6 · Formal Framework: The Mathematical Shell

This section gives the production formulas in symbolic form. Numerical values of all FA-related parameters are in Appendix B; RI parameters are symbolic throughout and are not disclosed. The product codes are labels bound to symbols in Table 11 (Appendix A); formulas are stated entirely in symbols.

6.1 · Indicator space and the layered observation state

Let $\mathcal{U}$ be the universe of AI Originators, $\mathcal{I}(t) \subseteq \mathcal{U}$ the included set at time $t$ (the reference set that defines normalization extremes and the centering mean), and $\mathcal{S}(t) \subseteq \mathcal{U} \setminus \mathcal{I}(t)$ the shadow set (evaluated with the same parameters, displayed but excluded from normalization extremes and from the centering mean). The system is a layered observation state. Each layer is a distinct kind of object with its own versioning, and the pipeline (Section 9.2) is a deterministic map between layers:

• L1 — raw observations. For each Originator $i$ and source $s$, a raw observation $x_{i,s}(t)$ with an observation time, a quality state, and a source version. L1 is the only layer that touches external reality directly; it is never retroactively edited, only corrected by new observations.
• L2 — derived indicators. A registry of derived series $y_k(t)$ computed from L1 by stated formulas under the unified scale contract (Section 6.3) and the missing-value taxonomy (Section 6.4). Examples: the seven capability inputs, the ecosystem sources, the commerce factors, the settled FA blocks, and the RI track decomposition (Table 4).
• L3 — published indexes. The index family (Section 6.7): a small set of public artifacts computed from L2 by the composition formulas. This is the layer the presentation layer (Section 6.8) projects onto public artifacts.

Definition 6.1 (Full indicator state).

For each Originator $i$, let $x_i = (x_{i,1},\dots,x_{i,K})$ be its vector of L1 source inputs — raw measurements from the third-party and self-collected sources of Section 13. The full indicator state at time $t$ is the tuple of all raw inputs, normalized values, block composites, settled anchors, live signals, and derived index values for all $i \in \mathcal{U}$.

The layering is not cosmetic. It is what makes the system recomputable in principle: the published L2 formulas allow the L3 indexes to be checked against the L1 observations, without access to proprietary code. The layers also localize corruption and error (Section 6.2): a source outage at L1 degrades only the L2 indicators that consume it, and the audit protocol (Section 9.3) verifies each layer's invariants separately.

6.2 · Source mapping and circuit breakers

Every L2 indicator consumes a declared set of L1 sources, with weights and a replacement policy, so that the system's dependence on any single source is an explicit, auditable graph rather than an implicit default. Two mechanisms make the dependence safe. Independent circuit breakers. Each source has an independent quality gate (freshness, completeness, availability, schema validity, and jump checks). When a gate trips, the affected source's contribution is degraded or suspended — a missing or degraded source never zeroes a factor and never silently fills with a suspected bad value; the most recent valid observation continues with an alert (Section 9.1). Because breakers are per-source, an outage is isolated to the smallest fault domain possible, and the rest of the measurement continues. Weighted multi-source redundancy. Where several sources measure the same quantity (e.g., the seven capability benchmarks), the factor is a weighted mean, so no single source is decisive. This is the same redundancy logic as the composite indicator literature recommends against indicator-level sensitivity to any single input (Nardo et al., 2005; Saisana et al., 2005). The combination of circuit breakers and redundancy gives a quantitative guarantee.

Proposition 6.1 (Circuit-breaker dilution bound).

Let a factor $\hat{v}$ be a weighted mean $\sum_s w_s\, y_s$ over sources $s$, with $\sum_s w_s = 1$ and $y_s \in [0,1]$. If source $s_0$ is degraded (its observation dropped and the factor re-normalized over the remaining present sources), then the change in $\hat{v}$ satisfies

where $\hat v_{\mathrm{out}}$ is the factor recomputed over the surviving sources.

Proof. When $s_0$ is dropped, the surviving weights are re-normalized to sum to one, so $\hat v_{\mathrm{out}} = \sum_{s\ne s_0} \frac{w_s}{1-w_{s_0}} y_s$. The difference is $\hat v - \hat v_{\mathrm{out}} = w_{s_0}\bigl(y_{s_0} - \sum_{s\ne s_0}\frac{w_s}{1-w_{s_0}}y_s\bigr)$, and the bracket lies in $[-1,1]$ because it is the difference of two values in $[0,1]$. The bound follows.

∎

The bound is the formal content of “a single-source outage dilutes rather than zeroes a factor”: with the largest capability-source weight ($0.40$, Appendix B), dropping it changes that factor by at most $0.40/0.60 \approx 0.67$ in the worst case, and in practice far less because the surviving sources remain informative. The mechanism is the institutional guarantee that the system is not in competition with its sources and that no single data provider holds a veto (Section 7.2).

