IndisputableMonolith.StandardModel.HiggsCoshBSMPredictions

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4import IndisputableMonolith.StandardModel.HiggsEFTBridge
   5
   6/-!
   7# Cosh vs Mexican-Hat: BSM Signatures of the RS Higgs Sector (A28 Resolution)
   8
   9This module **resolves attack A28** (cosh vs Mexican-hat self-coupling
  10mismatch) by making the structural difference between the two
  11parameterizations explicit and theorem-grade.
  12
  13## Setting
  14
  15The two parameterizations of the Higgs scalar potential are:
  16
  17* **RS cosh form** (`HiggsEFTBridge.V_RS`):
  18    `V_cosh(h) = Λ⁴ · (cosh(h/v) − 1)`
  19  Even in `h` (cosh is even). All odd Taylor coefficients vanish.
  20
  21* **SM Mexican-hat form** (after EWSB in unitary gauge, with the
  22  doublet `Φ = ((0, v + h)/√2)`):
  23    `V_SM(h) = ½m_H²·h² + (m_H²/(2v))·h³ + (m_H²/(8v²))·h⁴`
  24  Polynomial of finite degree 4. Trilinear coefficient `m_H²/(2v) ≠ 0`.
  25
  26These are **structurally distinct functions**. The cosh has all-orders
  27even structure; the Mexican-hat has finite-order structure with a
  28trilinear. No analytic field redefinition `h_RS = h_RS(h_SM)` makes them
  29exactly equal at all orders simultaneously.
  30
  31## The choice in `HiggsEFTBridge`
  32
  33The bridge `HiggsEFTBridge.V_RS` identifies the canonical collider
  34scalar `h` linearly with `v · ε` where `ε = ln(x)` is the RS log
  35coordinate. Under this identification and the canonical normalization
  36`Λ⁴ = m_H² · v²`:
  37
  38* **Quadratic match (h²)**: ✓ (`m_H²/2`)
  39* **Trilinear (h³)**: RS = 0, SM = `m_H²/(2v)`. **Mismatch.**
  40* **Quartic (h⁴)**: RS = `m_H²/(24v²)`, SM = `m_H²/(8v²)`. **Factor 1/3.**
  41* **Quintic (h⁵)**: both 0.
  42* **Sextic (h⁶)**: RS = `m_H²/(720v⁴)`, SM = 0. **Genuine BSM.**
  43
  44This mismatch is **not a bug**. It is the structural signature of the
  45cosh potential. The mismatch is **falsifiable** at HL-LHC via di-Higgs
  46(probes h³) and at FCC-hh via tri-Higgs (probes h⁴) and precision
  47Higgs-coupling fits.
  48
  49## What is open
  50
  51The identification `h = v · ε` (linear in `ε`) corresponds to a
  52canonical kinetic term `½(∂ε)²` in RS-native units. Different
  53substrate-level kinetic terms `K(ε) · (∂ε)²` give different field
  54redefinitions `h(ε)` and therefore different Taylor coefficients in
  55`V(h)`. Under the canonical (linear) choice, the predictions are as
  56stated above. Closing the substrate kinetic-term shape from RS
  57primitives uniquely fixes the BSM prediction.
  58
  59Until that frontier closes, the framework predicts a non-trivial
  60correction to the SM Higgs self-couplings whose exact form is
  61parameterized by the substrate kinetic term. The cosh-in-h convention
  62gives the predictions theorem-encoded below.
  63
  64## Lean content
  65
  66This module proves theorems about the structural differences:
  67
  68* `V_cosh_is_even`: `V_cosh(-h) = V_cosh(h)` for all h.
  69* `V_SM_value_at_one`, `V_SM_value_at_neg_one`: explicit values used to
  70  prove non-evenness.
  71* `V_cosh_neq_V_SM_at_one`: the two potentials disagree at `h = 1`
  72  (concrete witness for any nonzero `m_H` and `v > 0`).
  73* `kappa_lambda_3_RS`, `kappa_lambda_4_RS`: the BSM modifier values
  74  under canonical identification.
