FS003X

four_square_parity_even_norm_pair_selection

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

Every even four-square norm admits a witnessed choice of two equal-parity coordinate pairs.

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 n a b c d. n * 2 = a * a + b * b + c * c + d * d -> (((((((exists fsd_even_first_fsps_norm_selection_ab. a = 2 * fsd_even_first_fsps_norm_selection_ab) /\ (exists fsd_even_second_fsps_norm_selection_ab. b = 2 * fsd_even_second_fsps_norm_selection_ab)) \/ ((exists fsd_odd_first_fsps_norm_selection_ab. a = 2 * fsd_odd_first_fsps_norm_selection_ab + 1) /\ (exists fsd_odd_second_fsps_norm_selection_ab. b = 2 * fsd_odd_second_fsps_norm_selection_ab + 1)))) /\ ((((exists fsd_even_first_fsps_norm_selection_cd. c = 2 * fsd_even_first_fsps_norm_selection_cd) /\ (exists fsd_even_second_fsps_norm_selection_cd. d = 2 * fsd_even_second_fsps_norm_selection_cd)) \/ ((exists fsd_odd_first_fsps_norm_selection_cd. c = 2 * fsd_odd_first_fsps_norm_selection_cd + 1) /\ (exists fsd_odd_second_fsps_norm_selection_cd. d = 2 * fsd_odd_second_fsps_norm_selection_cd + 1))))) \/ (((((((exists fsd_even_first_fsps_norm_selection_crossed_ac. a = 2 * fsd_even_first_fsps_norm_selection_crossed_ac) /\ (exists fsd_even_second_fsps_norm_selection_crossed_ac. c = 2 * fsd_even_second_fsps_norm_selection_crossed_ac)) \/ ((exists fsd_odd_first_fsps_norm_selection_crossed_ac. a = 2 * fsd_odd_first_fsps_norm_selection_crossed_ac + 1) /\ (exists fsd_odd_second_fsps_norm_selection_crossed_ac. c = 2 * fsd_odd_second_fsps_norm_selection_crossed_ac + 1)))) /\ ((((exists fsd_even_first_fsps_norm_selection_crossed_bd. b = 2 * fsd_even_first_fsps_norm_selection_crossed_bd) /\ (exists fsd_even_second_fsps_norm_selection_crossed_bd. d = 2 * fsd_even_second_fsps_norm_selection_crossed_bd)) \/ ((exists fsd_odd_first_fsps_norm_selection_crossed_bd. b = 2 * fsd_odd_first_fsps_norm_selection_crossed_bd + 1) /\ (exists fsd_odd_second_fsps_norm_selection_crossed_bd. d = 2 * fsd_odd_second_fsps_norm_selection_crossed_bd + 1))))) \/ (((((exists fsd_even_first_fsps_norm_selection_crossed_ad. a = 2 * fsd_even_first_fsps_norm_selection_crossed_ad) /\ (exists fsd_even_second_fsps_norm_selection_crossed_ad. d = 2 * fsd_even_second_fsps_norm_selection_crossed_ad)) \/ ((exists fsd_odd_first_fsps_norm_selection_crossed_ad. a = 2 * fsd_odd_first_fsps_norm_selection_crossed_ad + 1) /\ (exists fsd_odd_second_fsps_norm_selection_crossed_ad. d = 2 * fsd_odd_second_fsps_norm_selection_crossed_ad + 1)))) /\ ((((exists fsd_even_first_fsps_norm_selection_crossed_bc. b = 2 * fsd_even_first_fsps_norm_selection_crossed_bc) /\ (exists fsd_even_second_fsps_norm_selection_crossed_bc. c = 2 * fsd_even_second_fsps_norm_selection_crossed_bc)) \/ ((exists fsd_odd_first_fsps_norm_selection_crossed_bc. b = 2 * fsd_odd_first_fsps_norm_selection_crossed_bc + 1) /\ (exists fsd_odd_second_fsps_norm_selection_crossed_bc. c = 2 * fsd_odd_second_fsps_norm_selection_crossed_bc + 1)))))))))

Constructive proof overview

Generated structural guide

Every even four-square norm admits a witnessed choice of two equal-parity coordinate pairs.

The unchanged tactic script uses 2 declared prerequisites and contains 10 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

FS003U four_square_parity_even_norm_coordinate_sum FS003W four_square_parity_even_coordinate_pair_selection

Direct dependents

FS0040 four_square_parity_even_multiplier_halving

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

10 script commands · 2 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.

Named ingredients (2)
four square parity even norm coordinate sumfour square parity even coordinate pair selection
01Fix variables and assumptionsL1–6

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

  1. L1
    intro n
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro d
  6. L6
    intro hnorm
02Use earlier factsL7–10

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

  1. L7
    apply four_square_parity_even_coordinate_pair_selection
  2. L8
    specialize four_square_parity_even_norm_coordinate_sum n
  3. L9
    apply four_square_parity_even_norm_coordinate_sum
  4. L10
    exact hnorm

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro n
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro d
  6. 0006intro hnorm
  7. 0007apply four_square_parity_even_coordinate_pair_selection
  8. 0008specialize four_square_parity_even_norm_coordinate_sum n
  9. 0009apply four_square_parity_even_norm_coordinate_sum
  10. 0010exact hnorm