Research-Stack/0-Core-Formalism/lean/external/OTOM/BracketShellCount.lean

/-
BracketShellCount.lean - Bracket Approach to Shell Counting

Applies BraidBracket methodology to shell occupancy counting:
- Nuclear shell model: counting nucleons in energy levels
- Electron shells: counting electrons in orbitals  
- Compression shells: counting elements in hierarchical containers

Key insight: Shell counts form bracket bounds on admissible configurations.
-/

import Semantics.BraidBracket
import Semantics.ShellModel
import Semantics.DynamicCanal

namespace Semantics.BracketShellCount

open BraidBracket ShellModel DynamicCanal

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Shell Count Types
-- ═══════════════════════════════════════════════════════════════════════════

/-- Shell occupancy count with bracket bounds -/
structure ShellCount where
  level : Nat              -- Shell energy level (n)
  capacity : Nat           -- Maximum occupancy (2·(2·l+1) for orbitals)
  occupied : Nat           -- Current occupancy
  -- Bracket bounds derived from shell structure
  lowerBound : Fix16       -- Minimum admissible count (bracket lower)
  upperBound : Fix16       -- Maximum admissible count (bracket upper)
  gap : Fix16              -- Bracket gap (upper - lower)
  admissible : Bool        -- Whether count is within bracket
  deriving Repr, DecidableEq, BEq

namespace ShellCount

/-- Convert Nat to Fix16 (simple conversion for shell counts) -/
def natToFix16 (n : Nat) : Fix16 :=
  ⟨(n.toUInt32 * 0x10000).toUInt32⟩  -- Scale to Q16.16

/-- Empty shell count (zero occupancy) -/
def empty (capacity : Nat) : ShellCount :=
  ShellCount.mk 0 capacity 0 Fix16.zero (natToFix16 capacity) (natToFix16 capacity) true

/-- Full shell count (maximum occupancy) -/
def full (level : Nat) (capacity : Nat) : ShellCount :=
  ShellCount.mk level capacity capacity Fix16.zero (natToFix16 capacity) (natToFix16 capacity) true

/-- Compute bracket bounds from shell structure
    
    The bracket [lower, upper] bounds admissible occupancy based on:
    - Shell capacity (geometric constraint)
    - Pauli exclusion (fermionic constraint)  
    - Energy level (hierarchical constraint)
    -/
def computeBracket (level : Nat) (capacity : Nat) (occupied : Nat)
    (energy : Fix16) (spin : Fix16) : ShellCount :=
  let capFix := natToFix16 capacity
  let occFix := natToFix16 occupied
  
  -- Lower bound: 0 (empty shell always admissible)
  let lo := Fix16.zero
  
  -- Upper bound: capacity (Pauli exclusion)
  let up := capFix
  
  -- Gap: capacity - 0 = capacity
  let g := Fix16.sub up lo
  
  -- Admissibility: 0 ≤ occupied ≤ capacity
  let adm := occupied ≤ capacity
  
  ShellCount.mk level capacity occupied lo up g adm

/-- Add particle to shell (increment count) -/
def addParticle (sc : ShellCount) : ShellCount :=
  if sc.occupied < sc.capacity then
    computeBracket sc.level sc.capacity (sc.occupied + 1) Fix16.zero Fix16.zero
  else
    ShellCount.mk sc.level sc.capacity sc.occupied sc.lowerBound sc.upperBound sc.gap false  -- Overfull: violates bracket

/-- Remove particle from shell (decrement count) -/
def removeParticle (sc : ShellCount) : ShellCount :=
  if sc.occupied > 0 then
    computeBracket sc.level sc.capacity (sc.occupied - 1) Fix16.zero Fix16.zero
  else
    sc  -- Empty: no change

end ShellCount

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Shell System with Brackets
-- ═══════════════════════════════════════════════════════════════════════════

/-- System of shells with bracketed counts -/
structure ShellSystem where
  shells : List ShellCount
  totalParticles : Nat
  totalCapacity : Nat
  -- System-level bracket bounds
  systemLower : Fix16
  systemUpper : Fix16
  systemGap : Fix16
  systemAdmissible : Bool
  deriving Repr, DecidableEq, BEq

namespace ShellSystem

/-- Empty shell system -/
def empty : ShellSystem :=
  ShellSystem.mk [] 0 0 Fix16.zero Fix16.zero Fix16.zero true

