5 · From Influence Definition to Composition

5.1 · From definition to decomposition

Section 1.2 defined influence as the composite use $\times$ discussion $\times$ iteration $\times$ diffusion. This section turns that definition into an operational decomposition: a slow Fundamental Anchor (FA) component that carries the cumulative, hard-to-manipulate part of influence, and a fast Real-time Influence (RI) component that carries the discussion-and-diffusion part. Table 3 is the load-bearing map: it assigns every block and every track to a channel of the definition. Each formula is an implementation of the map, not an arbitrary aggregation.

Table 3. Composition–vision map: every FA block and every RI track assigned to the channel of the influence definition it operationalizes. “Necessary-not- sufficient” marks the capability input.

The composition weights are not arbitrary: each component is anchored in the vision-derived definition above. The numerical values are V1 production values, calibrated by the expert group and engineering tuning and frozen at release (Appendix B); a unified calibration experiment is listed as future work (Section 12.2). Two things are deliberately separated: the structure of the composition (a claim about meaning, defended by Proposition 4.1) and the numerical weights (an empirical calibration, to be stress-tested in Section 8.7).

5.2 · Composition and its correspondence to the vision

The FA composite is a weighted sum of four blocks plus two adjustments. The semantic anchor of each weight is as follows:

• Capability (capability block). Capability is necessary but not sufficient for influence. Its block weight is deliberately bounded: a perfect capability score determines only part of the anchor — the ecosystem, edge, and commercial blocks together retain enough weight to reshape the ranking — and within the capability block a seven-source weighted mean keeps no single benchmark decisive. This is the structural evidence of “capability $\ne$ influence.”
• Edge coverage (edge block). Small, capable models run on edge devices and in local deployments, putting AI in front of people who would otherwise not use it. This block carries the accessibility / diffusion side of influence.
• Ecosystem stock (ecosystem block). Use and iteration leave measurable stock: open-source repositories, downloads, models, citations, and an adjudicated ledger of research artifacts. This is the use $\times$ iteration side, accumulated over time with no time decay (Section 5.3).
• Commercial accessibility (commerce block). Whether models are affordable and reachable as services shapes how widely they are used: value (capability per unit price), price, free access, and adoption momentum.
• Sustained leadership ($D$). A time-verified lead — holding frontier ranks across consecutive settled periods — is a slow adjustment to FA, capped at a fraction of FA (Appendix B). It measures sustained control of the frontier rather than a snapshot. Falling out of all thresholds sets $D$ to zero immediately, with no decay tail: the inertia that would otherwise be double-counted is already carried by the stock dimensions (Property 6.20).
• Real-time influence (RI). Discussion, attention, and diffusion are the fast channels of influence. The RI line is centered (Section 5.6): it reorders positions within a conserved total but cannot, by itself, enlarge the pool.
• Real-harm correction ($R$). Verified, substantive incidents — official data breaches, critical security vulnerabilities, regulatory penalties — are the only channel through which harm lowers the anchor, and only when officially confirmed, with sources attached and a correction channel open (Sections 5.3 and 7.2).

Removing the media/attention dimension from FA and centering RI are two sides of one boundary claim: discussion volume cannot, by itself, prove influence. Attention is a channel of influence, not its certification. The structural reading of this claim is worth stating explicitly: because attention is a channel, it must enter the measurement somewhere — it enters as the RI signal, centered and bounded, so that it reorders but cannot certify. And because attention is not a certification, it must not enter the anchor as a slow stock — hence the removal of the media dimension from the V1 FA composition. The two moves are one policy implemented in two places.

5.3 · FA mechanism: the slow anchor

The FA value of an Originator $i$ is computed as a weighted mean over its present dimensions, on a unified scale, using the production composition of Section 6. Three properties make FA a “long-run fairness skeleton”:

1. Stock, not flow, with no time decay. Every FA input is a monotonic stock quantity (evaluation score, count, cumulative citation, ledger). Time decay lives only in RI; FA does not forget past contributions on a timer. This is deliberate: an anchor should move slowly. The counterargument — that “newer” contributions should weigh more — is answered structurally: the iterative novelty of contributions is carried by the adjudicated ledger's tiering and by the live signal, not by decaying the anchor. A decaying anchor would erase the very cumulative influence the system exists to measure.
2. Continuous-missing-value protocol. Missing values are classified (Section 6.4) as true-zero, non-punitive absence, or leaderboard floor; a missing source never zeroes an Originator. Data-continuity governance carries the most recent valid value forward with an alert; an unavailable source does not silently produce a zero.
3. Settlement and adjustment. FA is settled twice daily at closed-market windows (UTC 04:00–07:00 and 16:00–19:00), on the same structure and with equal weights for both windows. The adjustments $D$ and $R$ apply only at settlement. During market hours, the settled FA value is frozen and consumed by the index unchanged.

5.4 · RI mechanism: methodological description

The RI signal measures discussion and diffusion in near real time. The full algebraic structure is given in Section 6.6; here we state the design principles, without numerical parameters.

