Foundation Sociology

Foundation sociology

Foundation sociology in Recognition Science is the study of social gaps using the framework's universal measure of recognition cost. The core idea is that any difference between two groups, captured as a ratio m/e, carries a price. That price is not chosen freely; the framework's central theorem forces it to be J(x) = (x + 1/x)/2 - 1. When the ratio is 1, meaning the groups are equal, the cost is zero. When the ratio grows, the cost rises, and it rises faster for a deficit than for a surplus.

The framework's machine-checked library of formal theorems proves three general facts about this cost. First, the cost vanishes when the ratio equals 1, which is the equality case. Second, the cost is never negative for positive inputs, so no gap can have a negative price. Third, a specific threshold value, phi minus 3/2, is positive. This threshold, roughly 0.118, is the framework's candidate for the point where social change becomes likely.

In Recognition Science, the framework models revolution probability as spiking when the rate of change in a society crosses this threshold. The idea, recorded as a research note in the framework, is that a society tolerates slow change, but when the pace of improvement relative to what is needed exceeds the threshold, the cost of the gap becomes unstable. This is a hypothesis, not a proved result; the framework proves only the three general facts about the cost function.

What the framework does not do is apply the cost to any specific social variable. The definition of m and e, the two quantities being compared, is left open. The cost function is defined as J(m/e) without specifying what m and e mean in social terms. This is the framework's central limitation and its central opportunity: the mathematics is ready, but the social content must come from elsewhere.

The practical consequence is that foundation sociology provides a scaffold, not a finished theory. A researcher who can define m and e for a concrete social question, such as income inequality or political polarization, can plug those definitions into the framework and get a cost curve. The three proved facts guarantee that the curve will behave sensibly: zero at equality, nonnegative everywhere, and with a positive threshold for instability. The rest, including whether the threshold actually predicts revolutions, remains a target for empirical work.

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]

structure RevProb4Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The module does not prove that any particular social variable follows the cost function. The revolution threshold is a research note, not a proved theorem about actual societies. This page does not claim foundation sociology is a complete empirical theory.

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/Foundation/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: