Foundation Recognition Budget Unsaturated Budget Exponent Eq

The unsaturated remainder

The number phi, the golden ratio, satisfies phi^2 = phi + 1. In Recognition Science, the framework's library of machine-checked theorems uses powers of phi to describe a budget of recognition events. The declaration unsaturated_budget_exponent is a definition: it sets an integer exponent equal to the active edge budget minus the saturated budget exponent. Here active edge budget is the exponent 1, and saturated budget exponent is the exponent 45 at dimension three. The theorem unsaturated_budget_exponent_eq proves the value of this definition is -44.

The meaning is a bookkeeping identity. The total budget exponent is 1, and it splits into a saturated part with exponent 45 and an unsaturated remainder with exponent -44. Exponentiation turns this additive split into a multiplicative factorization: phi^1 equals phi^-44 times phi^45. The theorem budget_partition states this product equality, and total_budget_factorization names the two factors as matter content and consciousness ceiling. The unsaturated remainder phi^-44 is the candidate matter content, a number less than one, while the saturated part phi^45 is the consciousness ceiling, a number greater than one.

The declaration and its theorem do not claim that phi^-44 is the measured baryon asymmetry of the universe. The library proves that this formal matter content equals two other framework-internal definitions, one named eta_B_phi_scale and another eta_B_derived. Those equalities are algebraic identities within the framework, not measurements. The connection to observed cosmology is a prediction with a named falsifier: if the measured baryon-to-photon ratio does not match phi^-44, the identification fails. The framework also does not claim that the exponent -44 is derived from first principles alone; it follows from the gap value 45 at dimension three, which is itself a theorem of the framework's forcing chain.

What the declaration establishes is precise and limited. It fixes a definition, proves the exponent equals -44, proves the factorization phi = phi^-44 times phi^45, and proves the matter content is positive and less than one. These are formal results inside the framework, with no claim about measurement. The reader can see the exact algebraic structure, and the open question is whether the physical identification holds against data.

/-- At `D = 3`, the unsaturated budget exponent is `1 - 45 = -44`. -/
theorem unsaturated_budget_exponent_eq : unsaturated_budget_exponent = -44 := by
  unfold unsaturated_budget_exponent active_edge_budget saturated_budget_exponent GapDerivation.A
  have hg : (↑(GapDerivation.consciousnessGap GapDerivation.D) : ℤ) = 45 := by
    exact_mod_cast GapDerivation.gap_at_D3
  rw [hg]
  norm_num

/-- Exponent addition becomes multiplicative budget factorization after
exponentiation. -/
theorem budget_partition : matter_content * consciousness_ceiling = phi := by
  unfold matter_content consciousness_ceiling unsaturated_budget_exponent saturated_budget_exponent active_edge_budget
  exact GapDerivation.gap_balance

/-- The budget factorization matches the total budget form. -/
theorem total_budget_factorization : matter_content * consciousness_ceiling = total_budget := by
  rw [total_budget_eq]
  exact budget_partition

/-- Bridge to the recognition-science cosmology file: the matter content here
is exactly the existing `eta_B_phi_scale`. -/
theorem matter_content_matches_recognition_science :
    matter_content = IndisputableMonolith.Cosmology.BaryonAsymmetryExact.eta_B_phi_scale := by
  simpa [IndisputableMonolith.Cosmology.BaryonAsymmetryExact.eta_B_phi_scale] using matter_content_eq

/-- Bridge to the derived D=3 cosmology module. -/
theorem matter_content_matches_theta_module :
    matter_content = IndisputableMonolith.Cosmology.ThetaCritFromDimension.eta_B_derived := by
  simpa [IndisputableMonolith.Cosmology.ThetaCritFromDimension.eta_B_derived] using matter_content_eq

/-- Bridge to the recognition-science cosmology file: the matter content here
is exactly the existing `eta_B_phi_scale`. -/
theorem matter_content_matches_recognition_science :
    matter_content = IndisputableMonolith.Cosmology.BaryonAsymmetryExact.eta_B_phi_scale := by
  simpa [IndisputableMonolith.Cosmology.BaryonAsymmetryExact.eta_B_phi_scale] using matter_content_eq

What this page does not claim

The declaration does not claim phi^-44 is the measured baryon asymmetry of the universe. The theorem does not derive the exponent -44 from first principles alone; it depends on the gap value 45 at dimension three. The factorization phi = phi^-44 times phi^45 is not a statement about physical particles, only about formal powers of phi.

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/RecognitionBudget.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.

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