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200 lines
No EOL
6.7 KiB
Text
200 lines
No EOL
6.7 KiB
Text
/-
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CompressionYield.lean — Holographic compression yield theorem.
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Traditional compression minimizes bits per symbol along coordinate axes.
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Holographic compression packs N structures per coordinate by separating them
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in lambda-space. The total compression multiplier is the product of three
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independent factors:
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R_total = R_coord * N_lambda * R_shrink
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where:
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R_coord = coordinate-space compression ratio (existing methods, >= 1)
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N_lambda = number of resolvable threshold bands in Q0_16 space (>= 1)
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R_shrink = implosion-style dimensional collapse multiplier (>= 1)
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At Q0_16 resolution (32767 distinct values), if each structure needs a minimum
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lambda-separation of delta_lambda, the holographic multiplier N_lambda is
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bounded by max(1, floor(1 / delta_lambda)). This stacks multiplicatively with
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coordinate compression and implosion shrink.
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-/
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import Semantics.FixedPoint
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import Semantics.LogogramRotationLoop
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set_option linter.dupNamespace false
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namespace Semantics.CompressionYield
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open Semantics.FixedPoint (Q0_16 Q0_16.ofRawInt)
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open Semantics.LogogramRotationLoop (ThresholdBand inBand)
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/--
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The compression yield: three independent Nat multipliers that compose
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multiplicatively. All values are dimensionless integer ratios (>= 1).
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-/
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structure CompressionYield where
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coordinateRatio : Nat -- existing coordinate-compression ratio (>= 1)
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lambdaBands : Nat -- number of resolvable threshold bands (>= 1)
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shrinkRatio : Nat -- implosion dimensional-collapse ratio (>= 1)
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deriving Repr, DecidableEq, BEq, Inhabited
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/--
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Compute the total compression multiplier as a Nat:
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R_total = R_coord * N_lambda * R_shrink
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All values are clamped to >= 1 before multiplication.
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-/
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def totalMultiplier (y : CompressionYield) : Nat :=
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(if y.coordinateRatio = 0 then 1 else y.coordinateRatio) *
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(if y.lambdaBands = 0 then 1 else y.lambdaBands) *
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(if y.shrinkRatio = 0 then 1 else y.shrinkRatio)
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/-- The number of distinct values representable in Q0_16 (positive range). -/
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def q0_16_valueCount : Nat := 32768
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/--
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Compute the maximum number of non-overlapping threshold bands given a minimum
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required separation delta_lambda in Q0_16.
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Two bands [l1, u1] and [l2, u2] are non-overlapping when u1 <= l2.
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Given minimum separation delta (band width + gap), the max bands in [0, 1]
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is floor(1 / delta), or q0_16_valueCount when delta = 0.
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-/
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def maxLambdaBands (delta_lambda : Q0_16) : Nat :=
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if Q0_16.le delta_lambda Q0_16.zero then
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q0_16_valueCount
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else
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max 1 (Q0_16.one.val.toNat / delta_lambda.val.toNat)
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/--
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Theorem: For any minimum separation delta, the max number of bands is at most
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the Q0_16 value count.
