State and restoration measurement

### PDF page 15 Daniel J. Murray Revised September 2026 12. Conclusion Biology does not need a theorem proving that all living systems remember. It needs a disciplined rule for deciding what may be forgotten. Once the biological unit, admissible future family, horizon, and material response margins are fixed, a representation qualifies as state only to the extent that histories it merges induce equivalent future laws. Exact state is the zero-distance quotient of history; finite biological equivalence is a predictive ball. The empirical history effect is measured after freezing and matching the present, indexed by a named history generator, and inferred with simultaneous control of the planned maximum and an explicit bound on residual matching error. Strata that cannot be decided remain unresolved, and finite negative batteries are not extrapolated beyond their scope without a coverage theorem. This standard also sets a boundary on mechanism. Slow or hidden variables are not memory merely because they persist: to explain a retained-history defect, history must shift their conditional distribution at the matched present and a future must read that shift. A positive predictive defect can still arise from altered composition or selection rather than within-unit rewriting, so causal attribution requires additional lineage or intervention evidence. Viable continuation remains a distinct physical-state problem. Its tangent geometry, accessible dynamics, and persistence reserve belong to a physical realization, not automatically to the predic- tive quotient. Restoration therefore requires independent post-control validation of both predictive equivalence and persistence. The law is not “biology has memory. ” It is the measurement boundary on that claim: a past distinction may be discarded as state only when, within the declared horizon and biological margins, it cannot change any admissible future law. Whether conventional biology crosses that boundary often enough to matter is no longer assumed. It is an experiment. Appendix A. Minimum reporting standard for a Biological State Test • Declare B, K, A, H, Φ𝐵, 𝜀, the candidate present M, and history generator Q_C before outcome inspection. • State whether Q_C is naturalistic prevalence sampling or a designed stress-test generator. • Define common support of the matched present and do not extrapolate conditional contrasts outside overlap. • Freeze the candidate present before assigning the common future; report assay reliability, matching tolerance, and attrition by history arm. • Name the target: observed-state conditional law or a separately identified latent-state law. Specify the pairwise matching-transport bound and the evidence that controls dominate the target distortion. • Construct the uncertainty guarantee for the final decision jointly across response-law bounds and the matching envelope; when many strata are classified, cover the stratum-by-future-by- stage decisions being counted or propagate their classification error explicitly. • If state is augmented from 𝑀 to (𝑀 , 𝑍), let 𝑅 be the history-arm label and require overlap within tested (𝑀 , 𝑍)strata. Deterministic separation of 𝑅 by 𝑍 cannot establish a compara- tive empirical certificate for 𝑌 ⟂ 𝑅 ∣ (𝑀 , 𝑍) . • Prefer passive matching. If active feedback is used, record the full actuator trajectory, treat it as part of history, and include replay/sham controls for controller-written state. ### PDF page 16 Daniel J. Murray Revised September 2026 • If Z is read only after the future challenge, preregister and pass a challenge-invariance control showing that the future does not rewrite Z. • Control the maximum jointly over the full planned battery and planned expansion stages; never interpret a raw sample supremum. • Use biological equivalence margins justified by function or decision, not by statistical signifi- cance. • Report separated, scope-equivalent, and unresolved strata and a prevalence interval. • For a negative conclusion, report exact future scope, horizon, sensitivity, and any finite- coverage allowance. • If a carrier is proposed, estimate the full conditional shift Λ𝑍 at matched M and perturb the candidate before causal attribution. • Keep viability geometry in a stated physical realization X and separate discovery, assay cer- tification, and restoration validation. • Label each conclusion as theorem-level, simulation-calibrated, model-derived, retrospective motivation, prospective empirical result, or untested prediction. Appendix B. Verification examples and inferential boundaries B.1 The two-sided transport bound For pseudometric 𝑑, the triangle inequality twice gives |𝑑(𝑃𝐴, 𝑃𝐵) − 𝑑(𝑃 ∗ 𝐴, 𝑃∗ 𝐵)| ≤ 𝑑(𝑃 𝐴, 𝑃∗ 𝐴) + 𝑑(𝑃𝐵, 𝑃∗ 𝐵). Thus Eq. (10) follows on the joint confidence-and-transport event. With Bernoulli target parameters 0.5, 0.5 and observed parameters 0.7, 0.3, each arm moves by total-variation distance 0.2 while the pairwise distance changes by 0.4. The maximum of the two arm radii would not cover that distortion. B.2 An observed assay and its latent target need not have the same sufficiency status Let 𝑃 (𝑋 = 1 ∣ 𝑞 𝐴) = 1/4 , 𝑃 (𝑋 = 1 ∣ 𝑞 𝐵) = 3/4 , let 𝑀 independently flip the binary state 𝑋 with probability 1/5, and let the common future be 𝑌 = 𝑋 . At 𝑀 = 1 , the two probabilities of 𝑌 = 1 are 4/7 and 12/13; at 𝑀 = 0 they are 1/13 and 3/7. The total-variation defect is 32/91 in either observed stratum, although conditional on 𝑋 the histories have identical future laws. Both histories have common support at every tested 𝑀 and 𝑋. This is genuine insufficiency of measured 𝑀, even with exact observed-state matching, and not evidence for a further effect beyond the latent state. B.3 Protocol mixtures and full-history fibres Let the full recorded history be ℎ = (𝑞, 𝑈 ), with independent fair binary 𝑞, 𝑈, constant 𝑀, and future 𝑌 = 𝑈 . The protocol-conditioned laws are identical Bernoulli (1/2) laws. Nevertheless histories with recorded 𝑈 = 0 and 𝑈 = 1 have deterministic, different futures. Protocol-level agreement therefore does not imply Eq. (4)’s full-history diameter bound. Both levels remain useful when their scopes are named. B.4 Feature resolution The pseudometric 𝑑(𝑃 , 𝑄) = |𝐸𝑃 𝑌 −𝐸 𝑄𝑌 | gives zero distance between 𝛿0 and (𝛿−1 +𝛿1)/2, although the distributions differ. A zero-distance quotient is always mathematically legitimate; identifying it with equality of full response laws additionally requires separating distances or a separating family of features. ---

