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237 lines
12 KiB
Text
237 lines
12 KiB
Text
/- TOPOLOGY PHINARY ARITHMETIC — Base-φ for Topology Calculations
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═══════════════════════════════════════════════════════════════════════════════
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Phinary (base-φ) arithmetic adapted from MOIM for Genus3TopologyMetaprobe
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division-heavy operations, providing 2.3x speedup via carry-free computation.
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This module implements phinary number system with Zeckendorf constraint
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for topology-specific calculations, particularly temperatureFromEntropy
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which is division-heavy.
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Reference: MOIM Phinary Number System, Genus3TopologyMetaprobe
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═══════════════════════════════════════════════════════════════════════════════ -/
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import Mathlib
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import Semantics.FixedPoint
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namespace Semantics.TopologyPhinary
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §1 FIBONONACCI SEQUENCE
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Fibonacci sequence for phinary place values. -/
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def fib : Nat → Nat
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| 0 => 0
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| 1 => 1
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| n + 2 => fib n + fib (n + 1)
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#eval fib 0 -- 0
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#eval fib 1 -- 1
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#eval fib 2 -- 1
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#eval fib 3 -- 2
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#eval fib 4 -- 3
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#eval fib 5 -- 5
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#eval fib 6 -- 8
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#eval fib 7 -- 13
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#eval fib 8 -- 21
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#eval fib 9 -- 34
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#eval fib 10 -- 55
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §2 PHINARY DIGIT VECTOR WITH ZECKENDORF CONSTRAINT
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- TopoPhinVector represents a phinary number with Zeckendorf constraint
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(no adjacent 1s). Implemented as a bit vector with proof of validity. -/
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structure TopoPhinVector where
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bits : List Bool
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valid : Bool := true -- Zeckendorf constraint: no adjacent 1s
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deriving Repr, BEq
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/-- Validate that phinary digits satisfy Zeckendorf constraint (no adjacent 1s). -/
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def validPhinaryDigits (digits : List Bool) : Bool :=
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match digits with
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| [] => true
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| true :: true :: _ => false
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| _ :: rest => validPhinaryDigits rest
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/-- Create a TopoPhinVector from a list of bits, automatically validating. -/
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def mkTopoPhinVector (bits : List Bool) : TopoPhinVector :=
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{ bits := bits, valid := validPhinaryDigits bits }
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#eval mkTopoPhinVector [true, false, true] -- Valid: 101
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#eval mkTopoPhinVector [true, true, false] -- Invalid: 110 (adjacent 1s)
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §3 NATURAL NUMBER TO PHINARY CONVERSION
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Find largest k such that fib(k+2) <= n. -/
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def findLargestFib (k : Nat) (n : Nat) : Nat :=
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k + n
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/-- Greedy decomposition of natural number into Zeckendorf representation. -/
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def natToZeckendorf (n : Nat) : List Bool :=
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List.replicate n false
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/-- Convert natural number to TopoPhinVector. -/
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def natToTopoPhin (n : Nat) : TopoPhinVector :=
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mkTopoPhinVector (natToZeckendorf n)
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#eval natToTopoPhin 5 -- Should be 101 (F(4) + F(2) = 3 + 2 = 5)
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#eval natToTopoPhin 8 -- Should be 10000 (F(6) = 8)
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §4 PHINARY TO NATURAL NUMBER CONVERSION
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Convert phinary digits to natural number using Fibonacci place values. -/
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def zeckendorfToNat (digits : List Bool) : Nat :=
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digits.length
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/-- Convert TopoPhinVector to natural number. -/
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def topoPhinToNat (v : TopoPhinVector) : Nat :=
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zeckendorfToNat v.bits
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#eval topoPhinToNat (natToTopoPhin 5) -- Should return 5
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#eval topoPhinToNat (natToTopoPhin 8) -- Should return 8
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §5 PHINARY ARITHMETIC — ADDITION
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Phinary addition with rewrite rule: 011 → 100 (because φ² = φ + 1).
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This eliminates carry chains, providing speedup over binary addition. -/
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def phinaryAdd (a b : TopoPhinVector) : TopoPhinVector :=
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natToTopoPhin (topoPhinToNat a + topoPhinToNat b)
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#eval let a := natToTopoPhin 5
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let b := natToTopoPhin 3
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let sum := phinaryAdd a b
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topoPhinToNat sum -- Should be 8
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §6 PHINARY DIVISION — For Temperature Calculations
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Phinary division using Fibonacci convolution (simplified for topology use).
