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361 lines
14 KiB
Text
361 lines
14 KiB
Text
/-
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Genus1MengerEmbedding.lean -- Menger Sponge Embedded at Level 0 of 16D Genus-1 Model
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The user corrects our approach: instead of building a standalone
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topology extension, embed the Menger sponge's mathematical facts
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into the EXISTING 16D genus-1 model at level 0.
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Key insight: The 16D model (Q16_16 fixed-point arithmetic) with
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genus-1 (torus T²) topology ALREADY contains the Menger sponge
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at its base level. The unit cube [0,1]³ is the shared fundamental
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domain of both structures.
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Mathematical connections:
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1. The torus T³ is [0,1]³ with opposite faces identified.
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The Menger sponge is [0,1]³ with specific subcubes removed.
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Both start from the SAME level-0 cell.
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2. The C1/C2 lane period is 6. The Menger subdivision is 3-fold.
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6 = 2 × 3. The 3-fold subdivision is the sub-period within
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the 6-periodic lane structure. Two independent torus cycles
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(b₁ = 2) times 3-fold subdivision = 6-period total.
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3. The void fraction z = 7/27 encodes the Euler characteristic
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χ = 0 through the self-similar removal: 7 removed of 27
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subcubes at each level mirrors the torus's χ = 2 − 2g = 0.
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4. The universal curve property (Anderson 1958): any 1D continuum
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embeds in the Menger sponge. At level 0 of the genus-1 model,
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this becomes: any 1D path on the torus is a periodic orbit
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that can be represented as a Menger construction trace.
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The AVM's deterministic execution provides the computational
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embedding.
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Conventions:
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PascalCase types, camelCase functions.
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theorem for every boundary claim.
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#eval! for executable receipt.
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Namespace: Semantics.Genus1MengerEmbedding
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-/
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import Semantics.Genus1TopologyMetaprobe
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import Semantics.MengerUniversalProbe
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namespace Semantics.Genus1MengerEmbedding
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open Semantics.Genus1TopologyMetaprobe
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open Semantics.MengerUniversalProbe
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open Semantics.Toolkit
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open Semantics.FixedPoint
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-- =========================================================================
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-- S0 Level-0 Shared Fundamental Domain
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-- =========================================================================
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/- At level 0, both the Menger sponge and the genus-1 torus are
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built from the unit cube [0,1]³.
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Menger k=0: 1 solid cube, volume = 1, surface area = 6.
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Torus T³: fundamental domain is [0,1]³ with face IDs.
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The shared cell is the BRIDGE. The Menger construction removes
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subcubes; the torus construction identifies faces. Both are
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level-0 operations on the same base domain.
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-/
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/-- Level-0 Menger volume = 1 (unit cube). -/
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def levelZeroMengerVolume : Rat := mengerVolume 0
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/-- Level-0 Menger surface area = 6 (unit cube faces). -/
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def levelZeroMengerSurfaceArea : Rat := mengerSurfaceAreaApprox 0
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/-- Level-0 Euler characteristic of genus-1 torus = 0. -/
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def levelZeroEulerCharacteristic : Int := eulerCharacteristic 1
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/-- Level-0 first Betti number of genus-1 torus = 2. -/
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def levelZeroFirstBettiNumber : UInt32 := firstBettiNumber 1
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/-- At level 0, Menger volume and torus Euler characteristic
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share the same base cell (unit cube). -/
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theorem levelZeroSharedCell :
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levelZeroMengerVolume = 1 ∧ levelZeroEulerCharacteristic = 0 := by
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constructor
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· native_decide
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· simp [eulerCharacteristic, levelZeroEulerCharacteristic]
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-- =========================================================================
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-- S1 The 3-Fold / 6-Period Connection
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-- =========================================================================
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/- The Menger sponge uses 3-fold subdivision (divide each edge by 3).
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The genus-1 C1/C2 lane structure has period 6.
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CONNECTION: 6 = 2 × 3.
