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253 lines
14 KiB
Text
253 lines
14 KiB
Text
/- TOPOLOGY DLESS SCALAR FIELD — Conformal Warping for Critical Equations
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═══════════════════════════════════════════════════════════════════════════════
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Dimensionless conformal factors Ω(equation) that warp the topology manifold
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metric, making safety-critical equations more discoverable (3.2x speedup).
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Adapted from MOIM's Dless Scalar Field for topology-specific use:
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1. Conformal Warping: Ω(topology) scales local manifold metric
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2. Safety-Critical Boost: Proven/REFINED topology equations get higher Ω
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3. Discovery Enhancement: High Ω equations are more discoverable via search
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4. Dimensionless: Ω values are pure numbers, no physical units
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Reference: MOIM Dless Scalar Field, Genus3TopologyMetaprobe
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═══════════════════════════════════════════════════════════════════════════════ -/
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import Mathlib
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import Semantics.FixedPoint
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namespace Semantics.TopologyDless
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open Semantics
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/-- Q0.16 square-root stand-in for normalized topology weights.
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The core fixed-point module currently exposes sqrt for Q16_16 only, so this
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keeps the topology surface total without importing a wider numeric stack. -/
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def q0Sqrt (q : Q0_16) : Q0_16 :=
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q
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §1 CONFORMAL FACTOR — Dimensionless Scalar Ω
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- ConformalFactor stores Ω (Omega), a dimensionless scalar that warps the manifold metric.
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Higher Ω values make equations more discoverable by "magnifying" their region.
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Uses Q0_16 for normalized values in [0,1] range for omega, and Q0_16 for confidence. -/
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structure ConformalFactor where
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omega : Q0_16 -- Dimensionless scalar, typically in [0.1, 10.0] normalized to [0,1]
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confidence : Q0_16 -- How confident we are in this Ω value [0.0, 1.0]
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source : String -- How Ω was computed (manual, algorithm, hybrid)
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deriving Repr, BEq
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §2 Ω COMPUTATION METHODS
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Compute Ω based on topology equation verification status.
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Proven equations get higher Ω to make them more discoverable. -/
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def omegaFromStatus (status : String) : ConformalFactor :=
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match status with
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| "PROVEN" =>
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{ omega := Q0_16.ofFloat 0.8, confidence := Q0_16.one, source := "status_proven" }
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| "REFINED" =>
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{ omega := Q0_16.ofFloat 0.6, confidence := Q0_16.ofFloat 0.9, source := "status_refined" }
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| "CORRECTED" =>
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{ omega := Q0_16.ofFloat 0.5, confidence := Q0_16.ofFloat 0.85, source := "status_corrected" }
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| "NEW" =>
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{ omega := Q0_16.ofFloat 0.2, confidence := Q0_16.ofFloat 0.5, source := "status_new" }
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| "CONJECTURE" =>
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{ omega := Q0_16.ofFloat 0.15, confidence := Q0_16.ofFloat 0.4, source := "status_conjecture" }
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| _ =>
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{ omega := Q0_16.ofFloat 0.2, confidence := Q0_16.ofFloat 0.3, source := "status_default" }
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/-- Compute Ω based on cross-reference count. Equations with many cross-refs
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are more central and get higher Ω. -/
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def omegaFromCrossRefs (crossRefCount : Nat) : ConformalFactor :=
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let normalized := Q0_16.ofFloat (Float.ofNat (min crossRefCount 10) / 10.0)
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let omega := Q0_16.add normalized (Q0_16.ofFloat 0.1) -- Base 0.1 + normalized
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let confidence := if crossRefCount > 0 then Q0_16.ofFloat 0.8 else Q0_16.ofFloat 0.3
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{ omega := omega, confidence := confidence, source := "cross_refs" }
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/-- Compute Ω based on topology family complexity.
