= Certificate for [333/106]_q (first 12 coefficients) =

Input: the real number x = 333/106. 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: [3, 7, 15]
    even-length MGO form:       [3, 7, 14, 1]
    even-length form evaluates to the convergent 333/106

(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_4 = 1:  1
    fold in a_3 = 14:  q**14 + q**13 + q**12 + q**11 + q**10 + q**9 + q**8 + q**7 + q**6 + q**5 + q**4 + q**3 + q**2 + q + 1
    fold in a_2 = 7:  (q**21 + 2*q**20 + 3*q**19 + 4*q**18 + 5*q**17 + 6*q**16 + 7*q**15 + 7*q**14 + 7*q**13 + 7*q**12 + 7*q**11 + 7*q**10 + 7*q**9 + 7*q**8 + 7*q**7 + 6*q**6 + 5*q**5 + 4*q**4 + 3*q**3 + 2*q**2 + q + 1)/(q**21 + q**20 + q**19 + q**18 + q**17 + q**16 + q**15 + q**14 + q**13 + q**12 + q**11 + q**10 + q**9 + q**8 + q**7)
    fold in a_1 = 3:  (q**24 + 2*q**23 + 4*q**22 + 7*q**21 + 10*q**20 + 13*q**19 + 16*q**18 + 19*q**17 + 21*q**16 + 22*q**15 + 22*q**14 + 22*q**13 + 22*q**12 + 22*q**11 + 22*q**10 + 21*q**9 + 20*q**8 + 18*q**7 + 15*q**6 + 12*q**5 + 9*q**4 + 6*q**3 + 4*q**2 + 2*q + 1)/(q**21 + 2*q**20 + 3*q**19 + 4*q**18 + 5*q**17 + 6*q**16 + 7*q**15 + 7*q**14 + 7*q**13 + 7*q**12 + 7*q**11 + 7*q**10 + 7*q**9 + 7*q**8 + 7*q**7 + 6*q**6 + 5*q**5 + 4*q**4 + 3*q**3 + 2*q**2 + q + 1)
    [333/106]_q (the convergent, whose Taylor expansion gives [x]_q) = (q**24 + 2*q**23 + 4*q**22 + 7*q**21 + 10*q**20 + 13*q**19 + 16*q**18 + 19*q**17 + 21*q**16 + 22*q**15 + 22*q**14 + 22*q**13 + 22*q**12 + 22*q**11 + 22*q**10 + 21*q**9 + 20*q**8 + 18*q**7 + 15*q**6 + 12*q**5 + 9*q**4 + 6*q**3 + 4*q**2 + 2*q + 1)/(q**21 + 2*q**20 + 3*q**19 + 4*q**18 + 5*q**17 + 6*q**16 + 7*q**15 + 7*q**14 + 7*q**13 + 7*q**12 + 7*q**11 + 7*q**10 + 7*q**9 + 7*q**8 + 7*q**7 + 6*q**6 + 5*q**5 + 4*q**4 + 3*q**3 + 2*q**2 + q + 1)
    Taylor coefficients: 1 + q + q^2 + q^10 + 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] [333/106]_q at q=1 is 333/106, the ordinary value 333/106
    [pass] the Taylor expansion of the exact rational function and the truncated series agree on q^0..q^11: [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0]
    summary: verified: truncation stable to 12, shift law [x+1]=q[x]+1, q=1 matches 333/106, 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.
