FS003F · theorem body

four_square_prime_modular_seed_multiple

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

Any witnessed solution of x² + y² + 1 = p·k gives an actual four-square representation of the prime multiple.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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

forall p x y k. x * x + y * y + 1 = p * k -> (exists fsl_a_seed_multiple fsl_b_seed_multiple fsl_c_seed_multiple fsl_d_seed_multiple. (p * k) = fsl_a_seed_multiple * fsl_a_seed_multiple + fsl_b_seed_multiple * fsl_b_seed_multiple + fsl_c_seed_multiple * fsl_c_seed_multiple + fsl_d_seed_multiple * fsl_d_seed_multiple)

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

none

In local proof propositions

none
Exact expanded first-order statement
forall p x y k. x * x + y * y + 1 = p * k -> (exists fsl_a_seed_multiple fsl_b_seed_multiple fsl_c_seed_multiple fsl_d_seed_multiple. (p * k) = fsl_a_seed_multiple * fsl_a_seed_multiple + fsl_b_seed_multiple * fsl_b_seed_multiple + fsl_c_seed_multiple * fsl_c_seed_multiple + fsl_d_seed_multiple * fsl_d_seed_multiple)

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

FS001I four_square_descent_prime_from_modular_seed_and_step FS0034 four_square_prime_from_bounded_strict_descent_and_seed

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

11 script commands · 3 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.

01Fix variables and assumptionsL1–5

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro p
  2. L2
    intro x
  3. L3
    intro y
  4. L4
    intro k
  5. L5
    intro hseed
02Construct an explicit witnessL6–9

Supply the displayed value, then prove that it has the required property.

  1. L6
    exists x
  2. L7
    exists y
  3. L8
    exists 1
  4. L9
    exists 0
03Calculate and transport equalitiesL10–11

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L10
    rewrite <- hseed
  2. L11
    simp

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro p
  2. 0002intro x
  3. 0003intro y
  4. 0004intro k
  5. 0005intro hseed
  6. 0006exists x
  7. 0007exists y
  8. 0008exists 1
  9. 0009exists 0
  10. 0010rewrite <- hseed
  11. 0011simp