Provedattentionlinear-informationlower-bounds

Bounded hidden blocks after any global rank-m summary

For all $\htmlData{mx=1}{B, m, r \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=20}{S : (\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{B}} \mathrel{\htmlData{mx=9}{\to}} \htmlData{mx=6}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=5}{\htmlData{mx=4}{m} + 1}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=7}{\mathbb{R}}) \mathrel{\htmlData{mx=19}{\to}} \htmlData{mx=18}{\htmlData{mx=10}{_{l}}\,\htmlData{mx=12}{\left[ \htmlData{mx=11}{\mathbb{R}} \right]}\,\htmlData{mx=15}{\htmlData{mx=13}{\operatorname{Fin}}\,\htmlData{mx=14}{r}} \mathrel{\htmlData{mx=17}{\to}} \htmlData{mx=16}{\mathbb{R}}}}$ , if $\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{finrank}}(\htmlData{mx=1}{\mathbb{R}}, \htmlData{mx=2}{\operatorname{range}})} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{m}$ , then

\begin{aligned} & \htmlData{mx=0}{\exists} z : \htmlData{mx=3}{\htmlData{mx=1}{\operatorname{Fin}}\,\htmlData{mx=2}{B}} \mathrel{\htmlData{mx=10}{\to}} \htmlData{mx=7}{\htmlData{mx=4}{\operatorname{Fin}}\,\htmlData{mx=6}{\htmlData{mx=5}{m} + 1}} \mathrel{\htmlData{mx=9}{\to}} \htmlData{mx=8}{\mathbb{R}}, \\ & \quad (\htmlData{mx=18}{\forall b, j,\ \htmlData{mx=16}{\left\lvert \htmlData{mx=15}{\htmlData{mx=12}{z}(\htmlData{mx=13}{b}, \htmlData{mx=14}{j})} \right\rvert} \mathrel{\htmlData{mx=17}{\leq}} 1}) \\ & \quad \htmlData{mx=11}{\land} (\htmlData{mx=26}{\forall b,\ \htmlData{mx=25}{\exists j,\ \htmlData{mx=23}{\left\lvert \htmlData{mx=22}{\htmlData{mx=19}{z}(\htmlData{mx=20}{b}, \htmlData{mx=21}{j})} \right\rvert} \mathrel{\htmlData{mx=24}{=}} 1}}) \\ & \quad \htmlData{mx=11}{\land} (\htmlData{mx=44}{\forall c : \htmlData{mx=41}{\htmlData{mx=39}{\operatorname{Fin}}\,\htmlData{mx=40}{B}} \mathrel{\htmlData{mx=43}{\to}} \htmlData{mx=42}{\mathbb{R}},\ \htmlData{mx=37}{\htmlData{mx=27}{S}(\htmlData{mx=36}{b, j \mapsto \htmlData{mx=35}{\htmlData{mx=30}{\htmlData{mx=28}{c}(\htmlData{mx=29}{b})} \cdot \htmlData{mx=34}{\htmlData{mx=31}{z}(\htmlData{mx=32}{b}, \htmlData{mx=33}{j})}}})} \mathrel{\htmlData{mx=38}{=}} 0}) \end{aligned}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables: B, m, r range over natural numbers: the statement must hold for every choice. ((B m r : ℕ))
finrank: Module.finrank: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Module.finrank)
≤At most: finrank(ℝ, range) is less than or equal to m. (Module.finrank ℝ S.range ≤ m)
Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application: Fin applied to B. Lean writes f x; papers write f(x). (Fin B)
⟹Implies: If Fin(m + 1), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin (m+1) → ℝ)
+Addition: m plus 1. (m+1)
range: S.range: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (S.range)
RReal numbers: The real numbers: every point on the number line. (ℝ)
[a, b]List: The list of ℝ, in order. ([ℝ])
∀For all: The statement after the comma holds for every b, j. (∀ b j, |z b j| ≤ 1)
|x|Absolute value: The size of z(b, j) without its sign. (|z b j|)
∧And: Both hold: ∀ b, ∃ j, |z(b, j)| = 1, and ∀ c, S(b, j ↦ c(b) · z(b, j)) = 0. ((∀ b, ∃ j, |z b j|=1) ∧ (∀ c : Fin B → …)
∃There exists: At least one j makes the statement after the comma true. (∃ j, |z b j|=1)
=Equality: |z(b, j)| and 1 are the same. (|z b j|=1)
x ↦ f(x)Function (λ-abstraction): The function sending b, j to c(b) · z(b, j). Lean writes fun x => …; papers write x ↦ …. (fun b j => c b*z b j)
·Multiplication: c(b) times z(b, j). (c b*z b j)

theorem attention-frontier-global-summary-hidden-blocks (B m r : ℕ)
    (S : (Fin B → Fin (m+1) → ℝ) →ₗ[ℝ] (Fin r → ℝ))
    (hrank : Module.finrank ℝ S.range ≤ m) :
    ∃ z : Fin B → Fin (m+1) → ℝ,
      (∀ b j, |z b j| ≤ 1) ∧ (∀ b, ∃ j, |z b j|=1) ∧
      (∀ c : Fin B → ℝ, S (fun b j => c b*z b j)=0)

The actual global summary rank constructs normalized hidden directions independently in every block. All independently rescaled directions have summary exactly zero. Kernel existence and boundedness are derived, not assumed. This formal component still needs the exponential interpolation bridge for the main attention frontier.

How it was proved

Proved: 8 October 2026
By: Ajax
Checked by: Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms: Classical.choice, Quot.sound, propext
Proof: 17 lines of Lean

Bounded hidden blocks after any global rank-m summary

How it was proved

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