/-- Add shell to system -/
def addShell (sys : ShellSystem) (capacity : Nat) : ShellSystem :=
  let newShell := ShellCount.empty capacity
  let newShells := newShell :: sys.shells
  let newTotalCap := sys.totalCapacity + capacity
  
  -- Recompute system bracket
  let sysLower := Fix16.zero
  let sysUpper := natToFix16 newTotalCap
  let sysGap := Fix16.sub sysUpper sysLower
  
  ShellSystem.mk newShells sys.totalParticles newTotalCap sysLower sysUpper sysGap true

/-- Fill shell at index (add particle) -/
def fillShell (sys : ShellSystem) (idx : Nat) : ShellSystem :=
  match sys.shells.get? idx with
  | none => sys  -- Invalid index
  | some shell =>
    let newShell := shell.addParticle
    let newShells := sys.shells.set idx newShell
    let newTotal := sys.totalParticles + 1
    
    -- Check system admissibility
    let sysAdm := newTotal ≤ sys.totalCapacity
    
    ShellSystem.mk newShells newTotal sys.totalCapacity sys.systemLower sys.systemUpper sys.systemGap sysAdm

/-- Compute total bracket from individual shell brackets -/
def computeSystemBracket (sys : ShellSystem) : ShellSystem :=
  -- Sum individual gaps (bracket algebra)
  let totalGap := sys.shells.foldl (fun acc s => 
    Fix16.add acc s.gap) Fix16.zero
  
  -- System bounds: [0, totalCapacity]
  let sysLower := Fix16.zero
  let sysUpper := natToFix16 sys.totalCapacity
  
  ShellSystem.mk sys.shells sys.totalParticles sys.totalCapacity sysLower sysUpper totalGap (sys.totalParticles ≤ sys.totalCapacity)

end ShellSystem

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Nuclear Shell Model Application
-- ═══════════════════════════════════════════════════════════════════════════

/-- Nuclear shell: 2·(2·j+1) capacity for each j level -/
def nuclearShellCapacity (j : Nat) : Nat :=
  2 * (2 * j + 1)  -- 2j+1 magnetic substates × 2 for proton/neutron

/-- Magic numbers: closed shell configurations -/
def magicNumbers : List Nat :=
  [2, 8, 20, 28, 50, 82, 126]  -- Standard nuclear magic numbers

/-- Create nuclear shell system with magic number closure -/
def nuclearShellSystem : ShellSystem :=
  let sys := ShellSystem.empty
  -- Add shells up to magic number 126
  let capacities := [2, 6, 12, 8, 22, 32, 44]  -- Cumulative capacities
  capacities.foldl (fun sys cap => sys.addShell cap) sys

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Theorems: Bracket Conservation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Theorem: Shell count always stays within bracket bounds -/
theorem shellCountWithinBracket (sc : ShellCount) :
    sc.admissible → 
    let occFix := natToFix16 sc.occupied
    sc.lowerBound.raw ≤ occFix.raw ∧ occFix.raw ≤ sc.upperBound.raw := by
  intro hAdm
  simp [ShellCount.computeBracket]
  exact ⟨by positivity, Nat.le_iff_eq_or_lt.mp hAdm⟩

/-- Theorem: Adding particle preserves bracket if not full -/
theorem addParticlePreservesBracket (sc : ShellCount) :
    sc.occupied < sc.capacity → 
    (sc.addParticle).admissible = true := by
  intro hNotFull
  simp [ShellCount.addParticle, ShellCount.computeBracket]
  exact hNotFull

/-- Theorem: System admissibility iff total ≤ capacity -/
theorem systemAdmissibleIff (sys : ShellSystem) :
    sys.systemAdmissible ↔ sys.totalParticles ≤ sys.totalCapacity := by
  unfold ShellSystem.systemAdmissible
  cases sys
  simp

/-- Theorem: Gap conservation across shell system -/
theorem gapConservation (sys : ShellSystem) :
    let sysGap := sys.systemGap
    let sumGaps := sys.shells.foldl (fun acc s => Fix16.add acc s.gap) Fix16.zero
    sysGap = sumGaps := by
  unfold ShellSystem.systemGap
  cases sys
  simp

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Verification Examples
-- ═══════════════════════════════════════════════════════════════════════════

-- Verification examples skipped due to Fix16 conversion dependencies
-- TODO(lean-port): Add proper #eval witnesses after Fix16 integration

end Semantics.BracketShellCount