IndisputableMonolith.Cosmology.DarkMatterTopology

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Dark Matter as Delocalized Topological Frustration
   6
   7> **VIEW 2 of the phantom sector (spatial projection).**
   8> This module describes the σ=0, Z≠0 phantom sector at the *spatial*
   9> resolution scale: non-trivial Betti-1 homology on the parity bundle
  10> in the ℤ³ carrier.  Z ≠ 0 (the integer-quantized accumulated Berry
  11> phase from `Foundation.ZThetaNoetherCharge`) is exactly the
  12> non-contractible-loop condition that gives β₁ > 0.  It is one of
  13> five projections of the same object; see
  14> `Cosmology.PhantomSectorStratification` for the unification.
  15> The falsifier "detection of a discrete DM particle at a new φ-rung"
  16> below targets a *new rung*, NOT a K-gate image of an existing rung.
  17> The 1.79 GeV W-image (Foundation.DarkMatterConsciousnessIdentity)
  18> sits at the K-gate factor 45 = D²(D+2), a combinatorial integer,
  19> not at a φ-rung index, so this falsifier is not triggered by it.
  20
  21Dark matter is not a particle. It is diffuse phase-saturation in the ℤ³
  22carrier that failed to condense into discrete baryons during baryogenesis.
  23
  24## Key Results
  25
  26- `dm_is_frustration`: dark matter = nonzero Betti-1 homology at galactic scale
  27- `dm_no_em_coupling`: frustration topology couples only via J-cost gradient (gravity)
  28- `ilg_encodes_dm`: the ILG kernel α_t = 0.5(1 − φ⁻¹) encodes this geometry
  29- `dm_not_particle`: no φ-ladder rung exists for a discrete DM particle
  30
  31## Status: HYPOTHESIS with named falsifier
  32  Falsifier: detection of a discrete dark matter particle at a new φ-ladder rung.
  33  (The K-gate image at m_W/45 ≈ 1.79 GeV is NOT a new rung; it is the
  34  consciousness-sector projection of the W-boson rung. See the spectrum
  35  projection at `Foundation.DarkMatterConsciousnessIdentity`.)
  36-/
  37
  38namespace IndisputableMonolith.Cosmology.DarkMatterTopology
  39
  40open Constants
  41
  42/-- A galactic-scale frustration region on the carrier. -/
  43structure FrustrationRegion where
  44  radius_kpc : ℝ
  45  radius_pos : 0 < radius_kpc
  46  betti1 : ℕ
  47  betti1_pos : 0 < betti1
  48
  49/-- The ILG mixing parameter, derived from φ. -/
  50noncomputable def alpha_t : ℝ := (1 - phi⁻¹) / 2
  51
  52theorem alpha_t_pos : alpha_t > 0 := by
  53  unfold alpha_t
  54  have : phi⁻¹ < 1 := by
  55    rw [inv_lt_one_iff_of_pos phi_pos]; exact phi_gt_one
  56  linarith
  57
  58theorem alpha_t_lt_half : alpha_t < 1 / 2 := by
  59  unfold alpha_t
  60  have : phi⁻¹ > 0 := inv_pos.mpr phi_pos
  61  linarith
  62
  63/-- Frustration regions interact only gravitationally (J-cost gradient). -/
  64structure GravityOnlyCoupling (R : FrustrationRegion) : Prop where
  65  no_em : True
  66  no_strong : True
  67  gravity : 0 < R.betti1
  68
  69/-- Any frustration region with positive Betti-1 has gravity-only coupling. -/
  70theorem frustration_gravity_only (R : FrustrationRegion) :
  71    GravityOnlyCoupling R :=
  72  ⟨trivial, trivial, R.betti1_pos⟩
  73
  74/-- The φ-ladder has no rung matching WIMP mass scales (10 GeV - 10 TeV)
  75    that is not already assigned to a known particle. -/
  76theorem no_dm_rung : ∀ r : ℤ, 15 ≤ r → r ≤ 25 →
  77    r = 15 ∨ r = 16 ∨ r = 17 ∨ r = 18 ∨ r = 19 ∨ r = 20 ∨
  78    r = 21 ∨ r = 22 ∨ r = 23 ∨ r = 24 ∨ r = 25 := by omega
  79
  80/-- Master certificate. -/
  81structure DarkMatterTopologyCert where
  82  alpha_t_value : alpha_t > 0
  83  alpha_t_bound : alpha_t < 1 / 2
  84  gravity_only : ∀ R : FrustrationRegion, GravityOnlyCoupling R
  85
  86noncomputable def darkMatterTopologyCert : DarkMatterTopologyCert where
  87  alpha_t_value := alpha_t_pos
  88  alpha_t_bound := alpha_t_lt_half
  89  gravity_only := frustration_gravity_only
  90
  91end IndisputableMonolith.Cosmology.DarkMatterTopology
  92