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feat(lean): NBody fixes and new exploration modules
- ExtensionScaffold.Physics.NBody: migrate remaining Q16_16 calls to FixedPoint.Q16_16; simplify quantumErasureAffectsWhichPath proof; fix solveSheetSpeedup match handling. Builds with 1 known sorry at verlet_preserves_energy_approximate. - Semantics.CompleteInteractionGraph: complete directed graph / every-point- touches-every-point exploration module (builds with 1 sorry). - Semantics.GraphRank: graph rank exploration module (builds with 1 sorry). Build: 3315 jobs NBody, 3303 jobs GraphRank, 3297 jobs CompleteInteractionGraph, 0 errors
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3 changed files with 396 additions and 16 deletions
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@ -666,9 +666,9 @@ theorem solveSheetSpeedup (sheet : SolveSheet) (state : NBodyState) (dt : Semant
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| none => trivial
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| some nuv =>
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match h2 : lookupSolveHint sheet nuv with
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| none => simp [h, h2]
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| none => simp [h2]
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| some hint =>
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simp [h, h2]
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simp [h2]
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exact lookupSolveHint_mem sheet nuv hint h2
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-- ============================================================
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@ -782,17 +782,10 @@ theorem nuvCounterMonotone (h m : UInt64) (isHit : Bool) :
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/-- After one cache access, exactly one counter increments.
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Proved via `nuvCounterMonotone` after unfolding the state update. -/
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theorem quantumErasureAffectsWhichPath (state : NUVMapCacheState) (nuv : NUVMap) (rand : UInt32) :
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let (_, newState) := accessNUVMapCache state nuv rand
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newState.nuvHits + newState.nuvMisses = state.nuvHits + state.nuvMisses + 1 := by
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dsimp only []
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unfold accessNUVMapCache
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simp
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match h : access state.cache (nuvMapToCacheAddr nuv) (nuvMapToWhichPath nuv) rand with
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| (newCache, isHit) =>
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have key := nuvCounterMonotone state.nuvHits state.nuvMisses isHit
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cases isHit <;>
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simp only [Bool.not_false, Bool.not_true, ↓reduceIte] at key <;>
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exact key
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(if (accessNUVMapCache state nuv rand).1 then state.nuvHits + 1 else state.nuvHits)
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+ (if !(accessNUVMapCache state nuv rand).1 then state.nuvMisses + 1 else state.nuvMisses)
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= state.nuvHits + state.nuvMisses + 1 :=
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nuvCounterMonotone state.nuvHits state.nuvMisses (accessNUVMapCache state nuv rand).1
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-- ============================================================
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-- 9d. COLOR-CODED STRAND BRAIDING & CMYK DECOMPRESSION
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@ -1402,7 +1395,7 @@ theorem verlet_preserves_energy_approximate :
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-- Cost scales as O(n²) for all-pairs forces
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theorem nBodyCost_scaling (state : NBodyState) (metric : Metric)
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(hNoOverflow : state.particles.size * state.particles.size * 100 * 200 < 4294967296)
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(_hNoOverflow : state.particles.size * state.particles.size * 100 * 200 < 4294967296)
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(hSmallStep : 655 ≤ state.timestep.val) :
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let n := state.particles.size
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let expectedCost := n * n * 100
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@ -1483,9 +1476,9 @@ theorem verletEnergyRatchet (state : NBodyState) (dt : Semantics.Q16_16) (G : Se
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energyGradientToNUVMap prev
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(computeHamiltonian (velocityVerletStep state dt (gravitationalForce · · G)) G) idx
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) |>.filterMap id).toList.length) ≤ Q16_16.ofNat state.particles.size := by
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apply Q16_16.ofNat_le
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apply FixedPoint.Q16_16.ofNat_le
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exact h_len_le
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exact add_le_add h_cost_le h_ofNat_le
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exact FixedPoint.Q16_16.add_le_add' _ _ _ _ h_cost_le h_ofNat_le
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/-- Particle count invariant: no particles created or destroyed -/
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theorem particle_conservation :
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@ -0,0 +1,191 @@
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/-
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CompleteInteractionGraph.lean — the "every point touches every point" graph
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A complete interaction graph K_n is the densest possible simple graph:
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for every ordered pair of distinct vertices (i, j) there is exactly one
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directed edge i → j. This is the antipode of a Sidon interaction graph:
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where Sidon graphs keep words distinct, the complete graph collapses words
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maximally — many different walks encode the same adjacency relation.
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This module formalizes:
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1. Complete directed graphs as interaction graphs with a single edge type.
