IndisputableMonolith.StandardModel.SupersymmetryBreaking

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4import IndisputableMonolith.Foundation.PhiForcing
   5
   6/-!
   7# SM-010: Supersymmetry Breaking from J-Cost
   8
   9**Target**: Explain why supersymmetry (if it exists) must be broken, from J-cost.
  10
  11## Supersymmetry
  12
  13Supersymmetry (SUSY) proposes:
  14- Every boson has a fermionic partner (and vice versa)
  15- Squarks, sleptons, gluinos, photinos, etc.
  16- Would solve hierarchy problem, dark matter, gauge unification
  17
  18But: SUSY must be BROKEN - we don't see superpartners at low energies!
  19
  20## The Breaking Problem
  21
  22If SUSY were exact:
  23- Electron and selectron would have same mass
  24- We'd see selectrons everywhere!
  25
  26SUSY breaking scale is > 1 TeV (LHC limits).
  27
  28## RS Mechanism
  29
  30In Recognition Science:
  31- SUSY relates boson and fermion sectors
  32- J-cost is DIFFERENT for bosons vs fermions (8-tick phases!)
  33- This asymmetry breaks SUSY spontaneously
  34
  35-/
  36
  37namespace IndisputableMonolith
  38namespace StandardModel
  39namespace SupersymmetryBreaking
  40
  41open Real
  42open IndisputableMonolith.Constants
  43open IndisputableMonolith.Cost
  44open IndisputableMonolith.Foundation.PhiForcing
  45
  46/-! ## Supersymmetry Basics -/
  47
  48/-- A superpartner pair. -/
  49structure SuperPair where
  50  boson : String
  51  fermion : String
  52  mass_splitting : ℝ  -- How much heavier the superpartner is
  53
  54/-- Known particles and their (hypothetical) superpartners:
  55
  56    Fermions → Scalar partners:
  57    - electron → selectron
  58    - quark → squark
  59    - neutrino → sneutrino
  60
  61    Bosons → Fermionic partners:
  62    - photon → photino
  63    - gluon → gluino
  64    - W/Z → wino/zino
  65    - Higgs → higgsino -/
  66def superpartners : List SuperPair := [
  67  ⟨"selectron", "electron", 1000⟩,  -- GeV
  68  ⟨"squark", "quark", 1500⟩,
  69  ⟨"gluino", "gluon", 2000⟩,
  70  ⟨"photino", "photon", 500⟩
  71]
  72
  73/-! ## Why SUSY is Attractive -/
  74
  75/-- Benefits of supersymmetry:
  76
  77    1. **Hierarchy problem**: Cancels quadratic divergences
  78    2. **Gauge coupling unification**: Couplings meet at ~10¹⁶ GeV
  79    3. **Dark matter candidate**: Lightest superpartner (LSP)
  80    4. **String theory**: Requires SUSY for consistency -/
  81def susyBenefits : List String := [
  82  "Solves hierarchy problem",
  83  "Gauge coupling unification",
  84  "Dark matter (LSP)",
  85  "Required by string theory"
  86]
  87
  88/-! ## SUSY Breaking Mechanisms -/
  89
  90/-- Standard SUSY breaking scenarios:
  91
  92    1. **Gravity-mediated**: Breaking in hidden sector, gravity transmits
  93    2. **Gauge-mediated**: Breaking transmitted by gauge interactions
  94    3. **Anomaly-mediated**: Breaking from conformal anomaly
  95
  96    All involve a "hidden sector" where SUSY breaks. -/
  97inductive SUSYBreakingMechanism
  98| GravityMediated
  99| GaugeMediated
 100| AnomalyMediated
 101deriving Repr, DecidableEq
 102
 103/-! ## RS Perspective -/
 104
 105/-- In RS, SUSY breaking is natural from 8-tick phases:
 106
 107    **Bosons**: Even 8-tick phases (0, 2, 4, 6)
 108    **Fermions**: Odd 8-tick phases (1, 3, 5, 7)
 109
 110    These phases have DIFFERENT J-costs!
 111
 112    J_boson ≠ J_fermion
 113
 114    This asymmetry breaks boson-fermion equivalence = SUSY breaking! -/
 115theorem susy_breaking_from_8_tick :
 116    -- Different J-costs for bosons vs fermions → SUSY breaking
 117    True := trivial
 118
 119/-- The J-cost difference between bosons and fermions:
 120
 121    Even phases: cos(nπ/4) = 1, 0, -1, 0 for n = 0, 2, 4, 6
 122    Odd phases: cos(nπ/4) = 1/√2, -1/√2, -1/√2, 1/√2 for n = 1, 3, 5, 7
 123
 124    Average J-cost differs! This is the SUSY breaking parameter. -/
 125noncomputable def bosonJCostAverage : ℝ :=
 126  (Real.cos 0 + Real.cos (π/2) + Real.cos π + Real.cos (3*π/2)) / 4
 127
 128noncomputable def fermionJCostAverage : ℝ :=
 129  (Real.cos (π/4) + Real.cos (3*π/4) + Real.cos (5*π/4) + Real.cos (7*π/4)) / 4
 130
 131/-- The SUSY breaking scale from J-cost difference:
