FS001O

four_square_descent_norm_bound_forces_smaller_multiplier

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

A centered quaternion norm k·r strictly below k² forces its quotient r to be strictly smaller than k.

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 k r z. k * r = z -> (exists gap. gap + S z = k * k) -> exists gap. gap + S r = k

Constructive proof overview

Generated structural guide

A centered quaternion norm k·r strictly below k² forces its quotient r to be strictly smaller than k.

The unchanged tactic script uses 3 declared prerequisites and contains 22 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

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

le_or_lt Stable theorem; checked-use authorized mul_le_mul_left Stable theorem; checked-use authorized lt_not_le Stable theorem; checked-use authorized

Direct dependents

FS000N four_square_odd_prime_bounded_modular_seed FS0024 four_square_descent_odd_centered_strict_step

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

22 script commands · 5 reading checkpoints · 1 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro k
  2. L2
    intro r
  3. L3
    intro z
  4. L4
    intro hnorm
  5. L5
    intro hbound
02Use earlier factsL6–7

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

  1. L6
    specialize le_or_lt k
  2. L7
    specialize le_or_lt r
03Separate the logical casesL8–9

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

  1. L8
    cases le_or_lt
  2. L9
    exfalso
04Establish hscaledL10–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul left.

  1. L10
    have hscaled : exists t. t + k * k = k * r
  2. L11
    specialize mul_le_mul_left k
  3. L12
    specialize mul_le_mul_left r
  4. L13
    specialize mul_le_mul_left k
  5. L14
    apply mul_le_mul_left
  6. L15
    exact le_or_lt_left
  7. L16
    rewrite hnorm at hscaled
  8. L17
    specialize lt_not_le z
  9. L18
    specialize lt_not_le (k * k)
  10. L19
    apply lt_not_le
05Use earlier factsL20–22

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

  1. L20
    exact hbound
  2. L21
    exact hscaled
  3. L22
    exact le_or_lt_right

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro k
  2. 0002intro r
  3. 0003intro z
  4. 0004intro hnorm
  5. 0005intro hbound
  6. 0006specialize le_or_lt k
  7. 0007specialize le_or_lt r
  8. 0008cases le_or_lt
  9. 0009exfalso
  10. 0010have hscaled : exists t. t + k * k = k * r
  11. 0011specialize mul_le_mul_left k
  12. 0012specialize mul_le_mul_left r
  13. 0013specialize mul_le_mul_left k
  14. 0014apply mul_le_mul_left
  15. 0015exact le_or_lt_left
  16. 0016rewrite hnorm at hscaled
  17. 0017specialize lt_not_le z
  18. 0018specialize lt_not_le (k * k)
  19. 0019apply lt_not_le
  20. 0020exact hbound
  21. 0021exact hscaled
  22. 0022exact le_or_lt_right