Gravity Ilgreal Exponent Enhancement Ilgreal Exponent Enhancement Cert

The real-exponent certificate

The declaration ILGRealExponentEnhancementCert is a bundled certificate in the Recognition Science framework's machine-checked library of formal theorems. It packages four proven facts about a specific radial weight function, written w_real(R, r0, α) = 1 + C · (R/r0)^α, where R is a radial distance, r0 a reference scale, α a real exponent, and C a positive constant. The certificate itself is a structure: a named collection of statements that must all hold together, like a contract that any implementation must satisfy.

The four facts are proved as theorems. First, the weight is always greater than 1 for positive distances and scales, meaning the enhancement is a genuine amplification. Second, the weight is strictly increasing in R, so farther out means stronger enhancement. Third, the weight grows without bound as R goes to infinity, so the enhancement diverges at large radii. Fourth, a velocity-squared term V_bar_sq is always dominated by the enhanced version w_real · V_bar_sq, a Newtonian-domination property. The certificate theorem ilgRealExponentEnhancementCert_holds proves that a certificate satisfying all these conditions actually exists, with zero axioms beyond the standard three and zero unfinished proofs.

The exponent α is locked to the value 1 − 1/φ, where φ is the golden ratio, placing it in the open interval (0, 1). The proofs use real exponentiation via Real.rpow, so they cover the continuous real-exponent case rather than only integer powers. The strict monotonicity and unboundedness results require α > 0, a condition stated explicitly in each theorem. The positivity and above-one results hold for any real α, but the definition itself does not require positivity, so the theorems carefully include the hypothesis.

What the certificate does not claim is essential to state plainly. It does not assert that the formula w_real describes actual gravitational physics; that is a separate empirical question. It does not claim the enhancement matches any measured rotation curve or gravitational observation. It does not claim the exponent 1 − 1/φ is derived from first principles within the framework; the exponent is locked by an external identification, not forced by the theorems. The certificate is purely structural: it proves mathematical properties of a defined function, nothing more.

The practical consequence is that any future work using this enhancement can rely on the four facts as machine-checked guarantees, without re-proving them. The certificate is a reusable building block for further theorems, not a physical prediction. A reader should understand it as a precise mathematical object with a narrow scope, valuable for what it proves and equally valuable for what it honestly leaves open.

theorem enhancement_real_above_one (R r0 α : ℝ)
    (hR : 0 < R) (hr0 : 0 < r0) :
    1 < w_real R r0 α := by
  unfold w_real
  have hC : 0 < C_lock := C_lock_pos
  have hd : 0 < R / r0 := div_pos hR hr0
  have hpow : 0 < (R / r0) ^ α := Real.rpow_pos_of_pos hd α
  have : 0 < C_lock * (R / r0) ^ α := mul_pos hC hpow
  linarith

theorem enhancement_real_strict_mono (R₁ R₂ r0 α : ℝ)
    (hR₁ : 0 < R₁) (hR₂ : R₁ < R₂) (hr0 : 0 < r0) (hα : 0 < α) :
    w_real R₁ r0 α < w_real R₂ r0 α := by
  unfold w_real
  have hC : 0 < C_lock := C_lock_pos
  have h1 : 0 < R₁ / r0 := div_pos hR₁ hr0
  have h2 : 0 < R₂ / r0 := div_pos (lt_trans hR₁ hR₂) hr0
  have hd : R₁ / r0 < R₂ / r0 := by
    have hinv : 0 < r0⁻¹ := inv_pos.mpr hr0
    have : R₁ * r0⁻¹ < R₂ * r0⁻¹ := mul_lt_mul_of_pos_right hR₂ hinv
    simpa [div_eq_mul_inv] using this
  have hpow_lt : (R₁ / r0) ^ α < (R₂ / r0) ^ α :=
    Real.rpow_lt_rpow (le_of_lt h1) hd hα
  have hmul_lt : C_lock * (R₁ / r0) ^ α < C_lock * (R₂ / r0) ^ α :=
    mul_lt_mul_of_pos_left hpow_lt hC
  linarith

