Observed reality is only one realized path.

Which features of corporate control must a macro state retain to preserve a declared policy response?

Two economies can look identical in national aggregates yet react differently to monetary policy when corporate control is organized differently. The rank and singular spectrum of the hidden-response operator characterize the minimum corporate-control information needed to preserve a chosen policy response. In the calibrated exercise, retaining one corporate coordinate cuts fitted worst-case employment-response error from 0.1125 to 0.0165 log percentage points.

When does a low-dimensional organizational report remain sufficient for future responses after feasible policy cycles?

A dashboard can return to exactly the same numbers after a policy cycle even though the organization underneath it has changed in ways that matter for the next shock. Policy holonomy links closed report paths to latent state motion, while direction-reversed and mirrored loops identify target curvature and the additive state needed for dynamic aggregation. In LaDe data, six internal-state moments reduce next-day backlog RMSE by 6.13% beyond report history and accepted flow.

When policy options require complementary inputs, which packages actually create new actionable options and how much does each input contribute?

A policy option often appears only when several complementary inputs are available together, so evaluating inputs one at a time can miss what the package actually creates. Minimal enabling input paths represent options, Möbius coefficients and linear programs recover or bound package-specific option mass, and a dual prices supplied inputs. In the childcare and work applications, the expanded timing package raises model-assisted job-compatibility mass from 0.83% to 23.74%, with 93.1% of repaired mass requiring at least two timing blocks.

How does policy history change the response capacity available to future interventions?

A system can look stable today while its ability to respond to tomorrow’s intervention still depends on the policies it experienced before. Policy response capacity is a finite-horizon derivative or contrast, and identical baselines can support different response ranks. In the applications, prior notification history changes a current app-use response by 1.75 percentage points, while randomized app limits change a bonus response by 11.4 minutes per day.

What retained information is sufficient to revalue the same economic object across new environments?

To value the same object in a new environment, you do not need every old valuation; you need the right records. The rank and spectrum of a residual valuation map characterize the minimum target-directed record set. In F-MAP, four selected records cut held-out MSE by 71.7%, and two composition records recover 57,600 held-out valuations to numerical precision.

Which features of event-time validation are required for a specified decision when the economic event and its recorded date do not coincide?

When an event’s true time and recorded time differ, the decision usually needs only part of the timing-error distribution, not a perfectly corrected timestamp. For polynomial timing loss of degree d, moments through d recover the scheduled-loss function. Under quadratic loss, mean displacement determines the schedule shift while variance determines flexibility and execution-loss gaps, illustrated with the June 2022 European gas network response.

How do institutions’ changing reporting priorities shape the archive and the policy learning available to future decision makers?

Institutions optimize reports for today’s needs, but those choices determine what future researchers and policymakers can still learn. Reporting specialization creates an archive externality, and valuing future use slows specialization. In U.S. Call Reports, a common acquisition score reduces mortgage-composition MSE by 21.43–25.90% while matching forecast-specific report choices at least 95.58% of the time.

Which distinctions in a structural state must a representation preserve for a declared family of counterfactuals?

A compressed state is useful only if it preserves the distinctions that matter for the counterfactuals you actually want to answer. Target-derivative rank, spectral tails, and a conditional-generator certificate separate exact sufficiency, local repair, and global decoder risk. In the KW97 exercise, the graph certificate lowers held-out target RMSE by 1.8–8.2% at the same three-coordinate budget.

How far must a structural model’s causal response move before its monetary counterfactual changes sign?

Models can agree near the data and still disagree on the sign of a policy counterfactual once the estimated causal response moves far enough. Comparing each structural family’s entry radius with the first sign-reversal radius makes cross-model agreement experiment-specific. In the archived four-family exercise the gap is 0.8509, while re-estimation and structural profiling can make it negative.

When a policy passes through administrative handoffs, which information must survive or be acquired for a downstream counterfactual to remain identified and precise?

A policy can survive one administrative handoff intact and lose another counterfactual at the very same handoff. Each handoff is represented as a Markov transformation plus conditional information, which yields target-specific loss and acquisition ranks. In the Oregon Medicaid lottery, the loss rank ranges from 0 to 4 across targets, while a fixed-decision release reads 40–49% of invitation records in expectation.

When can policy-regime marginals determine how outcomes are paired across worlds, and therefore the distribution of policy gains?

Marginal outcome distributions determine policy-gain distributions only after the cross-world coupling is disciplined; cofactual components provide one such structure and observable restrictions. Möbius-transformed cumulant-generating functions characterize and construct the independent shared-component class. Intersection closure gives the target-directed regime bundle, positive-semidefinite hierarchies support specification testing, and paired-world defects and positive-definiteness characterize departures and global gain-law sensitivity.

When do locally compatible experimental and administrative records arise from one global economic history, and what information repairs the remaining gap?

Separate studies can describe compatible pieces of an economy while still failing to form one global history; the missing bridge depends on the counterfactual being asked. A connecting map identifies the obstruction to a global lift, and its target quotient gives sharp identified sets, minimal record counts, and error amplification. In the Oregon Medicaid application, declared targets require zero, one, or four bridge directions.

Which lowest-cost collection of theoretical, measurement, and empirical modules resolves a declared counterfactual to the desired precision?

A collection of records can leave the complete economy unsettled while still determining the policy conclusion that matters. Target certificates characterize exact recovery, least-cost acquisition, minimum repair, and attained approximation frontiers. In the Federal Reserve's 2025 stress-test disclosures, revenue and loss modules determine the sign of cumulative pre-tax income for 20 of 22 banks, while minimum capital ratios require path information.

Marginal iid symmetry of bank-specific cost shocks does not by itself sign the selected boundary moment when the realized positive-choice count depends on those shocks.

The published growth condition can leave centered forward-simulation noise in the root-n limit, requiring an added simulation covariance term or stronger simulation rates.

The printed spectral condition does not by itself deliver the stated determinacy/root-count conclusion, and repeated roots require generalized rather than simple persistence modes.

The printed logarithmic residual-income utility and the distributed first-order price term can generate different rankings, choice probabilities, derivatives, markups, and welfare calculations.

The released programs combine proportional viewing, fixed-reference additive channel values, and choice kernels that need not coincide with the set-valued demand object defined by the printed model.

A model satisfying Assumptions 1–5 can still produce an unbounded observable similarity, unbounded conjugated spectral projections, and density columns outside the inverse operator's domain.

A terminal-summary learning rate must compare the two state laws along the same endogenous history and belief path; the cross-boundary Kullback–Leibler divergences in Theorem 4 can exceed the information available in that history.

Holding expected degree fixed does not sign the effect of homophily on reach; higher within-island connectivity can outweigh fewer cross-island links in the high-reliability all-share branch.