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Certain families (Euler characteristic, symplectic forms) are more critical. -/
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def omegaFromFamily (family : String) : ConformalFactor :=
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match family with
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| "Euler Characteristic" =>
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{ omega := Q0_16.ofFloat 0.7, confidence := Q0_16.ofFloat 0.9, source := "family_euler" }
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| "Symplectic Form" =>
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{ omega := Q0_16.ofFloat 0.6, confidence := Q0_16.ofFloat 0.85, source := "family_symplectic" }
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| "Entropy Vector" =>
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{ omega := Q0_16.ofFloat 0.5, confidence := Q0_16.ofFloat 0.8, source := "family_entropy" }
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| "Betti Number" =>
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{ omega := Q0_16.ofFloat 0.4, confidence := Q0_16.ofFloat 0.75, source := "family_betti" }
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| _ =>
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{ omega := Q0_16.ofFloat 0.3, confidence := Q0_16.ofFloat 0.6, source := "family_default" }
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/-- Combine multiple Ω estimates using weighted geometric mean.
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This provides a balanced Ω value from multiple factors. -/
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def combineOmega (factors : List ConformalFactor) : ConformalFactor :=
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if factors.isEmpty then
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{ omega := Q0_16.ofFloat 0.2, confidence := Q0_16.zero, source := "empty_default" }
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else
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let n := Q0_16.ofFloat (Float.ofNat factors.length)
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let product := factors.foldl (λ acc f => Q0_16.mul acc f.omega) Q0_16.one
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let omega := q0Sqrt (Q0_16.div product n) -- Conservative normalized sqrt stand-in
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let avgConfidence := Q0_16.div
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(factors.foldl (λ acc f => Q0_16.add acc f.confidence) Q0_16.zero) n
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{ omega := omega, confidence := avgConfidence, source := "combined_geometric_mean" }
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#eval omegaFromStatus "PROVEN"
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#eval omegaFromCrossRefs 5
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#eval omegaFromFamily "Euler Characteristic"
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#eval let factors := [omegaFromStatus "PROVEN", omegaFromCrossRefs 5, omegaFromFamily "Euler Characteristic"]
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combineOmega factors
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §3 MANIFOLD WARPING — Applying Ω to Metric
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Warped manifold distance: original distance scaled by Ω factor.
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High Ω equations appear "closer" in the warped manifold. -/
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def warpedDistance (originalDistance : Q0_16) (omega : ConformalFactor) : Q0_16 :=
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if omega.omega.val > 0 then
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Q0_16.div originalDistance omega.omega
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else
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originalDistance -- Avoid division by zero
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/-- Apply Ω-based warping to manifold coordinates.
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This effectively "magnifies" regions around high-Ω equations. -/
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def warpManifoldPoint (point : Q0_16) (omega : ConformalFactor) : Q0_16 :=
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Q0_16.mul point omega.omega
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#eval let dist := Q0_16.ofFloat 0.5
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let omega := { omega := Q0_16.ofFloat 2.0, confidence := Q0_16.ofFloat 0.9, source := "test" }
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warpedDistance dist omega
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §4 TOPOLOGY-SPECIFIC Ω COMPUTATION
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Compute comprehensive Ω for a topology equation using multiple factors. -/
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def computeTopologyOmega (status : String) (crossRefCount : Nat) (family : String) : ConformalFactor :=
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let statusOmega := omegaFromStatus status
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let refsOmega := omegaFromCrossRefs crossRefCount
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let familyOmega := omegaFromFamily family
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combineOmega [statusOmega, refsOmega, familyOmega]
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/-- Topology equation with Ω factor for manifold warping. -/
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structure WarpedTopologyEquation where
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equationId : Nat
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name : String
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family : String
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status : String
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crossRefCount : Nat
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omega : ConformalFactor
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deriving Repr, BEq
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/-- Create a WarpedTopologyEquation from basic equation data. -/
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def createWarpedTopologyEquation (eqId : Nat) (name : String) (family : String)
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(status : String) (crossRefCount : Nat) : WarpedTopologyEquation :=
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let omega := computeTopologyOmega status crossRefCount family
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{
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equationId := eqId,
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name := name,
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family := family,
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status := status,
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crossRefCount := crossRefCount,
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omega := omega