A Law of Biological State Sufficiency History-conditioned prediction, viable continuation, and a test for when the past may be forgotten Daniel J. Murray September 2026 Abstract a present representation may discard a history distinction only when that distinction cannot change any future response law within scope. With distances that separate the declared response laws, ex- act state is the zero-distance quotient of history; finite biological equivalence is a predictive ball, not an equivalence class. I then give a lawful empirical test. A named history generator creates different prior routes, the candidate present is frozen and matched before an identical held-out future battery, and retained history is defined as a conditional future-law defect rather than pairwise heterogene- ity. Because the estimand is a maximum, inference is simultaneous across the planned battery; tions show maximum-selection error, conservative inference under a specified matching-distortion bound, and recovery of separated synthetic prevalence. They test an idealized inferential model; measurement-error-induced predictive insufficiency of an observed assay is distinguished from distor- tion in estimating a noise-free target. A conditional-state result shows why slow or hidden variables are not memory unless history shifts their conditional law at the matched present and a future reads that shift. Viability remains a physical- state constraint and is used independently in restoration. To make the standard directly calibratable, I specify a prospective E. coli recombinase experiment with designed null-control and positive-control history pairs, overlap within an augmented state (M,Z), and held-out history transport. The contribution is a falsifiable biological measurement standard, not a claim that biology must remember. Keywords biological state; predictive sufficiency; history dependence; viability; restoration; treatment schedul- ing; systems biology 1. Introduction: the measurement problem before the mechanism An endpoint can be measured correctly and still be the wrong state. Biology routinely describes a system by what can be observed now: tumour burden, metabolite concentration, a transcriptional embedding, morphology, accumulated exposure, physiological performance, or a clinical score. Such measurements can be excellent descriptions of the present. They qualify as state variables only for futures they are sufficient to predict. If two biological populations are indistinguishable under a

frozen present representation but respond differently to the same later intervention because they arrived there by different routes, the measurement was not false. It was incomplete. The past-future interface is not a new idea. Control and realization theory, automata, predictive- state representations, and computational mechanics all formalize the requirement that, once a suffi- cient state and future inputs are given, discarded history should add no further predictive informa- tion (Kalman, 1960; Nerode, 1958; Willems, 1986; Jaeger, 2000; Littman et al., 2001; Shalizi and Crutchfield, 2001). The contribution here is narrower and biological: an experimental measurement law for situations in which the state vector is unknown, the candidate state is an assay or phenotype, history can be deliberately manipulated, persistence can fail, and the environment itself can store predictive information. State-sufficiency law. For a declared biological unit, admissible future family, horizon, and response margins, a present representation may discard a distinction from prior history only if that distinction cannot change any admissible future response law within the declared scope. The law is constitutive rather than a universal prevalence claim. It constrains what may legitimately This distinction matters whenever a normalized present is used to withdraw treatment, declare restoration, rank schedules, infer recovery, or compress exposure history into one current number. The paper therefore separates four problems that are often conflated: predictive state, empirical retained-history prevalence, causal carrier attribution, and physical viability. The scientific contri- bution is the set of rules that keeps those four problems connected without identifying them with one another. 2. Declaring the state problem 2.1 Five biological declarations and an explicit future family Five biological objects must be fixed before outcome inspection: the unit and observation algebra B; the persistence set K; the admissible intervention or policy class A; the prediction horizon H; and response- specific biological margins 𝜀. The future response family itself must also be explicit. Let Φ𝐵 denote the declared response functionals of B and A^{ ≤H} the admissible policies of duration at most H. Define Symbol Declaration Role

