= Certificate for [1/2]_q (first 12 coefficients) =

Input: the real number x = 1/2. The continued fraction is truncated to the depth that locks the first 12 coefficients (MGO Proposition 1.1).

(a) continued fraction and even-length MGO form
    regular continued fraction: [0, 2]
    even-length MGO form:       [0, 2]
    even-length form evaluates to the convergent 1/2

(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).
    innermost term a_2 = 2:  (q + 1)/q
    fold in a_1 = 0:  q/(q + 1)
    [1/2]_q (the convergent, whose Taylor expansion gives [x]_q) = q/(q + 1)
    Taylor coefficients: q - q^2 + q^3 - q^4 + q^5 - q^6 + q^7 - q^8 + q^9 - q^10 + q^11 + O(q^12)

(c) independent cross-checks
    [pass] the first 12 coefficients are unchanged when 16 are asked for
    [pass] [x+1]_q computed from its own continued fraction equals q*[x]_q + 1
    [pass] [1/2]_q at q=1 is 1/2, the ordinary value 1/2
    [pass] the Taylor expansion of the exact rational function and the truncated series agree on q^0..q^11: [0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]
    summary: verified: truncation stable to 12, shift law [x+1]=q[x]+1, q=1 matches 1/2, 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.
