# Conventions used by the website

The supplied 7 September 2026 manuscript is preserved byte for byte. These implementation notes identify the formulas used by the demonstration; they are not an author-issued corrected edition.

## Response interval in Remark 3.3

From the displayed matrix, h1=f2/2 and h2=2f1+f2/2. Therefore q1*q2=1+f2/(4*f1), with 2<f2/f1<4. The default response is f=(1,3). These formulas agree with Theorem 3.1(iv) and the m=2 case of Theorem 4.5. Remark 3.3 instead prints a different interval and objective; those lines are not used.

## Affinity normalization

We use Definition 2.2: A_w=sum_i w_i log(j_i_plus/j_i_minus). When w=N*g, A_w=N*A_g. Thus exp(A_w), without a further N, is the product of the rate ratios raised to w. Theorem 5.15's prose and some other passages include an extra N; the implementation uses the definition and product formula. The additional teaching fixture has N=2 and checks this distinction.

## Closed-class indicators

Proposition 5.11's proof says each closed-class indicator is a fixed vector of the routing matrix. That need not hold when transient states feed a closed class. The script checks P=[[0,1/2,1/2],[0,1,0],[0,0,1]]: (0,1,0) is not fixed, while (1/2,1,0) is. The stated kernel-dimension conclusion is consistent with the corresponding hitting-probability vectors. The site does not present this finite check as a proof of the universal theorem.

No Lean sources were supplied or compiled. The Python transcript covers exact example certificates, coefficient identities and bounded numerical checks.