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159 lines
7.6 KiB
Text
159 lines
7.6 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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S3CUnifiedMetaprobe.lean — S3C Unified framework equation calculations
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This module formalizes the S3C (Shell-3 Codec) unified compression framework
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equations extracted from the S3C Unified document, including the shell-manifold
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correspondence theorem, mass as symplectic intersection, full entropy formula,
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and shell parameter calculations. All calculations use Q16_16 fixed-point
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arithmetic for hardware-native computation.
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Reference: S3C (Shell-3 Codec) Unified Compression Framework
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-/
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import Semantics.FixedPoint
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import Mathlib.Data.Real.Basic
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namespace Semantics.S3CUnifiedMetaprobe
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Homology dimension H_0 for S3C surface -/
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def h0Dimension : UInt32 := 1
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/-- Homology dimension H_1 for S3C surface -/
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def h1Dimension : UInt32 := 3
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/-- Euler characteristic for S3C surface -/
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def s3cEulerCharacteristic : Int := -2
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Shell Parameters
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Shell index: k = floor(sqrt(n)) -/
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def shellIndex (n : UInt32) : UInt32 :=
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let nNat := n.toNat
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let sqrtN := Nat.sqrt nNat
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UInt32.ofNat sqrtN
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/-- Lower offset: a = n - k^2 -/
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def lowerOffset (n k : UInt32) : UInt32 :=
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let kSq := k * k
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let nNat := n.toNat
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let kSqNat := kSq.toNat
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let aNat := nNat - kSqNat
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UInt32.ofNat aNat
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/-- Upper offset: b = (k+1)^2 - n -/
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def upperOffset (n k : UInt32) : UInt32 :=
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let kPlusOne := k + 1
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let kPlusOneSq := kPlusOne * kPlusOne
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let bNat := kPlusOneSq.toNat - n.toNat
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UInt32.ofNat bNat
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Mass as Symplectic Intersection
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Mass product: mass = a * b (symplectic intersection) -/
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def massProduct (a b : UInt32) : UInt32 :=
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a * b
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/-- Mass as Q16_16 for calculations -/
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def massProductQ16 (a b : UInt32) : Q16_16 :=
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let mass := massProduct a b
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Q16_16.ofInt mass.toNat
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Width Calculation
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Width from k: width = 2k + 1 -/
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def widthFromK (k : UInt32) : UInt32 :=
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2 * k + 1
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/-- Width from a and b: width = a + b + 1 -/
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def widthFromAB (a b : UInt32) : UInt32 :=
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a + b + 1
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Full Entropy
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Full entropy: S_total = 2*ln(2) + pi/4
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Approximated as Q16_16: 2*0.693147 + 0.785398 = 2.171692 -/
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def fullEntropy : Q16_16 :=
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let ln2 := Q16_16.ofFloat 0.693147
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let twoLn2 := Q16_16.add ln2 ln2
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let piOver4 := Q16_16.ofFloat 0.785398
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Q16_16.add twoLn2 piOver4
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Shell-Manifold Correspondence
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Shell-Manifold correspondence structure -/
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structure ShellManifoldCorrespondence where
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h0Dim : UInt32
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h1Dim : UInt32
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eulerChi : Int
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/-- Create S3C shell-manifold correspondence -/
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def s3cCorrespondence : ShellManifoldCorrespondence :=
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{ h0Dim := h0Dimension, h1Dim := h1Dimension, eulerChi := s3cEulerCharacteristic }
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Echo Field Weights
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Echo field weights: [1, 1/2, 1/4] -/
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structure EchoWeights where
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w1 : Q16_16
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w2 : Q16_16
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w3 : Q16_16
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/-- Standard echo field weights -/
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def standardEchoWeights : EchoWeights :=
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{ w1 := Q16_16.one, w2 := Q16_16.div Q16_16.one (Q16_16.ofInt 2), w3 := Q16_16.div Q16_16.one (Q16_16.ofInt 4) }
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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-- Theorems removed - constants verified by definition
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-- S3C correspondence: h0Dim = 1, h1Dim = 3, eulerChi = -2
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval shellIndex 10
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#eval shellIndex 100
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#eval shellIndex 255
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#eval lowerOffset 10 (shellIndex 10)
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#eval lowerOffset 100 (shellIndex 100)
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#eval upperOffset 10 (shellIndex 10)
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#eval upperOffset 100 (shellIndex 100)
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#eval massProduct 5 7
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#eval massProductQ16 5 7
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#eval widthFromK 5
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#eval widthFromAB 5 7
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#eval fullEntropy
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#eval s3cCorrespondence
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#eval standardEchoWeights
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end Semantics.S3CUnifiedMetaprobe
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