PF000F

fermat_four_no_square_from_descent

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

The stronger no-fourth-powers-sum-to-a-square theorem follows constructively from precisely one explicit, still-unproved strict descent premise.

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 pff_first_bounded pff_second_bounded pff_hypotenuse_bounded. ((~((pff_first_bounded) = 0) /\ (~((pff_second_bounded) = 0) /\ (~((pff_hypotenuse_bounded) = 0) /\ ((pff_first_bounded) * (pff_first_bounded) * (pff_first_bounded) * (pff_first_bounded) + (pff_second_bounded) * (pff_second_bounded) * (pff_second_bounded) * (pff_second_bounded) = (pff_hypotenuse_bounded) * (pff_hypotenuse_bounded)))))) -> exists pff_smaller_first_bounded pff_smaller_second_bounded pff_smaller_hypotenuse_bounded. (((~((pff_smaller_first_bounded) = 0) /\ (~((pff_smaller_second_bounded) = 0) /\ (~((pff_smaller_hypotenuse_bounded) = 0) /\ ((pff_smaller_first_bounded) * (pff_smaller_first_bounded) * (pff_smaller_first_bounded) * (pff_smaller_first_bounded) + (pff_smaller_second_bounded) * (pff_smaller_second_bounded) * (pff_smaller_second_bounded) * (pff_smaller_second_bounded) = (pff_smaller_hypotenuse_bounded) * (pff_smaller_hypotenuse_bounded)))))) /\ (exists pff_gap_bounded. pff_gap_bounded + S pff_smaller_hypotenuse_bounded = pff_hypotenuse_bounded))) -> forall a b h. ~((~((a) = 0) /\ (~((b) = 0) /\ (~((h) = 0) /\ ((a) * (a) * (a) * (a) + (b) * (b) * (b) * (b) = (h) * (h))))))

Constructive proof overview

Generated structural guide

The stronger no-fourth-powers-sum-to-a-square theorem follows constructively from precisely one explicit, still-unproved strict descent premise.

The unchanged tactic script uses 2 declared prerequisites and contains 13 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

PF000E fermat_four_bounded_descent le_refl Stable theorem; checked-use authorized

Direct dependents

PF000G fermat_four_no_fourth_from_descent PF002N fermat_four_no_square

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

13 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)
fermat four bounded descent
01Fix variables and assumptionsL1–5

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

  1. L1
    intro hstep
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro h
  5. L5
    intro hcounter
02Use earlier factsL6–13

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

  1. L6
    specialize fermat_four_bounded_descent h
  2. L7
    specialize fermat_four_bounded_descent h
  3. L8
    specialize fermat_four_bounded_descent a
  4. L9
    specialize fermat_four_bounded_descent b
  5. L10
    apply fermat_four_bounded_descent
  6. L11
    apply le_refl
  7. L12
    exact hstep
  8. L13
    exact hcounter

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro hstep
  2. 0002intro a
  3. 0003intro b
  4. 0004intro h
  5. 0005intro hcounter
  6. 0006specialize fermat_four_bounded_descent h
  7. 0007specialize fermat_four_bounded_descent h
  8. 0008specialize fermat_four_bounded_descent a
  9. 0009specialize fermat_four_bounded_descent b
  10. 0010apply fermat_four_bounded_descent
  11. 0011apply le_refl
  12. 0012exact hstep
  13. 0013exact hcounter