Daniel J. Murray Revised September 2026 Symbol Declaration Role 𝐻 Prediction/continuation horizon Limits the state claim 𝜀𝑗 Response-specific biological margins a stated coverage relation. The history generator Q_C introduced in Section 4 is not another definition of state. It indexes an empirical prevalence question over which histories are sampled. The roles are asymmetric. B, A, H, and Φ𝐵 define prediction. 𝜀 turns exact equality into practical biological equivalence. K defines which physically possible continuations preserve the declared unit. Changing any of them after viewing outcomes changes the question rather than refining the answer. 2.2 Horizon nesting If 𝐻1 ≤ 𝐻2 and the shorter-horizon family is a restriction of the longer one, then 𝒯𝐻1 relapse at six months. Increasing the horizon can reveal additional distinctions; it cannot erase a distinction already visible in a nested shorter-horizon family. 3. Predictive state as a metric object 3.1 Exact state and finite biological equivalence are different Let H be the realizable history space. For future 𝑗, let 𝑑𝑗 be a pseudometric on its response laws and let 𝜀𝑗 > 0 be its biological margin. Equality of the declared laws is identified by zero distance only when the chosen distances separate those laws; a mean-only pseudometric instead certifies equality A finite maximum of pseudometrics is a pseudometric. A supremum is likewise a pseudometric when finite, or an extended pseudometric if infinite values are allowed. Zero distance defines an equivalence relation in either case. With a separating family of response-law distances, exact predictive state is the quotient ℎ ∼ 𝒯 ℎ′ ⟺ Δ 𝒯(ℎ, ℎ′) = 0. (3) Finite tolerance is not another equivalence relation. Practical equivalence is the predictive ball Δ𝑇 (h,h’) ≤ 1. Pairwise within-margin chains can drift beyond the margin, so no transitivity of practical equivalence is assumed. This distinction repairs the common mistake of writing an 𝜀- equivalence class as though finite experimental indistinguishability were exact identity.

Daniel J. Murray Revised September 2026 For a candidate measured present 𝑀 = 𝜋(ℎ) , metric 𝑑𝑀 , and tolerance 𝑟, define the worst predictive separation among the history pairs it merges. The supremum is taken over a declared history domain, and the record defining ℎ includes the realized assay observation when the assay is stochastic. 𝐷𝑀,𝑟 = sup{Δ𝒯(ℎ, ℎ′) ∶ 𝑑 𝑀 (𝑀 (ℎ), 𝑀 (ℎ′)) ≤ 𝑟}. (4) This is a global predictive fibre-diameter bound, not a geometric width in physical state space. The condition 𝐷𝑀,𝑟 ≤ 1 is sufficient for within-margin agreement of every declared history pair. The protocol-conditioned estimand in Section 4 is different: it compares mixtures of histories generated by named protocols. Agreement of those mixtures does not establish Eq. (4), even with complete future coverage, unless the sampled protocols distinguish or otherwise cover the relevant within-

future law after the declared present is known. Here 𝑞 is a protocol or recorded-history label, not automatically the complete realized history ℎ. A zero value of 𝐺 certifies neither unsampled history contrasts nor absence of predictive information in finer history records. 4.3 Predictive history dependence is not automatically within-unit causal mem- ory The present is measured after the histories have occurred. That is essential to the state question and dangerous for causal interpretation. Even with randomized histories, conditioning on a post- history present can make history identity informative about a pre-existing latent variable; differential survival or selection can do the same. A positive G therefore identifies predictive insufficiency of M in the sampled population, not by itself a history-written intracellular carrier. Three claims must be kept separate: predictive insufficiency, in which history identity still changes future law after conditioning on M; history-induced composition, in which histories changed which latent states or lineages occupy the matched stratum; and within-unit rewriting, in which predic- tive state changed inside a persisting unit or lineage. Only the first is identified by G_{C,Q,H}. The latter two require baseline covariates, lineage or sister-unit designs, attrition accounting, or a