/-- Asymptotic divergence of the real-exponent enhancement. -/
theorem enhancement_real_unbounded (r0 α : ℝ)
    (hr0 : 0 < r0) (hα : 0 < α)
    (M : ℝ) (hM : 0 < M) :
    ∃ R : ℝ, 0 < R ∧ M < w_real R r0 α := by
  unfold w_real
  -- Want C_lock * (R/r0)^α > M, i.e. (R/r0)^α > M/C_lock.
  set u : ℝ := M / C_lock + 1 with hu_def
  have hC : 0 < C_lock := C_lock_pos
  have hu_pos : 0 < u := by
    have h₁ : 0 < M / C_lock := div_pos hM hC
    have : 0 < M / C_lock + 1 := by linarith
    simpa [hu_def] using this
  -- Choose y so that y^α = u, namely y = u^(1/α).
  have hα_ne : α ≠ 0 := ne_of_gt hα
  set y : ℝ := u ^ (1 / α) with hy_def
  have hy_pos : 0 < y := by
    have : 0 < u ^ (1 / α) := Real.rpow_pos_of_pos hu_pos (1 / α)
    simpa [hy_def] using this
  have hyα : y ^ α = u := by
    have h_inv : (1 / α) * α = 1 := by
      field_simp
    have hmul := Real.rpow_mul (le_of_lt hu_pos) (1 / α) α
    -- hmul : u ^ ((1/α) * α) = (u ^ (1/α)) ^ α
    rw [hy_def, ← hmul, h_inv, Real.rpow_one]
  -- Set R = r0 * y; then (R/r0)^α = y^α = u, and C_lock * u = M + C_lock > M.
  refine ⟨r0 * y, mul_pos hr0 hy_pos, ?bound⟩
  have hratio : (r0 * y) / r0 = y := by field_simp
  have hRpow : ((r0 * y) / r0) ^ α = u := by rw [hratio]; exact hyα
  have hCu : C_lock * u = M + C_lock := by
    have : C_lock * (M / C_lock + 1) = M + C_lock := by field_simp
    simpa [hu_def] using this
  have hCpos : 0 < C_lock := hC
  have : C_lock * ((r0 * y) / r0) ^ α = M + C_lock := by
    rw [hRpow]; exact hCu
  linarith

/-- Newtonian-domination at the real-exponent level. -/
theorem ilg_real_velocity_sq_dominates_newtonian
    (V_bar_sq R r0 α : ℝ)
    (hVb : 0 ≤ V_bar_sq) (hR : 0 < R) (hr0 : 0 < r0) :
    V_bar_sq ≤ w_real R r0 α * V_bar_sq := by
  have hw : 1 < w_real R r0 α := enhancement_real_above_one R r0 α hR hr0
  have hwle : 1 ≤ w_real R r0 α := le_of_lt hw
  have : V_bar_sq * 1 ≤ V_bar_sq * w_real R r0 α :=
    mul_le_mul_of_nonneg_left hwle hVb
  linarith [mul_comm V_bar_sq (w_real R r0 α)]

theorem ilgRealExponentEnhancementCert_holds : Nonempty ILGRealExponentEnhancementCert :=
  ⟨{ C_pos := C_lock_pos
     enhancement_pos := enhancement_real_pos
     enhancement_above_one := enhancement_real_above_one
     enhancement_strict_mono := enhancement_real_strict_mono
     enhancement_unbounded := enhancement_real_unbounded
     newtonian_dominated := ilg_real_velocity_sq_dominates_newtonian }⟩

What this page does not claim

The certificate does not claim the formula w_real describes actual gravitational physics or matches any measured observation. The exponent 1 − 1/φ is locked by identification, not derived from first principles within the framework. The certificate does not assert that the enhancement applies to any specific physical system or regime.

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/Gravity/ILGRealExponentEnhancement.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: