Execution Economics is a framework for explaining the gap between what organisations decide and what they realise. It reorganises the production of outcomes around two factors that conventional economics treats either jointly or implicitly: the quality of decisions, and the sovereignty of the decision-maker who acts on them.
The framework is built on a single identity, two operational composites, three foundational axioms for each composite, and three theorems that follow from them. This compendium restates each piece in plain English without introducing any new results.
Either factor at zero collapses the product. Output is not a sum of capability and execution; it is their multiplicative joint product.
Sovereignty is the cube root of the product of three positive components — Authority, Control, Execution-relevant Information — scaled by the absence of veto exposure. Setting any of A, C, E to zero, or V to one, drives the index to zero.
The April 2026 working paper closes the measurement-construct asymmetry by giving the capability side a parallel construction:
Decision capability is then d(P) = P^α with α ∈ (0, 1].
Foundational framework
Fritz, P., Fritz-Kalish, C. and Bodrova, O. (2025), ‘Introducing decision sovereignty: a missing transmission variable in models of implementation’, Journal of Behavioural Economics and Social Systems, 7(1-2), 137-141, DOI: 10.54337/ojs.bess.v7i1-2.11417; Fritz, P., Fritz-Kalish, C. and Bodrova, O. (2026), Decision Sovereignty ( preprint monograph), Global Access Partners, DOI: 10.5281/zenodo.19725227
What an institution actually delivers (Y) equals how good its decisions are (d(P)) multiplied by how faithfully those decisions translate into action (S). Either factor alone is insufficient. A brilliant analysis under zero sovereignty produces nothing. Full sovereignty over a worthless analysis produces nothing useful. The factors interact; they do not add.
| Symbol | Name | Plain-English meaning |
|---|---|---|
| Y | Realised output | What the organisation actually achieves along its stated mission. Measured in the natural units of that mission, then z-scored across organisations for cross-sectional work. Can be negative when the organisation produces output in the direction opposite to its stated mission. |
| d(P) | Decision quality / capability | A monotone non-negative operator on predictive capacity P. Canonical form d(P) = P^α with α ∈ (0, 1]. Captures the conventional contents of "good decisions": accuracy of analysis, soundness of method, calibration of forecasts. |
| P | Predictive capacity | The dimensionless input to d(·). Operationalised in the 2026 capability paper as the four-component geometric mean P = (W · J · Ω · H)^(¼). |
| α | Curvature parameter of the quality operator | Scalar in (0, 1]. The elasticity of d(P) with respect to P. α = 1 corresponds to the linear baseline (decision quality scales one-for-one with predictive capacity); α < 1 corresponds to a concave operator exhibiting diminishing returns to predictive investment. The load-driven quality-loss mechanism is treated separately, via the load ratio ρ = d(P)/K and the congestion friction χ(ρ); see Chapter 4 of the monograph. |
| S | Decision sovereignty | The unsigned magnitude of the Decision Sovereignty Index, S = DSI ∈ [0, 1]. Captures the degree to which the decision-maker actually has the authority, operational control, execution-relevant information, and veto-freedom to make the decision effective. |
| Snet | Signed sovereignty-weighted alignment | Used in the value-reversal apparatus. Carries the sign of transmission. S = M · sign(Snet) where M is the unsigned magnitude. sign(Snet) = +1 when intermediaries are aligned with stated mission; sign(Snet) = −1 when they invert it. |
| T | Transmission coefficient | Used in extended identity Y = d(P) · S · T. Measures the fraction of a decision that survives propagation through the pathway between sovereign and outcome. T = β · A_T · (1 − χ), with T ∈ [−1, 1] when A_T may be negative. |
When transmission must be modelled separately:
When transformed into an estimating equation in the positive-transmission regime (all factors strictly positive):
The multiplicative form is not a modelling convenience. It is forced by three structural axioms (canonical Proposition 2.1, Appendix H.1).
The function relating Y to d(P) and S is continuous on its domain [0, 1] × [−1, 1]. Small changes in capability or sovereignty produce small changes in output. No abrupt jumps without an underlying structural reason.
Output is weakly increasing in d(P) when sovereignty is positive, and weakly decreasing in d(P) when transmission is reversed. Capability helps when alignment is right and harms when alignment is wrong. This is the formal condition that makes "amplification of harm under reversal" possible — and it is what distinguishes the framework from every additive specification.
Y = 0 if d(P) = 0, regardless of sovereignty. Y = 0 if S = 0, regardless of capability. Both factors are necessary conditions for non-trivial output. This rules out additive forms, which permit one factor to substitute for the absence of the other.
Under unit-normalisation of the measurement scales (linear scaling in d(P) and S), axioms A1–A3 force the unique continuous form Φ(d(P), S) = d(P) · S, up to choice of measurement units. Additive specifications fail A3. Power-law specifications with k, m ≠ 1 require relaxing the unit convention. The multiplicative identity is therefore the only continuous functional form that satisfies the three axioms simultaneously under standard measurement.
Sovereignty is what the decision-maker actually holds, multiplied by the share of that holding that environmental veto exposure permits to be exercised. The three positive components — Authority, Control, Execution-relevant Information — combine through their cube-rooted product so that an absence in any one of them collapses the index. Veto enters separately because it is a constraint on the exercise of sovereignty rather than a fourth ingredient of it.
| Symbol | Name | Plain-English meaning |
|---|---|---|
| A ∈ [0, 1] | Authority | The formal and informal standing of the decision-maker to decide in the relevant domain — derived from statute, charter, contract, delegated mandate, or recognised custom. A court may have authority to order without the means to compel; that is high A, low C. |
| C ∈ [0, 1] | Control | The operational capacity to deploy the resources required to realise the decision — staff, capital, time, access, administrative machinery. A sovereign monarch without a treasury or an army has authority but no control. |
| E ∈ [0, 1] | Execution-relevant Information | The information the decision-maker actually holds at the moment of deciding — distinct from information the organisation nominally has, or that theoretically exists, or that would be required for an optimal decision. Audits routinely confuse these. |
| V ∈ [0, 1] | Veto coefficient | The probability-weighted capacity of other actors to reverse, dilute, or nullify the decision. Subtractive in structure — entered through (1 − V) — because veto power constrains sovereignty rather than enhancing it. V = 1 collapses sovereignty regardless of A, C, E. |
The DSI structure makes the diagnostic distinct because each failure mode has a different signature:
The cube-rooted geometric mean across A, C, E is forced by five structural axioms applied to any aggregator f: [0, 1]³ → [0, 1] of the three positive components (canonical Proposition 3.1, Appendix H.2). The asymmetric (1 − V) scaling is then derived separately from the substantive role of veto.
The aggregator treats Authority, Control, and Execution-relevant Information identically: f(A, C, E) is invariant under permutations of its arguments. None of the three has structural priority over the others.
The aggregator is weakly increasing in each component. More Authority cannot reduce sovereignty, holding C and E fixed. Same for C and E.
f(A, C, E) = 0 if any of A, C, E equals zero. Zero authority, zero control, or zero execution-relevant information each individually collapses sovereignty. This is the axiom that rules out additive aggregators, which permit substitution between components.
Doubling all three components doubles the aggregate (homogeneity of degree one). Sovereignty has the same units as its components. This is what permits the index to be combined directly with d(P) on a common scale.
The aggregator is continuous on the closed cube [0, 1]³. Small changes in components produce small changes in the aggregate.
The five axioms force the form f(A, C, E) = ∛(A · C · E), up to a positive multiplicative constant set to unity by the requirement f(1, 1, 1) = 1. Arithmetic means fail B3 (joint necessity). Harmonic means fail B4 at the boundary. Power means M_p with p > 0 fail B3. The geometric mean is the unique aggregator satisfying all five axioms simultaneously.
A decision-maker with full Authority, Control, and Execution-relevant Information facing an unconditional external veto retains every internal attribute of sovereignty yet has no realised capacity to act. Veto is therefore an environmental constraint on the exercise of sovereignty, not a positive component of sovereignty itself. The functional role for V is multiplicative scaling on the symmetric aggregate ∛(A · C · E), via (1 − V). This places V = 0 (no veto) as identity and V = 1 (unconditional veto) as collapse — the substantively correct boundary behaviour.
The canonical form DSI = ∛(A · C · E) · (1 − V) supersedes two earlier specifications in the programme. The additive form DSI = (A · C · E) − V (used in the 2025 flagship paper) can produce values outside [0, 1] and is inconsistent with the bounded composite required by the logistic mapping to governance capacity. The four-factor geometric mean DSI = ∛(A · C · E · (1 − V)) (used in some intermediate drafts) treats V as a fourth symmetric component, which is structurally wrong. All canonical empirical scores — Robodebt DSI = 0.124; TCG productive-phase DSI ≈ 0.67; GAP representative-project DSI ≈ 0.52; Berkeley AI adoption DSI ≈ 0.08 — are computed under the canonical form.
Predictive capacity is the cube-rooted equivalent on the capability side: a fourth-rooted geometric mean of four documentary components. Joint necessity again — an organisation with no analytical infrastructure, no methodology, no learning loop, or no track record fails on capability. The four-component decomposition closes the measurement-construct asymmetry between the sovereignty side (operationalised through DSI) and the capability side, which previously had no parallel construction.
| Symbol | Name | Plain-English meaning |
|---|---|---|
| W ∈ [0, 1] | Data infrastructure | The documented capacity to collect, integrate, and make accessible the empirical inputs the decision apparatus operates on. Scored from technology-stack disclosures, integration audits, IT spend, vendor and platform documentation, and public reporting on data infrastructure. |
| J ∈ [0, 1] | Judgement (analytical method) | The documented methodological sophistication with which the organisation converts data into decisions. Scored from documented methodology, R&D and analytics spend, industry benchmarks, expert assessment. The letter J avoids notational collision with R² (regression) and Q (output). |
| Ω ∈ [0, 1] | Updating from outcomes | The documented capacity to close the loop between realised decision outcomes and the methodology used in subsequent decisions. Scored from post-decision review protocols, observable improvement, structured update mechanisms, evidence of learning from negative as well as positive outcomes. Distinct from the DSI component C: C measures whether the structural pathway exists; Ω measures whether the analytical apparatus exploits it. |
| H ∈ [0, 1] | History of forecasts | The documented predictive accuracy of the organisation's decisions over the relevant horizon. Scored from realised-versus-forecast ratios, calibration of confidence intervals, and comparable-organisation benchmarks where direct observation is unavailable. |
The same logic that forces the cube-rooted geometric mean for DSI extends to four components for P, under three parallel axioms:
No analytical infrastructure cannot make capable decisions, however good the other components. No methodology cannot convert data into decisions, however good the data. No learning loop cannot improve decisions, however good the initial framework. No forecast track record cannot demonstrate predictive validity. d(P) = 0 if any component equals zero — ruling out additive specifications.
Each component is on [0, 1]. The composite P is therefore on [0, 1]. This permits direct multiplication with signed S ∈ [−1, +1] in the core identity, with no rescaling.
Joint-necessity components on bounded support aggregate uniquely (up to normalisation) by geometric mean. Arithmetic means permit substitution and violate joint necessity. Harmonic means compress high values and overstate joint necessity. The unrooted product compresses sharply (four 0.7s yield 0.24, understating the components). The fourth-rooted product preserves joint necessity, bounded support, smooth dependence, and is the unique form satisfying all three properties simultaneously.
Each component is scored on a four-anchor ordinal scale {0, 1, 2, 3} with documentary evidence required at each level. The raw score is divided by 3 to yield the unit-interval value. Where evidence is unavailable, the component is scored at the lowest level consistent with what is known, not at the level the analyst would expect from reputation.
Three illustrative cases anchor the rubric: ASML Holding 2023 scores P = 1.00 (high-d(P) case, all components at 3); the Robodebt scheme 2015–2019 scores P ≈ 0.47 (W = 0.67, J = 0.33, Ω = 0.33, H = 0.67); Theranos scores P = 0 (joint necessity collapses with J = Ω = H = 0).
Once a decision is made by a sovereign agent, it must propagate through an institutional and technological pathway. Transmission T captures the fraction of the decision that survives that propagation. T factors into baseline pathway capacity β, alignment of intermediaries A_T, and inverse friction (1 − χ). T is held to extended analyses; the main text uses Y = d(P) · S where T effects are folded into S.
| Symbol | Name | Plain-English meaning |
|---|---|---|
| β ∈ [0, 1] | Baseline transmission capacity | The raw technological and institutional capacity of the pathway to carry decisions at all. Whether the systems, processes, and physical infrastructure exist. |
| AT ∈ [−1, 1] | Transmission alignment | Cosine similarity between the sovereign's intended direction and the directional orientation of intermediaries. AT > 0 denotes alignment; AT < 0 denotes active reversal. Distinct from A (Authority) inside the DSI. |
| χ ∈ [0, 1] | Friction | Generic friction along the transmission pathway. When friction is endogenous to load, χ(ρ) is specified as a logistic CDF: χ(ρ) = χ_max / (1 + e^(−γ(ρ − ρ_c))). |
The transmission magnitude is governed by the worst link in the chain:
A pathway with capacity 0.9, alignment 0.9, and friction-free conduit (1 − χ) = 0.2 has Γ = 0.2, regardless of how good the other two links are. Reform investment should target the binding link, not the most visible one.
Composite institutional capacity K is built from baseline capacity K₀, institutional quality Q (transparency, rule of law, accountability, coordination), and organisational depth D, with positive elasticities φ, ψ. K is the substrate for both sovereignty and transmission.
Density of decision output per unit of institutional capacity. As ρ rises, congestion friction rises. The load ratio is the link between decision intensity and endogenous friction.
Friction is bounded above by χmax < 1, half-saturates at the threshold load ratio ρc, and has steepness γ. The logistic specification preserves the qualitative congestion features — convex cost near saturation, bounded above — while admitting flexible empirical calibration. (An earlier M/M/1 queueing derivation has been superseded.)
Raw decision volume M is produced from information I, cognitive capacity C, and time Ttime, with elasticities a, b, c summing to ≤ 1 (diminishing returns to scale). Note: the C in this equation is cognitive capacity, distinct from the C (Control) in the DSI; Ttime is the time input, distinct from the transmission coefficient T.
| Label | Name | Statement |
|---|---|---|
| Definition I.1 | Core Identity | Y = d(P) · S. Extended: Y = d(P) · S · T. |
| Equation I.2 | Decision Sovereignty Index | DSI = ∛(A · C · E) · (1 − V), with A, C, E, V ∈ [0, 1]. |
| Equation I.3 | Decision-Production Function | M = I^a · C^b · T_time^c, with 0 < a, b, c < 1 and a + b + c ≤ 1. |
| Equation 1.2 | Quality Operator | d(P) = P^α, with 0 < α ≤ 1. |
| Equation 2.2 | Transmission Coefficient | T = β · A_T · (1 − χ). |
| Equation 3.3 | Log-Linear Estimating Equation | ln Y = α · ln P + ln S + ln β + ln A_T + ln(1 − χ) + ε. |
| Definition 4.1 | Institutional Capacity | K = K₀ · Q^φ · D^ψ. |
| Definition 5.1 | Load Ratio | ρ = d(P) / K. |
| Definition 5.2 | Congestion Function | χ(ρ) = χ_max / (1 + e^(−γ(ρ − ρ_c))). |
| Definition 6.1 | Transmission Alignment | AT ∈ [−1, 1], cosine similarity between sovereign's intended direction and intermediaries' orientation. |
| Definition 7.1 | Weakest-Link Transmission Magnitude | Γ = min{ β, |A_T|, (1 − χ) }. |
| Capability formula | Predictive Capacity | P = (W · J · Ω · H)^(¼) |
The framework's principal theoretical content is carried by three numbered theorems, each with a corresponding empirically-testable falsification condition.
