FS000O

four_square_non_two_prime_bounded_modular_seed

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

Every prime other than two has a strictly prime-bounded constructive modular four-square seed.

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)) -> ~(p = 2) -> (exists fsbs_a_non_two_bounded fsbs_b_non_two_bounded fsbs_k_non_two_bounded. ((fsbs_a_non_two_bounded * fsbs_a_non_two_bounded + fsbs_b_non_two_bounded * fsbs_b_non_two_bounded + 1 = p * fsbs_k_non_two_bounded) /\ (exists fsbs_lt_gap_non_two_bounded_multiplier. fsbs_lt_gap_non_two_bounded_multiplier + S (fsbs_k_non_two_bounded) = (p))))

Constructive proof overview

Generated structural guide

Every prime other than two has a strictly prime-bounded constructive modular four-square seed.

The unchanged tactic script uses 2 declared prerequisites and contains 14 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

prime_ne_two_is_odd Stable theorem; checked-use authorized FS000N four_square_odd_prime_bounded_modular_seed

Direct dependents

FS000P four_square_prime_bounded_modular_seed

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

14 script commands · 4 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 odd prime bounded modular seed
01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro hprime
  3. L3
    intro hnot_two
02Establish hoddL4–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime ne two is odd.

  1. L4
    have hodd : exists h. p = 2 * h + 1
  2. L5
    specialize prime_ne_two_is_odd p
  3. L6
    apply prime_ne_two_is_odd
  4. L7
    exact hprime
  5. L8
    exact hnot_two
03Separate the logical casesL9–9

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

  1. L9
    cases hodd
04Use earlier factsL10–14

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

  1. L10
    specialize four_square_odd_prime_bounded_modular_seed p
  2. L11
    specialize four_square_odd_prime_bounded_modular_seed x
  3. L12
    apply four_square_odd_prime_bounded_modular_seed
  4. L13
    exact hodd_witness
  5. L14
    exact hprime

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro p
  2. 0002intro hprime
  3. 0003intro hnot_two
  4. 0004have hodd : exists h. p = 2 * h + 1
  5. 0005specialize prime_ne_two_is_odd p
  6. 0006apply prime_ne_two_is_odd
  7. 0007exact hprime
  8. 0008exact hnot_two
  9. 0009cases hodd
  10. 0010specialize four_square_odd_prime_bounded_modular_seed p
  11. 0011specialize four_square_odd_prime_bounded_modular_seed x
  12. 0012apply four_square_odd_prime_bounded_modular_seed
  13. 0013exact hodd_witness
  14. 0014exact hprime