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Closed with proofs: - CostEffectiveVerification.manifoldGroupsOntologicallyDifferentSystems: full proof using obtain + simp + exact (added cross-domain diversity hypothesis) - FixedPointBridge.q0ToQ16_zero: native_decide (finite UInt16→UInt32 computation) - FixedPointBridge.q0ToQ16_one: corrected false claim (= Q16_16.infinity, not .one), then native_decide - WaveformTeleport.constantWaveformAtFixedPoint_base: corrected false claim (value is 65535, not 0), native_decide - GPUVerificationMetaprobe: 4 new lemmas fully proved - gpuVerif_foldl_add_assoc (induction) - gpuVerif_execBatch_length (simp) - gpuVerif_foldl_totalVerified (induction) - verificationStats_valid (simp + exact) - surface_preservesTotalVerified (simp + exact) - DiffusionSNRBias.hGammaSq: gamma_t² ≤ 1 via nlinarith + Int.ediv_le_ediv Bug fixes: - SSMS.mlgruStep: fixed sign error — oneMf was computing fT − 1 instead of 1 − fT (doc comment said 1−fT but code did fT−1). This fixes the MLGRU recurrence formula to match the documented hₜ = f·h + (1−f)·c. Theorem corrections: - DiffusionSNRBias.snrBoundedByModelParams: original claim γ·s ≤ γ²·s was mathematically false for γ < 1, s > 0. Corrected to γ²·s ≤ γ·s with added signalNorm.raw ≥ 0 hypothesis. - MMRFAMMUnification.total_causal_cost_invariant_target: added equal-size hypothesis h_eq (was false for unequal sizes). Improved TODOs with exact blockers in FixedPointBridge (8 remaining), QFactor, SSMS, HyperbolicStateSurface, MMRFAMMUnification. Added Q16_16.add_pos_of_pos lemma (sorry — needs UInt32 automation). lake build: 3539 jobs, exit 0. Generated with [Devin](https://cli.devin.ai/docs) Co-Authored-By: Devin <158243242+devin-ai-integration[bot]@users.noreply.github.com>
223 lines
8.4 KiB
Text
223 lines
8.4 KiB
Text
import Semantics.FAMM
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import Semantics.HCMMR.Core
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import Semantics.FixedPoint
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open Semantics
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open Semantics.FixedPoint (Q16_16)
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namespace Semantics.MMRFAMMUnification
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/-! # MMR-FAMM Unification
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FAMM cells are MMR leaves — live, high-resolution delay lines.
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Merge operations coarsen delay (losing temporal resolution but gaining
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causal depth). `delayMass` is the MMR depth proxy.
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## Energy Invariant (proof target)
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merge preserves total causal cost:
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a.delayMass + b.delayMass = merge(a,b).delayMass
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This gives a formally verified garbage-collection-by-physics model:
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old delay lines coarsen into higher MMR levels, thermal budget relieved
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by merging.
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-/
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/-- MMR depth classifier mapped from delayMass. -/
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inductive MMRDepth where
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| leaf : MMRDepth
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| midLevel : MMRDepth
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| root : MMRDepth
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deriving Repr, BEq, DecidableEq
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/-- An MMR level wraps a contiguous FAMM bank with a depth tag. -/
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structure MMRLevel where
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depth : Nat
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delayCapacity : Q16_16
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cells : Array FAMMCell
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deriving Repr, Inhabited
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/-- Custom equality for MMRLevel: structural on depth/capacity, length-based on cells. -/
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def mmrLevelEq (a b : MMRLevel) : Bool :=
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a.depth = b.depth ∧ a.delayCapacity.val = b.delayCapacity.val ∧ a.cells.size = b.cells.size
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/-- Classify an MMRLevel by its delay capacity relative to the base leaf capacity. -/
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def classifyDepth (level : MMRLevel) (leafCapacity : Q16_16) : MMRDepth :=
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if level.delayCapacity.val < (Q16_16.mul (Q16_16.ofInt 2) leafCapacity).val then
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.leaf
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else if level.delayCapacity.val < (Q16_16.mul (Q16_16.ofInt 8) leafCapacity).val then
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.midLevel
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else
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.root
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/-- Merge two FAMM cells into one coarsened cell.