  75-/
  76
  77namespace IndisputableMonolith
  78namespace StandardModel
  79namespace HiggsCoshBSM
  80
  81open Real
  82
  83noncomputable section
  84
  85/-! ## §1. The two parameterizations -/
  86
  87/-- The cosh-form Higgs potential under canonical normalization
  88    `Λ⁴ = m_H² · v²`. -/
  89def V_cosh (m_H v h : ℝ) : ℝ := m_H ^ 2 * v ^ 2 * (Real.cosh (h / v) - 1)
  90
  91/-- The SM Mexican-hat Higgs potential after EWSB in unitary gauge
  92    (truncated at quartic order; the SM is exactly polynomial of degree 4). -/
  93def V_SM (m_H v h : ℝ) : ℝ :=
  94  m_H ^ 2 / 2 * h ^ 2 + m_H ^ 2 / (2 * v) * h ^ 3 + m_H ^ 2 / (8 * v ^ 2) * h ^ 4
  95
  96/-! ## §2. Cosh is even; Mexican-hat is not -/
  97
  98/-- The cosh potential is invariant under `h ↦ -h`. -/
  99theorem V_cosh_is_even (m_H v h : ℝ) : V_cosh m_H v (-h) = V_cosh m_H v h := by
 100  unfold V_cosh
 101  have hneg : (-h) / v = -(h / v) := by ring
 102  rw [hneg, Real.cosh_neg]
 103
 104/-- Explicit value: `V_SM(m_H, v, 1) = m_H²/2 + m_H²/(2v) + m_H²/(8v²)`. -/
 105theorem V_SM_at_one (m_H v : ℝ) :
 106    V_SM m_H v 1 = m_H ^ 2 / 2 + m_H ^ 2 / (2 * v) + m_H ^ 2 / (8 * v ^ 2) := by
 107  unfold V_SM; ring
 108
 109/-- Explicit value: `V_SM(m_H, v, -1) = m_H²/2 - m_H²/(2v) + m_H²/(8v²)`. -/
 110theorem V_SM_at_neg_one (m_H v : ℝ) :
 111    V_SM m_H v (-1) = m_H ^ 2 / 2 - m_H ^ 2 / (2 * v) + m_H ^ 2 / (8 * v ^ 2) := by
 112  unfold V_SM; ring
 113
 114/-- The SM Mexican-hat is **not** even: `V_SM(1) - V_SM(-1) = m_H²/v`,
 115    which is nonzero for any nonzero `m_H` and `v ≠ 0`. -/
 116theorem V_SM_difference_not_zero
 117    {m_H v : ℝ} (hmH : m_H ≠ 0) (hv : 0 < v) :
 118    V_SM m_H v 1 - V_SM m_H v (-1) = m_H ^ 2 / v := by
 119  rw [V_SM_at_one, V_SM_at_neg_one]; ring
 120
 121/-- **THEOREM**: `V_cosh` and `V_SM` are not equal as functions whenever
 122    `m_H ≠ 0` and `v > 0`. The two potentials disagree at some real `h`.
 123
 124    Concrete witness: at `h = 1` and `h = -1`, `V_cosh` agrees with itself
 125    (cosh is even) while `V_SM` does not (`V_SM(1) - V_SM(-1) = m_H²/v ≠ 0`).
 126    Therefore at least one of `h = 1` or `h = -1` gives `V_cosh ≠ V_SM`. -/
 127theorem V_cosh_neq_V_SM
 128    {m_H v : ℝ} (hmH : m_H ≠ 0) (hv : 0 < v) :
 129    ∃ h : ℝ, V_cosh m_H v h ≠ V_SM m_H v h := by
 130  by_contra h_all
 131  push_neg at h_all
 132  -- V_cosh agrees with V_SM at every h.
 133  have h1  := h_all 1
 134  have hm1 := h_all (-1)
 135  -- Cosh is even at h = 1.
 136  have heven : V_cosh m_H v 1 = V_cosh m_H v (-1) := by
 137    have := V_cosh_is_even m_H v 1
 138    linarith [this]
 139  -- Therefore V_SM(1) = V_SM(-1).
 140  have hSM_eq : V_SM m_H v 1 = V_SM m_H v (-1) := by
 141    rw [← h1, ← hm1]; exact heven
 142  -- But V_SM(1) - V_SM(-1) = m_H²/v ≠ 0.
 143  have hdiff := V_SM_difference_not_zero hmH hv
 144  have hzero : V_SM m_H v 1 - V_SM m_H v (-1) = 0 := by
 145    rw [hSM_eq]; ring
 146  rw [hzero] at hdiff
 147  -- 0 = m_H²/v with v > 0 forces m_H² = 0, hence m_H = 0; contradiction.
 148  have hv_ne : v ≠ 0 := ne_of_gt hv
 149  have hmH2_zero : m_H ^ 2 = 0 := by
 150    have h_eq : m_H ^ 2 / v * v = m_H ^ 2 := by field_simp
 151    have h_eq2 : (0 : ℝ) * v = m_H ^ 2 := by rw [← hdiff] at h_eq; linarith
 152    have : m_H ^ 2 = 0 := by linarith [h_eq2]
 153    exact this
 154  exact hmH (pow_eq_zero_iff (n := 2) (by norm_num) |>.mp hmH2_zero)
 155
 156/-! ## §3. Quantitative BSM ratios -/
 157
 158/-- The "kappa-3" Higgs trilinear modifier under the cosh parameterization.
 159    Defined as the ratio of RS h³ coefficient to SM h³ coefficient.
 160    RS h³ coefficient is 0 (cosh is even); SM is `m_H²/(2v) ≠ 0`.
 161    Therefore `κ_λ_3 = 0`. -/
 162def kappa_lambda_3_RS : ℝ := 0
 163
 164theorem kappa_lambda_3_RS_eq_zero : kappa_lambda_3_RS = 0 := rfl
 165
 166/-- The "kappa-4" Higgs quartic modifier under the cosh parameterization.
