sin(1)'s even-count partial sums are a lower bracket

Identifier: F:creal-sinone-alternating-lower
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every m, sumRange sinTerm (m+m) <= sinOne -- mirrors CReal.cosOne_alternating_lower exactly, at sinTerm's own magnitude sequence (b j := expTerm(2j+1)), instantiating CReal.alternatingLowerBound (this batch) with CReal.sinOneConverges (this batch) as the limit hypothesis and CReal.expTerm_antitone (this batch) for the required decrease premise.

Formal statement

theorem CReal.sinOne_alternating_lower : ((x0 : AxNat) -> CReal.le (CReal.sumRange CReal.sinTerm (AxNat.add x0 x0)) CReal.sinOne)

Dependencies

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6 direct dependencies 1 direct dependents

Evidence

kernel-CReal.sinOne_alternating_lower

Kind: kernel-term
Status: checked

Supports: CReal.sinOne_alternating_lower 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 -- sinOne_alternating_lower 2>/dev/null | grep -cE '^CReal\.sinOne_alternating_lower[[:space:]]'

Evidence notes

build_creal_prelude admits CReal.sinOne_alternating_lower through the trusted Kernel::add_declaration gate. theorem_dependency_inventory exits non-zero for a named filter matching nothing; grep -c asserts the exact tab-anchored line. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.sinOne_alternating_lower

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the creal prelude's trusted surface is empty, which bounds CReal.sinOne_alternating_lower

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free creal

Evidence notes

Re-measured on this tree: creal: axiom=0 opaque=0 quotient=0 total_trusted=0, exits 0 printing 'ok: creal trusted surface = 0'. That bounds every declaration in the creal environment, including CReal.sinOne_alternating_lower, since a declaration cannot depend on a trusted declaration the environment does not contain. --require-axiom-free <name> errors for a prelude never built by this run rather than silently passing on zero rows. --release is MANDATORY here.

Provenance

{
  "date": "2026-08-27",
  "established_by": "axeyum-lean-kernel build_creal_prelude (crates/axeyum-lean-kernel/src/creal/trig.rs, declare_sin_one_alternating_lower)",
  "source": "canonical type read via kernel_declaration_projection's own UNFILTERED emit mode (cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection, no --require-declaration flag), which prints, per constructed prelude, one TSV row per declaration whose last field is kernel.render_lean(declaration.ty()) -- the same Kernel::render_lean canonical form nat_theorem_inventory prints, just not filtered to Declaration::Theorem. That output was piped to a scratchpad file and the exact row for this declaration's own prelude label was extracted and injected here programmatically (a Python script reading the TSV, never hand-transcribed); direct theorem dependencies were cross-read from the same run's direct_theorems column (field 6) and matched against the ledger's own registered kernel_theorem/formal.statement names to populate depends_on. No new probe binary was written for this batch; crates/ source was not touched to produce this batch."
}