import QFD.Gravity.TimeRefraction import QFD.Gravity.GeodesicForce import Mathlib.Analysis.SpecialFunctions.Exp import Mathlib.Analysis.Calculus.Deriv.Basic import Mathlib.Data.Real.Basic import Mathlib.Tactic noncomputable section namespace QFD.Nuclear open Real open QFD.Gravity /-! # Nuclear Binding from Time Refraction (No Filters) This file packages the nuclear "time cliff" mechanism in the same robust style as the no-Filters gravity rewrite: * No `Filter` usage * No `𝓝` notation * No `sqrt`-derivative machinery * Core calculus done via `HasDerivAt` witnesses Core model: * Soliton density: ρ(r) = A * exp( (-1/r₀) * r ) * Time potential: V(r) = -(c²/2) * κ * ρ(r) (exact, by `timePotential_eq`) * Radial force: F(r) = - dV/dr We prove: 1) ρ(r) > 0 for A>0 2) ρ decreases for r₀>0 3) well depth V(0) is explicit 4) dV/dr is explicit (positive), so F(r) is explicit (negative/attractive) -/ /-- Soliton density profile (exponential core), written to avoid division in the exponent. -/ def solitonDensity (A r₀ : ℝ) (r : ℝ) : ℝ := A * exp ((-1 / r₀) * r) /-- Convenience: nuclear context is just a `GravityContext` with κ interpreted as κₙ. -/ def ctxNuclear (c κₙ : ℝ) (hc : 0 < c) : GravityContext := { c := c, hc := hc, kappa := κₙ } /-- Nuclear time potential using the gravity timePotential (same equation, different regime). -/ def nuclearPotential (c κₙ A r₀ : ℝ) (hc : 0 < c) (r : ℝ) : ℝ := timePotential (ctxNuclear c κₙ hc) (solitonDensity A r₀) r /-- Nuclear radial force (1D radial proxy): F = - dV/dr. -/ def nuclearForce (c κₙ A r₀ : ℝ) (hc : 0 < c) (r : ℝ) : ℝ := radialForce (ctxNuclear c κₙ hc) (solitonDensity A r₀) r /-- Positivity of the soliton density when A>0. -/ lemma solitonDensity_pos {A r₀ : ℝ} (hA : 0 < A) (r : ℝ) : 0 < solitonDensity A r₀ r := by unfold solitonDensity exact mul_pos hA (exp_pos _) /-- Monotonicity: for r₀>0 and A>0, ρ(r) strictly decreases with r. (Equivalent to the "cliff": steep core, decays outward.) -/ lemma solitonDensity_decreasing {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀) {r₁ r₂ : ℝ} (h : r₁ < r₂) : solitonDensity A r₀ r₂ < solitonDensity A r₀ r₁ := by unfold solitonDensity have hneg : (-1 / r₀) < 0 := by -- since r₀>0, 1/r₀>0, so -1/r₀<0 have h_pos : 0 < (1 / r₀) := one_div_pos.mpr hr₀ -- -1/r₀ = -(1/r₀) < 0 calc (-1 / r₀) = -(1 / r₀) := by ring _ < 0 := neg_neg_of_pos h_pos have hlin : ((-1 / r₀) * r₂) < ((-1 / r₀) * r₁) := by -- multiplying by a negative constant reverses inequality exact (mul_lt_mul_of_neg_left h hneg) have hexp : exp ((-1 / r₀) * r₂) < exp ((-1 / r₀) * r₁) := by -- exp is strictly increasing exact (Real.exp_lt_exp).2 hlin -- multiply by A>0 preserves inequality exact (mul_lt_mul_of_pos_left hexp hA) /-- Exact closed form: V(r) = -(c²/2) * κₙ * ρ(r). -/ theorem nuclearPotential_eq (c κₙ A r₀ : ℝ) (hc : 0 < c) (r : ℝ) : nuclearPotential c κₙ A r₀ hc r = -(c ^ 2) / 2 * (κₙ * solitonDensity A r₀ r) := by unfold nuclearPotential -- `timePotential_eq` from the no-Filters gravity file simpa [ctxNuclear] using (timePotential_eq (ctx := ctxNuclear c κₙ hc) (rho := solitonDensity A r₀) (r := r)) /-- Well depth at the core: V(0) is explicit. -/ theorem wellDepth (c κₙ A r₀ : ℝ) (hc : 0 < c) : nuclearPotential c κₙ A r₀ hc 0 = -(c ^ 2) / 2 * (κₙ * A) := by -- use the exact form and simplify exp(0)=1 have := nuclearPotential_eq (c := c) (κₙ := κₙ) (A := A) (r₀ := r₀) (hc := hc) (r := 0) -- simplify solitonDensity at 0 simpa [solitonDensity] using this /-- A small compatibility lemma: `HasDerivAt (fun r => exp(a*r))` for constant `a`. -/ lemma hasDerivAt_exp_constMul (a r : ℝ) : HasDerivAt (fun x : ℝ => exp (a * x)) (exp (a * r) * a) r := by -- Using chain rule: exp' = exp, (a*x)' = a. have hid : HasDerivAt (fun x : ℝ => x) 1 r := by simpa using (hasDerivAt_id r) have hlin : HasDerivAt (fun x : ℝ => a * x) (a * 1) r := hid.const_mul a -- `Real.hasDerivAt_exp` is standard; if your snapshot names it differently, -- this is the single place to adjust. have hexp : HasDerivAt Real.exp (Real.exp (a * r)) (a * r) := by simpa using (Real.hasDerivAt_exp (a * r)) -- Compose exp ∘ (a*x) have hcomp : HasDerivAt (fun x : ℝ => exp (a * x)) (exp (a * r) * (a * 1)) r := hexp.comp r hlin -- normalize `a*1` simpa using hcomp /-- `HasDerivAt` witness for solitonDensity, using the stable exp-constMul lemma. -/ lemma hasDerivAt_solitonDensity' (A r₀ r : ℝ) : HasDerivAt (solitonDensity A r₀) (A * exp ((-1 / r₀) * r) * (-1 / r₀)) r := by unfold solitonDensity -- derivative of exp( (-1/r₀)*r ) is exp(...) * (-1/r₀) have hE : HasDerivAt (fun x : ℝ => exp ((-1 / r₀) * x)) (exp ((-1 / r₀) * r) * (-1 / r₀)) r := by simpa using (hasDerivAt_exp_constMul ((-1 / r₀)) r) -- scale by A have hScaled : HasDerivAt (fun x : ℝ => A * exp ((-1 / r₀) * x)) (A * (exp ((-1 / r₀) * r) * (-1 / r₀))) r := by exact hE.const_mul A -- normalize associativity simpa [mul_assoc] using hScaled /-- Exact derivative of the nuclear potential: dV/dr = (c²/2) κₙ * (A/r₀) * exp((-1/r₀)*r). -/ theorem nuclearPotential_deriv (c κₙ A r₀ : ℝ) (hc : 0 < c) (r : ℝ) : ∃ dV : ℝ, HasDerivAt (nuclearPotential c κₙ A r₀ hc) dV r ∧ dV = (c ^ 2) / 2 * κₙ * (A * exp ((-1 / r₀) * r) * (1 / r₀)) := by -- Start from the exact closed form: V = C * ρ with C = -(c^2)/2 * κₙ let C : ℝ := (-(c ^ 2) / 2) * κₙ have hVfun : (fun x => nuclearPotential c κₙ A r₀ hc x) = fun x => C * solitonDensity A r₀ x := by funext x -- use nuclearPotential_eq and fold C simp [nuclearPotential_eq, C, mul_assoc, mul_left_comm, mul_comm] -- derivative of solitonDensity have hρ : HasDerivAt (solitonDensity A r₀) (A * exp ((-1 / r₀) * r) * (-1 / r₀)) r := hasDerivAt_solitonDensity' (A := A) (r₀ := r₀) (r := r) -- scale by constant C have hCV : HasDerivAt (fun x => C * solitonDensity A r₀ x) (C * (A * exp ((-1 / r₀) * r) * (-1 / r₀))) r := hρ.const_mul C -- transport derivative back to nuclearPotential via hVfun refine ⟨C * (A * exp ((-1 / r₀) * r) * (-1 / r₀)), ?_, ?_⟩ · -- HasDerivAt goal -- rewrite function, then apply hCV simpa [hVfun] using hCV · -- simplify the algebra to the stated positive form (pull out minus signs) -- C = -(c^2)/2 * κₙ, and (-1/r₀) gives overall + (1/r₀) simp [C] ring /-- Exact nuclear force law (radial proxy): F(r) = -dV/dr = -(c²/2) κₙ * (A/r₀) * exp((-1/r₀)*r) -/ theorem nuclearForce_closed_form (c κₙ A r₀ : ℝ) (hc : 0 < c) (r : ℝ) : nuclearForce c κₙ A r₀ hc r = - (c ^ 2) / 2 * κₙ * (A * exp ((-1 / r₀) * r) * (1 / r₀)) := by -- `nuclearForce` is `radialForce` by definition, and `radialForce = - deriv V`. unfold nuclearForce -- Use the derivative witness from `nuclearPotential_deriv`, then rewrite `radialForce`. rcases nuclearPotential_deriv (c := c) (κₙ := κₙ) (A := A) (r₀ := r₀) (hc := hc) (r := r) with ⟨dV, hdV, hdV_eq⟩ -- `radialForce` is defined as `- deriv V`; `HasDerivAt.deriv` gives `deriv V r = dV`. have hderiv : deriv (nuclearPotential c κₙ A r₀ hc) r = dV := by simpa using hdV.deriv -- nuclearPotential unfolds to timePotential with ctxNuclear and solitonDensity have hVeq : nuclearPotential c κₙ A r₀ hc = timePotential (ctxNuclear c κₙ hc) (solitonDensity A r₀) := by rfl -- substitute and simplify rw [QFD.Gravity.radialForce, ← hVeq, hderiv, hdV_eq] ring /-! ## Future Work: Bound State Formalization The following structures document physics that requires additional Mathlib infrastructure (spectral theory, Sturm-Liouville) to formalize completely. -/ /-- Blueprint: Bound state existence requires proving the nuclear potential well admits normalizable solutions. This needs spectral theory. -/ structure BoundStateBlueprint where /-- The potential well depth from vacuum stiffness -/ well_depth : ℝ /-- Condition: well must be deep enough for bound states -/ depth_sufficient : well_depth > 0 /-- Blueprint: Force unification means EM and strong forces emerge from the same vacuum equations in different parameter regimes. -/ structure ForceUnificationBlueprint where /-- EM regime: low density, long range -/ em_regime : ℝ → Prop /-- Strong regime: high density, short range -/ strong_regime : ℝ → Prop /-- Same underlying equation governs both -/ unified_equation : ℝ → ℝ end QFD.Nuclear