FS004M

four_square_prime_modular_seed

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

Every prime, including the exceptional prime two, admits explicit constructive witnesses a²+b²+1=p·k without any supplementary hypothesis.

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.

Exact expanded first-order arithmetic statement

forall p. ((~(p = 1) /\ forall frm_prime_left_fsri_prime frm_prime_right_fsri_prime. p = frm_prime_left_fsri_prime * frm_prime_right_fsri_prime -> frm_prime_left_fsri_prime = 1 \/ frm_prime_right_fsri_prime = 1)) -> exists a b k. a * a + b * b + 1 = p * k

Constructive proof overview

Generated structural guide

Every prime, including the exceptional prime two, admits explicit constructive witnesses a²+b²+1=p·k without any supplementary hypothesis.

The unchanged tactic script uses 2 declared prerequisites and contains 16 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

eq_decidable Stable theorem; checked-use authorized FS004L four_square_non_two_prime_modular_seed

Direct dependents

FS0031 four_square_prime_from_strict_descent

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

Read the argument

Proof checkpoints

16 script commands · 7 reading checkpoints · 1 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.

Named ingredients (1)
four square non two prime modular seed
01Fix variables and assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro hprime
02Establish hcaseL3–6

Establish this local claim before using it. It is not an additional assumption.

  1. L3
    have hcase : p = 2 \/ ~(p = 2)
  2. L4
    specialize eq_decidable p
  3. L5
    specialize eq_decidable 2
  4. L6
    exact eq_decidable
03Separate the logical casesL7–7

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L7
    cases hcase
04Calculate and transport equalitiesL8–8

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

  1. L8
    rewrite hcase_left
05Construct an explicit witnessL9–11

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

  1. L9
    exists 1
  2. L10
    exists 0
  3. L11
    exists 1
06Calculate and transport equalitiesL12–12

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

  1. L12
    norm_num
07Use earlier factsL13–16

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

  1. L13
    specialize four_square_non_two_prime_modular_seed p
  2. L14
    apply four_square_non_two_prime_modular_seed
  3. L15
    exact hprime
  4. L16
    exact hcase_right

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro p
  2. 0002intro hprime
  3. 0003have hcase : p = 2 \/ ~(p = 2)
  4. 0004specialize eq_decidable p
  5. 0005specialize eq_decidable 2
  6. 0006exact eq_decidable
  7. 0007cases hcase
  8. 0008rewrite hcase_left
  9. 0009exists 1
  10. 0010exists 0
  11. 0011exists 1
  12. 0012norm_num
  13. 0013specialize four_square_non_two_prime_modular_seed p
  14. 0014apply four_square_non_two_prime_modular_seed
  15. 0015exact hprime
  16. 0016exact hcase_right