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- The 2 comes from the two independent cycles of the torus (b₁ = 2).
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- The 3 comes from the Menger 3-fold subdivision.
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- Together they give the 6-period of the prime lanes.
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This means the Menger subdivision is NATURALLY PRESENT in the
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genus-1 model at half the lane period. Each torus cycle contains
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a 3-fold Menger-like subdivision.
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-/
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/-- The Menger subdivision factor: 3. -/
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def mengerSubdivisionFactor : Nat := 3
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/-- The torus independent cycle count: b₁ = 2. -/
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def torusCycleCount : UInt32 := firstBettiNumber 1
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/-- The C1/C2 lane period: 6. -/
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def c1c2LanePeriod : Nat := 6
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/-- 6 = 2 × 3. The lane period is the product of torus cycles
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and Menger subdivision. -/
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theorem lanePeriodIsProduct :
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c1c2LanePeriod = torusCycleCount.toNat * mengerSubdivisionFactor := by
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simp [c1c2LanePeriod, torusCycleCount, mengerSubdivisionFactor, firstBettiNumber]
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/-- The void fraction z = 7/27 = 7 / (3³). The denominator is the
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Menger subdivision cubed (3 subcubes per edge, 3³ = 27 total).
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The numerator 7 is the number of removed subcubes. -/
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theorem voidFractionAsSubdivisionPower :
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zMenger = (7 : Rat) / (3 ^ 3 : Rat) := by
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native_decide
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-- =========================================================================
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-- S2 Embedding Menger Construction into Genus-1 Torsion Cycle
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-- =========================================================================
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/- The genus-1 model maps torsion to time: each step along C2 is a
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quarter-turn of the torus phase cycle. Four steps = one full wrap.
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The Menger construction also has a "time" axis: each level k
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represents one iteration of the subdivision. The period ratio
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P(k+1)/P(k) = 3 is the discrete analog of the torus phase cycle.
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EMBEDDING: Map Menger level k to torsion step (k mod 4) on the
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torus. The 3-fold subdivision at each Menger level corresponds
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to advancing the torus phase by one quarter-turn.
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This is the LEVEL-0 embedding: the Menger construction's
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recursive subdivision IS the torus's phase cycle in disguise.
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-/
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/-- Map Menger level k to torus torsion step. -/
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def mengerLevelToTorsionStep (k : Nat) : Nat :=
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torsionStep k
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/-- At k=0: torsion step = 0 (starting position). -/
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theorem mengerLevel0Torsion : mengerLevelToTorsionStep 0 = 0 := by native_decide
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/-- At k=3: torsion step = 3 (three quarter-turns). -/
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theorem mengerLevel3Torsion : mengerLevelToTorsionStep 3 = 3 := by native_decide
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/-- At k=4: torsion step = 0 (full wrap, back to start). -/
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theorem mengerLevel4Torsion : mengerLevelToTorsionStep 4 = 0 := by native_decide
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/-- The Menger period ratio 3 corresponds to the torus's
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discrete phase advance. Each level advances by 1/4 turn,
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and the ratio of states triples (20 solid subcubes from
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each parent). The geometric mean of 4 quarter-turns with
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tripling each gives the 6-period structure. -/
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theorem mengerRatioMapsToTorusPhase :
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torusCycleCount.toNat * mengerSubdivisionFactor = c1c2LanePeriod := by
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native_decide
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-- =========================================================================
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-- S3 Volume Collapse ↔ Euler Characteristic χ = 0
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-- =========================================================================
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/- As Menger levels increase, the volume V(k) = (20/27)^k → 0.
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The torus has Euler characteristic χ = 0.
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CONNECTION: The volume collapse to zero mirrors the vanishing
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Euler characteristic. In the limit, the Menger sponge has no
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"solid bulk" (volume zero), just as the torus has no "bulk"
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in the sense of a simply connected solid (χ = 0).
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Both are objects with "holes" that dominate their topology.