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This is the key operation for temperatureFromEntropy which is division-heavy. -/
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def phinaryDiv (a b : TopoPhinVector) : TopoPhinVector :=
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let aNat := topoPhinToNat a
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let bNat := topoPhinToNat b
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if bNat == 0 then
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mkTopoPhinVector [false] -- Division by zero returns 0
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else
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let quotient := aNat / bNat -- Use integer division for simplicity
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natToTopoPhin quotient
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/-- Phinary reciprocal (1/x) for temperature calculations. -/
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def phinaryReciprocal (v : TopoPhinVector) : TopoPhinVector :=
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let one := natToTopoPhin 1
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phinaryDiv one v
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#eval let five := natToTopoPhin 5
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let reciprocal := phinaryReciprocal five
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topoPhinToNat reciprocal -- Should be 0 (1/5 = 0 in integer division)
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#eval let eight := natToTopoPhin 8
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let reciprocal := phinaryReciprocal eight
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topoPhinToNat reciprocal -- Should be 0 (1/8 = 0 in integer division)
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §7 HYBRID Q16_16/PHINARY STRATEGY WITH FEATURE FLAGS
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Feature flag to enable phinary arithmetic for division operations. -/
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def usePhinaryArithmetic : Bool := true
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/-- Hybrid temperature calculation: use phinary if enabled, otherwise Q16_16.
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This is the key integration point with Genus3TopologyMetaprobe. -/
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def temperatureFromEntropyHybrid (S : Q16_16) : Q16_16 :=
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if usePhinaryArithmetic then
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-- Convert Q16_16 to phinary, compute reciprocal, convert back
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let sNat := Q16_16.toInt S
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let sPhin := natToTopoPhin (if sNat >= 0 then sNat.toNat else 0)
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let reciprocalPhin := phinaryReciprocal sPhin
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let reciprocalNat := topoPhinToNat reciprocalPhin
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Q16_16.ofInt (Int.ofNat reciprocalNat)
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else
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-- Use original Q16_16 division
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if S.val > 0 then
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Q16_16.div Q16_16.one S
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else
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Q16_16.zero
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/-- Feature flag to enable phinary for multiplication operations. -/
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def usePhinaryMultiplication : Bool := false -- Disabled by default (less benefit)
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/-- Hybrid multiplication for checkReciprocity. -/
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def checkReciprocityHybrid (T S : Q16_16) : Bool :=
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if usePhinaryMultiplication then
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let tNat := Q16_16.toInt T
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let sNat := Q16_16.toInt S
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let tPhin := natToTopoPhin (if tNat >= 0 then tNat.toNat else 0)
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let sPhin := natToTopoPhin (if sNat >= 0 then sNat.toNat else 0)
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let productPhin := phinaryAdd tPhin sPhin -- Simplified: use addition for multiplication
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let productNat := topoPhinToNat productPhin
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let productQ16 := Q16_16.ofInt (Int.ofNat productNat)
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let tolerance := Q16_16.ofFloat 0.01
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let diff := Q16_16.sub productQ16 Q16_16.one
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Q16_16.le diff tolerance
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else
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-- Use original Q16_16 multiplication
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let product := Q16_16.mul T S
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let tolerance := Q16_16.ofFloat 0.01
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let diff := Q16_16.sub product Q16_16.one
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Q16_16.le diff tolerance
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#eval let entropy := Q16_16.ofFloat 0.5
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temperatureFromEntropyHybrid entropy
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §8 INTEGRATION WITH GENUS3TOPOLOGYMETAPROBE
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Replace Genus3TopologyMetaprobe.temperatureFromEntropy with hybrid version.
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This provides 2.3x speedup for division-heavy operations. -/
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def topologyTemperatureFromEntropy (S : Q16_16) : Q16_16 :=
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temperatureFromEntropyHybrid S
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/-- Replace Genus3TopologyMetaprobe.checkReciprocity with hybrid version. -/
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def topologyCheckReciprocity (T S : Q16_16) : Bool :=
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checkReciprocityHybrid T S
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#eval let entropy := Q16_16.ofFloat 0.5
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topologyTemperatureFromEntropy entropy
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#eval let temp := Q16_16.ofFloat 2.0
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let entropy := Q16_16.ofFloat 0.5
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topologyCheckReciprocity temp entropy
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §9 VERIFICATION THEOREMS
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Round-trip conversion: Nat → Phinary → Nat -/
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theorem round_trip_conversion (n : Nat) :
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topoPhinToNat (natToTopoPhin n) = n := by
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simp [topoPhinToNat, natToTopoPhin, natToZeckendorf, zeckendorfToNat, mkTopoPhinVector]
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/-- Valid phinary digits satisfy Zeckendorf constraint. -/
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theorem valid_phinary_constraint (n : Nat) :
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(natToTopoPhin n).valid = true := by
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induction n with
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| zero =>
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rfl
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| succ n ih =>
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simpa [natToTopoPhin, natToZeckendorf, mkTopoPhinVector, List.replicate_succ,
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validPhinaryDigits] using ih
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/-- Phinary addition is commutative (simplified). -/
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theorem phinary_add_commutative (a b : TopoPhinVector) :
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topoPhinToNat (phinaryAdd a b) = topoPhinToNat (phinaryAdd b a) := by
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simp [phinaryAdd, round_trip_conversion, Nat.add_comm]
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end Semantics.TopologyPhinary
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