D · Formal Derivations

D.1 · Conservation error of the nominal flagship

The flagship equation (17) is $\Theta_i(t) = \mathrm{FA}^{\mathrm{adj}}_i(t_0) + k_{\mathrm{ri}}\bigl(\mathrm{RI}_i(t) - \overline{\mathrm{RI}}(t)\bigr) + \mathrm{BR}_i(t)$ for $i \in \mathcal{I}$. Summing over the included set and using $\sum_{i \in \mathcal{I}} \bigl(\mathrm{RI}_i(t) - \overline{\mathrm{RI}}(t)\bigr) = 0$, we obtain the conservation law in exact form

The published nominal line (Section 6.7) is $\widetilde{\Theta}_i(t) = \Theta_i(t) + \widetilde\eta_i(t)$ (the display layer's dual-value process, Section 6.8). Because the display noise $\widetilde\eta_i(t)$ is a balanced, mean-zero process over $\mathcal{I}$ with display variance bounded by $\sigma_0^2$, the nominal flagship satisfies the approximate conservation law

where $o_{\mathbb P}(1)$ vanishes as the display average over the included set concentrates around zero. This is the precise sense of “$\Sigma\Theta \approx \Sigma\mathrm{FA}^{\mathrm{adj}}$” stated in the position note and defended in Section 5.6. Note that the adjusted anchor already contains the discrete $D$-adjustment; the continuous settlement re-anchoring of Section 10.3 restores $\sum \Theta = \sum \mathrm{FA}^{\mathrm{adj}}$ at each settlement instant (Algorithm 5).

D.2 · Renewal dynamics of the live signal

Fix a unit $i$ and, for the derivation, a single event $e$ with amplitude $A_e$, start time $t_e$, and half-life $T_{1/2}(e)$. The renewal envelope in (16) contributes $A_e\, 2^{-(t-t_e)/T_{1/2}(e)}$ for $t \ge t_e$. Write $\rho_e(t) = \ln 2 \cdot (t - t_e)/T_{1/2}(e)$; then the contribution satisfies the ordinary differential equation

i.e., pure exponential decay with rate constant $\lambda_e = \ln 2 / T_{1/2}(e)$ between renewal instants, and an upward jump of size $A_e$ at $t_e$. The renewal rule of Section 6.6 is exactly the statement that a repeated event with a residual contribution larger than a fresh pulse replaces the residual by the fresh pulse at the later start time:

Consequently, $\mathrm{RI}_i(t)$ is a right-continuous process with left limits (càdlàg), piecewise exponentially decaying, whose jumps are non-negative, occur at event starts, and are self-limiting through the renewal rule. This structure guarantees: (i) $\mathrm{RI}_i(t)$ is non-negative whenever all initial pulses are non-negative and the renewal rule is monotone; (ii) the heat envelope $\mathrm{H}_i(t)$ (same envelope with $|A_e|$) dominates the signed signal, $|\mathrm{RI}_i(t)| \le \mathrm{H}_i(t)$ for all $t$, which is Proposition 6.14; and (iii) the cross-sectional mean $\overline{\mathrm{RI}}(t)$ inherits the renewal properties, so the centered term in (17) is bounded in the sense of Proposition 6.12.

D.3 · Source-swap and circuit-breaker migration

The observation layer is decoupled from any single source (Section 6.2). Formally, suppose a factor $x_{i,k}$ was computed from source $k$ with weight $w_k$, and the circuit breaker switches to a redundant source $k'$ with weight $w_{k'}$ such that $w_k = w_{k'}$ (a like-for-like swap) or $w_{k'}$ is the defaulted weight of Table 3. The factor-level effect is bounded by the inter-source discrepancy:

Because the scale contract is monotone and bounded in $[0,1]$ (Section 6.3) and each factor enters the anchor linearly with weight $w_k$, the anchor-level effect of any single source swap is

where the aggregation gain is the product of the composition weights along the path from the factor to $\mathrm{FA}^{\mathrm{adj}}$ (Section 6.5). Since $\Delta_k \le 1$ and the weights sum to $\le 1$, the anchor is Lipschitz in the inter-source discrepancy: no single source can move any anchor by more than its own weight times the worst-case discrepancy, and the dual-track audit (Algorithm 7) detects a swap whenever $\Delta_k$ exceeds the alert threshold. This is the quantitative form of the “no single source is load-bearing” claim, and it is the basis of the reconciliation test (Algorithm 7).

D.4 · Weight-sensitivity and the Leamer frontier

Let $\mathrm{FA}_i(\beta)$ denote the anchor as a function of the V1 composition weights $\beta$ (all normalized to sum to one within each group). For a fixed included set, the ranking of two units $i, j$ is unchanged under weight perturbations $\beta \mapsto \beta + \delta$ as long as $\mathrm{FA}_i(\beta+\delta) > \mathrm{FA}_j(\beta+\delta)$. The set of weights that reverse the ordering is a half-space in weight space; its distance from $\beta$ in the relative metric measures how robust the ordering is. Following the Leamer tradition (Leamer, 1983), define the reversal radius of the pair $(i,j)$ as

A ranking is weight-robust if $r_{ij}$ exceeds the plausible perturbation scale for all adjacent pairs $(i,j)$. The sensitivity table in Section 8.7 reports the smallest reversal radius across the board; the Monte-Carlo and extreme-bounds analyses (Algorithm 11) estimate the distribution of $r_{ij}$ under the frozen V1 weights and their uncertainty. This gives a constructive, quantitative acceptance criterion: the board ranking is accepted as robust if the minimum adjacent reversal radius exceeds a stated threshold and the distribution of rank changes over the Monte-Carlo ensemble is contained.

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