FS001K

four_square_lagrange_from_modular_seeds_and_strict_descent

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

All-natural Lagrange follows by checked construction from exactly two explicit remaining premises: modular square seeds and strict centered-quaternion multiplier descent.

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 fsd_seed_prime_universal. ((~(fsd_seed_prime_universal = 1) /\ forall frm_prime_left_fsd_seed_universal frm_prime_right_fsd_seed_universal. fsd_seed_prime_universal = frm_prime_left_fsd_seed_universal * frm_prime_right_fsd_seed_universal -> frm_prime_left_fsd_seed_universal = 1 \/ frm_prime_right_fsd_seed_universal = 1)) -> (exists fsd_seed_residue_universal. fsd_seed_prime_universal = 4 * fsd_seed_residue_universal + 3) -> exists fsd_seed_first_universal fsd_seed_second_universal fsd_seed_multiplier_universal. fsd_seed_first_universal * fsd_seed_first_universal + fsd_seed_second_universal * fsd_seed_second_universal + 1 = fsd_seed_prime_universal * fsd_seed_multiplier_universal) -> (forall fsd_prime_universal fsd_multiplier_universal. ((~(fsd_prime_universal = 1) /\ forall frm_prime_left_fsd_universal_prime frm_prime_right_fsd_universal_prime. fsd_prime_universal = frm_prime_left_fsd_universal_prime * frm_prime_right_fsd_universal_prime -> frm_prime_left_fsd_universal_prime = 1 \/ frm_prime_right_fsd_universal_prime = 1)) -> ~(fsd_multiplier_universal = 0) -> ~(fsd_multiplier_universal = 1) -> (exists fsl_a_fsd_universal_source fsl_b_fsd_universal_source fsl_c_fsd_universal_source fsl_d_fsd_universal_source. (fsd_prime_universal * fsd_multiplier_universal) = fsl_a_fsd_universal_source * fsl_a_fsd_universal_source + fsl_b_fsd_universal_source * fsl_b_fsd_universal_source + fsl_c_fsd_universal_source * fsl_c_fsd_universal_source + fsl_d_fsd_universal_source * fsl_d_fsd_universal_source) -> exists fsd_smaller_universal. (~(fsd_smaller_universal = 0) /\ ((exists fsd_gap_universal. fsd_gap_universal + S fsd_smaller_universal = fsd_multiplier_universal) /\ (exists fsl_a_fsd_universal_target fsl_b_fsd_universal_target fsl_c_fsd_universal_target fsl_d_fsd_universal_target. (fsd_prime_universal * fsd_smaller_universal) = fsl_a_fsd_universal_target * fsl_a_fsd_universal_target + fsl_b_fsd_universal_target * fsl_b_fsd_universal_target + fsl_c_fsd_universal_target * fsl_c_fsd_universal_target + fsl_d_fsd_universal_target * fsl_d_fsd_universal_target)))) -> forall n. (exists fsl_a_fsd_lagrange_result fsl_b_fsd_lagrange_result fsl_c_fsd_lagrange_result fsl_d_fsd_lagrange_result. (n) = fsl_a_fsd_lagrange_result * fsl_a_fsd_lagrange_result + fsl_b_fsd_lagrange_result * fsl_b_fsd_lagrange_result + fsl_c_fsd_lagrange_result * fsl_c_fsd_lagrange_result + fsl_d_fsd_lagrange_result * fsl_d_fsd_lagrange_result)

Constructive proof overview

Generated structural guide

All-natural Lagrange follows by checked construction from exactly two explicit remaining premises: modular square seeds and strict centered-quaternion multiplier descent.

The unchanged tactic script uses 2 declared prerequisites and contains 6 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

FS001J four_square_descent_three_mod_four_primes_from_seed_and_step FS003K four_square_lagrange_from_three_mod_four_primes

Direct dependents

none

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

6 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 descent three mod four primes from seed and stepfour square lagrange from three mod four primes
01Fix variables and assumptionsL1–2

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

  1. L1
    intro hseeds
  2. L2
    intro hstep
02Use earlier factsL3–6

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

  1. L3
    apply four_square_lagrange_from_three_mod_four_primes
  2. L4
    apply four_square_descent_three_mod_four_primes_from_seed_and_step
  3. L5
    exact hseeds
  4. L6
    exact hstep

Library-wide reading audit

Original exact command ledger · 6 lines
  1. 0001intro hseeds
  2. 0002intro hstep
  3. 0003apply four_square_lagrange_from_three_mod_four_primes
  4. 0004apply four_square_descent_three_mod_four_primes_from_seed_and_step
  5. 0005exact hseeds
  6. 0006exact hstep