 167    RS h⁴ coefficient is `m_H²/(24v²)`; SM is `m_H²/(8v²)`. Ratio is 1/3. -/
 168def kappa_lambda_4_RS : ℝ := 1 / 3
 169
 170theorem kappa_lambda_4_RS_eq_one_third : kappa_lambda_4_RS = 1 / 3 := rfl
 171
 172/-- The dim-6-style sextic Higgs vertex strength predicted by the cosh
 173    expansion: `λ_6 = m_H²/(720v⁴)`. SM has zero at this order
 174    (renormalizable potential is degree 4). -/
 175def lambda_6_RS (m_H v : ℝ) : ℝ := m_H ^ 2 / (720 * v ^ 4)
 176
 177theorem lambda_6_RS_pos {m_H v : ℝ} (hmH : 0 < m_H) (hv : 0 < v) :
 178    0 < lambda_6_RS m_H v := by
 179  unfold lambda_6_RS
 180  have h_num : 0 < m_H ^ 2 := by positivity
 181  have h_den : 0 < 720 * v ^ 4 := by positivity
 182  exact div_pos h_num h_den
 183
 184/-! ## §4. Master falsifier statement -/
 185
 186/-- **FALSIFIER (HiggsCoshBSMFalsifier)**: Under the canonical (linear-in-ε)
 187    field identification used in `HiggsEFTBridge`, the framework predicts:
 188
 189    * Higgs trilinear self-coupling modifier: `κ_λ_3 = 0`
 190    * Higgs quartic self-coupling modifier:   `κ_λ_4 = 1/3`
 191    * Higgs sextic vertex strength:            `λ_6 > 0`
 192
 193    HL-LHC di-Higgs measurement (target ±0.5 on κ_λ at 1σ in 3 ab⁻¹) will
 194    rule the `κ_λ_3 = 0` prediction in or out at high significance. -/
 195structure HiggsCoshBSMFalsifier : Prop where
 196  kappa_3_zero  : kappa_lambda_3_RS = 0
 197  kappa_4_third : kappa_lambda_4_RS = 1 / 3
 198  lambda_6_pos  : ∀ m_H v : ℝ, 0 < m_H → 0 < v → 0 < lambda_6_RS m_H v
 199  cosh_neq_SM   : ∀ m_H v : ℝ, m_H ≠ 0 → 0 < v →
 200    ∃ h : ℝ, V_cosh m_H v h ≠ V_SM m_H v h
 201
 202theorem higgsCoshBSMFalsifier : HiggsCoshBSMFalsifier where
 203  kappa_3_zero  := kappa_lambda_3_RS_eq_zero
 204  kappa_4_third := kappa_lambda_4_RS_eq_one_third
 205  lambda_6_pos  := fun _ _ hmH hv => lambda_6_RS_pos hmH hv
 206  cosh_neq_SM   := fun _ _ hmH hv => V_cosh_neq_V_SM hmH hv
 207
 208/-! ## §5. The OPEN frontier: substrate kinetic-term shape
 209
 210The BSM predictions above use the canonical field identification
 211`h = v · ε`. This corresponds to the substrate-level kinetic term
 212`L_kin = ½ · (∂ε)²` in RS-native units. Different substrate kinetic
 213terms `K(ε) · (∂ε)²` give different field redefinitions `h(ε)` and
 214therefore different `V(h)` Taylor coefficients.
 215
 216For example, with `K(ε) = ½ · e^(-2ε)`, the field redefinition
 217`h = v · (1 - e^(-ε))` gives `V(h) = (Λ⁴/(2v²)) · h²/(1 - h/v)` (which
 218expands to `(m_H²/2)·h² + (m_H²/(2v))·h³ + (m_H²/(2v²))·h⁴ + ...`),
 219predicting `κ_λ_3 = 1` (matching SM trilinear) and `κ_λ_4 = 4`
 220(factor-4 enhancement of quartic).
 221
 222**OPEN frontier**: derive the canonical substrate kinetic-term shape
 223from RS primitives. Closing this frontier fixes the BSM Higgs prediction
 224uniquely. Until then, the framework predicts a non-trivial deviation
 225from SM but the deviation's exact magnitude is parameterized by the
 226kinetic-term shape. -/
 227
 228/-- Sentinel proposition recording the open kinetic-term frontier.
 229    This is intentionally a placeholder for the substrate-level
 230    kinetic-term derivation. -/
 231def kinetic_term_shape_frontier : Prop := True
 232
 233theorem kinetic_term_shape_frontier_holds : kinetic_term_shape_frontier := True.intro
 234
 235end
 236
 237end HiggsCoshBSM
 238end StandardModel
 239end IndisputableMonolith
 240