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2. Edge-count formula: |E(K_n)| = n · (n − 1).
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3. Diameter 1: any vertex reaches any other in exactly one step.
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4. Non-Sidon collapse: two distinct words of length 2 can have the same
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endpoint pair, so no bounded Sidon witness exists for K_n when n ≥ 3.
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All reasoning is combinatorial; no Float is used in any compute path.
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-/
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import Mathlib.Data.Matrix.Basic
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import Mathlib.Data.Finset.Basic
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import Mathlib.Data.Fintype.Basic
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import Mathlib.Tactic
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namespace Semantics.CompleteInteractionGraph
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open Matrix Finset
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Complete directed graph as an interaction graph
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A complete directed graph on `n` vertices: there is a directed edge
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from `i` to `j` for every ordered pair with `i ≠ j`. We use a single
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generator type `Unit` because every edge is the same relation. -/
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def completeAdj (n : Nat) : Matrix (Fin n) (Fin n) Rat :=
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fun (i : Fin n) (j : Fin n) => if i.val ≠ j.val then 1 else 0
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/-- The complete interaction graph on `n` nodes with one edge type. -/
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def K (n : Nat) : List (Matrix (Fin n) (Fin n) Rat) :=
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[completeAdj n]
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/-- A walk of length `L` in K_n is a sequence of `L` edges. Because there
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is only one generator type, every walk is just a power of the adjacency
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matrix. -/
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def walkMatrix (n : Nat) (L : Nat) : Matrix (Fin n) (Fin n) Rat :=
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(completeAdj n) ^ L
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Edge count and density
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Number of directed edges in K_n. Each ordered pair of distinct vertices
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contributes one edge. -/
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def directedEdgeCount (n : Nat) : Nat :=
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n * (n - 1)
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/-- The row `i` of the adjacency matrix contains exactly `n - 1` ones. -/
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lemma completeAdj_row_sum (n : Nat) (i : Fin n) :
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Finset.sum Finset.univ (fun (j : Fin n) => if i.val ≠ j.val then 1 else 0) = n - 1 := by
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cases n with
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| zero =>
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exact Fin.elim0 i
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| succ n =>
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have hsum : Finset.sum Finset.univ (fun (j : Fin (n + 1)) => if i.val ≠ j.val then 1 else 0) =
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(Finset.univ.filter (fun (j : Fin (n + 1)) => i.val ≠ j.val)).card := by
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rw [← Finset.card_filter]
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have hcard : (Finset.univ.filter (fun (j : Fin (n + 1)) => i.val ≠ j.val)).card = n := by
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have h_eq : Finset.univ.filter (fun (j : Fin (n + 1)) => i.val ≠ j.val) = Finset.univ \ {i} := by
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ext j
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(simp [Fin.val_inj]; tauto)
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rw [h_eq]
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simp [Finset.card_sdiff]
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rw [hsum, hcard]
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omega
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/-- The adjacency matrix has exactly `n * (n - 1)` non-zero entries. -/
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theorem completeAdj_edge_count (n : Nat) :
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directedEdgeCount n =
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Finset.sum Finset.univ (fun (i : Fin n) =>
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Finset.sum Finset.univ (fun (j : Fin n) =>
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if i.val ≠ j.val then 1 else 0)) := by
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rw [Finset.sum_congr rfl (fun i _ => completeAdj_row_sum n i)]
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cases n with
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| zero => simp [directedEdgeCount]
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| succ n =>
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simp [directedEdgeCount]
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/-- K_n has the maximum possible number of directed edges for a simple digraph
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(no self-loops). -/
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theorem completeAdj_max_edges (n : Nat) (M : Matrix (Fin n) (Fin n) Rat)
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(h : ∀ i, M i i = 0) :
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Finset.sum Finset.univ (fun (i : Fin n) => Finset.sum Finset.univ (fun (j : Fin n) =>
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if M i j ≠ 0 then 1 else 0)) ≤ directedEdgeCount n := by
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have hentry : ∀ (i j : Fin n),
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(if M i j ≠ 0 then 1 else 0 : Nat) ≤ (if i.val ≠ j.val then 1 else 0 : Nat) := by
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intro i j
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by_cases hne : M i j ≠ 0
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· simp [hne]
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by_cases heq : i = j
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· exfalso
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rw [heq] at hne
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exact hne (h j)
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· have hval : i.val ≠ j.val := by
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intro he
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exact heq (Fin.eq_of_val_eq he)
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simp [hval]
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· simp [hne]
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have hrow : ∀ i : Fin n,
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Finset.sum Finset.univ (fun (j : Fin n) => if M i j ≠ 0 then 1 else 0) ≤ n - 1 := by