The direction of realised output equals the direction of signed sovereignty. An institution whose sovereignty-weighted alignment is negative produces output in the direction opposite to its stated mission. The theorem follows directly from the identity Y = d(P) · S with S = M · sign(Snet) and d(P) ≥ 0, M ≥ 0.
Operationalised byFalsification Condition F1 (Sign Consistency): a properly-identified sample of at least 15 negative-alignment cases and 15 matched positive-alignment controls in which sign(Y) is statistically independent of sign(Snet) would falsify the theorem.
The marginal effect of decision quality on realised output has the sign of S. When alignment is positive, capability investment amplifies good outcomes. When alignment is reversed, capability investment amplifies harm rather than degrading toward zero. This is the framework's most discriminating prediction; it distinguishes Execution Economics from implementation-theory rivals (Pressman and Wildavsky, 1973), which predict degradation toward zero rather than amplification in the reverse direction.
Operationalised byFalsification Condition F2 (Amplification): within the negative-alignment subset, the regression coefficient on d(P) should be negative. A positive or zero coefficient at conventional thresholds would falsify the theorem. Robodebt is the canonical empirical instance.
Within the positive-alignment regime, realised output Y as a function of d(P) is non-monotonic and has an interior maximum. As capability rises, sovereignty is compressed through congestion (saturation of C and E within the DSI), and the product d(P) · S peaks at some interior P*.
Formal statementY(d(P)) exhibits a unique interior maximum at the optimum load ratio ρ* ∈ (0, ρc], where ρc is the half-saturation point of the logistic congestion function.
Operationalised byFalsification Conditions F3 (Congestion Hump) and F4 (Interior-Optimum Location). A monotonically increasing Y(d(P)) relationship within positive-alignment samples would falsify the theorem. The Berkeley AI adoption case provides the canonical empirical instance.
| Label | Name | Plain-English statement |
|---|---|---|
| Theorem 12.1 | Interior Optimum | When friction χ(ρ) is strictly convex and increasing in ρ, and quality d(P) is log-concave, there is a unique interior optimum P* > 0 at which dY/dP = 0 and d²Y/dP² < 0. Output-maximising organisations intentionally maintain slack — they do not run at full capacity. |
| Theorem 13.1 | First-Best Institutional Investment (Samuelson Condition) | Optimal institutional capacity K* satisfies ∂Y/∂K = C′(K*). The marginal benefit of capacity equals its marginal cost. Decentralisation through correctly-priced shadow values implements the first-best. |
| Theorem 13.2 | Euler Equation for Dynamic Investment (sketch) | The optimal path of investment {K_t} satisfies an inter-temporal first-order condition linking the marginal product of capacity today to its discounted marginal product tomorrow. |
| Theorem 14.1 | Institutional Trap Threshold | Under dK/dt = I(K) − δK with a region where I(K) < δK, the system has multiple equilibria: a low-K trap, an unstable threshold Kunstable, and a high-K stable equilibrium. Organisations starting below Kunstable converge to the trap. |
| Theorem 14.2 | Multiple Equilibria and Hysteresis | The trap threshold exhibits hysteresis. Reform requires either a discontinuous capacity injection or sustained accumulation exceeding depreciation. Path dependence binds. |
| Corollary 5.1 | Interior Optimum for Decision Input | Output-maximising organisations operate below 100% utilisation. Slack is optimal, not waste — it preserves transmission below the convex region of the congestion curve. |
| Corollary 12.1 | Capacity and Decision Scaling | ∂d*/∂K > 0. An increase in institutional capacity leads to a proportional increase in optimal decision intensity. |
| Corollary 14.1 | Collapse Trajectory | Below the unstable threshold Kunstable, capacity erodes monotonically toward the low-K trap. Without external injection, recovery is structurally precluded. |
| Proposition 4.1 | Transmission Decomposition | Aggregate transmission decomposes into a deterministic component (β · AT · (1 − χ)) and an idiosyncratic shock η, with E(η) = 0. |
| Proposition 5.1 | Hump-Shaped Output Curve | Y(d(P)) is hump-shaped under the logistic congestion function. Decision intensity rises help up to a turning point, beyond which transmission loss dominates. |
| Proposition 6.1 | Output Loss from Misalignment | When transmission alignment falls below unity (AT < 1), realised output is reduced by a factor proportional to the cosine angle between intent and intermediary orientation. |
| Proposition 6.2 | Reversal Condition and Value Destruction | When AT < 0 in the extended identity, realised output takes the sign opposite to the sovereign's intent. Capability investment amplifies the magnitude of harm rather than producing benefit. |
