File size: 4,886 Bytes
fa128fd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | -- Layer 2: Arithmetic Invariant Search
-- Replaces LLM hypothesis generation.
-- Enumerates the space of arithmetic identities.
-- LiquidHaskell verifies invariants at compile time.
{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE RecordWildCards #-}
module Quantabeta.InvariantSearch where
import Data.List (nub, sortBy)
import Data.Ord (comparing, Down(..))
-- ─── Invariant expression grammar ────────────────────────────────────────────
data InvariantExpr
= HeckeCorr { level :: Int, weight :: Int }
| PartitionVol { window :: Int }
| RamanujanCong { modulus :: Int, residue :: Int }
| Compose InvariantExpr InvariantExpr
deriving (Eq, Show)
-- ─── Well-typed invariant check ──────────────────────────────────────────────
wellTyped :: InvariantExpr -> Bool
wellTyped HeckeCorr{..} = level > 0 && weight > 0 && weight `mod` 2 == 0
wellTyped PartitionVol{..} = window > 0 && window <= 252 -- max one trading year
wellTyped RamanujanCong{..} = modulus `elem` [5, 7, 11, 13] && residue < modulus
wellTyped (Compose a b) = wellTyped a && wellTyped b
-- ─── Arithmetic invariant verification ───────────────────────────────────────
-- | Ramanujan partition congruences (exact, no approximation)
ramanujanCongruence :: Int -> Int -> [Int] -> Bool
ramanujanCongruence m r pValues =
all (\k -> pValues !! (m * k + r) `mod` m == 0)
[0 .. (length pValues - r - 1) `div` m]
-- | Hecke eigenvalue Deligne bound: |a_p| <= 2 * p^((k-1)/2)
deligneBound :: Int -> Int -> Rational -> Bool
deligneBound p k a_p =
abs a_p <= 2 * fromIntegral p ^ ((k - 1) `div` 2 :: Int)
-- | Information coefficient positivity: IC > 0 implies signal has predictive power
icPositive :: [Rational] -> [Rational] -> Bool
icPositive signals returns =
let n = fromIntegral (length signals) :: Rational
meanS = sum signals / n
meanR = sum returns / n
num = sum (zipWith (\s r -> (s - meanS) * (r - meanR)) signals returns)
denS = sqrt $ fromRational $ sum (map (\s -> (s - meanS)^(2::Int)) signals) / n
denR = sqrt $ fromRational $ sum (map (\r -> (r - meanR)^(2::Int)) returns) / n
ic = if denS * denR == 0 then 0 else fromRational num / (denS * denR)
in ic > 0
-- ─── Invariant enumeration ────────────────────────────────────────────────────
candidateInvariants :: [InvariantExpr]
candidateInvariants =
-- Hecke operators at prime levels, even weights (modular forms)
[ HeckeCorr l w | l <- [2, 3, 5, 7, 11, 13, 17, 19], w <- [2, 4, 6, 8] ]
++
-- Ramanujan partition volatility windows
[ PartitionVol w | w <- [5, 10, 20, 63, 126, 252] ]
++
-- Ramanujan congruences (exact number-theoretic)
[ RamanujanCong m r | m <- [5, 7, 11], r <- [m-1] ]
-- ─── Main search ─────────────────────────────────────────────────────────────
-- | Search for valid arithmetic invariants over a feature set.
-- Filters to well-typed expressions only.
-- LiquidHaskell would verify IC > 0 at compile time with refinement types.
searchInvariants :: [InvariantExpr] -> [InvariantExpr]
searchInvariants features =
filter wellTyped
. nub
$ features ++ candidateInvariants
-- | Score invariants by algebraic complexity (prefer simpler)
scoreInvariant :: InvariantExpr -> Int
scoreInvariant HeckeCorr{..} = level + weight
scoreInvariant PartitionVol{..} = window
scoreInvariant RamanujanCong{..} = modulus
scoreInvariant (Compose a b) = scoreInvariant a + scoreInvariant b + 1
rankInvariants :: [InvariantExpr] -> [InvariantExpr]
rankInvariants = sortBy (comparing scoreInvariant)
-- ─── Connection to clojure-lisp-bridge ───────────────────────────────────────
-- | Convert invariant to SGML claim tag for claimguard oracle
toSGMLClaim :: InvariantExpr -> String
toSGMLClaim inv = unlines
[ "<claim>"
, " <source>" ++ show inv ++ "</source>"
, " <invariant_type>" ++ invariantType inv ++ "</invariant_type>"
, " <actor>quantabeta-invariant-search</actor>"
, "</claim>"
]
invariantType :: InvariantExpr -> String
invariantType HeckeCorr{} = "hecke-eigenvalue-correlation"
invariantType PartitionVol{} = "ramanujan-partition-volatility"
invariantType RamanujanCong{} = "ramanujan-congruence"
invariantType (Compose _ _) = "composed"
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