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-/
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theorem max_bands_bounded_by_value_count (delta : Q0_16) :
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maxLambdaBands delta ≤ q0_16_valueCount := by
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unfold maxLambdaBands q0_16_valueCount
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split
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· rfl
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· have h_div : Q0_16.one.val.toNat / delta.val.toNat ≤ 32768 := by
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have h1 : Q0_16.one.val.toNat = 32767 := rfl
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rw [h1]
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have h_le : 32767 / delta.val.toNat ≤ 32767 := Nat.div_le_self 32767 _
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omega
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omega
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/-- Canonical yield: DISH holographic scenario (10x coord, 3 bands, 2000x shrink). -/
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def dishHolographicYield : CompressionYield :=
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{ coordinateRatio := 10
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, lambdaBands := 3
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, shrinkRatio := 2000 }
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/-- Canonical yield: full-resolution bound (10x coord, 3000 bands, 2000x shrink). -/
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def fullLambdaYield : CompressionYield :=
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{ coordinateRatio := 10
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, lambdaBands := 3000
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, shrinkRatio := 2000 }
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/-- Baseline: coordinate + implosion only, no holographic packing. -/
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def baselineYield : CompressionYield :=
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{ coordinateRatio := 10
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, lambdaBands := 1
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, shrinkRatio := 2000 }
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/-- Three-structure rotation, no implosion shrink. -/
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def threeRotationYield : CompressionYield :=
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{ coordinateRatio := 10
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, lambdaBands := 3
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, shrinkRatio := 1 }
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/-- Logogram yield: 1 structure per band baseline. -/
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def logogramBaselineYield : CompressionYield :=
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{ coordinateRatio := 1
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, lambdaBands := 1
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, shrinkRatio := 1 }
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/-- Logogram yield: 3000 structures via holographic packing. -/
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def logogramHolographicYield : CompressionYield :=
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{ coordinateRatio := 1
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, lambdaBands := 3000
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, shrinkRatio := 1 }
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/- =======================================================================
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Theorems
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======================================================================= -/
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theorem dish_total_is_60000 :
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totalMultiplier dishHolographicYield = 60000 := by
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native_decide
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theorem baseline_total_is_20000 :
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totalMultiplier baselineYield = 20000 := by
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native_decide
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theorem full_lambda_total_is_60000000 :
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totalMultiplier fullLambdaYield = 60000000 := by
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native_decide
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theorem full_lambda_dominates_baseline :
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totalMultiplier fullLambdaYield > totalMultiplier baselineYield := by
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native_decide
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theorem full_lambda_dominates_dish :
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totalMultiplier fullLambdaYield > totalMultiplier dishHolographicYield := by
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native_decide
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theorem holographic_advantage_over_baseline_dish :
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totalMultiplier dishHolographicYield > totalMultiplier baselineYield := by
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native_decide
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theorem three_rotation_advantage :
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totalMultiplier threeRotationYield > totalMultiplier logogramBaselineYield := by
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native_decide
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theorem logogram_holographic_advantage :
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totalMultiplier logogramHolographicYield > totalMultiplier logogramBaselineYield := by
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native_decide
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theorem logogram_holographic_is_3000x :
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totalMultiplier logogramHolographicYield = 3000 * totalMultiplier logogramBaselineYield := by
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native_decide
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theorem multipliers_are_multiplicative (y : CompressionYield) :
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totalMultiplier y = (if y.coordinateRatio = 0 then 1 else y.coordinateRatio) *
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(if y.lambdaBands = 0 then 1 else y.lambdaBands) *
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(if y.shrinkRatio = 0 then 1 else y.shrinkRatio) :=
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rfl
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theorem delta_zero_gives_all_bands :
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maxLambdaBands Q0_16.zero = 32768 := by
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native_decide
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theorem delta_one_gives_one_band :
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maxLambdaBands Q0_16.one = 1 := by
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native_decide
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theorem delta_half_gives_two_bands :
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maxLambdaBands Q0_16.half = 2 := by
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native_decide
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theorem delta_small_gives_many_bands :
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maxLambdaBands (Q0_16.ofRawInt 0x0010) = 2047 := by
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native_decide
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/- =======================================================================
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#eval witnesses
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======================================================================= -/
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#eval totalMultiplier baselineYield
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#eval totalMultiplier dishHolographicYield
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#eval totalMultiplier fullLambdaYield
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#eval totalMultiplier threeRotationYield
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#eval totalMultiplier logogramBaselineYield
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#eval totalMultiplier logogramHolographicYield
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#eval maxLambdaBands Q0_16.zero
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#eval maxLambdaBands Q0_16.half
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#eval maxLambdaBands Q0_16.one
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#eval maxLambdaBands (Q0_16.ofRawInt 0x0010)
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end Semantics.CompressionYield |