6.3 · The unified scale contract and robust normalization

All factor values are normalized by one of four rules, with extremes defined only over the included set $\mathcal{I}$ (shadow units use the same extremes):

Every factor is normalized before any weighting (the “unified scale first” discipline). This is a deliberate departure from composite-indicator practice that normalizes at the end of aggregation: because every sub-factor is already in $[0,1]$ with the same semantics (fraction of the included-set maximum), weighted sums of sub-factors are comparisons of like quantities, and pathological behaviors such as negative composites or log-domain failures are impossible by construction. Robustness. Raw source values are often heavy-tailed and noisy. The design uses three robustness layers. First, the heavy-tail rule (4) is itself a compression that de-emphasizes extreme right tails — the appropriate treatment for skewed counts such as downloads and citations. Second, anomaly detection uses a robust scale (the median absolute deviation, scaled to estimate the standard deviation) rather than the raw standard deviation, following the recommendations of the robust-statistics literature (Rousseeuw and Croux, 1993; Huber and Ronchetti, 2009); a single contaminated observation inflates the MAD far less than it inflates the variance, so anomaly flags are not triggered by outliers themselves. Third, verified real jumps are never winsorized or smoothed: the robustness layers protect the scale from contamination, they do not suppress true dynamics (the data-continuity protocol of Section 9.1 makes this distinction explicit).

Proposition 6.2 (Robust scale is bounded under a single outlier).

Let $z$ be a sample of size $n$ and let $\mathrm{MAD}$ be its median absolute deviation from the median. Replacing one observation by an arbitrarily large value changes the robust scale $1.4826\,\mathrm{MAD}$ by a bounded amount, whereas the sample standard deviation can increase without bound.

Proof. The median of $z$ is the $\lceil n/2\rceil$-th order statistic; moving a single observation to $+\infty$ can shift the median by at most one position in the order statistics. The absolute deviations from the median therefore change by a bounded amount on a bounded number of points, so their median — and hence the MAD — is bounded. The standard deviation, by contrast, grows like the moved value divided by $\sqrt{n}$, which is unbounded.

∎

The proposition formalizes why the anomaly pipeline (Section 9.3) can flag contamination without being blinded by it: the scale that defines “anomalous” is robust to the very outlier being flagged.

Proposition 6.3 (Ratio invariance of linear factors).

For any factor under the linear normalization (3), a common proportional change $v_j \mapsto c\,v_j$ for all $j$, with $c>0$, leaves every normalized value and hence FA unchanged.

Proof. $\widehat{c\,v_j} = c v_j / \max_k (c v_k) = v_j / \max_k v_k$.

∎

Remark.

The heavy-tail rule (4) contains an additive $1$ inside the logarithm, so it is only approximately scale-invariant. Absolute progress is not suppressed by this design: it is reported through the absolute family (20)–(23), which is exactly the separation the paper argues for.

6.4 · Missing-value taxonomy and two-level renormalization

Missingness is the most common corruption vector in real measurement, and the system treats it with the formality it deserves, following the missing-data taxonomy of Rubin (Rubin, 1976; Little and Rubin, 2019). The design distinguishes four missingness states for the FA field, which refine the earlier trichotomy into a four-state taxonomy whose judgement is registered and audited:

• True zero ($P$-state). The object of measurement is participation itself, and the unit is present but has no observable activity of the measured kind (e.g., a present Originator with no official free API access, no open-source model, no edge-band coverage). Handling: the zero is counted with weight zero and always present; it is forbidden to re-normalize it away. A true zero is a fact about participation, and erasing it would reward absence.
• Numerical absence ($Z$-state). The source does not cover the unit, or the value is unknowable/unbounded (e.g., the source's editorial list does not include the unit; a data field is frozen; no routing exists). Handling: non-punitive renormalization over the present dimensions. This is the Rubin-compatible reading: if missingness is independent of the unobserved value given the observed, dropping the dimension and re-normalizing is unbiased.
• Leaderboard floor ($A$-state). The unit is below the printed cutoff of a ranked source; “it is not that it is absent, it is that it was not photographed onto the board.” Handling: value by the last readable rank, taken dynamically from the current board tail — never a hardcoded constant. This prevents the floor from becoming a privileged “absence discount.”
• Technical unavailability ($F$-state). The observation is missing for an infrastructural reason (source outage, schema failure, transmission error). Handling: the most recent valid value is carried forward with an alert and the true failure state recorded separately; the system never fabricates a new observation time and never fills a zero. This is the data-continuity guarantee of Section 9.1.

Two-level renormalization. The taxonomy acts at two levels. At the factor level, a unit with a missing factor is renormalized over its present factors within the block (a $Z$-state unit with no commerce coverage is measured on the other commerce factors). At the block level, a unit missing a whole block (e.g., no commerce data at all) is measured on the present blocks with the top-level weights renormalized. The two levels are applied in that order, and the result is a well-defined projection.

