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feat(lean): ProductSidon — injective product-space encoding from Sidon partition
Formalizes the core equation extracted from atproto's "no instances" principle:
α(x₁) + β(y₁) = α(x₂) + β(y₂) → x₁ = x₂ ∧ y₁ = y₂
Key additions (0 sorry):
- IsProductSidon: injectivity of joint encoding f(x,y) = α(x) + β(y)
- CrossDiffDisjoint: (Δ image α) ∩ (Δ image β) = {0}
- productSidon_injective: CrossDiffDisjoint → IsProductSidon
- isProductSidon_iff_crossDiffDisjoint: equivalence (given injective encodings)
- sidonPartition_implies_productSidon: Sidon set partition → ProductSidon pair
- atmosphere_sidon_principle: equal observables → same host/app decomposition
Extends sidon_diff_injective (E8Sidon §8) from intra-set to cross-set differences.
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01Xrttjg3619VRrMjxqUPUkL
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0-Core-Formalism/lean/Semantics/Semantics/ProductSidon.lean
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import Mathlib.Data.Int.Defs
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import Mathlib.Data.Finset.Basic
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import Semantics.E8Sidon
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/-! # Product Sidon Injectivity
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## The Equation
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Stripping away all social-network semantics from the atproto architecture,
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the central claim reduces to one equation:
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**If `α(x₁) + β(y₁) = α(x₂) + β(y₂)` then `x₁ = x₂` and `y₁ = y₂`.**
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This is exactly the injectivity condition on the joint encoding `f(x,y) = α(x) + β(y)`.
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## Sidon Connection
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The necessary and sufficient condition for this injectivity (given `α`, `β` injective)
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is that no nonzero difference from `image(α)` equals any nonzero difference from `image(β)`:
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**(Δ image(α)) ∩ (Δ image(β)) = {0}**
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This is the *cross-difference disjointness* condition. It is implied by
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`image(α) ∪ image(β)` being a Sidon set — a bridge to the E₈ Sidon framework
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in `Semantics.E8Sidon`.
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## Interpretation
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In atproto terms:
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- `α` encodes hosting providers (H)
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- `β` encodes app projections (A)
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- The encoding `f(H, A) = α(H) + β(A)` is injective iff cross-differences don't collide
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If the joint (H, A)-encoding is product-Sidon, then two observable states being
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equal forces the underlying host/app decompositions to be equal. This is the
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"no instances" property: data identity is primary, projections cannot masquerade
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as each other. The Sidon uniqueness condition *is* the decentralization invariant.
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## Relation to existing infrastructure
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- `sidon_diff_injective` (E8Sidon §8): differences within a *single* Sidon set are
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injective. `productSidon_injective` extends this to *cross*-differences between
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two sets.
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- `sidonPartition_implies_productSidon`: if a Sidon set is partitioned into two
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disjoint blocks used as encoding ranges, the cross-difference condition holds
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automatically.
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-/
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namespace Semantics.ProductSidon
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open Semantics.E8Sidon
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variable {X Y : Type*}
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/-! ## Core definitions -/
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/-- A pair of encodings `(α, β)` is *product-Sidon* if the joint encoding
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`f(x, y) = α(x) + β(y)` is injective as a function on `X × Y`. -/
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def IsProductSidon (α : X → ℤ) (β : Y → ℤ) : Prop :=
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Function.Injective (fun p : X × Y => α p.1 + β p.2)
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/-- Cross-difference disjointness: any α-difference that equals a β-difference
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must be zero. Equivalently, no nonzero α-difference equals any β-difference. -/
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def CrossDiffDisjoint (α : X → ℤ) (β : Y → ℤ) : Prop :=
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∀ x₁ x₂ : X, ∀ y₁ y₂ : Y,
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α x₁ - α x₂ = β y₂ - β y₁ → α x₁ = α x₂
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/-! ## The main theorem -/
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/-- **Product Sidon Injectivity**: given injective encodings `α`, `β` with
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cross-difference disjoint images, the joint encoding `f(x,y) = α(x) + β(y)`
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is injective.
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This is the key equation: `α(x₁) + β(y₁) = α(x₂) + β(y₂)` implies
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`x₁ = x₂` and `y₁ = y₂`. -/
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theorem productSidon_injective
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(α : X → ℤ) (β : Y → ℤ)
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(hα : Function.Injective α) (hβ : Function.Injective β)
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(hdisj : CrossDiffDisjoint α β) :
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IsProductSidon α β := by
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intro ⟨x₁, y₁⟩ ⟨x₂, y₂⟩ heq
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have heq' : α x₁ + β y₁ = α x₂ + β y₂ := heq
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have hdiff : α x₁ - α x₂ = β y₂ - β y₁ := by linarith
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have hαeq : α x₁ = α x₂ := hdisj x₁ x₂ y₁ y₂ hdiff
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have hβeq : β y₁ = β y₂ := by linarith
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exact Prod.ext (hα hαeq) (hβ hβeq)
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/-- The equivalence: `IsProductSidon` iff `CrossDiffDisjoint`
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(given injective encodings). -/
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theorem isProductSidon_iff_crossDiffDisjoint
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(α : X → ℤ) (β : Y → ℤ)
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(hα : Function.Injective α) (hβ : Function.Injective β) :
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IsProductSidon α β ↔ CrossDiffDisjoint α β := by
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constructor
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· intro hinj x₁ x₂ y₁ y₂ hdiff
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have heq : α x₁ + β y₁ = α x₂ + β y₂ := by linarith
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have hpair : (x₁, y₁) = (x₂, y₂) := hinj heq
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exact congr_arg α (Prod.ext_iff.mp hpair).1
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· exact productSidon_injective α β hα hβ
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/-! ## Bridge from Sidon sets -/
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/-- **Sidon Partition → Product Sidon**: if a Sidon set `S` is split into
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two disjoint parts `A` and `B`, and `α` maps into `A` while `β` maps into `B`,
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then `(α, β)` is product-Sidon.
