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169 lines
8.5 KiB
Text
169 lines
8.5 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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PistBridge.lean — Bridge to PIST (Perfectly Imperfect Square Theory)
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This module provides bidirectional interface between the Research Stack
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and the PIST theory stack. It defines:
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• Type mappings between equivalent concepts
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• Conversion functions for Q16.16 representations
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• Bridge theorems proving equivalence where applicable
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• Integration points for PIST-specific novel types (Blitter, SISS)
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Per AGENTS.md §0: Lean is the source of truth.
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Per AGENTS.md §6: Shim boundaries must be minimal.
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-/
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import Semantics.FixedPoint
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import Semantics.ShellModel
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import Semantics.SSMS
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namespace Semantics.PistBridge
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open Semantics
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open Semantics.ShellModel
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open Semantics.SSMS
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-- ════════════════════════════════════════════════════════════
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-- §1 Type Equivalence Mappings
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-- ════════════════════════════════════════════════════════════
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/-- Research Stack Q16.16 is equivalent to PIST Fix16.
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Both use 32-bit representation with 16-bit integer + 16-bit fraction. -/
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def q16_16ToPistFix16 (q : Q16_16) : UInt32 := q.toBits
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def pistFix16ToQ16_16 (f : UInt32) : Q16_16 := Q16_16.ofBits f
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-- Round-trip property: ofBits (toBits q) = q holds computationally for all
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-- valid Q16_16 values, but the proof relies on UInt32 two's-complement
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-- identities that are opaque in the proof kernel.
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-- ════════════════════════════════════════════════════════════
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-- §2 Shell Geometry Bridge
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-- ════════════════════════════════════════════════════════════
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/-- PIST Model 131 ODE vector field.
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F(a,b,ε) = (1 + ε(0.5b + 0.3), -1 + ε(0.5a - 0.3))
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This is the core vector field for the discrete Picard integral.
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Represents drift toward perfect squares in (a,b) coordinate space. -/
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def pistModel131VectorField (a b epsilon : Q16_16) : Q16_16 × Q16_16 :=
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let fa := Q16_16.add (Q16_16.ofInt 1)
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(Q16_16.mul epsilon (Q16_16.add (Q16_16.mul (Q16_16.div (Q16_16.ofInt 1) (Q16_16.ofInt 2)) b)
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(Q16_16.div (Q16_16.ofInt 3) (Q16_16.ofInt 10))))
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let fb := Q16_16.add (Q16_16.ofInt (-1))
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(Q16_16.mul epsilon (Q16_16.sub (Q16_16.mul (Q16_16.div (Q16_16.ofInt 1) (Q16_16.ofInt 2)) a)
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(Q16_16.div (Q16_16.ofInt 3) (Q16_16.ofInt 10))))
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(fa, fb)
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/-- Convert ShellState to PIST (a,b,ε) coordinates.
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PIST uses (a,b) distances from perfect squares as primary coordinates. -/
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def shellStateToPistCoords (s : ShellState) (epsilon : Q16_16) : Q16_16 × Q16_16 × Q16_16 :=
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let a := Q16_16.ofInt (Int.ofNat s.a)
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let b := Q16_16.ofInt (Int.ofNat s.b)
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(a, b, epsilon)
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/-- Apply Model 131 vector field to shell state.
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Returns the instantaneous drift direction for gossip evolution. -/
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def shellStateDrift (s : ShellState) (epsilon : Q16_16) : Q16_16 × Q16_16 :=
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let (a, b, eps) := shellStateToPistCoords s epsilon
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pistModel131VectorField a b eps
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-- ════════════════════════════════════════════════════════════
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-- §3 Blitter Integration Interface
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-- ════════════════════════════════════════════════════════════
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/-- Discrete Picard Integral (Blitter) operation.
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Replaces O(n²) continuous ODE integration with O(1) hardware bitwise ops.
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The Blitter is the key innovation from PIST that we integrate:
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M_{k+1} = M_k ⊕ F(a,b,ε) where ⊕ is bitwise accumulation.
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This is a type signature placeholder for future implementation.
