Gravity Ilgreal Exponent Enhancement Enhancement Real Strict Mono

The monotone weight

The declaration enhancement_real_strict_mono establishes a monotonicity property for a specific mathematical function used in a gravity model. The function, written w_real (a radial weight factor, a multiplier applied to a baseline value depending on distance), is defined as w_real(R, r0, α) = 1 + C · (R/r0)^α. Here R is a distance from a center, r0 is a reference distance, C is a positive constant, and α is a real exponent. The theorem states that if R₁ and R₂ are two distances with R₁ < R₂, and if the exponent α is positive, then w_real(R₁) < w_real(R₂). In plain language: as you move farther away, the weight factor strictly increases. It never stays flat and never decreases, provided the exponent is positive.

This property is part of a larger package of results in the framework's machine-checked library of formal theorems, all concerning the same weight function. The package also proves that the weight is always greater than 1 (enhancement_real_above_one), that it grows without bound as R increases (enhancement_real_unbounded), and that a velocity-squared term multiplied by this weight is never smaller than the original term (ilg_real_velocity_sq_dominates_newtonian). The strict monotonicity result is the one that guarantees the weight's behavior is orderly: it rises steadily, which is a useful structural fact for any model that treats this weight as a distance-dependent enhancement.

The theorem does not claim that this weight function is the correct description of gravity in the physical world. It does not assert that the exponent α equals any particular value, nor that the constant C has any specific magnitude; those are parameters of the model, not outputs of this theorem. The result also does not say anything about what happens when the exponent is zero or negative, since the proof explicitly requires α > 0. Finally, it does not claim that the weight function applies to all possible distances or all physical situations; it is a statement about a defined mathematical object, not a universal law of nature.

What the theorem does provide is a rigorous guarantee about the shape of a function. For anyone working with this model, it means the enhancement grows monotonically with distance, which is a concrete and checkable property. It is the kind of structural fact that makes further analysis possible, because it rules out oscillating or reversing behavior in the weight. The proof is machine-checked, meaning it has been verified by a computer program that checks every logical step, with no gaps and no unstated assumptions.

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

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

/-- 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

What this page does not claim

The theorem does not claim that the weight function is the correct description of gravity in the physical world. It does not assert that the exponent α equals any particular value, nor that the constant C has any specific magnitude. The result does not say anything about what happens when the exponent is zero or negative.

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.

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