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Pipeline: - q16_lut_vcn.py: Q16_16 LUT generation + VCN frame encoding (8 ops) - braid_vcn_encoder.py: Delta+RLE → RS ECC → ChaCha20 → VCN → MKV - braid_search.py: Sidon set slots, soliton search, QUBO optimization - test_braid_pipeline.py: 67 tests covering full round-trip WebGPU/Scripts: - braid_fft.wgsl: Cooley-Tukey radix-2 FFT on phase vectors - reed_solomon_vcn.py: Reed-Solomon ECC for VCN frame data - chacha20_braid.py: ChaCha20 encryption + key derivation - polynomial_commitment.py: KZG scheme for receipt verification Lean: - BraidBitwiseODE.lean: XOR crossing, O(1) integration, 2 proved theorems FPGA (Tang Nano 9K): - q16_lut_core.v: 8-op arithmetic, 2-stage pipeline, BRAM reciprocal - braid_crossing_core.v: 4-stage crossing residual, 7 Q16 instances - Testbenches with edge cases + VCD dumps
155 lines
7.2 KiB
Text
155 lines
7.2 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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BraidBitwiseODE.lean — Bitwise ODE integration for braid crossings
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Replaces continuous O(1) integration with XOR-based bitwise operations
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on Q16.16 fixed-point values. Pattern derived from PistBridge's BlitterState
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discrete Picard integral.
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Hierarchy: PistBridge BlitterState → BraidBitwiseODE crossing + integration
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Per AGENTS.md §0: Lean is the source of truth.
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-/
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import Semantics.FixedPoint
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import Semantics.BraidStrand
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namespace Semantics.BraidBitwiseODE
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open Semantics
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open Semantics.FixedPoint
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open Semantics.BraidStrand
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-- ════════════════════════════════════════════════════════════
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-- §1 Bitwise Crossing Operation
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-- ════════════════════════════════════════════════════════════
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/-- XOR-based braid crossing step.
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Models a crossing between two strands as a bitwise operation
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on their Q16.16 payloads. Parity of each strand determines
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the crossing sign (over/under).
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Pattern: M_{k+1} = M_k ⊕ F(a,b,ε) from PistBridge §3. -/
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def bitwiseCrossStep (a b : BraidStrand) : BraidStrand :=
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let pa := a.phaseAcc
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let pb := b.phaseAcc
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-- XOR the phase accumulations' bit patterns
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let newX := Q16_16.ofBits (pa.x.toBits ^^^ pb.x.toBits)
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let newY := Q16_16.ofBits (pa.y.toBits ^^^ pb.y.toBits)
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-- Swap parity to indicate crossing
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let newParity := !a.parity
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-- Accumulate residue: XOR with crossing signal
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let crossSignal := Q16_16.ofBits (a.residue.toBits ^^^ b.residue.toBits)
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{ phaseAcc := { x := newX, y := newY }
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, parity := newParity
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, slot := a.slot
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, residue := crossSignal
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, jitter := a.jitter
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, bracket := a.bracket }
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-- ════════════════════════════════════════════════════════════
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-- §2 O(1) ODE Integration Step
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-- ════════════════════════════════════════════════════════════
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/-- Discrete Picard integral: O(1) bitwise ODE integration.
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Replaces continuous ODE solver with a single bitwise accumulation.
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From PistBridge: M_{k+1} = M_k ⊕ (state + input) >> 16
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This is the Blitter operation: accumulate state and input via
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XOR with right-shift, giving a single-step ODE approximation. -/
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def bitwiseODEIntegrate (state : Q16_16) (input : Q16_16) : Q16_16 :=
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-- Combine state and input via addition, then XOR-accumulate
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let combined := Q16_16.ofNat ((state.toBits.toNat + input.toBits.toNat) % (2^32))
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-- Right-shift by 16 to extract the integer portion effect
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let shifted := Q16_16.ofBits ((combined.toBits.toNat >>> 16).toUInt32)
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-- XOR with original state for the Blitter accumulation
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Q16_16.ofBits (state.toBits ^^^ shifted.toBits)
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/-- Multi-step bitwise ODE integration.
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Applies the bitwise ODE integrator iteratively over a sequence
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of input values, accumulating into the state. -/
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def bitwiseODEIntegrateSeq (state : Q16_16) (inputs : List Q16_16) : Q16_16 :=
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inputs.foldl bitwiseODEIntegrate state
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-- ════════════════════════════════════════════════════════════
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-- §3 Braid Strand ODE Integration
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-- ════════════════════════════════════════════════════════════
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/-- Integrate a braid strand's phase accumulation via bitwise ODE.
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Applies the discrete Picard integral to both x and y components
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of the strand's phase vector. -/
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def integrateStrand (strand : BraidStrand) (input : Q16_16) : BraidStrand :=
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let newX := bitwiseODEIntegrate strand.phaseAcc.x input
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let newY := bitwiseODEIntegrate strand.phaseAcc.y input
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{ strand with phaseAcc := { x := newX, y := newY } }
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/-- Two-strand crossing with integrated ODE step.
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Combines bitwiseCrossStep with ODE integration in a single
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atomic operation: cross, then integrate the result. -/
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def crossingODEStep (a b : BraidStrand) : BraidStrand × BraidStrand :=
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let crossed := bitwiseCrossStep a b
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let integrated := integrateStrand crossed a.residue
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(integrated, crossed)
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-- ════════════════════════════════════════════════════════════
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-- §4 Fixed-Point Invariants
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-- ════════════════════════════════════════════════════════════
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/-- The bitwise ODE integrator preserves Q16.16 range.
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Since all operations (XOR, shift, ofBits) produce valid Q16_16
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values, the output is always in the representable range. -/
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theorem bitwise_ode_preserves_range (state input : Q16_16) :
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q16MinRaw ≤ (bitwiseODEIntegrate state input).toInt ∧
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(bitwiseODEIntegrate state input).toInt ≤ q16MaxRaw := by
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unfold bitwiseODEIntegrate
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exact ⟨(Q16_16.ofBits _).property.left, (Q16_16.ofBits _).property.right⟩
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/-- Bitwise cross step preserves strand slot identity.
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The slot field is carried through from the first strand. -/
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theorem cross_step_preserves_slot (a b : BraidStrand) :
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(bitwiseCrossStep a b).slot = a.slot := by
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unfold bitwiseCrossStep
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rfl
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-- ════════════════════════════════════════════════════════════
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-- §5 Convergence (Placeholder)
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-- ════════════════════════════════════════════════════════════
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/-- TODO(lean-port): Prove that bitwise ODE integration converges
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for bounded inputs. This requires showing that repeated application
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of `bitwiseODEIntegrate` reaches a fixed point (absorbing state).
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Conjecture: for any state s and bounded input sequence,
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bitwiseODEIntegrateSeq reaches a fixed point after at most
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32 steps (the bit width of Q16.16). -/
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theorem bitwise_ode_correct (state : Q16_16) (h : state.toInt = 0) :
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bitwiseODEIntegrate state state = state := by
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unfold bitwiseODEIntegrate
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simp [h, Q16_16.ofBits, Q16_16.toBits, Q16_16.ofNat]
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sorry -- TODO(lean-port): requires Q16.16 bit-level identity proof
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-- ════════════════════════════════════════════════════════════
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-- §6 Verification Examples
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-- ════════════════════════════════════════════════════════════
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#eval bitwiseODEIntegrate (Q16_16.ofInt 2) (Q16_16.ofInt 3)
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#eval bitwiseODEIntegrate (Q16_16.ofInt 0) (Q16_16.ofInt 0)
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end Semantics.BraidBitwiseODE
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