Gravity Ilgreal Exponent Enhancement Enhancement Real Unbounded

The unbounded enhancement

In the Recognition Science framework, gravity is described through a ledger, a discrete record of recognition events, and the framework's library of machine-checked formal theorems contains a module about a radial enhancement factor. The object in question is a simple mathematical function: w_real(R, r0, α) = 1 + C · (R/r0)^α, where R is a distance from a source, r0 is a reference distance, α is a real exponent, and C is a positive constant. This is not a physical law by itself; it is a definitional choice, a model of how a correction factor might scale with distance.

The theorem named enhancement_real_unbounded establishes a property of this defined function. It proves that for any positive reference distance r0, any positive exponent α, and any positive bound M, there exists a distance R such that the value of w_real exceeds M. In plain language, as R grows without bound, the enhancement factor also grows without bound. The proof is a theorem in the framework's library, verified with no unproved assumptions, and it relies on the positivity of the exponent and the positivity of the constant C. The theorem is a statement about the real numbers, not about the physical universe.

The same module proves three companion facts about w_real: it is always greater than 1, it is strictly increasing as R increases, and it dominates the Newtonian velocity-squared term everywhere. These facts, together with the unboundedness, are bundled into a single certificate structure. The certificate is a formal object that packages these properties for downstream use in the library. It is a structural result about the model, not an empirical claim.

What the declaration does not claim is equally precise. It does not claim that the function w_real is the correct description of gravity in the actual universe. It does not claim that the exponent α must be the specific value 1 − 1/φ, where φ is the golden ratio, although the module notes that this locked real exponent is the case of interest. The theorem holds for any positive α, and it does not select one. It does not claim that the unbounded growth is physically realized; it claims only that the defined function has this mathematical property. The physical bridge, from the ledger to actual gravitational phenomena, is a separate question that the theorem does not address.

The consequence for a reader is a clear picture of what the framework's library has and has not established. The library has a proved theorem about a specific mathematical function: it grows without limit. The library has not proved that this function is gravity. The distinction between a theorem about a model and a claim about the world is the boundary that the declaration respects.

/-- 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

/-- The real-exponent radial weight, with `α : ℝ` (no positivity required
    by the definition itself). -/
def w_real (R r0 α : ℝ) : ℝ := 1 + C_lock * (R / r0) ^ α

/-- 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 w_real is the correct description of gravity in the actual universe. The theorem does not claim that the unbounded growth is physically realized. The theorem does not claim that α must be the specific value 1 − 1/φ.

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

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