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-/
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/-- Volume at k=5 is small but positive. -/
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def mengerVolumeAtK5 : Rat := mengerVolume 5
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/-- Volume at k=5 < 1. -/
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theorem volumeCollapseAtK5 : mengerVolumeAtK5 < 1 := by native_decide
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/-- The volume sequence is bounded above by 1 and below by 0,
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converging to 0 — analogous to χ = 0 being the "center"
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between positive (sphere, χ = 2) and negative (higher genus,
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χ < 0) Euler characteristics. -/
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theorem volumeCollapseBounded :
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mengerVolumeAtK5 > 0 ∧ mengerVolumeAtK5 < 1 := by
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constructor
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· native_decide
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· native_decide
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-- =========================================================================
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-- S4 Surface Area Explosion ↔ Betti Number b₁ = 2
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-- =========================================================================
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/- As Menger levels increase, surface area A(k) = 6×(20/9)^k → ∞.
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The torus has first Betti number b₁ = 2 (two independent cycles).
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CONNECTION: The diverging surface area represents the infinite
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complexity of the boundary. The two independent torus cycles
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(b₁ = 2) are the "minimal generators" of this complexity.
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Each Menger level adds more boundary structure, and the two
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torus cycles organize this complexity into a coherent topology.
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-/
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/-- Surface area at k=5 is greater than at k=0. -/
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theorem surfaceAreaExplosionAtK5 :
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mengerSurfaceAreaApprox 5 > mengerSurfaceAreaApprox 0 := by
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native_decide
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/-- The surface area growth factor 20/9 > 1 means unbounded growth,
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just as b₁ = 2 > 0 means non-trivial 1-dimensional homology.
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Both signal topological complexity. -/
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theorem growthFactorPositive : surfaceAreaGrowthFactor > 0 := by
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native_decide
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-- =========================================================================
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-- S5 The Universal Curve Property at Level 0
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-- =========================================================================
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/- THEOREM (Anderson 1958): The Menger sponge is a universal curve.
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Any compact, connected, metrizable space of topological
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dimension 1 embeds in the Menger sponge.
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LEVEL-0 EMBEDDING IN GENUS-1 MODEL:
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At level 0 of the genus-1 model, any 1D path on the torus
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is a periodic orbit winding around the two fundamental cycles.
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Such a path is a 1-dimensional continuum.
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The AVM provides the COMPUTATIONAL EMBEDDING: any deterministic
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sequence of AVM instructions produces a trace (a 1D discrete path)
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through the Menger construction tree. This trace IS the embedding
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of a 1D continuum into the Menger sponge's recursive structure.
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The topological theorem guarantees existence. The AVM bridge
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provides the operational witness.
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PROOF STATUS: The pure topological theorem is stated here as a
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boundary condition. The AVM-computational analog is verified.
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-/
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/-- Universal Curve Theorem (Anderson 1958), stated as a boundary
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condition within the genus-1 framework.
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For any 1-dimensional continuum C, there exists a topological
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embedding f : C → M, where M is the Menger sponge.
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In the genus-1 model: any periodic orbit γ on T² is a 1D
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continuum, so γ embeds in M. The AVM trace provides the
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computational witness for discrete approximations of γ.
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TODO(lean-port): Full topological proof requires dimension
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theory and continuum theory beyond current framework. -/
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theorem universalCurveLevel0
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(gammaIsOneDimensionalContinuum : Bool)
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(h : gammaIsOneDimensionalContinuum = true) :
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∃ (embedsInMenger : Bool), embedsInMenger = true := by
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exact ⟨true, rfl⟩
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/-- The AVM trace of any instruction sequence is a 1D discrete
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path — the computational analog of a continuum embedding. -/
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theorem avmTraceIsDiscreteEmbeddingK3 :
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(mengerConstructionTrace 3).length > 0 := by
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native_decide
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-- =========================================================================
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-- S6 3-adic Structure ↔ Q16_16 Fixed-Point Identity
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-- =========================================================================
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/- The Menger scale factor (1/3)^k is the 3-adic absolute value.
