FS000P

four_square_prime_bounded_modular_seed

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Every prime, including two, has actual witnesses a²+b²+1=p·k with the constructive strict bound k<p.

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_fsbs_prime frm_prime_right_fsbs_prime. p = frm_prime_left_fsbs_prime * frm_prime_right_fsbs_prime -> frm_prime_left_fsbs_prime = 1 \/ frm_prime_right_fsbs_prime = 1)) -> (exists fsbs_a_prime_bounded fsbs_b_prime_bounded fsbs_k_prime_bounded. ((fsbs_a_prime_bounded * fsbs_a_prime_bounded + fsbs_b_prime_bounded * fsbs_b_prime_bounded + 1 = p * fsbs_k_prime_bounded) /\ (exists fsbs_lt_gap_prime_bounded_multiplier. fsbs_lt_gap_prime_bounded_multiplier + S (fsbs_k_prime_bounded) = (p))))

Constructive proof overview

Generated structural guide

Every prime, including two, has actual witnesses a²+b²+1=p·k with the constructive strict bound k<p.

The unchanged tactic script uses 2 declared prerequisites and contains 20 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 FS000O four_square_non_two_prime_bounded_modular_seed

Direct dependents

FS0035 four_square_prime_from_bounded_strict_descent

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

20 script commands · 10 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 bounded 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 equalitiesL8–9

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

  1. L8
    rewrite hcase_left
  2. L9
    rewrite hcase_left
05Construct an explicit witnessL10–12

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

  1. L10
    exists 1
  2. L11
    exists 0
  3. L12
    exists 1
06Separate the logical casesL13–13

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

  1. L13
    split
07Calculate and transport equalitiesL14–14

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

  1. L14
    norm_num
08Construct an explicit witnessL15–15

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

  1. L15
    exists 0
09Calculate and transport equalitiesL16–16

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

  1. L16
    norm_num
10Use earlier factsL17–20

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

  1. L17
    specialize four_square_non_two_prime_bounded_modular_seed p
  2. L18
    apply four_square_non_two_prime_bounded_modular_seed
  3. L19
    exact hprime
  4. L20
    exact hcase_right

Library-wide reading audit

Original exact command ledger · 20 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. 0009rewrite hcase_left
  10. 0010exists 1
  11. 0011exists 0
  12. 0012exists 1
  13. 0013split
  14. 0014norm_num
  15. 0015exists 0
  16. 0016norm_num
  17. 0017specialize four_square_non_two_prime_bounded_modular_seed p
  18. 0018apply four_square_non_two_prime_bounded_modular_seed
  19. 0019exact hprime
  20. 0020exact hcase_right