function itself, not of any social phenomenon. The same three facts are stated once in a shared template module and copied verbatim into 2383 sibling modules, each with a different subject name but the same generic content.

In Recognition Science, the framework models the cost of recognition as forced by five conditions, and the unique cost function is J(x) = (x + 1/x)/2 − 1. The sociology module applies this formula to a ratio m/e without specifying what the numerator and denominator represent. The structure City_Size_RS4 packages the three proved facts as a certificate, and the certificate is shown to exist. That is the entire mathematical content: the cost is zero at equality, nonnegative for positive inputs, and the threshold is positive.

What this means in plain language is that the module is a template, not a theory of cities. It establishes that if you define a social quantity as a ratio of two positive numbers, the framework's cost function will give a nonnegative number that vanishes when the two are equal. But the interesting question, what m and e are for a city, is left open. The module records the intention, not the result.

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

This module does not prove any statement specific to cities or societies. The cost function's uniqueness theorem is not proved in this module. No claim is made about what m or e should be for a real city.

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/Physics/Sociology.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: