FS0032

four_square_lagrange_from_strict_descent

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

The all-natural Lagrange four-square theorem follows from exactly one remaining explicit hypothesis: uniform strict prime-multiple 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_prime_bridge fsd_multiplier_bridge. ((~(fsd_prime_bridge = 1) /\ forall frm_prime_left_fsd_bridge_prime frm_prime_right_fsd_bridge_prime. fsd_prime_bridge = frm_prime_left_fsd_bridge_prime * frm_prime_right_fsd_bridge_prime -> frm_prime_left_fsd_bridge_prime = 1 \/ frm_prime_right_fsd_bridge_prime = 1)) -> ~(fsd_multiplier_bridge = 0) -> ~(fsd_multiplier_bridge = 1) -> (exists fsl_a_fsd_bridge_source fsl_b_fsd_bridge_source fsl_c_fsd_bridge_source fsl_d_fsd_bridge_source. (fsd_prime_bridge * fsd_multiplier_bridge) = fsl_a_fsd_bridge_source * fsl_a_fsd_bridge_source + fsl_b_fsd_bridge_source * fsl_b_fsd_bridge_source + fsl_c_fsd_bridge_source * fsl_c_fsd_bridge_source + fsl_d_fsd_bridge_source * fsl_d_fsd_bridge_source) -> exists fsd_smaller_bridge. (~(fsd_smaller_bridge = 0) /\ ((exists fsd_gap_bridge. fsd_gap_bridge + S fsd_smaller_bridge = fsd_multiplier_bridge) /\ (exists fsl_a_fsd_bridge_target fsl_b_fsd_bridge_target fsl_c_fsd_bridge_target fsl_d_fsd_bridge_target. (fsd_prime_bridge * fsd_smaller_bridge) = fsl_a_fsd_bridge_target * fsl_a_fsd_bridge_target + fsl_b_fsd_bridge_target * fsl_b_fsd_bridge_target + fsl_c_fsd_bridge_target * fsl_c_fsd_bridge_target + fsl_d_fsd_bridge_target * fsl_d_fsd_bridge_target)))) -> forall n. (exists fsl_a_fslb_natural_result fsl_b_fslb_natural_result fsl_c_fslb_natural_result fsl_d_fslb_natural_result. (n) = fsl_a_fslb_natural_result * fsl_a_fslb_natural_result + fsl_b_fslb_natural_result * fsl_b_fslb_natural_result + fsl_c_fslb_natural_result * fsl_c_fslb_natural_result + fsl_d_fslb_natural_result * fsl_d_fslb_natural_result)

Constructive proof overview

Generated structural guide

The all-natural Lagrange four-square theorem follows from exactly one remaining explicit hypothesis: uniform strict prime-multiple descent.

The unchanged tactic script uses 2 declared prerequisites and contains 4 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

FS0031 four_square_prime_from_strict_descent FS003J four_square_lagrange_from_all_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

4 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 prime from strict descentfour square lagrange from all primes
01Fix variables and assumptionsL1–1

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

  1. L1
    intro hstep
02Use earlier factsL2–4

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

  1. L2
    apply four_square_lagrange_from_all_primes
  2. L3
    apply four_square_prime_from_strict_descent
  3. L4
    exact hstep

Library-wide reading audit

Original exact command ledger · 4 lines
  1. 0001intro hstep
  2. 0002apply four_square_lagrange_from_all_primes
  3. 0003apply four_square_prime_from_strict_descent
  4. 0004exact hstep