Proposition 6.4 (Two-level renormalization is a projection).

Let $w_1,\dots,w_K$ be positive block weights with $\sum w_k = 1$, and let a unit be present in a nonempty subset $P$ of blocks. The two-level renormalized composite $\sum_{k\in P} \frac{w_k}{\sum_{j\in P} w_j}\, \hat v_k$ is the orthogonal projection of the full composite onto the subspace of present blocks, in the weighted $\ell_2$ sense, and it lies in $[0,1]$ whenever the present block values do.

Proof. The weighted projection minimizes $\sum_k w_k (u_k - \hat v_k)^2$ over $u$ supported on $P$; the first-order conditions give $u_k = \hat v_k$ for $k \in P$ and the projection is exactly the re-normalized weighted mean. Convexity of the sum gives $[0,1]$ containment.

∎

The four-state taxonomy implements the design principle “a missing source never zeroes an Originator and never rewards absence.” The distinction between the four states is a judgment, and the judgment is registered (the classification of each factor for each unit is a versioned, auditable table, Section 10.2), so that a reader can in principle see why each missing value was handled the way it was. This is the operational form of the missing-data discipline of Rubin (Rubin, 1976): the mechanism of missingness is modeled explicitly rather than assumed away.

6.5 · The FA composition

Blocks. Four blocks are computed per Originator:

In (7), $\hat x_{i,k}$ are linear-normalized benchmark scores and $w_k$ the source weights. In (8), $a^{\ast}_{i,c}$ is the capability of the unit's best model in parameter band $c$ (bands defined by total parameters; missing band contributes $0$ and is always present in the mean — a true zero under Section 6.4, because a missing edge band is a fact about participation). In (9), the developer block is a heavy-tail mean over five ecosystem sources and the academic block is a mean of a heavy-tail citation ripple and a linear ledger of adjudicated artifacts: $\mathrm{acad}_i = \delta_{\mathrm{rip}}\,\mathrm{ripple}_i + \delta_{\mathrm{led}}\,\mathrm{ledger}_i$. In (10), $\mathrm{value}$ is capability per unit blended price (with the same-model pairing rule — the capability score and the price must come from the same flagship model, so that “value” is not a category error mixing two different models), $\mathrm{price}$ is inverse price on a heavy-tail scale, $\mathrm{free}$ is boolean (true-zero discipline), and $\mathrm{momentum}$ is linearly normalized token adoption with a leaderboard-floor treatment for non-listed units. Composite. The two vision groups (capability and ecosystem) are combined with group normalization, so that within-group block weights act within the group and top-level weights allocate between groups:

The group-normalized form means the top-level weights allocate between the two vision groups while the block weights allocate within them. With the V1 values (Appendix B) the closed form is $\mathrm{FA}_i = 100\,(0.55\,\mathrm{cap}_i + 0.05\,\mathrm{edge}_i + 0.25\,\mathrm{eco}_i + 0.15\,\mathrm{com}_i)$, and the reading of FA is “percentage of the best-weighted combination in the included set.”

Proposition 6.5 (FA monotonicity in its blocks).

For fixed normalization extremes and fixed membership, $\mathrm{FA}_i$ is non-decreasing in each block $\mathrm{cap}_i,\mathrm{edge}_i,\mathrm{eco}_i,\mathrm{com}_i$ and in each within-block factor, and strictly increasing in any block whose weight is positive and whose value increases while all others are held fixed.

Proof. Every block enters (13) through nonnegative weights, and each block is a nonnegative weighted mean of its factors, so the composite is a nonnegative linear combination of factor values. A nonnegative linear combination is non-decreasing in each argument, and strictly increasing when the coefficient is positive.

∎

Monotonicity is the minimal sanity property a composite must have: it guarantees that no unit is penalized for improving on any measured dimension, so the composition is a proper score. It is worth stating because composite-indicator weighting schemes can violate it (Nardo et al., 2005) (e.g., through ratios in the denominator whose monotonicity is not checked); the production formula is verified to satisfy it by construction. Adjustments. The settled anchor consumed by the index is

where $D_i \in [0,D_{\mathrm{cap}}]$ is the sustained-leadership adjustment and $0 \le R_i \le R_{\mathrm{cap}}\mathrm{FA}_i$ the substantive-incident deduction (Appendix B). $D$ is computed per board $d$ with threshold set $Y_d$ from the number of consecutive settled periods held within threshold $y$, $\mathrm{streak}_{d,y}$:

with tier weights $w_y$ and leaderboard weights $\beta_d$ over the five ranked leaderboards (Table 19). Falling out of all thresholds on all boards sets $D$ to zero immediately (no decay tail); re-entry accumulates from the current tier. The immediate reset is the double-counting discipline of Property 6.20: brand, mind-share, and lock-in inertia are already carried by the stock dimensions (capability and ecosystem slow variables), so a decay tail would count the same inertia twice. $R$ applies only to officially confirmed substantive incidents, carries sources, and exposes a correction channel; because the incident may already have depressed the RI signal, the $R$ settlement nets out the portion already counted through RI (one piece of evidence is never counted twice).