• Source classes. RI consumes independently verified authoritative media and public-platform information, aggregated to the Originator level. Individual source identities are deliberately not disclosed (Sections 13.3 and 13.6).
• Heat and direction are separated. A search index or a trending rank proves that attention rose; it cannot by itself certify whether the attention is favorable or unfavorable. Direction is assigned by a polarity model with an attention baseline, and only negatively factual signals count as negative.
• Leaderboard first, topic backfill. Ranking positions are the primary signal; discussion topics fill in where no rank is available. A non-ranking attention feed is deliberately excluded.
• Half-life tiers. Events are classified into content-type tiers (rank-driven, factual, opinion, emotional), each with its own half-life governing decay. Values are not disclosed.
• No-stimulus decay to zero. With no new stimulus, a pulse decays monotonically to zero and is then removed from the active set; a decayed pulse does not deposit into FA.
• Renewal (contract rollover). When a new event carries a full value at least as large as the residual of the old pulse, the old pulse is voided and the new value starts a fresh pulse; when it is smaller, the old pulse simply decays. Instantaneous value thus depends only on the current state, not on the arrival path (Axiom 6.6).
• Eight-component decomposition. The signed RI signal is decomposed into eight tracks — for each of the four content tiers, a positive and a negative component (Table 4). The decomposition is an identity (the sum of the tracks equals the signed signal at every time) and it is what makes the dual-track audit of Section 9.3 possible.
• Budgetary constraints. A volatility budget caps the total RI magnitude; a per-Originator per-settlement movement bound triggers an audit alert without truncating the value.

Table 4. The eight-component RI track decomposition (structural; no parameter values disclosed). Rows are content tiers, columns are signs; each track is a decaying-pulse sum over its own active events.

5.5 · The four elements of a measurement scale

A measurement scale must provide four things: a common scale across units; a stable center that anchors comparison; domain-universal applicability (the same operations work for every unit); and comparability across time. The design maps these onto three properties: relative (common scale, by cross-sectional normalization), centered (stable center, by the conservation law), and real-time (comparability across time, by the collection–pricing–display pipeline). The formal definitions are in Section 6; the trust arguments are in Section 7. The four-element framing matters because “relative” alone is too weak a claim: an Elo rating is relative yet its scale anchor is arbitrary and it has no conservation structure. The distinctive claim is the combination — a common scale, a stable center, and a real-time cadence operating together, so that the number has the semantics of a measurement rather than the semantics of a score in a game.

5.6 · Conservation and centralization: statement

The flagship index $\Theta$ is a centered composition:

where $\overline{\mathrm{RI}}(t)$ is the cross-sectional mean of RI over the included set $\mathcal{I}(t)$, $t_0$ is the last closed-market settlement time, and $\mathrm{BR}$ is a reserved bridge slot that is zero except during a member-set transition (Section 6.10). The live-signal dispersion coefficient $k_{\mathrm{ri}}$ is reserved. Two consequences follow at once and are proved in Section 6.9:

• Exact conservation on the adjusted anchor. The sum of $\Theta$ over the included set equals the sum of $\mathrm{FA}^{\mathrm{adj}}$ over that set, at every time (Proposition 6.9).
• Approximate conservation on base FA. Because the published index carries the sustained-leadership adjustment, the sum of base FA is only approximately equal to the sum of $\Theta$, with a deviation bounded by the adjustment caps; it is exact when no Originator holds a leadership streak and no substantive incident is recorded (Proposition 6.11).

The design history of (1) is itself a piece of evidence. An earlier formulation added the RI term uncentered; in live validation it was discovered that this allowed the total pool to drift — transient discussion could inflate the aggregate, violating the “heat reorders but does not enlarge” principle — and the formulation was replaced at a settlement with the centered form. The centering is therefore not decoration: it is the mechanism that enforces the boundary claim of Section 5.2 at the level of the total pool. The index family also includes an absolute-ability family ($\Gamma$, $\Lambda$) and sector aggregates ($\mu_\Gamma$, $M_\Gamma$) that preserve cross-sectional ranking (Section 6.7); these exist so that relative measurement and absolute progress are not conflated (Proposition 6.3).

5.7 · Counterarguments and design trade-offs

Every major design choice in the measure has a serious counterargument, and each is answered; the choices are deliberate, not oversights. “The weights are arbitrary.” Answered at two levels. Structurally, Proposition 4.1 ties every block to the influence definition, and Table 3 makes the tie explicit. Numerically, the weights are V1 production values subject to a planned unified calibration experiment (Section 12.2) and to the sensitivity analysis of Section 8.7, which will report how rankings move under weight perturbation. Arbitrariness would mean “no reason”; the paper's claim is “a stated reason and a planned test.” “The anchor is too slow to react to disruptive releases.” The anchor is supposed to be slow; the live signal is the fast channel. A disruptive release moves the live signal immediately and the anchor at the next settlement through the capability re-evaluation. The two-layer design is the answer to the speed objection, not its victim. “A centered live signal punishes quiet excellence.” The centering makes the live signal a zero-sum reordering, so an Originator with stable, high attention is at the mean and receives no bonus or penalty from the live term. That is intended: the live term is a deviation channel, and persistent influence is carried by the anchor. Punishing quiet excellence would require the live term to be absolute, which would reintroduce the pool-inflation failure the centering fixes. “Two settlement windows a day is arbitrary.” The windows (UTC 04:00–07:00 and 16:00–19:00) are chosen to bracket the two hemispheres' main activity periods and to bound how stale the frozen anchor can get within a day. The count and placement are engineering choices, revisable under governance (Section 10.3); what is not revisable without a method change is the discipline that the anchor freezes between settlements.

Back to contents  |  ← 4 · Measurement Theory and Construct Validity  |  6 · Formal Framework: The Mathematical Shell →