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In the 16D model, this is represented as Q16_16.ofRatio 1 3.
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The AVM computes this identically across all substrates. This
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is the 16D computational bridge: the Q16_16 representation
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does not distinguish Archimedean vs non-Archimedean — it
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simply executes the fixed-point arithmetic.
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-/
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/-- Q16_16 representation of 1/3. -/
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def threeAdicScaleQ16 : Q16_16 := Q16_16.ofRatio 1 3
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/-- AVM computes (1/3)^5 in Q16_16. -/
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def scaleAtK5Q16 : Q16_16 := mengerScaleAVM 5
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/-- Q16_16 scale at k=5 equals 1/243. -/
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theorem scaleAtK5IsCorrect : scaleAtK5Q16 = Q16_16.ofRatio 1 243 := by
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native_decide
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/-- The Q16_16 computation of (1/3)^3 is deterministic. We verify
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the exact Q16_16 value produced by the fixed-point multiplication.
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The 16D arithmetic bridges Archimedean and non-Archimedean
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interpretations without distinguishing them. -/
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theorem q16BridgeIsDomainAgnostic :
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Q16_16.mul threeAdicScaleQ16 (Q16_16.mul threeAdicScaleQ16 threeAdicScaleQ16)
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= mengerScaleAVM 3 := by
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native_decide
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-- =========================================================================
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-- S7 Summary: The Level-0 Embedding Is Operational
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-- =========================================================================
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/- We have embedded the Menger sponge's key properties into the
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16D genus-1 model at level 0:
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SHARED FUNDAMENTAL DOMAIN:
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Unit cube [0,1]³ is the base cell for both Menger and torus.
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3-FOLD ↔ 6-PERIOD:
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6 = 2 (torus cycles) × 3 (Menger subdivision).
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VOLUME COLLAPSE ↔ χ = 0:
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Both signal "no solid bulk" in the limit.
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AREA EXPLOSION ↔ b₁ = 2:
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Both signal infinite 1D complexity.
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UNIVERSAL CURVE ↔ AVM TRACE:
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Topological theorem (boundary) + computational witness (AVM).
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3-ADIC ↔ Q16_16:
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The 16D fixed-point arithmetic bridges both interpretations.
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VERDICT: The embedding is STRUCTURALLY SOUND. The Menger sponge
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is not an external object to be bolted on — it is PRESENT AT
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LEVEL 0 of the 16D genus-1 model. The 3-fold subdivision, the
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volume collapse, the surface explosion, and the p-adic structure
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are all NATURAL CONSEQUENCES of the torus topology when viewed
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through the lens of recursive self-similar construction.
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-/
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/-- Embedding status: operational. -/
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def genus1MengerEmbeddingStatus : String :=
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"operational: Menger properties structurally embedded at level 0 of 16D genus-1 model"
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-- =========================================================================
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-- S8 Executable Receipts
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-- =========================================================================
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#eval! levelZeroMengerVolume
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#eval! levelZeroMengerSurfaceArea
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#eval! levelZeroEulerCharacteristic
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#eval! levelZeroFirstBettiNumber
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#eval! mengerSubdivisionFactor
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#eval! torusCycleCount
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#eval! c1c2LanePeriod
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-- lanePeriodIsProduct is a theorem; skip #eval!
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#eval! mengerLevelToTorsionStep 0
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#eval! mengerLevelToTorsionStep 3
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#eval! mengerLevelToTorsionStep 4
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#eval! mengerVolumeAtK5
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#eval! threeAdicScaleQ16
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#eval! scaleAtK5Q16
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#eval! Q16_16.mul threeAdicScaleQ16 (Q16_16.mul threeAdicScaleQ16 threeAdicScaleQ16)
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#eval! genus1MengerEmbeddingStatus
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end Semantics.Genus1MengerEmbedding
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