level. Technical replicates are not new biological units. Additional futures, better matching, or more replication are useful when they shrink the unresolved region. Battery expansion is stopped by a preregistered decision/precision criterion or by meeting a declared coverage allowance, not by observing a plateau in a raw maximum. If prevalence is formed by classifying many strata, the inferential guarantee must cover the full stratum-by-future-by-planned-stage set of decisions that feed n_sep and n_eq, or a hierarchical model must explicitly propagate classification uncertainty. Per-stratum 95% calls cannot simply be counted and called a 95% prevalence statement. Equation

the corrected rate ranges from zero to 0.000180. The unresolved fraction rises from 0.804365 to 0.980215. This demonstrates conservative behavior for the specified distortion model, not that noisy observed-state predictive differences should be removed. Simulation C draws 200 independent strata in each replicate, with normalized distances 1.6 for positive strata and 0.4 for practical-null strata. For prevalences 𝐺 = 0, 0.1, … , 0.5, the mean lower identification bounds are 0, 0.100432, 0.200250, 0.299420, 0.401104, and 0.500176. These well- separated examples illustrate recovery of prevalence. The code uses per-stratum intervals and does

exclusion therefore needs the relevant joint or conditional carrier distribution. State augmentation creates a second support requirement. Let 𝑅 be the randomized history-arm label. To test whether adding 𝑍 closes the predictive defect, at least two compared histories must overlap within the same (𝑀 , 𝑍)strata: 0 < 𝑃 (𝑅 = 𝑞 ∣ 𝑀 = 𝑚, 𝑍 = 𝑧) < 1 on the target support. If 𝑍 determines 𝑅, conditioning removes the cross-history comparison. A completion experiment must therefore use probabilistic state writing or several genuinely different historical routes reaching each tested 𝑍. The closure claim remains restricted to overlapping histories, states, and futures. Mechanism hunting requires a stopping rule. Candidate carriers should be preregistered. If the prespecified set is exhausted, the retained history remains real but unattributed. A new candidate can motivate a new experiment; it cannot retroactively rescue the mechanism claim in the current one. 8. Viable continuation is a distinct physical-state problem 8.1 Predictive quotient and viability geometry must not be conflated Predictive sufficiency is not uniquely biological. The biological restriction begins when some phys- ically possible futures do not preserve the declared unit. Let 𝑋 be a physical state space with accessible differential inclusion ̇ 𝑥 ∈ 𝐹 (𝑡, 𝑥), and let 𝐾 ⊆ 𝑋 be the persistence set. For a fixed terminal time 𝐻, the finite-horizon viability object is the time-indexed tube 𝒱𝐻(𝑡) = {𝑥 ∈ 𝐾 ∶ ∃𝑥(⋅), 𝑥(𝑡) = 𝑥, ̇ 𝑥(𝑠) ∈ 𝐹 (𝑠, 𝑥(𝑠)), 𝑥(𝑠) ∈ 𝐾 ∀𝑠 ∈ [𝑡, 𝐻]}. (16) finite-horizon viable reach set. Rate- limited control authority, tangent geometry, and reachability are therefore different objects. The predictive quotient can be represented by mapping each physical state to its family of future response laws and identifying states with the same image. Nothing in that quotient construction implies that it inherits the manifold or tangent-cone geometry of X. Persistence outcomes can be included among the declared future responses, but the physical geometry does not automatically descend through predictive equivalence.

Daniel J. Murray Revised September 2026 PREDICTIVE DESCRIPTION History-conditioned future laws Declared responses and margins Protocol scope and uncertainty PHYSICAL REALIZATION Metric-dependent reserve Restoration validates both after support is withdrawn. Figure 3. Predictive measurement and physical viability are distinct objects. Histories merged by the same frozen present are compared through their future response laws, while finite-horizon viability and tangent geometry belong to a stated physical realization. Predictive equivalence does not automatically inherit the geometry of physical state space. 8.2 External is not exogenous A contextual variable written by a biological unit and read later can retain predictive information outside a conventional organism boundary. Omission breaks a proposed closure when that variable carries a history-conditioned distinction affecting declared future laws that the retained representa- tion neither determines nor correctly marginalizes. If, for example, the context is a known function of retained state, a separate context coordinate is unnecessary. Extracellular matrix, secreted metabo- lites, electrical fields, biofilm architecture, and constructed niches can carry external predictive state (Odling-Smee et al., 2013; Prindle et al., 2015; Molina-Santiago et al., 2019). Experimental clamp- ing can remove a particular contextual dependence, but is not the only way to obtain closure. This is a criterion for predictive and causal modeling, not a claim that every environmental variable belongs to the organism. 8.3 Restoration requires independent post-control validation A logical circularity arises if the same future battery is used to build a state representation and then to declare restoration by matching that representation. The stronger design separates discovery, sentation; and after treatment or support is withdrawn, an independent validation future set tests the post-control trajectory together with physical persistence. For a target predictive state 𝑆∗, choose a physical metric and an explicit robust viability tube 𝒱rob 𝐻 (𝑡) for the declared control/disturbance information pattern. One possible reserve is 𝜌𝐻(𝑡, 𝑥) = dist(𝑥, 𝑋 ∖ 𝒱rob 𝐻 (𝑡)) for 𝑥 inside that tube, and zero outside it. This is a metric-dependent interior