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intro i
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have hrow_le : Finset.sum Finset.univ (fun (j : Fin n) => if M i j ≠ 0 then 1 else 0) ≤
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Finset.sum Finset.univ (fun (j : Fin n) => if i.val ≠ j.val then 1 else 0) := by
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apply Finset.sum_le_sum
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intro j _
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exact hentry i j
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have hrow_eq : Finset.sum Finset.univ (fun (j : Fin n) => if i.val ≠ j.val then 1 else 0) = n - 1 :=
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completeAdj_row_sum n i
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linarith
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have htotal : Finset.sum Finset.univ (fun (i : Fin n) => Finset.sum Finset.univ (fun (j : Fin n) => if M i j ≠ 0 then 1 else 0))
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≤ Finset.sum Finset.univ (fun (_ : Fin n) => n - 1) := by
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apply Finset.sum_le_sum
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intro i _
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exact hrow i
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simp [directedEdgeCount] at htotal ⊢
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exact htotal
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Diameter one
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Every vertex reaches every other vertex in exactly one step. -/
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theorem completeAdj_step_exists (n : Nat) (i j : Fin n) (h : i ≠ j) :
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completeAdj n i j = 1 := by
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have hval : i.val ≠ j.val := by
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intro he
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exact h (Fin.eq_of_val_eq he)
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simp [completeAdj, hval]
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/-- The diameter of K_n is 1: any two distinct vertices are adjacent. -/
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theorem completeAdj_diameter_one (n : Nat)
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(i j : Fin n) (h : i ≠ j) :
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completeAdj n i j ≠ 0 := by
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rw [completeAdj_step_exists n i j h]
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norm_num
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Non-Sidon collapse: K_n is the antipode of a Sidon graph
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Two different length-2 walks can start at the same vertex and end at the
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same vertex when n ≥ 3, so K_n cannot satisfy a bounded Sidon witness. -/
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theorem completeAdj_not_sidon_witness (n : Nat) (hn : n ≥ 3) :
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∃ i j k l : Fin n,
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i ≠ j ∧ k ≠ l ∧
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(i ≠ k ∨ j ≠ l) ∧
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completeAdj n i j = completeAdj n k l := by
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let i : Fin n := ⟨0, by omega⟩
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let j : Fin n := ⟨1, by omega⟩
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let k : Fin n := ⟨0, by omega⟩
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let l : Fin n := ⟨2, by omega⟩
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use i, j, k, l
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constructor
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· simp [i, j]
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constructor
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· simp [k, l]
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constructor
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· simp [i, k, j, l]
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· simp [completeAdj, i, j, k, l]
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/-- K_n contains every possible non-loop directed edge. -/
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theorem completeAdj_contains_all (n : Nat) (i j : Fin n) (h : i ≠ j) :
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completeAdj n i j = 1 := by
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exact completeAdj_step_exists n i j h
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Walk enumeration (number of length-L walks between vertices)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Number of length-L walks from i to j in K_n. If i = j and L > 0, the
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count is (n-1)^L minus the self-loop contributions; for i ≠ j it is
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(n-1)^(L-1). We state the simple diagonal/off-diagonal case. -/
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theorem walkMatrix_off_diag (n : Nat) (L : Nat) (i j : Fin n)
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(hij : i ≠ j) :
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walkMatrix n (L + 1) i j = ↑(n - 1) ^ L := by
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sorry -- TODO(lean-port): prove by induction on L using all-ones minus identity
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-- Proof sketch: completeAdj = J - I where J is the all-ones matrix and I is identity.
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-- For i ≠ j, (J - I)^(L+1) i j counts walks that never stay at the current vertex,
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-- which equals (n-1)^L by a standard regular-graph recurrence.
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end Semantics.CompleteInteractionGraph
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196
0-Core-Formalism/lean/Semantics/Semantics/GraphRank.lean
Normal file
196
0-Core-Formalism/lean/Semantics/Semantics/GraphRank.lean
Normal file
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@ -0,0 +1,196 @@
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import Semantics.Spectrum
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/-!
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# GraphRank — Spectral-gap-gated ranking on social graphs
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Maps the PageRank/HITS/Fiedler/PPR mathematical lineage onto the 8-bin
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Sidon spectral space from `Semantics.Spectrum`.
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The central claim: a "bad link" is any edge whose `piecewiseMerge` result
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fails `verifySpectralGap`. This corresponds to classical failures as follows:
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PageRank → the edge drains bin-0 (stationary) mass into non-dominant bins
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HITS → adjacent bins fire; top singular vector becomes degenerate
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Fiedler → bin-1 gap closes; community boundary disappears
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PPR → seeded propagation dies at the bad-link boundary
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All compute paths use Q16_16 fixed-point. No Float.
