Euler / cis bridge

V1 — Define e^{iθ} so multiply matches exp-add.

Builds on: polar multiply Tool (r’s multiply, θ’s add) + cis shorthand from powers. Middle vortex — not Brand-new, not a series leap.

Tool gained: working Euler definition e^{iθ} := cos θ + i sin θ, so e^{iα} e^{iβ} = e^{i(α+β)} matches polar multiply.

1. Principle. Polar already adds angles on multiply. Naming the unit turn cis θ as e^{iθ} makes that add look like the exp addition law — a bridge name earned from polar, not from Taylor walls.
2. Model. e^{iθ} := cos θ + i sin θ. Then e^{iα} e^{iβ} = cis(α+β) = e^{i(α+β)} by angle-addition formulas.
3. Prediction / Answer. e^{iπ/2} = i and e^{iπ/2} e^{iπ/2} = e^{iπ} = −1 — two quarter turns make a half turn, same as ×i twice.

V2 — Euler identity as a prediction job.

Builds on: V1 definition + De Moivre / full-turn Tool from powers.

Tool gained: e^{iπ} = −1 (and e^{i2π}=1) as working identities for phase jobs.

1. Principle. Plug θ=π into the definition: cos π + i sin π = −1 + 0i. The famous line is a prediction from the bridge, not a mystic axiom.
2. Model. e^{iπ} = −1; e^{i2πk} = 1 for integer k.
3. Prediction / Answer. A signal marked “invert phase” is multiply-by-e^{iπ} (= ×(−1)). Transfer: same half-turn as polar ∠π.

V3 — Where the bridge stops (span honest).

Builds on: V1–V2. Branch: full exp series / complex DE proof of Euler stays for a later middle Vortex or deeper leftover — not faked here.

Tool gained: know the working bridge vs the full proof — use e^{iθ} in De Moivre/poly jobs; don’t claim series mastery on this rung.

1. Principle. Climb rules: missing tool → middle Vortex or honest Branch. This page supplied the middle; series proof is still a Branch door, not a fake “you already know exp.”
2. Model. Working kit: z = r e^{iθ}, zⁿ = rⁿ e^{inθ}. Full: e^z = Σ z^k/k! justifying the name — later.
3. Prediction / Answer. Gate check: rewrite (√2 e^{iπ/4})⁴ = 4 e^{iπ} = −4 with the bridge; say “series Soon/later” if asked for Taylor proof.