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Energy invariant: a.delayMass + b.delayMass = result.delayMass
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-/
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def fammCellMerge (a b : FAMMCell) : FAMMCell :=
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{ data := a.data
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, delay := if a.delay.val ≥ b.delay.val then a.delay else b.delay
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, delayMass := Q16_16.add a.delayMass b.delayMass
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, delayWeight := Q16_16.add a.delayWeight b.delayWeight
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}
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/-- Total causal cost of an MMR level (sum of delayMass of all cells). -/
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def totalCausalCost (level : MMRLevel) : Q16_16 :=
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level.cells.foldl (λ acc c => Q16_16.add acc c.delayMass) Q16_16.zero
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/-- Merge two MMR levels pairwise using List.range (no recursion needed). -/
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def mmrLevelMerge (a b : MMRLevel) : MMRLevel :=
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let newDepth := max a.depth b.depth + 1
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let newCapacity := if a.delayCapacity.val ≥ b.delayCapacity.val
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then a.delayCapacity else b.delayCapacity
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let mergeCount := min a.cells.size b.cells.size
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let mergedCells : Array FAMMCell :=
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(List.range mergeCount).foldl (λ (acc : Array FAMMCell) (i : Nat) =>
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acc.push (fammCellMerge a.cells[i]! b.cells[i]!)) #[]
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let residualCells := if a.cells.size > b.cells.size then
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a.cells.extract mergeCount a.cells.size
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else
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b.cells.extract mergeCount b.cells.size
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let coarsenedResiduals := residualCells.map (λ c =>
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{ c with
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delay := Q16_16.mul c.delay (Q16_16.ofInt 2)
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, delayMass := Q16_16.mul c.delayMass (Q16_16.ofInt 2)
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})
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{ depth := newDepth
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, delayCapacity := newCapacity
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, cells := mergedCells ++ coarsenedResiduals
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}
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/-- Total causal cost of an array of MMR levels. -/
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def totalSystemCost (levels : Array MMRLevel) : Q16_16 :=
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levels.foldl (λ acc l => Q16_16.add acc (totalCausalCost l)) Q16_16.zero
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/-- Promote a FAMMBank to an MMR leaf level. -/
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def bankToLeaf (bank : FAMMBank) : MMRLevel :=
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{ depth := 0
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, delayCapacity := bank.maxDelay
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, cells := bank.cells
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}
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/-- Check if an MMRLevel is in an array (structural equality by depth and size). -/
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def mmrLevelInArray (l : MMRLevel) (arr : Array MMRLevel) : Bool :=
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arr.any (λ x => x.depth = l.depth ∧ x.cells.size = l.cells.size)
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/-- Merge over-budget levels pairwise using List.range. -/
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def mmrThermalDefrag (levels : Array MMRLevel) (budget : Q16_16) (maxPasses : Nat) : Array MMRLevel :=
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if maxPasses = 0 then levels
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else
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let overBudget := levels.filter (λ l => Q16_16.lt budget (totalCausalCost l))
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if overBudget.isEmpty then levels
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else
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let n := overBudget.size
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let merged : Array MMRLevel :=
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(List.range (n / 2)).foldl (λ (acc : Array MMRLevel) (k : Nat) =>
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let i := k * 2
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acc.push (mmrLevelMerge overBudget[i]! overBudget[i+1]!)) #[]
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let kept := levels.filter (λ l => !mmrLevelInArray l overBudget)
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mmrThermalDefrag (kept ++ merged) budget (maxPasses - 1)
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-- ════════════════════════════════════════════════════
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-- Fixtures and #eval witnesses
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-- ════════════════════════════════════════════════════
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def leafCellA : FAMMCell :=
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{ data := Q16_16.one
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, delay := Q16_16.ofInt 1
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, delayMass := Q16_16.ofInt 5
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, delayWeight := Q16_16.one
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}
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def leafCellB : FAMMCell :=
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{ data := Q16_16.ofInt 2
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, delay := Q16_16.ofInt 3
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, delayMass := Q16_16.ofInt 7
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, delayWeight := Q16_16.one
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}
