FP0005

prime_field_residue_congruence_transport

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Balanced congruence transports canonical residues; no quotient oracle is assumed.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

This checkpoint constructs prime-order fields (k=1), with genuine finite arithmetic tables, cardinality, and characteristic. Inversion is proved only for nonzero elements; the table's zero entry is a zero-to-zero convention. G091 for every prime power p^k, with an irreducible polynomial of degree k, remains open. No extension-field construction or G091 closure is claimed.

Exact theorem in conservative defined notation

∀ p. ∀ n. ∀ m. ∀ r. ModEq(p,n,m) → CanonicalModularResidue(p,m,r) → CanonicalModularResidue(p,n,r)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

CanonicalModularResidue(m,a,r) · 2ModEq(m,a,b) · 1

Actual proof prerequisites

mod_eq_trans · checked external prerequisite
Original expanded first-order statement 
forall p n m r. (exists pfa_offset_left_transport pfa_offset_right_transport. (n) + (p) * pfa_offset_left_transport = (m) + (p) * pfa_offset_right_transport) -> (((exists pfa_gap_transport_sourcebound. pfa_gap_transport_sourcebound + S (r) = (p)) /\ ((exists pfa_offset_left_transport_sourcecongruence pfa_offset_right_transport_sourcecongruence. (m) + (p) * pfa_offset_left_transport_sourcecongruence = (r) + (p) * pfa_offset_right_transport_sourcecongruence)))) -> (((exists pfa_gap_transport_resultbound. pfa_gap_transport_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_transport_resultcongruence pfa_offset_right_transport_resultcongruence. (n) + (p) * pfa_offset_left_transport_resultcongruence = (r) + (p) * pfa_offset_right_transport_resultcongruence))))

Complete tactic proof in conservative notation

All 16 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

16 script commands · 3 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro m
  4. L4
    intro r
  5. L5
    intro hmod
  6. L6
    intro hr
02Separate the logical casesL7–8

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

  1. L7
    cases hr
  2. L8
    split
03Use earlier factsL9–16

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

  1. L9
    exact hr_left
  2. L10
    specialize mod_eq_trans (p)
  3. L11
    specialize mod_eq_trans (n)
  4. L12
    specialize mod_eq_trans (m)
  5. L13
    specialize mod_eq_trans (r)
  6. L14
    apply mod_eq_trans
  7. L15
    exact hmod
  8. L16
    exact hr_right

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro m
  4. 0004intro r
  5. 0005intro hmod
  6. 0006intro hr
  7. 0007cases hr
  8. 0008split
  9. 0009exact hr_left
  10. 0010specialize mod_eq_trans (p)
  11. 0011specialize mod_eq_trans (n)
  12. 0012specialize mod_eq_trans (m)
  13. 0013specialize mod_eq_trans (r)
  14. 0014apply mod_eq_trans
  15. 0015exact hmod
  16. 0016exact hr_right