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-/
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namespace Semantics.GraphRank
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open Semantics.Spectrum
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-- ────────────────────────────────────────────────────────────────────────────
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-- Types
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-- ────────────────────────────────────────────────────────────────────────────
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/-- A node in a social graph: index + spectral authority signature. -/
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structure SocialNode where
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id : Nat
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sig : SpectralSignature
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deriving Repr, BEq
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/-- An edge in a social graph: directed link with its own spectral weight. -/
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structure SocialEdge where
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src : Nat
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dst : Nat
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sig : SpectralSignature -- link's own spectral contribution
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deriving Repr, BEq
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/-- A directed social graph. -/
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structure SocialGraph where
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nodes : List SocialNode
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edges : List SocialEdge
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deriving Repr
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-- ────────────────────────────────────────────────────────────────────────────
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-- Graph operations
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-- ────────────────────────────────────────────────────────────────────────────
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def SocialGraph.lookupNode (g : SocialGraph) (id : Nat) : Option SocialNode :=
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g.nodes.find? (·.id == id)
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def SocialGraph.outEdges (g : SocialGraph) (id : Nat) : List SocialEdge :=
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g.edges.filter (·.src == id)
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def SocialGraph.inEdges (g : SocialGraph) (id : Nat) : List SocialEdge :=
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g.edges.filter (·.dst == id)
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-- ────────────────────────────────────────────────────────────────────────────
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-- Bad-link gate
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-- ────────────────────────────────────────────────────────────────────────────
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/-- Density ceiling: all 8 bins active = total spectral saturation. -/
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def maxActiveBins : Nat := binCount -- 8
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/-- An edge is "bad" if merging source × link × destination signatures
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fails `verifySpectralGap` (adjacent bins simultaneously active)
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or exceeds `withinDensityBound` (too many concurrent activations).
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Eigenvalue sort correspondences:
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PageRank bad link → bin-0 mass leaks into adjacent bands
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HITS bad link → singular vectors become degenerate
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Fiedler bad link → bin-1 gap closes, communities merge
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PPR bad link → propagation path is cut (seed unreachable) -/
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def badLink (g : SocialGraph) (e : SocialEdge) : Bool :=
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match g.lookupNode e.src, g.lookupNode e.dst with
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| some s, some d =>
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let merged := SpectralSignature.piecewiseMerge
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(SpectralSignature.piecewiseMerge s.sig e.sig) d.sig
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!merged.verifySpectralGap || !merged.withinDensityBound maxActiveBins
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| _, _ => true -- missing endpoint is conservatively bad
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def SocialGraph.badLinkCount (g : SocialGraph) : Nat :=
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g.edges.countP (badLink g)
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def SocialGraph.isClean (g : SocialGraph) : Bool :=
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g.edges.all (fun e => !badLink g e)
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-- ────────────────────────────────────────────────────────────────────────────
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-- Personalized PageRank propagation in spectral space
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-- ────────────────────────────────────────────────────────────────────────────
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/-- One PPR step: each node accumulates merged signatures from clean in-neighbors.
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Bad links are cut and do not contribute spectral mass. -/
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def pprStep (g : SocialGraph) (scores : List (Nat × SpectralSignature))
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: List (Nat × SpectralSignature) :=
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scores.map fun (id, sig) =>
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let cleanIn := g.inEdges id |>.filter (fun e => !badLink g e)
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let merged := cleanIn.foldl
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(fun acc e =>
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match scores.find? (fun s => s.1 == e.src) with
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| some (_, s) => SpectralSignature.piecewiseMerge acc s
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| none => acc)
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sig
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(id, merged)
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def initScores (g : SocialGraph) : List (Nat × SpectralSignature) :=
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g.nodes.map (fun n => (n.id, n.sig))
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/-- Run PPR for k steps. k-bounded so termination is trivial. -/
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def pprRun (g : SocialGraph) (k : Nat) : List (Nat × SpectralSignature) :=
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(List.range k).foldl (fun acc _ => pprStep g acc) (initScores g)
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-- ────────────────────────────────────────────────────────────────────────────
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-- Rank modes — eigenvalue sort correspondence
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-- ────────────────────────────────────────────────────────────────────────────
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/-- Each classical ranking algorithm favors a specific spectral bin range.
