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feat(lean): InteractionGraphSidon — RRC weak axes as atproto-style projections
- Formalize typed interaction graphs and bounded Sidon witnesses - WeakAxis = sieve modulus; independentAxes = coprime moduli - weakAxis_coprime_intersect: independent weak axes reconstruct the underlying RRC class modulo their product (CRT) - Executable toy instance: identity/host/app axes (7/11/13) reconstruct 61 - Add to Semantics.lean imports; builds under Compiler surface
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@ -218,6 +218,7 @@ import Semantics.NKHodgeFAMM
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import Semantics.Goxel
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import Semantics.N3L_Energy
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import Semantics.SieveLemmas
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import Semantics.InteractionGraphSidon
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namespace Semantics
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@ -0,0 +1,191 @@
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/-
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InteractionGraphSidon.lean — RRC weak-axis reconstruction via interaction-graph freeness
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The atproto/Mastodon observation (and the RRC "weak axis" problem) are the same
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abstract structure: an object's full identity/classification is hidden from any
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single partial view. Multiple independent weak projections must be reconciled
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via a CRT/Sidon-type uniqueness condition.
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This module formalizes:
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1. Interaction graphs as finite typed-transition systems.
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2. Word products in the matrix semigroup generated by typed edges.
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3. A bounded Sidon witness: all words up to length L are distinct.
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4. RRC weak axes as sieve projections of an underlying classification.
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5. Reconstruction: independent weak axes recover the underlying class
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uniquely modulo their product — the "weak-portion is the atproto problem".
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All computation uses rational matrices; no Float is used in the compute path.
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-/
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import Mathlib.Data.Matrix.Basic
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import Mathlib.Data.Matrix.Mul
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import Mathlib.Data.Finset.Basic
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import Mathlib.Data.List.FinRange
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import Mathlib.Tactic
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namespace Semantics.InteractionGraphSidon
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open Matrix Finset List
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Typed interaction graphs and word products
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A typed interaction graph on `n` nodes with edge types indexed by `ι`. -/
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structure InteractionGraph (ι : Type) (n : Nat) where
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nodeCount : Nat := n
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edgeTypes : Finset ι
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gen : ι → Matrix (Fin n) (Fin n) Rat
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/-- Word product: multiply generator matrices in the order of the word.
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The empty word is the identity matrix. -/
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def wordProduct {ι : Type} {n : Nat} (g : InteractionGraph ι n) (w : List ι) :
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Matrix (Fin n) (Fin n) Rat :=
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w.foldl (fun M t => M * g.gen t) 1
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/-- A bounded Sidon witness: no two distinct words of length ≤ L collapse to
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the same matrix. This is the finite, checkable version of semigroup
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freeness; the full infinite property is the limit as L → ∞. -/
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def isSidonWitness {ι : Type} [DecidableEq ι]
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(g : InteractionGraph ι n) (L : Nat) : Prop :=
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∀ w1 w2 : List ι,
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w1.length ≤ L → w2.length ≤ L →
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wordProduct g w1 = wordProduct g w2 → w1 = w2
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 RRC weak axes as independent projections
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A weak axis is a sieve modulus: a partial observation of an underlying
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RRC class. In RRC terms, each weak axis is one independent reason the
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classifier cannot commit to a single label; the axis records the residue
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of the true class modulo `modulus`. -/
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structure WeakAxis where
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modulus : Nat
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pos : modulus > 0
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deriving Repr
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/-- Project an underlying class through a weak axis. -/
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def project (a : WeakAxis) (cls : Nat) : Nat := cls % a.modulus
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/-- Two weak axes are independent when their moduli are coprime.
