FS0011

four_square_conjugate_coordinate_square_transport

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

Each exact conjugate absolute coordinate independently satisfies its constructive natural square/cross-term balance.

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 a b c d e f g h m0 m1 m2 m3. ((((a * e + b * f + c * g + d * h) = (0) + m0) \/ ((0) = (a * e + b * f + c * g + d * h) + m0)) /\ ((((a * f + c * h) = (b * e + d * g) + m1) \/ ((b * e + d * g) = (a * f + c * h) + m1)) /\ ((((a * g + d * f) = (c * e + b * h) + m2) \/ ((c * e + b * h) = (a * g + d * f) + m2)) /\ (((a * h + b * g) = (d * e + c * f) + m3) \/ ((d * e + c * f) = (a * h + b * g) + m3))))) -> ((((a * e + b * f + c * g + d * h) * (a * e + b * f + c * g + d * h) + (0) * (0)) = (((m0) * (m0)) + ((a * e + b * f + c * g + d * h) * (0) + (0) * (a * e + b * f + c * g + d * h)))) /\ ((((a * f + c * h) * (a * f + c * h) + (b * e + d * g) * (b * e + d * g)) = (((m1) * (m1)) + ((a * f + c * h) * (b * e + d * g) + (b * e + d * g) * (a * f + c * h)))) /\ ((((a * g + d * f) * (a * g + d * f) + (c * e + b * h) * (c * e + b * h)) = (((m2) * (m2)) + ((a * g + d * f) * (c * e + b * h) + (c * e + b * h) * (a * g + d * f)))) /\ (((a * h + b * g) * (a * h + b * g) + (d * e + c * f) * (d * e + c * f)) = (((m3) * (m3)) + ((a * h + b * g) * (d * e + c * f) + (d * e + c * f) * (a * h + b * g)))))))

Constructive proof overview

Generated structural guide

Each exact conjugate absolute coordinate independently satisfies its constructive natural square/cross-term balance.

The unchanged tactic script uses 1 declared prerequisite and contains 27 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

FS000B four_square_absolute_square_balance

Direct dependents

FS0012 four_square_signed_conjugate_quaternion

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

27 script commands · 8 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 absolute square balance
01Fix variables and assumptionsL1–10

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro e
  6. L6
    intro f
  7. L7
    intro g
  8. L8
    intro h
  9. L9
    intro m0
  10. L10
    intro m1
02Fix variables and assumptionsL11–13

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

  1. L11
    intro m2
  2. L12
    intro m3
  3. L13
    intro habsolute
03Separate the logical casesL14–17

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

  1. L14
    cases habsolute
  2. L15
    cases habsolute_right
  3. L16
    cases habsolute_right_right
  4. L17
    split
04Use earlier factsL18–19

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

  1. L18
    apply four_square_absolute_square_balance
  2. L19
    exact habsolute_left
05Separate the logical casesL20–20

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

  1. L20
    split
06Use earlier factsL21–22

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

  1. L21
    apply four_square_absolute_square_balance
  2. L22
    exact habsolute_right_left
07Separate the logical casesL23–23

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

  1. L23
    split
08Use earlier factsL24–27

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

  1. L24
    apply four_square_absolute_square_balance
  2. L25
    exact habsolute_right_right_left
  3. L26
    apply four_square_absolute_square_balance
  4. L27
    exact habsolute_right_right_right

Library-wide reading audit

Original exact command ledger · 27 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro g
  8. 0008intro h
  9. 0009intro m0
  10. 0010intro m1
  11. 0011intro m2
  12. 0012intro m3
  13. 0013intro habsolute
  14. 0014cases habsolute
  15. 0015cases habsolute_right
  16. 0016cases habsolute_right_right
  17. 0017split
  18. 0018apply four_square_absolute_square_balance
  19. 0019exact habsolute_left
  20. 0020split
  21. 0021apply four_square_absolute_square_balance
  22. 0022exact habsolute_right_left
  23. 0023split
  24. 0024apply four_square_absolute_square_balance
  25. 0025exact habsolute_right_right_left
  26. 0026apply four_square_absolute_square_balance
  27. 0027exact habsolute_right_right_right