6.6 · The RI renewal envelope

The live signal is an additive set of decaying pulses:

where each event $e$ has full amplitude $A_e$, start time $t_e$, and a half-life $T_{1/2}(e)$ from its content-type tier. Events are classified into four tiers (rank-driven, factual, opinion, emotional); tier half-lives are not disclosed.

Axiom 6.6 (State pricing).

The instantaneous value $\mathrm{RI}_i(t)$ depends only on the current envelope state, not on the arrival path.

The Axiom is enforced by the renewal rule: if a new event carries a full value $\ge$ the residual of the existing pulse for the same primary key $(\mathrm{org},\ \mathrm{source},\ \mathrm{title})$, the old pulse is voided and the new full value starts a fresh pulse; otherwise the old pulse decays untouched. Deduplication by primary key prevents the same post from being counted twice across snapshots. Two consequences of state pricing are worth spelling out because they distinguish the design from a moving-average filter. First, the value of rank 1 is the same whether the unit climbed to it or jumped to it: the arrival path is priced out of the signal by construction. Second, the renewal log (the list of voided pulses and their full values) is a deterministic replay ledger: given the log, any past instantaneous value can be recomputed exactly (Section 9.3). Rank-driven amplitudes use an attention baseline: $A_e = \operatorname{clip}\!\bigl(h_{\mathrm{raw}}/h_{\mathrm{ref}},\, h_{\mathrm{cap}}\bigr) \, w_{\mathrm{src}}\, |p|$, with polarity $p \in [-1,1]$; when no heat value is available the amplitude falls back to $\propto 1/\sqrt{\mathrm{rank}}$. Factual posts use $A_e = B_{\mathrm{base}}\, w_{\mathrm{role}}\, w_{\mathrm{kind}}\, p$, with an attention baseline (being discussed is itself weakly positive; only negatively factual signals are negative). During closed-market windows the signal only decays: no injection, no renewal, and settlement applies injected events at the next opening. The design deliberately does not use an exponentially weighted moving average; the renewal envelope is the system's answer to the cold-start and path-dependence defects of fixed-window exponential smoothing (Section 5.4).

Lemma 6.7 (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 $t$.

Proof. $t-t_e \ge 0$, so the exponent is $\le 0$ and the value lies in $(0,1]$, with equality at $t=t_e$. Differentiating in $t$ gives $-\ln 2 \cdot T_{1/2}(e)^{-1} \cdot 2^{-(t-t_e)/T_{1/2}(e)} < 0$.

∎

Lemma 6.8 (Renewal jump magnitude).

Under the renewal rule, a single renewal jump of pulse $e$ equals $A_e - r_e(t)$, where $A_e$ is the new full value and $r_e(t)$ the residual of the voided pulse at the renewal time.

Proof. Before the renewal the instantaneous contribution is the residual $r_e(t)$; after the renewal it is $A_e$ (the new pulse starts at full value with $t_e=t$), and the renewal rule triggers exactly when $A_e \ge r_e(t)$, so the jump is nonnegative and equals $A_e - r_e(t)$.

∎

6.7 · The index family

The flagship composition (1) and the family surrounding it are:

In (20), $C_i$ is the field-normalized capability level

where $x_{i,k}$ is unit $i$'s raw capability datum on field $k$, $M_k$ the field's normalization constant, $\mathcal F_k$ the set of units active in field $k$, and $w_k$ the capability-source weights of (7); a unit is normalized over its active fields, so the definition is the two-level projection of Proposition 6.4 applied to the raw capability stock. Unlike the relative flagship, which uses included-set maxima at each settlement, the constants $M_k$ are fixed across settlements; this is what lets $\Gamma$ register simultaneous proportional progress by all units (the time-axis channel discussed below). The family is organized by semantics, and the semantics are what the code–symbol binding in Appendix A (Table 11) records: $\Theta$ is the relative flagship (code 8100), $\mathrm{FA}^{\mathrm{adj}}$ the transparent anchor (8101), $\mathrm{RI}$ the live signal (8102), $\mathrm{H}$ the absolute discussion heat (8111), $\Gamma$ the absolute capability composite (8200), $\Lambda$ the inflation-deflated absolute composite (8300), and $\mu_\Gamma, M_\Gamma$ the sector aggregates of $\Gamma$ (9200). The regional variants (codes CN1111, US1111, CN3333, US3333, CN5555, US5555) are country subsets: for each country, the Top-1 (1111), Top-3 (3333), and Top-5 (5555) composites over the country's Originators ordered by $\Theta$. They exist so that a national audience can read its own landscape without losing the cross-sectional discipline of the full set. The nominal line $\widetilde{\Theta}_i(t)$ adds a zero-mean display-noise process to the actual line $\Theta_i(t)$; the process is deterministic pseudo-random (hash-seeded), has constant relative amplitude, and is forced to zero at the close of each window, so that the closing nominal equals the closing actual (Proposition 6.17). The absolute-heat line $\mathrm{H}$ is pure actual: it carries no display noise, because its volatility is itself the signal. Two properties of the family deserve emphasis. First, the absolute family ($\Gamma$, $\mu_\Gamma$, $M_\Gamma$) is what carries time-axis information that the relative flagship deliberately does not: because FA uses a cross-sectional maximum normalization, simultaneous proportional progress by all members leaves the relative index unchanged, and the absolute family is where such progress is visible (Proposition 6.3). Second, the deflated composite $\Lambda$ multiplies the whole market by a common factor $C(t)$; it is the only family member that carries a slowly moving industry-level level, and because the factor is common, it preserves every cross-sectional ranking (Proposition 6.16). The two properties together implement the paper's central separation: relative measurement for fairness across units, absolute measurement for honesty about progress over time, never conflated.

6.8 · The presentation layer as a controlled projection

Definition 6.2 (Presentation-layer mapping).

A presentation-layer mapping is a function $\Pi$ from the full indicator state $\xi(t)$ (Definition 6.1) to a public artifact $p \in \mathcal{P}$, where $\mathcal{P}$ is a small set of artifacts: a single number, a single line, and a consistent standard.

The presentation layer is not a simplification of the science; it is a well-defined, deterministic function of the state, and its fidelity boundary is the communication anchor: the artifact must preserve the ranking and the relative order of magnitude of the underlying state, must not reorder it, and must declare any aggregation it performs. Every simplification in the presentation layer must pass a “does not violate scientific fairness” check: it may omit detail, but it must not distort the ordering that the full state certifies. The honest claim for the reader of the artifact is see the pattern, not master the technology. This distinction is analyzed further in Section 7.4. The presentation layer has a second, subtler job: it must present the same state honestly to two very different audiences at two different time scales. The settlement audience needs the exact, recomputable value (the actual line); the intraday audience needs a responsive, readable line without being misled into believing that micro-movements carry micro-signal. The dual-value process is the answer: the nominal line is the actual line plus a display-noise process that is (i) zero-mean, (ii) deterministic pseudo-random (hash-seeded, hence recomputable and auditable), (iii) bounded in amplitude, and (iv) forced to zero at each close, so the settlement value is always the actual value. The display process is a presentation-layer concern and is parameterized in Appendix B (Table 22); it is never allowed to contaminate the actual line, and the actual line is the only line that feeds settlement and audit. The presentation layer's fidelity claim is therefore two-part and both parts are checkable: the mapping is a deterministic function of the state (a reader can in principle verify that the artifact is the declared function of the declared inputs), and the mapping preserves ordering and magnitude (the “does not violate scientific fairness” check of Section 8.5).

6.9 · Invariants: propositions and proofs

The invariants below are the testable heart of the system. They are stated as propositions because each is a precise claim that can in principle be checked against public data, and they are proved because the system's credibility rests on the reader being able to re-derive them rather than take them on faith.

Proposition 6.9 (Exact conservation of the adjusted pool).

For every time $t$,

Proof. Sum (17) over the included set. The centered term vanishes because $\sum_{i\in\mathcal{I}}(\mathrm{RI}_i - \overline{\mathrm{RI}}) = \sum_i \mathrm{RI}_i - |\mathcal{I}| \cdot \frac{1}{|\mathcal{I}|}\sum_j \mathrm{RI}_j = 0$, by the definition of the mean. The bridge slot $\mathrm{BR}$ is zero outside a member-set transition (Section 6.10), and when active is engineered to sum to zero over the included set at every time (Proposition 6.19). The result follows.

∎

This is the mathematical mechanism behind “heat reorders positions but does not enlarge the total pool.” An arithmetic re-computation on a settlement tick confirmed the centered term vanishes to machine precision.

Corollary 6.10 (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.

Proof. Immediate from the proof of Proposition 6.9 and the definition of $\overline{\mathrm{RI}}$.

∎

Proposition 6.11 (Approximate conservation of the base pool).

Let $D_{\mathrm{cap}}$ and $R_{\mathrm{cap}}$ be the per-Originator caps on $D$ and $R$, both relative to FA (Appendix B). Then

In the special case where no Originator holds a leadership streak ($D \equiv 0$) and no substantive incident is recorded ($R \equiv 0$), the sums are equal.

