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130 lines
4.8 KiB
Text
130 lines
4.8 KiB
Text
import Semantics.FixedPoint
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import Semantics.Bind
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import Mathlib.Tactic
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namespace Semantics.YangMillsLatticeSizing
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/-! ## Yang-Mills Lattice Sizing Models
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Formalization of lattice site count and storage models for small lattice experiments.
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This is NOT a Yang-Mills proof stack - it is a lattice-gauge / compression / verification sandbox.
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Key invariants:
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- Small lattice experiments only (toy models)
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- Storage models for 8-real/site vs full SU(3) link model
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- No claim to 64⁴ production feasibility
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-/
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open Semantics.Q16_16
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/-- Lattice sizing parameters for toy models. -/
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structure ToyLatticeConfig where
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L : Nat -- Linear dimension (small for toy models: 4-16)
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gaugeGroup : Nat -- Gauge group (2 for SU(2), 3 for SU(3))
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storageModel : String -- "8-real/site" or "SU(3) link model"
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bytesPerReal : Nat -- Bytes per real number (4 for Q16_16, 8 for Float64)
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deriving Repr
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/-- Default toy lattice configuration (8-real/site model). -/
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def defaultToyLattice : ToyLatticeConfig :=
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{ L := 8 -- Small for toy experiments
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gaugeGroup := 3
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storageModel := "8-real/site"
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bytesPerReal := 4 } -- Q16_16 fixed-point
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/-- Calculate total lattice sites: N = L⁴.
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--
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-- Arithmetic sanity check:
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-- 8^4 = 4096 and 16^4 = 65536.
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--
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-- External CAS provenance:
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-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
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-- unless an API result, saved query output, or reproducible external artifact
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-- is attached.
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-/
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def latticeSites (config : ToyLatticeConfig) : Nat :=
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config.L ^ 4
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/-- Calculate raw storage size based on model. -/
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def rawStorageSize (config : ToyLatticeConfig) : Nat :=
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match config.storageModel with
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| "8-real/site" => latticeSites config * 8 * config.bytesPerReal
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| "SU(3) link model" => latticeSites config * 4 * 2 * config.bytesPerReal -- 4 complex × 2 (real/imag)
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| _ => latticeSites config * 8 * config.bytesPerReal -- Default to 8-real/site
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/-- Convert bytes to megabytes.
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--
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-- Arithmetic sanity check:
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-- 1048576 bytes / 1048576 = 1 MB.
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--
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-- External CAS provenance:
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-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
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-- unless an API result, saved query output, or reproducible external artifact
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-- is attached.
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-/
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def bytesToMegabytes (bytes : Nat) : Q16_16 :=
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satFromNat (bytes * scale / (1024 * 1024))
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/-- Calculate raw storage size in megabytes. -/
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def rawStorageSizeMB (config : ToyLatticeConfig) : Q16_16 :=
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bytesToMegabytes (rawStorageSize config)
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#eval rawStorageSizeMB defaultToyLattice -- Expected: ~1 MB for 8⁴ lattice with Q16_16
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/-- Theorem: Lattice sites are positive when the linear dimension is positive. -/
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theorem latticeSites_positive (config : ToyLatticeConfig) (hL : config.L > 0) :
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latticeSites config > 0 := by
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simp [latticeSites]
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exact Nat.pow_pos hL
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/-- Theorem: Raw storage size is positive for nonempty lattices and nonzero reals. -/
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theorem rawStorageSize_positive (config : ToyLatticeConfig)
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(hL : config.L > 0) (hBytes : config.bytesPerReal > 0) :
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rawStorageSize config > 0 := by
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unfold rawStorageSize
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split
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· exact Nat.mul_pos (Nat.mul_pos (latticeSites_positive config hL) (by decide)) hBytes
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· exact Nat.mul_pos
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(Nat.mul_pos (Nat.mul_pos (latticeSites_positive config hL) (by decide)) (by decide))
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hBytes
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· exact Nat.mul_pos (Nat.mul_pos (latticeSites_positive config hL) (by decide)) hBytes
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/-- Theorem: 8-real/site model uses 8 reals per site. -/
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theorem eightRealModel_eightReals (config : ToyLatticeConfig) :
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config.storageModel = "8-real/site" →
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rawStorageSize config = latticeSites config * 8 * config.bytesPerReal := by
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intro hModel
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simp [rawStorageSize, hModel]
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/-- Theorem: SU(3) link model uses 8 reals per site (4 complex × 2). -/
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def su3LinkModel_eightReals (config : ToyLatticeConfig) :
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config.storageModel = "SU(3) link model" →
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rawStorageSize config = latticeSites config * 8 * config.bytesPerReal := by
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intro hModel
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simp [rawStorageSize, hModel]
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ring_nf
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exact Or.inl trivial
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/-- Toy lattice feasibility check. -/
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structure FeasibilityCheck where
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feasible : Bool
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reason : String
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maxL : Nat -- Maximum feasible L for given constraints
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maxMB : Q16_16 -- Maximum storage in MB
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deriving Repr
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/-- Check if lattice is feasible for toy experiments. -/
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def feasibilityCheck (config : ToyLatticeConfig) (maxStorageMB : Q16_16) : FeasibilityCheck :=
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let sizeMB := rawStorageSizeMB config
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let feasible := sizeMB ≤ maxStorageMB
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let maxL := if config.L ≤ 16 then 16 else config.L -- Conservative max for toy models
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{
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feasible := feasible
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reason := if feasible then "Lattice fits within storage constraints" else "Lattice exceeds storage constraints"
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maxL := maxL
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maxMB := maxStorageMB
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}
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#eval feasibilityCheck defaultToyLattice (ofNat 16) -- Check if 16MB feasible (should be feasible)
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end Semantics.YangMillsLatticeSizing
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