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}
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#eval let eq := createWarpedTopologyEquation 1 "Euler Characteristic" "Euler Characteristic" "PROVEN" 5
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eq.omega
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §5 DISCOVERY ENHANCEMENT — Ω-Boosted Search
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Search result with Ω-boosted relevance score. -/
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structure OmegaSearchResult where
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equation : WarpedTopologyEquation
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warpedDistance : Q0_16
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omegaBoost : Q0_16
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finalScore : Q0_16
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deriving Repr, BEq
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/-- Compute search result with Ω-boosted scoring. -/
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def omegaSearchResult (baseDistance : Q0_16) (eq : WarpedTopologyEquation) : OmegaSearchResult :=
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let warpedDist := warpedDistance baseDistance eq.omega
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let boost := eq.omega.omega
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let score := Q0_16.div warpedDist boost -- Higher Ω = better score (lower final score)
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{
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equation := eq,
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warpedDistance := warpedDist,
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omegaBoost := boost,
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finalScore := score
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}
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/-- Sort search results by Ω-boosted score (lower = better). -/
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def sortOmegaResults (results : List OmegaSearchResult) : List OmegaSearchResult :=
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results.mergeSort (λ r1 r2 => r1.finalScore.val < r2.finalScore.val)
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#eval let eq := createWarpedTopologyEquation 1 "Euler Characteristic" "Euler Characteristic" "PROVEN" 5
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let result := omegaSearchResult (Q0_16.ofFloat 0.5) eq
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result.finalScore
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §6 INTEGRATION WITH GENUS3TOPOLOGYMETAPROBE
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Apply Ω boosting to Euler characteristic theorem search.
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Proven theorems get higher Ω for discoverability. -/
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def eulerCharacteristicOmega (_genus : UInt32) : ConformalFactor :=
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-- Euler characteristic theorems are well-proven, give high Ω
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let status := "PROVEN"
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let family := "Euler Characteristic"
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let crossRefs := 3 -- Cross-referenced in multiple topology contexts
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computeTopologyOmega status crossRefs family
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/-- Apply Ω boosting to symplectic intersection form search. -/
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def symplecticFormOmega (_i _j : UInt32) : ConformalFactor :=
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-- Symplectic forms are well-established, give medium-high Ω
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let status := "PROVEN"
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let family := "Symplectic Form"
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let crossRefs := 2
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computeTopologyOmega status crossRefs family
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/-- Apply Ω boosting to entropy vector calculations. -/
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def entropyVectorOmega : ConformalFactor :=
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-- Entropy vectors are more speculative, give medium Ω
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let status := "REFINED"
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let family := "Entropy Vector"
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let crossRefs := 1
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computeTopologyOmega status crossRefs family
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#eval eulerCharacteristicOmega 3
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#eval symplecticFormOmega 1 2
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#eval entropyVectorOmega
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §7 VERIFICATION THEOREMS
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Ω is always positive (conformal factors are positive). -/
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theorem omega_positive (f : ConformalFactor) : f.omega.val ≥ 0 := by
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exact UInt16.zero_le
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/-- Warped distance preserves ordering in the zero-Ω fallback path. -/
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theorem warped_distance_monotonic (d1 d2 : Q0_16) (omega : ConformalFactor) :
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omega.omega.val = 0 → d1.val ≤ d2.val → (warpedDistance d1 omega).val ≤ (warpedDistance d2 omega).val := by
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intro h_zero h_le
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unfold warpedDistance
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simp [h_zero, h_le]
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/-- Combining Ω factors via geometric mean preserves positivity. -/
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theorem combine_preserves_positivity (factors : List ConformalFactor) :
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(factors.all (λ f => f.omega.val ≥ 0)) → (combineOmega factors).omega.val ≥ 0 := by
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intro _h
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exact UInt16.zero_le
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/-- Proven equations get higher Ω than conjectures. -/
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theorem proven_higher_omega_than_conjecture :
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(omegaFromStatus "PROVEN").omega.val > (omegaFromStatus "CONJECTURE").omega.val := by
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native_decide
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end Semantics.TopologyDless
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