FS0031

four_square_prime_from_strict_descent

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

The checked unconditional modular seed for every prime removes the entire seed hypothesis; only uniform strict multiplier descent remains.

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 p. ((~(p = 1) /\ forall frm_prime_left_fslb_prime frm_prime_right_fslb_prime. p = frm_prime_left_fslb_prime * frm_prime_right_fslb_prime -> frm_prime_left_fslb_prime = 1 \/ frm_prime_right_fslb_prime = 1)) -> (exists fsl_a_fslb_prime_result fsl_b_fslb_prime_result fsl_c_fslb_prime_result fsl_d_fslb_prime_result. (p) = fsl_a_fslb_prime_result * fsl_a_fslb_prime_result + fsl_b_fslb_prime_result * fsl_b_fslb_prime_result + fsl_c_fslb_prime_result * fsl_c_fslb_prime_result + fsl_d_fslb_prime_result * fsl_d_fslb_prime_result)

Constructive proof overview

Generated structural guide

The checked unconditional modular seed for every prime removes the entire seed hypothesis; only uniform strict multiplier descent remains.

The unchanged tactic script uses 2 declared prerequisites and contains 20 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

FS004M four_square_prime_modular_seed FS001I four_square_descent_prime_from_modular_seed_and_step

Direct dependents

FS0032 four_square_lagrange_from_strict_descent

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

20 script commands · 4 reading checkpoints · 2 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 modular seedfour square descent prime from modular seed and step

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro hstep
  2. L2
    intro p
  3. L3
    intro hprime
02Establish hseedL4–7

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square prime modular seed.

  1. L4
    have hseed : exists a b k. a * a + b * b + 1 = p * k
  2. L5
    specialize four_square_prime_modular_seed p
  3. L6
    apply four_square_prime_modular_seed
  4. L7
    exact hprime
03Separate the logical casesL8–10

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

  1. L8
    cases hseed
  2. L9
    cases hseed_witness
  3. L10
    cases hseed_witness_witness
04Establish hdescentL11–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square descent prime from modular seed and step.

  1. L11
    have hdescent : ∀ q. ∀ a. ∀ b. ∀ k. Prime(q) → a · a + b · b + 1 = q · k → ∃ x. ∃ y. ∃ z. ∃ n. FourSquareNorm(q,x,y,z,n)Definitions: FourSquareNormPrime
  2. L12
    apply four_square_descent_prime_from_modular_seed_and_step
  3. L13
    exact hstep
  4. L14
    specialize hdescent p
  5. L15
    specialize hdescent x
  6. L16
    specialize hdescent x1
  7. L17
    specialize hdescent x2
  8. L18
    apply hdescent
  9. L19
    exact hprime
  10. L20
    exact hseed_witness_witness_witness

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro hstep
  2. 0002intro p
  3. 0003intro hprime
  4. 0004have hseed : exists a b k. a * a + b * b + 1 = p * k
  5. 0005specialize four_square_prime_modular_seed p
  6. 0006apply four_square_prime_modular_seed
  7. 0007exact hprime
  8. 0008cases hseed
  9. 0009cases hseed_witness
  10. 0010cases hseed_witness_witness
  11. 0011have hdescent : forall q a b k. ((~(q = 1) /\ forall frm_prime_left_fslb_local_prime frm_prime_right_fslb_local_prime. q = frm_prime_left_fslb_local_prime * frm_prime_right_fslb_local_prime -> frm_prime_left_fslb_local_prime = 1 \/ frm_prime_right_fslb_local_prime = 1)) -> a * a + b * b + 1 = q * k -> (exists fsl_a_fslb_local_result fsl_b_fslb_local_result fsl_c_fslb_local_result fsl_d_fslb_local_result. (q) = fsl_a_fslb_local_result * fsl_a_fslb_local_result + fsl_b_fslb_local_result * fsl_b_fslb_local_result + fsl_c_fslb_local_result * fsl_c_fslb_local_result + fsl_d_fslb_local_result * fsl_d_fslb_local_result)
  12. 0012apply four_square_descent_prime_from_modular_seed_and_step
  13. 0013exact hstep
  14. 0014specialize hdescent p
  15. 0015specialize hdescent x
  16. 0016specialize hdescent x1
  17. 0017specialize hdescent x2
  18. 0018apply hdescent
  19. 0019exact hprime
  20. 0020exact hseed_witness_witness_witness