E · Proof Corpus

E.1 · Propositions

• Prop. 1 (Interior membership addition preserves existing normalizations). An admission that does not redefine a normalization extreme leaves every previously included unit's normalized values — and hence FA — unchanged; the index can grow without “restating” the past.
• Prop. 2 (Construct–indicator consistency of the influence definition). The four-channel definition of social influence (use $\times$ discussion $\times$ iteration $\times$ diffusion) maps injectively to the four composition blocks; the operationalization preserves the construct's meaning.
• Prop. 3 (Circuit-breaker dilution bound). When a degraded source is dropped and the factor is re-normalized over surviving sources, the factor changes by at most $w_{s_0}/(1-w_{s_0})$, the ratio of the dropped weight to the surviving weight.
• Prop. 4 (Robust scale is bounded under a single outlier). The robust scale $1.4826\,\mathrm{MAD}$ changes by a bounded amount under a single extreme outlier, whereas the sample standard deviation can increase without bound.
• Prop. 5 (Ratio invariance of linear factors). A common proportional change to all raw values leaves every linear-normalized factor — and hence FA — unchanged.
• Prop. 6 (Two-level renormalization is a projection). The two-level renormalization of missing values is the orthogonal projection of the full composite onto the subspace of present blocks in the weighted $\ell_2$ sense, and stays in $[0,1]$ when the present block values do.
• Prop. 7 (FA monotonicity in its blocks). FA is non-decreasing in every block and in every within-block factor, and strictly increasing in any block with positive weight.
• Prop. 8 (Exact conservation of the adjusted pool). At every time, $\sum_{i\in\mathcal{I}}\Theta_i(t) \equiv \sum_{i\in\mathcal{I}}\mathrm{FA}^{\mathrm{adj}}_i(t_0)$; the live term reallocates but never changes the pool total.
• Prop. 9 (Approximate conservation of the base pool). The relative gap between $\sum\Theta_i$ and $\sum\mathrm{FA}_i$ is bounded by $D_{\mathrm{cap}} + R_{\mathrm{cap}}$; in the current configuration with no streak and no incident, the sums are equal.
• Prop. 10 (Well-posedness and boundedness). Under the unified scale contract and the missing-value taxonomy, every factor and composite lies in $[0,1]$; hence $\mathrm{FA}_i \in [0,100]$ for all $i$.
• Prop. 11 (State-pricing consequences). Under the state-pricing axiom and the renewal rule: (a) a renewal jump equals full value minus residual; (b) a fresh event enters at full value with zero response lag; (c) at most one active pulse exists per primary key and the renewal log is a deterministic replay ledger.
• Prop. 12 (Audit inequality). For every Originator and time, $\mathrm{H}_i(t) \ge |\mathrm{RI}_i(t)|$ over the same envelope set — the heat line is an upper bound on the signed signal.
• Prop. 13 (Monotonicity of sector aggregates, conditional). Holding membership, scale, and method fixed, if every member's $\Gamma$ is non-decreasing then the sector mean and maximum are non-decreasing.
• Prop. 14 (Rank invariance under the deflator). Multiplication by a common positive deflator preserves cross-sectional ranking and relative gaps of $\Theta$ at every time.
• Prop. 15 (Nominal display is drift-free and converges). The display-noise process has zero mean and is forced to zero at the close of each window; the nominal line does not drift from the actual line and equals it at the close.
• Prop. 16 (Centering preserves cross-sectional ordering of the anchor). With equal live signals, flagship ordering is determined entirely by anchor ordering; the live term cannot flip it.
• Prop. 17 (BR transition convergence). Under (32): the bridge is zero at time $H$ and thereafter; the published path converges to the new fair value; the bridge never deposits into FA and never enters the RI feed; with no mechanical change no bridge is generated.

E.2 · Lemmas, corollary, and property

• Lemma A (Decay kernel positivity). For $t \ge t_e$ and $T_{1/2}(e)>0$, the kernel $2^{-(t-t_e)/T_{1/2}(e)}$ lies in $(0,1]$ and is strictly decreasing in $t$.
• Lemma B (Renewal jump magnitude). A single renewal jump of pulse $e$ equals $A_e - r_e(t)$, the new full value minus the residual of the voided pulse at the renewal time; the jump is therefore non-negative.
• Corollary A (Zero-sum character of the live term). The live term $k_{\mathrm{ri}}(\mathrm{RI}_i - \overline{\mathrm{RI}})$ has zero sum over the included set at every time; it reallocates within the pool and never changes the pool total.
• Property A (Zero-reset of sustained leadership). Dropping out of all leadership thresholds sets $D = 0$ immediately with no decay tail; re-entry accumulates from the current tier. No double-counting of inertia that the stock dimensions already carry.

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