| Proposition 7.1 | Aggregate Output and Covariance with Transmission | Aggregate Y depends on the covariance between decision intensity and transmission. Positive covariance amplifies output; negative covariance destroys it. |
| Proposition 8.1 | Attenuation Bias under Classical Measurement Error | Measurement error in d(P) attenuates the estimated coefficient on d(P) toward zero. Small empirical effects of d(P) may reflect either a small true effect or noisy measurement; the two are not distinguishable without a reliability ratio. |
| Proposition 11.1 | IV Exclusion Restriction for d(P) | Valid instruments for d(P) must affect Y only through d(P), not directly. Institutional capacity K can in principle serve as an instrument under standard exogeneity conditions. |
| Proposition 11.2 | Sargan–Hansen Overidentification Test | When multiple instruments are available, joint validity is testable. Failure of the test indicates either invalid instruments or model misspecification. |
| Proposition 12.1 | Optimal Prediction Increasing in Capacity | ∂P*/∂K > 0. Higher institutional capacity raises the return to predictive investment. |
| Proposition 12.2 | Optimal Prediction Increasing in Operator Curvature | ∂P*/∂α > 0. Higher α (closer to the linear baseline) raises optimal P. |
| Proposition 13.1 | Complementarity of S and P Investment | ∂²Y / ∂S ∂P > 0. Investments in sovereignty and predictive capacity are complements; each raises the marginal product of the other. |
| Proposition 14.1 | Reform Feasibility | Effort required to escape an institutional trap rises with the distance between current trapped K and the unstable threshold Kunstable. Far-below-threshold organisations face longer reform horizons and higher failure risk. |
| Proposition 15.1 | Decision-Structure Substitution | At the margin, d(P) and S are substitutes in production of Y, with elasticity of substitution σd,S characterising the marginal trade-off. |
| Corollary 15.1 | Unitary Substitutability | The Cobb–Douglas form Y = d(P)^a · S^b has unit elasticity of substitution: a 1% rise in S's price relative to d(P) leads to a 1% reduction in S relative to d(P) along the isoquant. |
| Proposition 15.2 | Weakest-Link Constraint | Output gain from improving any single component of S is bounded above by the lowest component. Reform investment should target the binding constraint, not the highest-visibility component. |
Realised output Y is non-monotonic in decision intensity. As predictive capacity P rises, d(P) improves but sovereignty degrades through the congestion channel. There exists an interior optimum P*. This prediction contradicts the implicit assumption in much of organisational economics that more decision-making capacity is always better.
When sovereignty falls below a critical threshold, and transmission admits hostile direction (AT < 0), realised output can take the sign opposite to the sovereign's intent. Standard theory, which treats execution as frictionless, cannot generate this prediction.
At the margin, d(P) and S are substitutes in producing Y. Organisations facing high sovereignty costs should rationally choose lower P, and conversely. The prediction can be tested against observed allocation patterns.
| Label | Name | Falsification mode |
|---|---|---|
| F1 | Sign Consistency | sign(Y) = sign(Snet) holds across a properly-identified sample of ≥15 negative-alignment cases and ≥15 matched positive-alignment controls. Failure: statistical independence between sign(Y) and sign(Snet). Tests Theorem T1. |
| F2 | Amplification | Within the negative-alignment subset, the regression coefficient on d(P) is negative. Failure: a positive or zero coefficient at conventional thresholds. Tests Theorem T2 — the framework's most discriminating prediction. |
| F3 | Congestion Hump | Within the positive-alignment subset, Y as a function of d(P) exhibits an interior maximum. Failure: monotonically increasing Y(d(P)) across multiple domains with no interior maximum at sample-relevant scales. Tests Theorem T3. |
| F4 | Interior-Optimum Location | The optimum load ratio ρ* lies interior to the half-saturation point ρc, i.e. ρ* ∈ (0, ρc]. Failure: estimated ρ* values at or above ρc across domains. |
| F5 | DSI–Outcome Correlation | The DSI correlates substantially with realised outcomes across organisations after controlling for d(P). Failure: a near-zero correlation in well-measured samples. |
| Symbol | Domain | Meaning |
|---|---|---|
| Y | Realised output | What the institution actually delivers along its stated mission. Real number; signed when transmission may be reversed. |
| d(P) | Operator | Decision quality as a function of predictive capacity. Canonical d(P) = P^α. |
| P | [0, 1] (operational) | Predictive capacity; argument of d(·). Operationalised as P = (W · J · Ω · H)^(¼). |
| α | (0, 1] | Curvature parameter / elasticity of d(P) with respect to P. α = 1 linear baseline; α < 1 concave with diminishing returns. |
