Every unit mod n has a multiplicative order (Int.order_exists)

Identifier: F:int-multiplicative-order-exists
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every n : Nat with 0 < n and every a : Int coprime to n, there is a k with `Int.IsOrder (ofNat n) a k` -- that is, k is positive, a^k is congruent to 1 mod n, and no smaller positive exponent is. The order is picked out by BOUNDED SEARCH, not defined as a function: `Nat.lnp_bounded_search` at the predicate `Q j := a^(succ j) = 1 (mod n)` with bound `Nat.totient n`. Two things make that search legitimate and both are load-bearing. The search's decidability hypothesis -- the ONLY thing separating it from the excluded middle it would otherwise imply -- is discharged by `Int.eq_em`, because `Int.ModEq n x y` is a Definition unfolding to `Eq Int (emod x n) (emod y n)`, so deciding it is deciding an integer equation and nothing classical enters. The bound is Euler's theorem, `Int.euler_totient_theorem`, together with `Nat.totient_eq_zero` making the totient a successor once 0 < n. The predicate is stated at `succ j` rather than at j deliberately: at j = 0 the condition a^0 = 1 holds for EVERY a, so a search on the unshifted predicate would always answer 0. What this does NOT claim: no computable function is produced, no efficiency claim is made, and both hypotheses are load-bearing (without coprimality Euler's theorem does not apply and the search has no bound). Uniqueness is a separate result, `Int.order_unique`.

Formal statement

theorem Int.order_exists : ((x0 : AxNat) -> ((x1 : Int) -> ((x2 : AxNat.lt AxNat.zero x0) -> ((x3 : Int.Coprime x1 (Int.ofNat x0)) -> Exists.{1} AxNat (fun (x4 : AxNat) => Int.IsOrder (Int.ofNat x0) x1 x4)))))

Dependencies

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8 direct dependencies 0 direct dependents

Evidence

kernel-Int.order_exists

Kind: kernel-term
Status: checked

Supports: Int.order_exists is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- Int.order_exists 2>/dev/null | grep -cE '^Int\.order_exists[[:space:]]'

Evidence notes

Two independent failure modes, so the exit status depends on the finding rather than on the run completing: theorem_dependency_inventory exits non-zero when a NAMED filter matches nothing, and grep -c exits 1 printing 0 when the anchored line is absent. Anchored with [[:space:]], never \t -- in a scripted (GNU) grep \t is a literal t. grep -c rather than grep -q, which would SIGPIPE the producer under pipefail. --release is MANDATORY: this tool builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-Int.order_exists

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the integer prelude admits no Axiom, Opaque or Quotient declaration, which bounds Int.order_exists.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free integer

Evidence notes

--require-axiom-free exits non-zero when the named prelude's trusted surface (Axiom + Opaque + Quotient) is not empty, and errors rather than silently passing for a prelude the run never built. A declaration cannot depend on a trusted declaration the environment does not contain, so an empty integer surface bounds every declaration in it. The per-declaration figure is the footprint column of kernel_declaration_projection, measured 0 for this row.

concrete-Int.order_exists

Kind: instance-pin
Status: checked

Supports: The definitions this statement is about compute the intended values at concrete small arguments, and the discriminating negative cases FAIL: ord_8(3) = 2 (not phi(8) = 4), ord_7(3) = 6, and 3 is refutably NOT a primitive root mod 8.

Checker command

cargo test --release -p axeyum-lean-kernel --lib int_prelude::mult_order_tests -- --test-threads=4

Evidence notes

The trusted gate cannot tell you a Definition is wrong -- it type-checks either way -- so Int.IsOrder and Int.IsPrimitiveRoot are pinned by reduction at concrete small arguments. Each battery asserts BOTH what must hold and what must fail, so no run can be vacuous: three_mod_eight_is_killed_by_exactly_the_even_exponents requires hits at m in {0,2,4} and misses at {1,3,5} (the misses are the negative control for Int.pow_modeq_one_of_dvd); is_order_unfolds_to_the_intended_conjunction requires the definition to be def_eq to the intended conjunction and NOT def_eq to the variant with the positivity comparison transposed; three_has_multiplicative_order_two_mod_eight builds a witness for IsOrder 8 3 2 and a REFUTATION of IsOrder 8 3 4, which only the minimality conjunct separates. Confirm a NONZERO test count (6).

Provenance

{
  "date": "2026-09-04",
  "curation": "curated",
  "established_by": "axeyum-lean-kernel build_int_prelude (crates/axeyum-lean-kernel/src/int_prelude/mult_order.rs)",
  "source": "formal.statement is Kernel::render_lean of the declared type, taken verbatim from the unfiltered emit of `cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection`; axiom_footprint is that same row's footprint column, measured 0. The prose statement and the \"what this does NOT claim\" clause are hand-written by the `primitive-roots` lane and are NOT generator output. Design record: ADR-1598.",
  "prior_art": "Classical; the multiplicative order of a unit modulo n is Gauss, Disquisitiones Arithmeticae (1801), art. 45-56. The statement here is the constructive existence form over this kernel's own `Int.ModEq` relation and `Nat.totient`, not a transcription of any particular modern formalization."
}