Mathematical and LaTeX debugging ledger

• Script alphabet. The source loads mathrsfs for every || expression.

• Rate versus retarded kernel. The biological rate and the analytic retarded distribution are distinguished by context and the explicit macro $R_{\mathrm{ret}}$.

• Coordinate kernel versus metric operator. $L_{\mathrm{coord}}$ and $L_{\mathrm{gr}}$ have different types.

• Fresh-bath preparation. The collision argument includes a bath state with zero $X_{\mathrm{B}}$ mean; the unitary alone does not determine its reduced channel.

• Hamiltonian units. The generalized code Hamiltonian is $\hslash h$, so $-\mathrm{i}[h,S]$ has rate units.

• Exact gap versus enclosure. The formula is exact for rectangular families. For correlated calibration sets it is a certified lower bound.

• Code-only reduction trajectories. The two-eigenvalue collapse statement is restricted to the code. The leakage level adds another $Z$-eigenvalue if admitted.

• Phase noise and bath spectra. White-force calculations are separate from the finite thermal oscillator spectrum and use a specified Markov process.

• Completion sign. The uniqueness proof uses $(c+1){\mathcal{C}}_{3,\mathrm{D}}=0$ and a nonzero-family hypothesis, not an implication from admission to nonvanishing.

• Weighted density representative. Moving a divergence before ramp multiplication changes the action by ramp-derivative terms. The selected representative is retained.

• Both roots. The coordinate and scalar legs are transposed before introducing the collar root. The raw advanced root is not falsely assigned nonnegative-time support.

• Tree versus cycle. Balanced tree composition is proved by leaf elimination. It is not promoted to a cyclic two-cross-line theorem.

• Derivative-order claims. The source-specific order-two incidence is a record for the selected density, not a universal bound for arbitrary second-jet cubic densities.

• Inherited source components. Unavailable tensor values are not invented to make a displayed kernel look complete.

• Historical H2 and current I3/J2. I2 advances the internal amplitude stop. It does not define the full characteristic product.

• Presentation identities. The null identity $q{\delta }^{\prime }+\delta =0$ is included. A candidate must kill the entire reconstruction kernel, not one selected null presentation.

• Contact support before and after propagation. The off-contact obstruction is stated on its actual graph domain. A propagated contact can have larger support.

• Normal jets. Finite normal order is not a finite-dimensional tangential normalization space.

• Pushforward map. The printed simple wavefront bound is for a projection submersion with proper support.

• Carrier ideal. An on-shell name does not establish a two-sided quantum ideal. The exact inherited full graded carrier is not replaced by the abstract even-sector interface.