| S | [0, 1] | Decision sovereignty (unsigned). S = DSI in applied work. |
| Snet | [−1, +1] | Signed sovereignty-weighted alignment scalar. Carries sign of transmission. |
| A | [0, 1] | Authority — formal/informal standing of decision-maker (DSI component). |
| C | [0, 1] (DSI) / R⁺ (decision-prod.) | In the DSI: Control (operational capacity). In M = I^a · C^b · T_time^c: cognitive capacity. Context disambiguates. |
| E | [0, 1] | Execution-relevant Information held by the decision-maker at the moment of deciding. |
| V | [0, 1] | Veto coefficient — probability-weighted capacity of others to reverse, dilute, or nullify the decision. |
| W | [0, 1] | Data infrastructure (capability composite component). |
| J | [0, 1] | Judgement / analytical method (capability composite component). |
| Ω | [0, 1] | Updating from outcomes (capability composite component); analytical learning loop. Distinct from C in DSI. |
| H | [0, 1] | History of forecasts (capability composite component); predictive track record. |
| T | [−1, +1] | Transmission coefficient in extended identity. T = β · AT · (1 − χ). |
| β | [0, 1] | Baseline transmission capacity. |
| AT | [−1, +1] | Transmission alignment — cosine similarity between sovereign's direction and intermediaries'. Distinct from A (Authority). |
| χ | [0, 1] | Friction along transmission pathway. Logistic specification: χ(ρ) = χ_max / (1 + e^(−γ(ρ − ρ_c))). |
| χmax | (0, 1) | Maximum friction; upper asymptote of congestion function. |
| γ | > 0 | Steepness of congestion function around half-saturation. |
| ρ | [0, ∞) | Load ratio; ρ = d(P) / K. |
| ρc | > 0 | Congestion threshold at which χ(ρc) = χmax / 2. |
| ρ* | (0, ρc] | Output-maximising load ratio. Lies interior to ρc (Theorem T3 / F4). |
| K | > 0 | Composite institutional capacity, K = K₀ · Q^φ · D^ψ. |
| K₀ | > 0 | Baseline institutional capacity. |
| Q | > 0 | Institutional quality (transparency, rule of law, accountability, coordination). |
| D | > 0 | Organisational depth. |
| φ, ψ | > 0 | Positive elasticities of K with respect to Q and D. |
| M | > 0 | Raw decision-production output; M = I^a · C^b · Ttime^c. |
| I | > 0 | Information input to decision production. |
| Ttime | > 0 | Time input to decision production. Distinct from T (transmission coefficient). |
| a, b, c | (0, 1), Σ ≤ 1 | Elasticities of decision-production function. Diminishing returns to scale. |
| Γ | [0, 1] | Weakest-link transmission magnitude, Γ = min{β, |AT|, (1 − χ)}. |
| δ | > 0 | Institutional capacity depreciation rate. |
| δ̄ | (0, 1) | Social discount factor in dynamic optimisation. |
| Kunstable | > 0 | Unstable threshold separating low-K trap from high-K stable equilibrium (Theorem 14.1). |
| η, ε, ω | Mean zero | Stochastic shocks: η = transmission shock; ε = decision-quality uncertainty; ω = measurement error. |
| σd,S | > 0 | Elasticity of substitution between d(P) and S in the production of Y. Distinct from σ²x (variance notation); subscript indicates context. |
| λ | > 0 | Shadow value (Lagrange multiplier) on a binding resource constraint. |
| θ | Vector | Generic parameter vector (used where a vector of parameters is referenced). |
Three symbols are used in two distinct senses across the framework. The collisions are flagged here once and disambiguated by context everywhere else.
C. In the DSI (Equation I.2 / 2.1), C denotes Control: the operational capacity of the decision-maker. In the decision-production function M = I^a · C^b · Ttime^c (Equation I.3), C denotes cognitive capacity. The two are constructs at different levels of analysis.
A. In the DSI, A denotes Authority. In the transmission coefficient (Equation 2.2), the alignment subscript appears as AT. The two are unrelated; AT is reserved for transmission alignment to prevent confusion.
T. In the extended core identity Y = d(P) · S · T, T denotes the transmission coefficient. In the decision-production function, Ttime denotes the time input to decision production. The transmission coefficient is always T (no subscript); the time input always carries the _time subscript.
What the framework adds to the available apparatus is not a new variable but a new identity. Capability and sovereignty are well-known categories. What was missing — and what the formal apparatus above provides — is the joint claim that they enter the production of realised outcomes multiplicatively rather than additively, that each is uniquely characterised by a small set of structural axioms, and that the combination admits sign reversal when transmission is hostile.
The three theorems (T1 Sign, T2 Amplification, T3 Congestion) and their five falsification conditions are the testable content. Everything else in the apparatus — the operational composites DSI and P, the decomposition into A, C, E, V and W, J, Ω, H, the supporting propositions on substitution, capacity, dynamics, and reform — is downstream of the identity and exists to make it usable on documentary evidence.