Proof. From (14), $\mathrm{FA}^{\mathrm{adj}}_i = \mathrm{FA}_i(1 + D_i) - R_i$. Summing over the included set and using Proposition 6.9, $\sum \Theta = \sum \mathrm{FA} + \sum \mathrm{FA} D - \sum R$. Since $0 \le D_i \le D_{\mathrm{cap}}$ and $0 \le R_i \le R_{\mathrm{cap}}\mathrm{FA}_i$, the per-Originator deviation from base FA is bounded by $D_{\mathrm{cap}}\mathrm{FA}_i + R_{\mathrm{cap}}\mathrm{FA}_i$; dividing by $\sum \mathrm{FA}$ gives (26). When $D\equiv 0$ and $R\equiv 0$, the deviation vanishes.

∎

The “approximately” in the paper's conservation claim is this proposition, and it is stated this precisely because the published index carries the adjustments. A reader who checks the conservation against the documented FA methodology and the published snapshot (Appendix G) is checking exact conservation against $\mathrm{FA}^{\mathrm{adj}}$ and approximate conservation against $\mathrm{FA}$, with the bound given here.

Proposition 6.12 (Well-posedness and boundedness).

Under the unified scale contract (3)–(6) and the missing-value taxonomy (Section 6.4), every factor value and every composite value lies in $[0,1]$; consequently $\mathrm{FA}_i \in [0,100]$ for all $i$.

Proof. By induction over the composition tree. Linear and heavy-tail normalizations map to $[0,1]$ (the maximum is attained in $\mathcal{I}$; $\log_{10}$ is monotone with $\log 1 = 0$); the inverse rule preserves non-negativity before normalizing; boolean factors are in $\{0,1\}$; the missing-value protocol assigns either $0$ (true zero, weight-preserved), a re-normalized presence, or a leaderboard floor in $[0,1]$. Weighted arithmetic means of values in $[0,1]$ stay in $[0,1]$.

∎

Boundedness is not a technicality; it is what makes the scale interpretable (as “percentage of the best-weighted combination”) and what makes the audit inequalities of the live signal meaningful. A composite that could take arbitrary values would carry no such semantics.

Remark 6.2 (Pool boundedness and the balance reading).

The boundedness of Proposition 6.12 is per-Originator; it extends at once to the pool. Write $n = |\mathcal{I}|$ for the membership size, and $S_b=\sum_{i \in \mathcal{I}} b_i$ for each block $b\in\{\mathrm{cap},\mathrm{edge},\mathrm{eco},\mathrm{com}\}$. Under the scale contract (linear (3), heavy-tail (4), and boolean (6) forms, with the inverse and floor conventions used in the commerce block), every normalized factor value lies in $[0,1]$ and the value $1.0$ is attained by at least one included unit, so each factor's column sum lies in $[1,n]$. Each block is a weighted mean of its factors, so $S_b\in[1,n]$. Writing the closed form (13) with block weights $\beta_{\mathrm{cap}},\beta_{\mathrm{edge}},\beta_{\mathrm{eco}},\beta_{\mathrm{com}}$ — summing to one and fixed in Appendix B — as $\mathrm{FA}_i = 100 \sum_b \beta_b\, b_i$ and summing over $\mathcal{I}$ gives

The lower bound is attained when a single unit leads every factor; the upper bound when every included unit attains the leader's value on every factor. An equivalent form exposes the frontier. Let $\hat x_{i,f}\in[0,1]$ be the normalized value of factor $f$ for Originator $i$, and $\alpha_f$ its composite weight (block weight times within-block weight, so $\sum_f \alpha_f = 1$). Then

Since for every factor at least one included unit attains $1.0$, $G\in[0,n-1]$: the pool is a pooled distance from the composite frontier. For the current membership ($n=10$) the bounds read $[100,1000]$, i.e. $\sum_{i\in\mathcal{I}}\mathrm{FA}_i = 1000 - 100\,G$ with $G\in[0,9]$. Read as a share, the pool over $100n$ is the fraction of the composite frontier jointly held. A single-actor monopoly driving every factor to its own maximum pushes the pool toward $100$; a perfectly competitive field bunched at the frontier pushes it toward $100n$. The pool is thus an industry-balance indicator; the current instance is reported with the snapshot (Appendix G). By Proposition 6.9, the Eins Index pool equals the adjusted-anchor pool; when the $D$ adjustment vanishes ($D\equiv 0$, $R\equiv 0$) this equals the FA pool exactly, and Proposition 6.11 bounds the relative gap otherwise. The balance reading therefore transfers to the Index. The ceiling scales as $100\,|\mathcal{I}|$, so the membership-invariant reading is the share of the maximal pool, not the absolute ceiling.

Proposition 6.13 (State-pricing consequences).