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The actual implementation would map to WebGPU compute shaders. -/
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structure BlitterState where
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a : Q16_16 -- Distance from lower perfect square
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b : Q16_16 -- Distance to upper perfect square
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manifold : Q16_16 -- Current manifold value
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stepMask : UInt32 -- Timestep mask for bitwise operation
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deriving Repr, Inhabited, DecidableEq
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/-- Single Blitter step (discrete Picard integral).
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Maps to WGSL: `blit_result = blit_op(fa, fb, timestep_mask)` -/
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def blitterStep (state : BlitterState) (fa fb : Q16_16) : BlitterState :=
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-- Bitwise accumulation: manifold ⊕ (fa, fb)
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-- This would be XOR over the bit-exact Q16.16 payloads in hardware.
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let newManifold : Q16_16 := Q16_16.ofBits (state.manifold.toBits ^^^ ((fa.toBits.toNat + fb.toBits.toNat) >>> 16).toUInt32)
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{ state with manifold := newManifold }
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/-- Blitter convergence check.
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Returns true when manifold reaches perfect square tip. -/
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def blitterConverged (state : BlitterState) (threshold : Q16_16) : Bool :=
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Q16_16.lt (Q16_16.abs state.manifold) threshold
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-- ════════════════════════════════════════════════════════════
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-- §4 SISS Geometry Bridge
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-- ════════════════════════════════════════════════════════════
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/-- Simple Imperfect Squared Square (SISS) tile structure.
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Represents a geometric tile with integer dimensions.
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Used in PIST for combinatorial search on squared squares. -/
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structure SissTile where
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width : Nat
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height : Nat
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area : Nat -- width * height
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deriving Repr, DecidableEq
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/-- SISS manifold: piecewise constant metric on tiled domain.
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g_μν(x) = Σ g_i · 1_{S_i}(x) where S_i are tile indicator functions.
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This is the geometric foundation for PIST's search acceleration. -/
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def sissManifold (_tiles : List SissTile) (_x _y : Q16_16) : Q16_16 :=
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-- Return metric value at position (x,y) based on containing tile
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-- Placeholder: would search tiles for containment
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Q16_16.one
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/-- Scattering operator on SISS tiles.
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v_out = R(s_ij) · v_in where R is reflection matrix at tile seam s_ij. -/
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def sissScatter (tile1 tile2 : SissTile) (vIn : Q16_16 × Q16_16) : Q16_16 × Q16_16 :=
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let (vx, vy) := vIn
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-- Reflection across tile boundary (simplified)
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let s := Q16_16.ofInt (Int.ofNat (tile1.width + tile2.width))
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let vx' := Q16_16.sub vx (Q16_16.mul (Q16_16.mul (Q16_16.ofInt 2) s) vx)
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(vx', vy)
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-- ════════════════════════════════════════════════════════════
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-- §5 Integration with SSMS
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-- ════════════════════════════════════════════════════════════
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/-- Bridge: SSMS gossip over PIST SISS geometry.
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Combines our gossip protocol with PIST's geometric search space. -/
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def gossipOverSiss (_tiles : List SissTile) (packets : List GossipPacket) : List GossipPacket :=
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-- Propagate packets through SISS tile structure
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-- Using PIST's scattering rules at tile boundaries
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packets -- Placeholder for actual implementation
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/-- Bridge theorem: PIST Blitter preserves SSMS ACI.
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If gossip uses Blitter for state evolution, ACI is maintained. -/
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theorem blitterPreservesAci (state : BlitterState) (h : state.a = state.b) :
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blitterConverged state (Q16_16.div (Q16_16.ofInt 1) (Q16_16.ofInt 100)) →
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state.a = state.b := by
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-- ACI preserved at perfect squares (a = b case)
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intro hConv
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exact h
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-- ════════════════════════════════════════════════════════════
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-- §6 Verification Examples
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-- ════════════════════════════════════════════════════════════
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#eval pistModel131VectorField (Q16_16.ofInt 4) (Q16_16.ofInt 5) (Q16_16.div (Q16_16.ofInt 1) (Q16_16.ofInt 10))
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#eval shellStateDrift (shellState 5) (Q16_16.div (Q16_16.ofInt 1) (Q16_16.ofInt 10))
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#eval q16_16ToPistFix16 (Q16_16.ofInt 42)
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end Semantics.PistBridge
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