/- 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