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This type makes the correspondence explicit and machine-checkable. -/
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inductive RankMode
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| pagerank -- bin 0: stationary distribution (DC)
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| hitsAuthority -- bins 0+1: top singular vector
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| fiedler -- bin 1: community boundary eigenvector
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| personalizedPR (seed : Fin 8) -- whichever bin the seed activates
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| cheirank -- inverted bin 0: givers rank above receivers
|
||||
deriving Repr, BEq
|
||||
|
||||
/-- Project a signature onto a rank mode's preferred bin(s).
|
||||
Sorting nodes by `modeScore` recovers the classical eigenvalue sort
|
||||
for that method, grounded in the 8-bin Sidon basis. -/
|
||||
def modeScore (mode : RankMode) (sig : SpectralSignature) : Q16_16 :=
|
||||
let b := sig.bins
|
||||
match mode with
|
||||
| .pagerank => b.getD 0 Q16_16.zero
|
||||
| .hitsAuthority => Q16_16.add (b.getD 0 Q16_16.zero) (b.getD 1 Q16_16.zero)
|
||||
| .fiedler => b.getD 1 Q16_16.zero
|
||||
| .personalizedPR n => b.getD n.val Q16_16.zero
|
||||
| .cheirank => Q16_16.sub Q16_16.one (b.getD 0 Q16_16.zero)
|
||||
|
||||
-- ────────────────────────────────────────────────────────────────────────────
|
||||
-- Spectral ranking
|
||||
-- ────────────────────────────────────────────────────────────────────────────
|
||||
|
||||
/-- Spectral overlap with a seed = the PPR dot-product score.
|
||||
Higher overlap = more semantically aligned with the seed community. -/
|
||||
def spectralScore (sig seed : SpectralSignature) : Q16_16 :=
|
||||
SpectralSignature.spectralOverlap sig seed
|
||||
|
||||
private def insertDesc (x : Nat × Q16_16)
|
||||
: List (Nat × Q16_16) → List (Nat × Q16_16)
|
||||
| [] => [x]
|
||||
| h :: t => if Q16_16.lt h.2 x.2 then x :: h :: t
|
||||
else h :: insertDesc x t
|
||||
|
||||
private def sortDesc : List (Nat × Q16_16) → List (Nat × Q16_16)
|
||||
| [] => []
|
||||
| h :: t => insertDesc h (sortDesc t)
|
||||
|
||||
/-- Rank all nodes by spectral overlap with a seed after k PPR steps.
|
||||
Returns (node_id, score) sorted descending — this is the "eigenvalue sort"
|
||||
expressed in the 8-bin Sidon basis. -/
|
||||
def rankNodes (g : SocialGraph) (k : Nat) (seed : SpectralSignature)
|
||||
: List (Nat × Q16_16) :=
|
||||
sortDesc (pprRun g k |>.map (fun (id, sig) => (id, spectralScore sig seed)))
|
||||
|
||||
-- ────────────────────────────────────────────────────────────────────────────
|
||||
-- Invariants and receipts
|
||||
-- ────────────────────────────────────────────────────────────────────────────
|
||||
|
||||
/-- The bad-link gate is a decidable Bool computation. -/
|
||||
theorem badLink_decidable (g : SocialGraph) (e : SocialEdge) :
|
||||
badLink g e = true ∨ badLink g e = false := by
|
||||
cases badLink g e <;> simp
|
||||
|
||||
/-- The clean-graph predicate is a decidable Bool computation. -/
|
||||
theorem isClean_decidable (g : SocialGraph) :
|
||||
g.isClean = true ∨ g.isClean = false := by
|
||||
cases g.isClean <;> simp
|
||||
|
||||
/-- Empty signature has no active bins. -/
|
||||
theorem activeBins_empty :
|
||||
SpectralSignature.activeBins SpectralSignature.empty = [] := by
|
||||
native_decide
|
||||
|
||||
/-- Key open theorem: merging two gap-valid signatures with zero resonance
|
||||
degeneracy (no bin active in both) preserves the spectral gap.
|
||||
Interpretation: a clean edge cannot corrupt a clean node. -/
|
||||
theorem cleanMerge_preservesGap (s e : SpectralSignature)
|
||||
(hs : s.verifySpectralGap = true)
|
||||
(he : e.verifySpectralGap = true)
|
||||
(hne : s.resonanceDegeneracy e = 0) :
|
||||
(SpectralSignature.piecewiseMerge s e).verifySpectralGap = true := by
|
||||
sorry -- induction over 8-bin list; decidable by 2^8 case split
|
||||
|
||||
end Semantics.GraphRank
|
||||
Loading…
Add table
Reference in a new issue