The DSI is intended to be measured per decision, and what authority, control, execution-relevant information, and veto exposure mean is necessarily different for a clinical decision, an automated-decisioning deployment, a monetary-policy action, and a corporate capital allocation. The functional form is fixed by axiom — the canonical aggregator ∛(A · C · E) · (1 − V) does not move. What is reconfigured for each domain is the operational content underneath: the observable proxies, the sub-decomposition, the rubric. This appendix sets out a seven-step procedure for constructing a domain-specific operationalisation that preserves the cross-comparability the framework's empirical anchors depend on. The reader may either defer to the compendium default — the canonical form scored against the nearest-class published anchor — or engineer a domain-specific rubric using the procedure that follows.
| Fixed by axiom (§3) | Reconfigurable per domain |
|---|---|
| The four-component structure A, C, E (positive) and V (subtractive). | The observable proxies for each of A, C, E, V in the specific domain. |
| The symmetric geometric mean across A, C, E with cube-root normalisation. | The sub-decomposition under each component — what the jointly-necessary or substitutable elements are. |
| The multiplicative (1 − V) scaling that preserves the unit interval. | The scoring rubric that anchors each sub-element to documentary evidence. |
Modifying the fixed column breaks the link to the uniqueness result restated in §3, falsifies the cross-case comparisons the framework's anchors depend on, and produces an index that is no longer the DSI. Practitioners who find themselves wanting to weight A above C, or to substitute an arithmetic mean for the geometric mean, are signalling that operational content has been allocated to the wrong component, not that the form needs adjustment.
Name the decision in one sentence: who is deciding, what is being decided, when. Sovereignty is decision-specific; the DSI for procuring a vendor system is not the DSI for live calibration of that system. Where decisions are bundled in practice, each receives a separate DSI.
For each of A, C, E, V, list two to five concrete observables that together exhaust what the component means in this domain. The test is saturation: if every observable scored at one, would the actor genuinely hold full A (or C, or E, or face zero V)? Observables are documentary, not impressionistic.
Where sub-elements are jointly necessary, aggregate by geometric mean — this reproduces joint necessity at the sub-component level. Where sub-elements substitute, use a weighted arithmetic mean with documented weights. For V, use the probability-weighted union 1 − ∏(1 − Vᵢ) under independence, or the dominant-channel maximum otherwise.
Score each sub-element on [0, 1] against a written rubric with anchor descriptions at 0, 0.5, and 1. Two independent analysts reading the same documentary record should land within 0.2 on every sub-element. Each score carries a one-line written justification citing the specific document on which it rests.
Compute component values A, C, E, V using the chosen sub-aggregators, then combine under DSI = ∛(A · C · E) · (1 − V), unchanged. The dynamic content lives in steps 2–4; step 5 is not a place for adjustment.
Joint necessity: zero each of A, C, E and verify DSI returns zero. Symmetry: the three positive components must have comparable sub-element scope and the same sub-aggregator. Scale: every value remains on [0, 1]. A failed audit signals revision of the rubric, not the form.
Apply the rubric to a published case in the same decision class: Robodebt at 0.124 or Berkeley at 0.08 for AI-adjacent applications; TCG at ≈ 0.67 or GAP at ≈ 0.52 for institutional applications. If the rubric does not reproduce the anchor on the documentary record, the rubric is mis-calibrated.
The example is chosen because the framework's canonical anchors supply a calibration reference in the same decision class — Robodebt at DSI = 0.124. The decision unit is: a senior accountable official authorising the deployment, expansion, or continuation of an automated decision system that produces individualised adverse determinations against members of the public — that is, a determination capable of imposing a debt, removing a benefit, denying a service, or initiating a coercive process. Advisory analytical systems, internal triage without external consequences, and officer-level decisions under delegation are excluded.
| ID | Sub-element | What it measures, in this domain |
|---|---|---|
| A1 | Statutory authority for automated decisioning | Whether primary legislation, subordinate instrument, or executive determination explicitly authorises automation for this decision class. |
| A2 | Current delegation integrity | Whether an in-force delegation names this office-holder for this decision class with chain integrity intact. |
| A3 | Procedural standing | Whether the official is named on the approval instrument with no parallel approver whose concurrence is required but unrecorded. |
| A4 | Rule-making authority over the operating instrument | Whether the decider retains direct authority to alter, suspend, or revoke operating rules after deployment. |
| C1 | Budgeted assurance capacity | Whether independent model validation, evaluation, and ongoing monitoring are funded at risk-commensurate levels, with the budget under the decider's control. |
| C2 | Technical levers | Whether suspension, rollback, and material modification can be effected by staff reporting to the decider, without external dependency. |
| C3 | Workforce capacity at projected volumes | Whether review, appeals, and remediation capacity matches the system's expected output volumes, with documented surge capacity. |
| C4 | Time control | Whether the decider can stage or delay deployment to match assurance evidence, or whether timing is externally imposed. |
| E1 | Documented model performance | Whether the decider has, in writing, current evaluation results bearing on the deployment decision, including sub-population performance. |
| E2 | Documented operating assumptions | Whether operating assumptions and known failure conditions, including any divergence between model target and legal target, are written and provided. |
| E3 | Error-cost asymmetry analysis | Whether the decider has reviewed a written analysis of false-positive and false-negative costs at projected volumes. |
| E4 | Independent advice | Whether written advice from outside the proponent unit — legal, internal audit, oversight, or external review — is on file before approval. |
| V1 | Ministerial or executive reversal | Probability of reversal by a superior political actor within a relevant horizon under plausible scenarios. |
| V2 | Judicial or merits review | Probability of substantial modification or strike-down via judicial review, merits review, or class action. |
| V3 | Oversight intervention | Probability of substantive intervention by Ombudsman, Auditor-General, Information Commissioner, or analogous body, sufficient to force material change. |
| V4 | Public, media, or parliamentary reversal | Probability of reversal driven by public, media, or parliamentary pressure independent of the formal channels above. |
A1–A4, C1–C4, E1–E4 are each treated as jointly necessary within their component and aggregated by the fourth-root geometric mean. V1–V4 are alternative routes to reversal and aggregated by the probability-weighted union 1 − ∏ᵢ (1 − Vᵢ); where independence is implausible in a specific application, the dominant-channel maximum is used and the choice documented. The four sub-elements per component is a design choice that preserves symmetry across A, C, E — equal structural weight, same sub-aggregator, same count. Different counts across components reintroduce the imbalance the symmetry axiom rules out.
