Signatures $(1,0,\dots)$ and $(1,2,0,\dots)$ give $K=N$ and $K=N^2$

Proved

HardyFiveAxioms.signature_classical_quantum

by Lucas · Oct 2, 2026 · Mathlib 0df444a (Lean v4.33.1)

combinatoricsmathematical-physicsquantum-foundations

For every $N\in\mathbb N$ , with $K_x(N)=\sum_k\binom Nkx_k$ as in hardy2001_signature:

K_{(1)}(N)=N,\qquad K_{(1,2)}(N)=N+2\binom N2=N^2 .

These are the classical signature $x_{\rm Classical}=(1,0,0,\dots)$ and the quantum signature $x_{\rm Quantum}=(1,2,0,0,\dots)$ of Section 5.

Preamble

import Mathlib
import Definitions.Def_hardy2001_signature

Formal statement

namespace HardyFiveAxioms

/-- Hardy 2001, Section 5 (after Eq. (28)): the classical signature `(1, 0, 0, …)` gives
`K = N` and the quantum signature `(1, 2, 0, 0, …)` gives `K = N²`. -/
theorem signature_classical_quantum (N : ℕ) :
    dofOfSignature [1] N = N ∧ dofOfSignature [1, 2] N = N ^ 2 := by sorry

end HardyFiveAxioms

Source

L. Hardy, *Quantum Theory From Five Reasonable Axioms*, arXiv:quant-ph/0101012v4 (2001), https://arxiv.org/abs/quant-ph/0101012, p. 8, Section 5, Eq. (28) and the following sentence

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic)

NON-BLIND READ-BACK — NOT INDEPENDENT TESTIMONY. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statement below, with full knowledge of the source paper and of the intended meaning. It was not produced by a blind, independent auditor, so it may share the drafter's blind spots. Reviewers must not treat it as independent confirmation of faithfulness; compare the Lean code against the source directly.

For every natural number $N$ : $\binom N1\cdot1=N$ , and $\binom N1\cdot1+\binom N2\cdot2=N^2$ . Equivalently, with the signature function $K_x(N)=\sum_{k=1}^{L}\binom Nkx_k$ of hardy2001_signature, $K_{[1]}(N)=N$ and $K_{[1,2]}(N)=N^2$ , as natural numbers.

Human review

Endorsed by Shuze Chen · Oct 8, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 8, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Signatures (1,0,… )(1,0,\dots)(1,0,…) and (1,2,0,… )(1,2,0,\dots)(1,2,0,…) give K=NK=NK=N and K=N2K=N^2K=N2

Signatures $(1,0,\dots)$ and $(1,2,0,\dots)$ give $K=N$ and $K=N^2$