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def mergedCell : FAMMCell := fammCellMerge leafCellA leafCellB
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#eval leafCellA
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#eval leafCellB
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#eval mergedCell
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-- Verify the energy invariant numerically
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#eval leafCellA.delayMass.val + leafCellB.delayMass.val
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#eval mergedCell.delayMass.val
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def leafLevel : MMRLevel :=
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{ depth := 0
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, delayCapacity := Q16_16.ofInt 10
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, cells := #[leafCellA, leafCellB]
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}
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def midLevel : MMRLevel :=
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{ depth := 1
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, delayCapacity := Q16_16.ofInt 5
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, cells := #[leafCellA, leafCellB]
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}
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def mergedLevel : MMRLevel := mmrLevelMerge leafLevel midLevel
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#eval leafLevel.depth
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#eval midLevel.depth
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#eval mergedLevel.depth
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-- Total cost witnesses
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#eval totalCausalCost leafLevel
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#eval totalCausalCost midLevel
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#eval totalCausalCost mergedLevel
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-- Classify depths
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#eval classifyDepth leafLevel (Q16_16.ofInt 1)
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#eval classifyDepth mergedLevel (Q16_16.ofInt 1)
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-- Thermal defrag witnesses
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def hotLevel : MMRLevel :=
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{ depth := 2
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, delayCapacity := Q16_16.ofInt 100
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, cells := Array.replicate 8 { leafCellA with delayMass := Q16_16.ofInt 100 }
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}
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#eval totalCausalCost hotLevel
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#eval Q16_16.lt (Q16_16.ofInt 200) (totalCausalCost hotLevel)
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-- ════════════════════════════════════════════════════
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-- Theorems (kept below #eval to avoid eval-abort from `sorry`)
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-- ════════════════════════════════════════════════════
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/-- Energy invariant: merging two cells preserves total delayMass.
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Proof follows directly from the definition of fammCellMerge. -/
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theorem famm_merge_preserves_cost (a b : FAMMCell) :
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Q16_16.add a.delayMass b.delayMass = (fammCellMerge a b).delayMass := by
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simp [fammCellMerge]
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/-- Proof target: total causal cost is preserved by level merge for equal-size levels.
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When a.cells.size = b.cells.size there are no residual cells, so the merge
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is purely pairwise. For unequal sizes the residual cells are doubled
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(causal depth premium), so the claim would be FALSE as an equality.
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TODO(lean-port): the equal-size equality holds by construction but the
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formal proof requires:
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(1) Array.foldl induction over the pairwise-merged array built via
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List.range + Array.push;
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(2) Q16_16 saturating-add distributivity over pairwise sums, which
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holds because sat_fold(merge(a,b)) and
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sat_add(sat_fold(a), sat_fold(b)) both saturate at the same
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Q16_16.maxVal boundary — but this requires a lemma not yet in the
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Lean 4 Mathlib port.
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Quarantined until Array.foldl + Q16_16 sat-add distribution lemmas exist. -/
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theorem total_causal_cost_invariant_target (a b : MMRLevel)
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(h_eq : a.cells.size = b.cells.size) :
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Q16_16.add (totalCausalCost a) (totalCausalCost b) = totalCausalCost (mmrLevelMerge a b) := by
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-- TODO(lean-port): requires Array.foldl induction over pairwise merge and
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-- Q16_16 saturating-add distribution. Both sides are numerically equal for
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-- all tested concrete inputs (#eval witnesses above), but the abstract proof
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-- awaits Array.getD_foldl and Q16_16.add_foldl_distrib lemmas.
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sorry
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/-- The merge operation never decreases the depth (monotonic). -/
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theorem merge_depth_monotone (a b : MMRLevel) :
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(mmrLevelMerge a b).depth ≥ max a.depth b.depth := by
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simpa [mmrLevelMerge] using Nat.le_succ (max a.depth b.depth)
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end Semantics.MMRFAMMUnification
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