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Explain the Lean theorem `kepler_selection_principle` in module `IndisputableMonolith.Papers.DraftV1`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

Big AI job. Grok 4.3 reads the canon and writes a Lean-grounded derivation; usually 20 seconds to 2 minutes. Your answer will appear below.
confidence: high in recognition cached

The declaration kepler_selection_principle appears in module IndisputableMonolith.Papers.DraftV1.

(1) In plain English it states that the apsidal angle function equals exactly 2π if and only if the dimension parameter D equals 3.

(2) In Recognition Science it supplies the algebraic core of the Kepler non-precession constraint (K), showing that closed orbits without precession occur uniquely at D = 3 and thereby contributing one of the three selection principles (T/K/S) that together force three-dimensional space.

(3) The formal statement is a theorem taking a natural number D and asserting the biconditional apsidalAngle D = 2 * Real.pi ↔ D = 3. The forward direction assumes the equality, introduces x = sqrt(4 - D), cancels the nonzero factor 2π to obtain x = 1, squares both sides under the non-negativity check 4 - D ≥ 0, and concludes D = 3 by injectivity of the cast from ℕ to ℝ. The reverse direction simply substitutes D = 3 into the definition of apsidalAngle and simplifies.

(4) Visible dependencies inside the supplied source are the definition of apsidalAngle in the same module, the arithmetic lemmas syncPeriod_eq_mul and synchronization_selection_principle used elsewhere in the file, and the re-export of AlexanderDualityForCircleHypothesis from IndisputableMonolith.Foundation.AlexanderDuality. The proof uses only Mathlib Real arithmetic and contains no sorry or external axioms.

(5) The declaration does not prove the physical origin of the apsidal-angle formula (the substitution of the Green-kernel power law is assumed, not derived here), does not establish the full dimensional-rigidity theorem that combines (T), (K) and (S), and does not address empirical comparison with observed planetary orbits.

cited recognition theorems

outside recognition

Aspects Recognition does not yet address:

  • Physical derivation of the apsidalAngle formula from the Green-kernel power law
  • Empirical validation against solar-system data
  • Integration with the complete (T/K/S) rigidity theorem beyond the forward direction shown here

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.