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*Proof*: Sidon says `α(x₁) + β(y₁) = α(x₂) + β(y₂)` implies
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`{α(x₁), β(y₁)} = {α(x₂), β(y₂)}`. Since `A ∩ B = ∅`, we cannot have
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`α(x₁) = β(y₂)` (different parts), so `α(x₁) = α(x₂)` and `β(y₁) = β(y₂)`. -/
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theorem sidonPartition_implies_productSidon
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(S A B : Finset ℤ)
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(hS : IsSidonSet S)
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(hAB_union : A ∪ B = S)
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(hAB_disj : Disjoint A B)
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(α : X → ℤ) (β : Y → ℤ)
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(hα_range : ∀ x, α x ∈ A)
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(hβ_range : ∀ y, β y ∈ B)
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(hα_inj : Function.Injective α)
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(hβ_inj : Function.Injective β) :
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IsProductSidon α β := by
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intro ⟨x₁, y₁⟩ ⟨x₂, y₂⟩ heq
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have heq' : α x₁ + β y₁ = α x₂ + β y₂ := heq
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-- All four values lie in S
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have hαx₁S : α x₁ ∈ S := hAB_union ▸ Finset.mem_union_left B (hα_range x₁)
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have hβy₁S : β y₁ ∈ S := hAB_union ▸ Finset.mem_union_right A (hβ_range y₁)
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have hαx₂S : α x₂ ∈ S := hAB_union ▸ Finset.mem_union_left B (hα_range x₂)
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have hβy₂S : β y₂ ∈ S := hAB_union ▸ Finset.mem_union_right A (hβ_range y₂)
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-- Apply the Sidon property
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rcases hS (α x₁) hαx₁S (β y₁) hβy₁S (α x₂) hαx₂S (β y₂) hβy₂S heq' with
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(⟨h1, h2⟩ | ⟨h1, h2⟩)
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· -- Case 1: α x₁ = α x₂ ∧ β y₁ = β y₂ → done by injectivity
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exact Prod.ext (hα_inj h1) (hβ_inj h2)
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· -- Case 2: α x₁ = β y₂ → contradicts A ∩ B = ∅
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exfalso
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have hαx₁_in_B : α x₁ ∈ B := by rw [h1]; exact hβ_range y₂
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exact Finset.disjoint_left.mp hAB_disj (hα_range x₁) hαx₁_in_B
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/-! ## The Atmosphere Sidon Principle -/
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/-- **Atmosphere Sidon Principle**: if the (host, app) joint encoding is
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product-Sidon, then any two equal observables arose from the same
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host/app decomposition.
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`β h₁ + γ a₁ = β h₂ + γ a₂ → h₁ = h₂ ∧ a₁ = a₂`
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This is the formal content of "there are no instances" in atproto:
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data identity `D` is primary; if two projection paths
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`π_{A₁} ∘ σ_{H₁}(D)` and `π_{A₂} ∘ σ_{H₂}(D)` yield the same
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observable state, the paths themselves must be equal.
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The Sidon uniqueness condition *is* the decentralization invariant. -/
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theorem atmosphere_sidon_principle
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{Host App : Type*}
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(β : Host → ℤ) (γ : App → ℤ)
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(hProductSidon : IsProductSidon β γ)
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{h₁ h₂ : Host} {a₁ a₂ : App}
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(hobs : β h₁ + γ a₁ = β h₂ + γ a₂) :
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h₁ = h₂ ∧ a₁ = a₂ :=
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Prod.ext_iff.mp (hProductSidon hobs)
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/-- **Corollary**: equal observables under product-Sidon encoding imply
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the first coordinates (hosts) are equal. -/
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theorem atmosphere_host_eq
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{Host App : Type*}
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(β : Host → ℤ) (γ : App → ℤ)
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(hProductSidon : IsProductSidon β γ)
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{h₁ h₂ : Host} {a₁ a₂ : App}
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(hobs : β h₁ + γ a₁ = β h₂ + γ a₂) :
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h₁ = h₂ :=
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(atmosphere_sidon_principle β γ hProductSidon hobs).1
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/-- **Corollary**: equal observables under product-Sidon encoding imply
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the second coordinates (apps) are equal. -/
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theorem atmosphere_app_eq
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{Host App : Type*}
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(β : Host → ℤ) (γ : App → ℤ)
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(hProductSidon : IsProductSidon β γ)
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{h₁ h₂ : Host} {a₁ a₂ : App}
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(hobs : β h₁ + γ a₁ = β h₂ + γ a₂) :
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a₁ = a₂ :=
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(atmosphere_sidon_principle β γ hProductSidon hobs).2
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/-! ## Symmetric version -/
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/-- Product-Sidon is symmetric: if `(α, β)` is product-Sidon, then so is `(β, α)`.
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(Swapping host and app roles doesn't break injectivity.) -/
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theorem isProductSidon_symm (α : X → ℤ) (β : Y → ℤ)
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(h : IsProductSidon α β) : IsProductSidon β α := by
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intro ⟨y₁, x₁⟩ ⟨y₂, x₂⟩ heq
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have heq' : α x₁ + β y₁ = α x₂ + β y₂ := by linarith
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have hpair : (x₁, y₁) = (x₂, y₂) := h heq'
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exact Prod.ext (Prod.ext_iff.mp hpair).2 (Prod.ext_iff.mp hpair).1
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end Semantics.ProductSidon
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