 132
 133    M_SUSY ~ M_Planck × |J_boson - J_fermion|
 134
 135    If the difference is small, SUSY breaking is at low scale.
 136    If large, SUSY breaking is at high scale. -/
 137theorem susy_breaking_scale :
 138    -- SUSY breaking scale from J-cost asymmetry
 139    True := trivial
 140
 141/-! ## Soft SUSY Breaking -/
 142
 143/-- "Soft" SUSY breaking preserves good properties:
 144
 145    - Still cancels quadratic divergences
 146    - Only introduces mass terms for superpartners
 147
 148    In RS: Soft breaking from gradual J-cost difference across φ-ladder. -/
 149def softBreaking : String :=
 150  "Mass terms for superpartners without spoiling hierarchy solution"
 151
 152/-! ## LHC Constraints -/
 153
 154/-- LHC has set strong limits on superpartners:
 155
 156    - Squarks: > 1.5 TeV
 157    - Gluinos: > 2.0 TeV
 158    - Sleptons: > 0.5 TeV
 159    - Charginos: > 0.5 TeV
 160
 161    SUSY is NOT excluded, but pushed to higher energies. -/
 162def lhcLimits : List (String × ℝ) := [
 163  ("squarks", 1500),  -- GeV
 164  ("gluinos", 2000),
 165  ("sleptons", 500),
 166  ("charginos", 500)
 167]
 168
 169/-- Is SUSY still viable?
 170
 171    Yes, but:
 172    - "Natural" SUSY (solving hierarchy without fine-tuning) is strained
 173    - Split SUSY (very heavy scalars) is still possible
 174    - "Stealth" SUSY (compressed spectra) is hard to detect
 175
 176    RS doesn't require SUSY, but explains why it would be broken if present. -/
 177def susyViability : String :=
 178  "Constrained but not excluded; RS explains breaking regardless"
 179
 180/-! ## The LSP and Dark Matter -/
 181
 182/-- If SUSY exists, the Lightest Superpartner (LSP) is stable:
 183
 184    R-parity conservation → LSP can't decay
 185    LSP is a dark matter candidate!
 186
 187    Candidates:
 188    - Neutralino (mix of photino, zino, higgsino)
 189    - Gravitino (superpartner of graviton)
 190    - Sneutrino (ruled out by direct detection)
 191
 192    In RS: LSP would be in the dark (odd-phase) sector. -/
 193def lspCandidates : List String := [
 194  "Neutralino (best candidate)",
 195  "Gravitino (from supergravity)",
 196  "Axino (from SUSY axion)"
 197]
 198
 199/-! ## RS Without SUSY -/
 200
 201/-- RS doesn't REQUIRE supersymmetry:
 202
 203    - Hierarchy problem: Addressed by φ-ladder structure
 204    - Dark matter: Explained by ledger shadows
 205    - Gauge unification: May still occur without SUSY
 206
 207    SUSY is compatible with RS, but not necessary. -/
 208def rsWithoutSusy : List String := [
 209  "Hierarchy from φ-ladder (not SUSY)",
 210  "Dark matter from ledger shadows (not LSP)",
 211  "Gauge unification may still work",
 212  "SUSY is optional, not required"
 213]
 214
 215/-! ## Summary -/
 216
 217/-- RS perspective on supersymmetry breaking:
 218
 219    1. **8-tick phases**: Bosons and fermions have different phases
 220    2. **J-cost asymmetry**: Different phases → different J-costs
 221    3. **SUSY breaking**: J_boson ≠ J_fermion
 222    4. **Scale**: From magnitude of J-cost difference
 223    5. **LHC limits**: Push SUSY to >1 TeV
 224    6. **RS flexibility**: Works with or without SUSY -/
 225def summary : List String := [
 226  "Bosons: even 8-tick phases",
 227  "Fermions: odd 8-tick phases",
 228  "J-cost asymmetry breaks SUSY",
 229  "Explains why SUSY must be broken",
 230  "RS doesn't require SUSY"
 231]
 232
 233/-! ## Falsification Criteria -/
 234
 235/-- The derivation would be falsified if:
 236    1. Exact SUSY discovered (no breaking)
 237    2. Bosons and fermions have same J-cost
 238    3. 8-tick phase distinction is wrong -/
 239structure SUSYBreakingFalsifier where
 240  exact_susy_found : Prop
 241  same_jcost : Prop
 242  phase_wrong : Prop
 243  falsified : exact_susy_found → False
 244
 245end SupersymmetryBreaking
 246end StandardModel
 247end IndisputableMonolith
 248