Under Axiom 6.6 and the renewal rule, (a) a single-step renewal jump equals full value minus residual; (b) a fresh event enters at full value with zero response lag; (c) at any instant at most one active pulse exists per primary key $(\mathrm{org},\mathrm{source},\mathrm{title})$, and the renewal log is a deterministic replay ledger.

Proof. (a) The instantaneous value before the renewal is the residual $r$; after the renewal it is the full value $A$, by the rule that $A \ge r$ triggers replacement; the jump is $A - r$ (Lemma 6.8). (b) A fresh pulse has $t_e = t$ so the exponential factor is $1$ (Lemma 6.7). (c) The renewal rule modifies in place and never appends a second pulse for the same key; the log therefore replays deterministically.

∎

Proposition 6.14 (Audit inequality).

For every Originator $i$ and time $t$, over the same envelope set,

Proof. Both sums run over the same active set with the same kernels. By the triangle inequality, $\left|\sum_e A_e\,w_e\right| \le \sum_e |A_e|\,w_e$ with $w_e = 2^{-(t-t_e)/T_{1/2}(e)} > 0$, which is exactly $|\mathrm{RI}| \le \mathrm{H}$.

∎

This inequality is the testable invariant behind the dual-track audit: the absolute-heat line is always at least as large in magnitude as the signed live line, and any violation is a data-integrity alarm. Because $\mathrm{H}$ and $\mathrm{RI}$ are computed from the same envelope set by formulas that differ only in the absolute value, the audit is a strong check — it is not a bound that could be satisfied vacuously; it holds tick by tick and its failure is diagnostic (Section 9.3).

Proposition 6.15 (Monotonicity of sector aggregates, conditional).

Holding membership, scale, and method fixed, if every member's $\Gamma$ is non-decreasing over a window, then the sector mean $\mu_\Gamma$ and the sector maximum $M_\Gamma$ ((23)) are non-decreasing over that window.

Proof. The mean is a positive linear combination of the members' $\Gamma$; the maximum is an order-preserving map. Both are non-decreasing functions of non-decreasing arguments. The premise fails whenever membership changes, which is a real constituency change rather than a method artifact; the sector aggregates do not claim unconditional monotonicity (Section 6.10).

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Proposition 6.16 (Rank invariance under the deflator).

Multiplication by a common positive deflator $C(t)$ as in (21) preserves the cross-sectional ranking and all relative gaps of $\Theta$ at every time.

Proof. $C(t)$ factors out of every ratio and every comparison: for $i,j$, $\frac{\Lambda_i}{\Lambda_j} = \frac{\Theta_i}{\Theta_j}$ and $\Lambda_i > \Lambda_j \iff \Theta_i > \Theta_j$.

∎

Proposition 6.17 (Nominal display is drift-free and converges).

The display-noise process $x(t)$ has zero mean and is forced to zero at the close of each window; the nominal line therefore does not drift systematically from the actual line, and at the close nominal $\equiv$ actual.

Proof. $x(t)$ is a deterministic pseudo-random, hash-seeded discrete Ornstein–Uhlenbeck process (Uhlenbeck and Ornstein, 1930) with zero-mean increments, so $\mathbb{E}[x(t)] = 0$ by construction. The taper window multiplies both new innovations and existing stock by $(t_{\mathrm{close}} - t)/N_{\mathrm{taper}}$, driving $x \to 0$ as $t \to t_{\mathrm{close}}$; the deterministic seeding makes the entire nominal series recomputable.

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The OU process is used in its classical role as the standard model of a mean- reverting, bounded random walk; the innovation here is not the process but its discipline: deterministic seeding (recomputable), zero mean (drift-free), and forced close convergence (settlement integrity).

Proposition 6.18 (Centering preserves cross-sectional ordering of the anchor).

For two Originators $i,j$ with equal live signals $\mathrm{RI}_i(t) = \mathrm{RI}_j(t)$, the flagship ordering $\Theta_i(t) \ge \Theta_j(t)$ is determined entirely by the anchor ordering $\mathrm{FA}^{\mathrm{adj}}_i(t_0) \ge \mathrm{FA}^{\mathrm{adj}}_j(t_0)$; the live term cannot flip it.

Proof. If $\mathrm{RI}_i = \mathrm{RI}_j$ then the centered terms are equal (both equal $k_{\mathrm{ri}}(\mathrm{RI}_i - \overline{\mathrm{RI}})$), and with $\mathrm{BR}$ zero the difference $\Theta_i - \Theta_j = \mathrm{FA}^{\mathrm{adj}}_i - \mathrm{FA}^{\mathrm{adj}}_j$.

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The proposition states the boundary of the live signal's authority: discussion can reorder units with different live signals, but among units with the same live signal the anchor's ordering is preserved exactly. The live signal is a deviation mechanism, not a certification mechanism — the same boundary claim as Section 5.2, now at the level of a theorem.

