Physics Black Hole Mass Gap From Phi

The mass gap and its ledger

Black holes are observed in three separated mass ranges: stellar-mass black holes from about 3 to 100 solar masses, intermediate-mass black holes from 100 to 10^5 solar masses, and supermassive black holes from 10^6 to 10^10 solar masses. The gaps between these ranges are not empty by accident; they reflect how black holes form. Stellar black holes come from single stars collapsing, while supermassive ones grow by mergers and accretion. The intermediate range is sparsely populated because few known processes produce an object that heavy directly.

In Recognition Science, the framework models the cost of recognizing one mass scale from another using a function J(x) = (x + 1/x)/2 - 1, which is proved to be the unique cost satisfying five plain conditions. The module defines domainCost as J(m/e), where m is a candidate mass and e is a reference mass. This cost vanishes when m equals e, and it is nonnegative for all positive inputs. A threshold constant phi - 3/2 is also defined, where phi is the golden ratio, and proved to be positive.

What the module proves, in plain language, is three general facts about this cost function: it is zero at equality, never negative, and the golden-ratio threshold is positive. These are theorems in the machine-checked library of formal theorems. What it does not prove is anything specific to black holes. The definition of domainCost uses the ratio m/e without linking m or e to any physical mass scale. The paragraph in the source that mentions the observed ranges is a research note, not a result.

The observed ratios do align with the golden ratio in a suggestive way: 100/3 is about 33, close to phi^7.6; 10^5/100 is 1000, close to phi^14.4; and 10^10/10^6 is 10^4, close to phi^19.2. These are empirical coincidences, not derivations. The framework has not yet defined what m and e are in terms of black hole physics. Until it does, the mass gap remains an observed pattern, not a forced consequence.

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]

What this page does not claim

The module does not prove any specific property of black hole masses. The observed ratios are empirical coincidences, not derived results. The golden-ratio threshold has no physical interpretation yet.

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/BlackHoleMassGapFromPhi.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: