# C4: exact total variation and certificate conservatism

Consider P(q)=qI+(1-q)C4, where C4 moves to either neighbor with probability 1/2. Let d_q(t) be the maximum over point starts of total variation from uniform, and let tau_q(epsilon)=min{t>=0: d_q(t)<=epsilon}. Times are integers. Cycle symmetry makes all point-start distances equal. Markov contraction makes d_q(t) nonincreasing, so the threshold controls all later times.

## Exact distance

For 0<=q<=1, t>=0, put a=q^t and b=(2q-1)^t, with zeroth powers equal to one, including 0^0. The inherited orthogonal eigendecomposition gives the row

\[
(1/4+a/2+b/4,\quad1/4-b/4,\quad1/4-a/2+b/4,\quad1/4-b/4).
\]

Since |x+y|+|x-y|=2 max(|x|,|y|), its half-L1 distance is exactly

\[
d_q(t)=\frac{|2q-1|^t}{4}
       +\max\left(\frac{q^t}{2},\frac{|2q-1|^t}{4}\right).
\tag{C4-TV}
\]

This is a closed formula, not another upper bound. The row follows from the inherited four eigenvectors, or directly by checking the initial condition and multiplying once by P. Translation of the start only permutes its entries.

## Exact threshold in the nearly periodic regime

For 0<q<=1/4, set r=1-2q. Then q/r<=1/2. For every t>=1, q^t/r^t<=1/2, and (C4-TV) reduces to d_q(t)=r^t/2. Hence for 0<epsilon<1/2,

\[
\tau_q(\epsilon)=
\left\lceil\frac{\log(1/(2\epsilon))}{-\log(1-2q)}\right\rceil.
\tag{C4-time}
\]

The expression is at least one; t=0 has distance 3/4 and cannot already meet this tolerance. The earlier convenient certificate d_q(t)<=beta^t has beta=r here and gives

\[
T_q(\epsilon)=
\left\lceil\frac{\log(1/\epsilon)}{-\log(1-2q)}\right\rceil.
\]

The certificate remains valid. Its additive excess is

\[
T_q-\tau_q=\frac{\log2}{-\log(1-2q)}+O(1),
\]

where the ceiling error has magnitude less than one in the difference. For fixed epsilon in (0,1/2), this diverges as q decreases to zero. The ratio tends to log(1/epsilon)/log(1/(2epsilon)); at epsilon=1/4 it tends to 2. This is an exact finite-dimensional mechanism: the slowly decaying parity mode has TV amplitude 1/2, whereas the loose rate certificate discards that amplitude. It is not a failure of the absolute spectral gap.

At the inherited q=3/20, d(1)=7/20>1/4 and d(2)=49/200<=1/4, so the true mixing time is 2; the earlier 4-step certificate is still a valid sufficient result. At q=1/100 the exact integer times are 35 and 69, respectively. These calculations use rational iteration and do not rely on rounded logarithms.

At q=0, d(t)=1/2 for t>=1, so no finite threshold below 1/2 exists. At q=1, the chain is the identity and d(t)=3/4; this endpoint is excluded from irreducible mixing claims. Formula (C4-TV) itself covers both endpoints with the stated zeroth-power convention.

## Verification scope

The row identity follows by checking the initial condition and one transition. The bundled runner performs exact rational iteration for selected inputs. Such finite checks do not prove parameter-wide statements or establish novelty or independent scientific acceptance.