= Certificate for the q-integer [5]_q =

Input: the integer n = 5.

(a) continued fraction and even-length MGO form
    regular continued fraction: [5]
    even-length MGO form:       [5]
    even-length form evaluates to the convergent 5

(b) MGO formula folded step by step
    odd positions carry [a]_q with q^a above; even positions
    carry [a]_(q^-1) with q^-a above (MGO eqn 1.1).
    [5]_q is the Gauss q-integer 1 + q + ... + q^(n-1):  q**4 + q**3 + q**2 + q + 1
    [5]_q = q**4 + q**3 + q**2 + q + 1
    Taylor coefficients: 1 + q + q^2 + q^3 + q^4 + O(q^12)

(c) independent cross-checks
    [pass] [5]_q at q=1 is 5, the ordinary value 5
    [pass] the Taylor expansion of the exact rational function and the truncated series agree on q^0..q^11: [1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]
    summary: verified: q=1 matches 5, exact = truncated to 12

References
    S. Morier-Genoud and V. Ovsienko, q-deformed rationals and
    q-continued fractions, Forum Math. Sigma 8 (2020), e13.
    Per-function theorem and check mapping: docs/CORRECTNESS.md.

This is a human-auditable derivation, not a formal machine proof;
every line above is checkable by hand.
