cosOne is at most 4 -- a loose, uniform bound, not a sharp estimate

Identifier: F:creal-cosone-le-four
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

CReal.le CReal.cosOne 4. TRUE and kernel-verified, but NOT sharp: cos(1) is approximately 0.5403, and this bound does not even pin the sign. The domination bound is reused unchanged from CReal.e (loose by about 180x at k=3), and composing it with the triangle inequality (abs_sumRange_le) discards the alternating series' sign cancellation entirely -- so the SAME uniform bound holds at every index n with no case split, unlike e_le_three's genuine mathematical kink at index 2. A tighter bound would need alternating-series pairing machinery this fact does not use.

Formal statement

theorem CReal.cosOne_le_four : CReal.le CReal.cosOne (CReal.mul (CReal.ofRat (Rat.normalize (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))) (AxNat.succ AxNat.zero) (AxNat.le.refl (AxNat.succ AxNat.zero)))) (CReal.ofRat (Rat.normalize (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))) (AxNat.succ AxNat.zero) (AxNat.le.refl (AxNat.succ AxNat.zero)))))

Dependencies

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32 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.cosOne_le_four

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.cosOne_le_four through the trusted Kernel::add_declaration gate, which re-checks the proof term against the stated type. theorem_dependency_inventory exits non-zero for a named filter matching nothing (a deleted theorem cannot read as a re-derived one), and grep -c (never -q) both consumes the pipe and asserts the exact tab-anchored line is present. Verified on this tree: the command prints exactly one matching line for CReal.cosOne_le_four. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.cosOne_le_four

Kind: exhaustive-enumeration
Status: checked

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

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.cosOne_le_four, 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_cos_one_le_four)",
  "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 dependency edges (where applicable) were cross-read from the same run's direct_theorems column and independently confirmed against theorem_dependency_inventory's own output on this tree. No new probe binary was written for this batch, since the in-tree tool already emits the canonical type; crates/ source was not touched to produce this batch."
}