FS003O · theorem body

four_square_prime_representation

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

Every natural prime has an unconditionally constructed four-square representation, using the checked bounded seed, all sixteen signed quaternion cases, and terminating multiplier descent.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Statement with defined notation

∀ p. Prime(p) → ∃ x. ∃ y. ∃ z. ∃ n. p = x · x + y · y + z · z + n · n

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0004 Prime

In local proof propositions

none
Exact expanded first-order statement
forall p. ((~(p = 1) /\ forall frm_prime_left_fslf_prime frm_prime_right_fslf_prime. p = frm_prime_left_fslf_prime * frm_prime_right_fslf_prime -> frm_prime_left_fslf_prime = 1 \/ frm_prime_right_fslf_prime = 1)) -> (exists fsl_a_fslf_prime fsl_b_fslf_prime fsl_c_fslf_prime fsl_d_fslf_prime. (p) = fsl_a_fslf_prime * fsl_a_fslf_prime + fsl_b_fslf_prime * fsl_b_fslf_prime + fsl_c_fslf_prime * fsl_c_fslf_prime + fsl_d_fslf_prime * fsl_d_fslf_prime)

Proof neighborhood

Direct theorem prerequisites

FS003M four_square_prime_from_odd_signed_quaternion FS0058 four_square_signed_centered_representation

Direct theorem dependents

FS003P four_square_lagrange

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

2 script commands · 1 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (2)
four square prime from odd signed quaternionfour square signed centered representation
01Use earlier factsL1–2

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L1
    apply four_square_prime_from_odd_signed_quaternion
  2. L2
    exact four_square_signed_centered_representation

Library-wide reading audit

Original defined command ledger · 2 lines
  1. 0001apply four_square_prime_from_odd_signed_quaternion
  2. 0002exact four_square_signed_centered_representation