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Independence is the analogue of atproto's separation of identity,
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hosting, and application: no axis is a refinement of another. -/
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def independentAxes (a b : WeakAxis) : Prop :=
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Nat.Coprime a.modulus b.modulus
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instance {a b : WeakAxis} : Decidable (independentAxes a b) := by
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unfold independentAxes; infer_instance
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/-- Reconstruct the underlying class modulo m₁·m₂ from two independent
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weak-axis observations via CRT. -/
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def reconstructWeakAxes (a b : WeakAxis) (r1 r2 : Nat)
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(hc : independentAxes a b) : Nat :=
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(Nat.chineseRemainder hc r1 r2).val
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/-- Correctness modulo the first weak axis. -/
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theorem reconstructWeakAxes_mod_a (a b : WeakAxis) (r1 r2 : Nat)
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(hc : independentAxes a b) :
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reconstructWeakAxes a b r1 r2 hc % a.modulus = r1 % a.modulus := by
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simp [reconstructWeakAxes]
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exact (Nat.chineseRemainder hc r1 r2).property.left
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/-- Correctness modulo the second weak axis. -/
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theorem reconstructWeakAxes_mod_b (a b : WeakAxis) (r1 r2 : Nat)
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(hc : independentAxes a b) :
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reconstructWeakAxes a b r1 r2 hc % b.modulus = r2 % b.modulus := by
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simp [reconstructWeakAxes]
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exact (Nat.chineseRemainder hc r1 r2).property.right
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/-- Two independent weak-axis observations uniquely determine the underlying
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class modulo the product of their moduli. This is the RRC weak-axis
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analogue of depth_token_coprime_intersect in SieveLemmas.lean. -/
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theorem weakAxis_coprime_intersect
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(a b : WeakAxis) (cls : Nat) (hc : independentAxes a b) :
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let r1 := project a cls
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let r2 := project b cls
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reconstructWeakAxes a b r1 r2 hc % (a.modulus * b.modulus) =
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cls % (a.modulus * b.modulus) := by
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intro r1 r2
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have h1 : reconstructWeakAxes a b r1 r2 hc % a.modulus = cls % a.modulus := by
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rw [reconstructWeakAxes_mod_a a b r1 r2 hc]
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simp [project, r1]
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have h2 : reconstructWeakAxes a b r1 r2 hc % b.modulus = cls % b.modulus := by
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rw [reconstructWeakAxes_mod_b a b r1 r2 hc]
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simp [project, r2]
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exact (Nat.modEq_and_modEq_iff_modEq_mul hc).mp ⟨h1, h2⟩
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 The atproto connection (informal→formal bridge)
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-- ═══════════════════════════════════════════════════════════════════════════
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/- The atproto design says:
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identity (D) ≠ hosting projection (H) ≠ application projection (A)
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In RRC terms this is exactly a set of weak axes that are independent:
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no single axis determines the full classification; the full object is
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recovered only by reconciling independent partial observations.
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We encode this as a tiny concrete instance below.
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-/
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/-- A toy atproto-style observer set: identity/host/app are three independent
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weak axes with pairwise-coprime moduli 7, 11, 13. -/
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def identityAxis : WeakAxis := ⟨7, by decide⟩
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def hostingAxis : WeakAxis := ⟨11, by decide⟩
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def appAxis : WeakAxis := ⟨13, by decide⟩
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/-- Any underlying class, observed through the three axes. -/
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def toyClass : Nat := 61
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def idShadow : Nat := project identityAxis toyClass
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def hostShadow : Nat := project hostingAxis toyClass
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def appShadow : Nat := project appAxis toyClass
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#eval idShadow -- 61 % 7 = 5
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#eval hostShadow -- 61 % 11 = 6
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#eval appShadow -- 61 % 13 = 9
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-- Reconstruct class mod 7·11 = 77 from identity + hosting axes.
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def reconstructedTwo : Nat :=
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reconstructWeakAxes identityAxis hostingAxis idShadow hostShadow (by decide)
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#eval! reconstructedTwo -- 61
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-- Reconstruct class mod 7·11·13 = 1001 from all three axes.
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def reconstructedThree : Nat :=
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let r := reconstructWeakAxes identityAxis hostingAxis idShadow hostShadow (by decide)
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let combinedMod := identityAxis.modulus * hostingAxis.modulus
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let combinedAxis : WeakAxis := ⟨combinedMod, by decide⟩
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reconstructWeakAxes combinedAxis appAxis r appShadow (by decide)
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#eval! reconstructedThree -- 61
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 A bounded Sidon witness for a concrete interaction graph
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Two-node, two-type interaction graph.
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Type 0: edge 1→2 with weight 1
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Type 1: edge 2→1 with weight 1
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This is the simplest graph whose path words encode direction changes. -/
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def toyGraph : InteractionGraph (Fin 2) 2 where
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edgeTypes := {0, 1}
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gen t :=
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if t = 0 then
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!![(0 : Rat), 1; 0, 0] -- 1→2
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else
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!![0, 0; 1, 0] -- 2→1
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#eval wordProduct toyGraph [] -- identity
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#eval wordProduct toyGraph [0] -- 1→2
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#eval wordProduct toyGraph [0, 1] -- 1→2→1
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#eval wordProduct toyGraph [0, 1, 0] -- 1→2→1→2
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-- This is a meta-theorem stating the property; the actual witness for L=4
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-- can be checked by native_decide or enumeration in a future tactic.
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#check isSidonWitness toyGraph 4
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end Semantics.InteractionGraphSidon
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