FS001H

four_square_descent_prime_from_strict_step

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

Every nonzero represented prime multiple descends all the way to a representation of the prime under the precise strict-step hypothesis.

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_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 p k. ((~(p = 1) /\ forall frm_prime_left_fsd_p frm_prime_right_fsd_p. p = frm_prime_left_fsd_p * frm_prime_right_fsd_p -> frm_prime_left_fsd_p = 1 \/ frm_prime_right_fsd_p = 1)) -> ~(k = 0) -> (exists fsl_a_fsd_multiple fsl_b_fsd_multiple fsl_c_fsd_multiple fsl_d_fsd_multiple. (p * k) = fsl_a_fsd_multiple * fsl_a_fsd_multiple + fsl_b_fsd_multiple * fsl_b_fsd_multiple + fsl_c_fsd_multiple * fsl_c_fsd_multiple + fsl_d_fsd_multiple * fsl_d_fsd_multiple) -> (exists fsl_a_fsd_prime_result fsl_b_fsd_prime_result fsl_c_fsd_prime_result fsl_d_fsd_prime_result. (p) = fsl_a_fsd_prime_result * fsl_a_fsd_prime_result + fsl_b_fsd_prime_result * fsl_b_fsd_prime_result + fsl_c_fsd_prime_result * fsl_c_fsd_prime_result + fsl_d_fsd_prime_result * fsl_d_fsd_prime_result)

Constructive proof overview

Generated structural guide

Every nonzero represented prime multiple descends all the way to a representation of the prime under the precise strict-step hypothesis.

The unchanged tactic script uses 2 declared prerequisites and contains 16 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

le_refl Stable theorem; checked-use authorized FS001G four_square_descent_strict_multiplier_bounded

Direct dependents

FS001I four_square_descent_prime_from_modular_seed_and_step

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

16 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 (1)
four square descent strict multiplier bounded
01Fix variables and assumptionsL1–6

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

  1. L1
    intro hstep
  2. L2
    intro p
  3. L3
    intro k
  4. L4
    intro hprime
  5. L5
    intro hnonzero
  6. L6
    intro hrepresented
02Use earlier factsL7–16

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

  1. L7
    specialize four_square_descent_strict_multiplier_bounded k
  2. L8
    specialize four_square_descent_strict_multiplier_bounded p
  3. L9
    specialize four_square_descent_strict_multiplier_bounded k
  4. L10
    apply four_square_descent_strict_multiplier_bounded
  5. L11
    specialize le_refl k
  6. L12
    exact le_refl
  7. L13
    exact hprime
  8. L14
    exact hnonzero
  9. L15
    exact hrepresented
  10. L16
    exact hstep

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro hstep
  2. 0002intro p
  3. 0003intro k
  4. 0004intro hprime
  5. 0005intro hnonzero
  6. 0006intro hrepresented
  7. 0007specialize four_square_descent_strict_multiplier_bounded k
  8. 0008specialize four_square_descent_strict_multiplier_bounded p
  9. 0009specialize four_square_descent_strict_multiplier_bounded k
  10. 0010apply four_square_descent_strict_multiplier_bounded
  11. 0011specialize le_refl k
  12. 0012exact le_refl
  13. 0013exact hprime
  14. 0014exact hnonzero
  15. 0015exact hrepresented
  16. 0016exact hstep