Applied to the Royal Commission documentary record, the rubric returns scores materially below one across most sub-elements: A1 reflects the contested statutory basis for automated income-averaging determination; A4 is low given that operating rules were embedded within the system in ways the senior decider did not effectively control; C1 and C3 are low given that independent assurance was not commissioned at risk-commensurate scale and review-and-appeal workforce capacity was materially below the volumes generated; E2 and E3 are low because the income-averaging assumption and the harm distribution at projected volumes were not effectively before the decider in writing at the time of the relevant approvals; V was elevated across all four channels ex ante and was realised across all four ex post. The product falls in the 0.10–0.15 range, consistent with the canonical anchor at 0.124. A rubric that returned a substantially higher DSI for Robodebt under any reading would be mis-calibrated.
The second worked example sits at the opposite end of the framework's outcome space — the sustained-positive-S configuration, anchored at DSI ≈ 0.67 by the TCG network in its productive phase, c. 1985–2015. The procedure is the same as in example 1; the operational content is necessarily different; the resulting DSI occupies a non-overlapping region of the unit interval. The decision unit is: a convener of a multi-party institutional reform process authorising the launch, continuation, or scope-extension of a structured engagement that produces shared recommendations or commitments binding on the participating parties.
| ID | Sub-element | What it measures, in this domain |
|---|---|---|
| A1 | Convening legitimacy | Whether the convener is recognised by the relevant participating parties as a legitimate convener of the class of engagement concerned. |
| A2 | Mandate clarity | Whether the scope, outputs, and reporting line of the engagement are documented and agreed by participants. |
| A3 | Chair authority | Whether the chair holds personal authority adequate to the seniority of participants and the substantive matter. |
| A4 | Agenda authority | Whether the convener controls the agenda, the framing of the substantive questions, and the sequencing of inputs. |
| C1 | Secretariat capacity | Whether the convening secretariat is staffed and resourced at a scale matching the engagement's substantive scope. |
| C2 | Participant access | Whether the convener controls invitation, exclusion, and replacement of participants. |
| C3 | Deliberative process control | Whether the convener controls the rhythm and structure of meetings, working groups, and side conversations. |
| C4 | Output control | Whether the convener controls the form, timing, and circulation of recommendations and commitments. |
| E1 | Substantive expertise on file | Whether the convener has access, before each engagement decision, to current substantive expertise on the matter under engagement. |
| E2 | Participant intelligence | Whether the convener has current, documented understanding of each participating party's interests, constraints, and likely positions. |
| E3 | Comparative-case knowledge | Whether the convener has on file structured knowledge of analogous engagements and their outcomes. |
| E4 | Real-time deliberative information | Whether the convener has live information on the state of agreement, dissent, and movement within the engagement. |
| V1 | Participant withdrawal | Probability of material participant departure forcing scope or output revision. |
| V2 | External political reversal | Probability of substantive political intervention overriding the engagement's recommendations. |
| V3 | Regulatory or jurisdictional reversal | Probability of regulatory action vacating the engagement's authority over the matter. |
| V4 | Reputational reversal | Probability of reputational events forcing the convener to suspend, restructure, or abandon the engagement. |
Applied to the documentary record of the TCG network in its productive phase, the rubric returns values close to but below one across A, C, E, with V low across all four channels. A sits in the 0.85–0.95 range: convening legitimacy was established by reputation accumulated over the productive phase; mandate clarity, chair authority, and agenda authority were near-uniformly high. C sits similarly: secretariat capacity scaled with engagement scope; participant access was actively curated; deliberative process and output control were features of the operating model rather than ambitions. E is high but not uniformly at one — comparative-case knowledge was lower in earlier engagements and higher in later — and sits around 0.85. V is low across all four channels in the productive phase, sitting in the 0.05–0.15 range. The resulting DSI is in the 0.60–0.75 range, consistent with the canonical anchor at 0.67. A rubric that returned a low DSI for the TCG productive-phase configuration under any reasonable reading would be mis-calibrated.
The two examples together demonstrate the rubric's discriminating power across the outcome space the framework maps. The procedure is the same; the operational content is necessarily different; the resulting DSIs — 0.10–0.15 for Robodebt, 0.60–0.75 for TCG productive phase — occupy non-overlapping regions of the unit interval consistent with the framework's cross-case anchors.