6.10 · The BR transition

The bridge slot $\mathrm{BR}_i(t)$ in (1) exists for exactly one purpose: to carry an Originator across a compositional transition of the reference set without letting the transition's mechanical arithmetic masquerade as a change in the measured quantity. The first and currently only confirmed type is the member-set transition BR-01. Let $I^-$ be the unit's formally published value before a member-set change, $I^{+}$ its fair value under the new membership with the new FA, and $I^{\mathrm{mech}}$ the counterfactual value that would have been published if the new membership were in force while the old FA were frozen. Define the mechanical and fundamental changes as

The bridge is initialized to cancel exactly the mechanical part and reveal exactly the fundamental part:

Thereafter the bridge decays along a deterministic path to zero:

so the published path starts at $I^- + \Delta_{\mathrm{fund}}$ and converges to the new fair value $I^{+}$.

Proposition 6.19 (BR transition convergence).

Under (32) with $\varphi$ continuous and $\varphi(H)=0$, (a) the bridge is zero at time $H$ and thereafter; (b) the published path converges to the new fair value $I^{+}$; (c) the bridge never deposits into FA and never enters the RI feed; and (d) if $G_{\mathrm{mech}} = 0$ (no mechanical change), no bridge is generated.

Proof. (a) By the terminal condition $\varphi(H)=0$. (b) Since $\mathrm{BR}(t) \to 0$, $I^{\mathrm{pub}}(t) \to I^{+}$; the initial value is $I^- + \Delta_{\mathrm{fund}}$ which need not equal $I^{+}$, so the path is finite-duration interpolation, and the terminal value is exact. (c) The bridge is a display-path device; the FA anchor is defined on the new membership independently, and the RI feed only accepts event pulses, which the bridge never creates. (d) $\mathrm{BR}(0) = -G_{\mathrm{mech}} = 0$, and the zero bridge stays zero.

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The engineering meaning of the proposition is the separation it enforces: a purely mechanical revaluation caused by the roster changing (new member redefines a normalization extreme, shifting everyone's level) is temporarily bridged so that the published path does not jump for an arithmetic reason, while the genuine fundamental change ($\Delta_{\mathrm{fund}}$) is revealed immediately and the published path terminates at the new fair value. The bridge is bounded, terminates, and never contaminates the anchor or the live feed. When the counterfactual $I^{\mathrm{mech}}$ cannot be reliably reconstructed, no un-audited bridge is generated (Section 10.5); the sector aggregates $\mu_\Gamma, M_\Gamma$ deliberately do not use BR-01, because a membership change alters the statistical population they describe and that change is real (Proposition 6.15).

Property 6.20 (Zero-reset of sustained leadership).

Dropping out of all leadership thresholds on all boards sets $D = 0$ immediately, with no decay tail; re-entry accumulates from the current tier. This prevents double-counting: brand, mind-share, and lock-in inertia are already carried by the stock dimensions (capability and ecosystem slow variables), so a decay tail would count the same inertia twice.

6.11 · Discussion of design choices

The invariants above are deliberate design choices; we state which and why. Why a centered composition? The conservation law (Propositions 6.9–6.11) is what turns a set of relative scores into a scale with a stable center: attention can reorder positions but cannot, in aggregate, manufacture total influence. This is the property that the paper identifies as the measurement-scale contribution of the system. The cost is that total-pool information is not carried by the flagship line; it is carried instead by the absolute family, which is the right place for it (Section 6.7). Why two temporally separated components? FA is a stock with no time decay; RI is a flow with mandatory decay. The separation is what keeps the anchor scrutable (slow, source-rich, public) while the live signal stays responsive (but manipulable, hence opaque). The two components answer two different questions — “what is the cumulative footprint?” and “what is happening right now?” — and forcing them into one homogeneous score would destroy both answers. Why the missing-value taxonomy is four-state, not binary? A binary “present/absent” collapses three genuinely different situations — the unit participates with zero activity (true zero), the source does not cover the unit (numerical absence), and the unit fell below the printed cutoff (leaderboard floor) — plus the infrastructural case (technical unavailability). Treating them identically would either reward absence (erasing true zeros) or punish it (zeroing a covered-but-unlisted unit); the four-state taxonomy is the minimal classification that avoids both errors, and its cost — judgment calls — is managed by registering every classification (Section 6.4). Tension with Goodhart. Any index can be gamed (Sections 1.2 and 7.5). The design does not pretend to eliminate gaming; it confines it. The conservation law bounds the aggregate damage of gaming the live signal; source diversity and replaceability raise the cost of gaming the anchor; the volatility budget bounds the damage of a single manipulated burst. These are engineering responses to a formal threat model, analyzed in Section 7.5.

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