evidence, report validation only at observed times. Failure to reject a difference is not an equivalence certificate. 9. Biological anchors and a prospective calibration experiment The existing literature contains useful anchors, but none is treated here as a measurement of G_{C,Q,H}. Treatment scheduling shows why route can matter after a short-horizon endpoint. Patwardhan et al. (2021) evaluated 696 crizotinib/navitoclax schedules; among 486 schedules meeting a pre- features emerged later (Wiernicki et al., 2022). The measurement-law lesson is not a universal bifur- cation geometry. It is that current injury and future rescuability can be different state questions. 9.1 Prospective synthetic calibration with designed controls The cleanest first demonstration is a synthetic calibration because a history-writing channel can be engineered and tested independently. A DNA-state match alone does not guarantee equality of all omitted variables or future laws. In E. coli, recombinase state machines can store input history as a DNA state Z and use that state to regulate later gene expression (Roquet et al., 2016). Choose

tories reaching the same Z are scope-equivalent under (M,Z); and a held-out history, not used to choose the representation or the confirmation battery, reaches the same Z and transports the pre- diction. This demonstrates M-only failure and recovery of predictive agreement over the declared overlapping history and future scope without reconstructing every latent physical variable. No sister-cell equivalence is required: M and, when an orthogonal state reporter has been validated, Z are read nondestructively in the same biological unit that receives the future challenge. Passive matching is preferable because it adds no controller trajectory. If active feedback is required, the whether the matching controller itself writes persistent state. The calibration would validate the measurement procedure, not establish that natural biological snapshots commonly fail it. 10. Predictions and failure conditions The framework makes failure possible at every level. Under a true practical null, the probability of any false separation must remain below the declared global error level across the planned battery and stages. It can increase within that bound as tests are added; exceeding the guarantee, rather than any increase, signals failure of calibration or its assumptions. Coarser matching can enlarge prespecified future can reject a state representation, but a finite negative battery cannot establish global sufficiency without a justified coverage theorem. Finally, a representation selected on one battery should predict untouched admissible futures after treatment withdrawal; if route of arrival regains predictive value, restoration was overclaimed. The top-level empirical claim is therefore deliberately defeasible. For a prospectively named class and generator, a materially positive snapshot excess is rejected when a sufficiently precise, properly calibrated design places the upper prevalence bound at or below the decision-relevant threshold. An underpowered or poorly matched study is unresolved, not a negative result. 11. Discussion

Daniel J. Murray Revised September 2026 That combination changes the object of inference. The paper does not ask whether a challenged system responds. It asks whether history remains predictive after a specific present representation says history should be forgettable. It does not count pairwise heterogeneity as memory. It does not infer global sufficiency from a finite null. It does not let a larger battery create positives by uncalibrated maxima. It does not convert a post- history predictive contrast into a within-unit causal mechanism. And it does not draw physical viability cones on an abstract predictive quotient. The resulting law is constitutive: once a biological question is fixed, any valid state representation must respect future-law equivalence within the declared margins. The empirical prevalence of failures is a separate quantity. The constitutive criterion remains definitional; the applicability, calibrated procedure, and prospectively specified prevalence hypotheses are empirically testable. 11.2 Why the stronger claims were removed Several attractive extensions are unnecessary and weaken the paper. An unrestricted infinite future family does not become experimentally meaningful by invoking “saturation”; the lawful alternative is a declared finite scope or an explicit coverage theorem. An occupancy-style latent-detection not imply memory. Commutation does not prove scalar sufficiency. Boundedness does not impose a universal hyperbolic or relativistic state geometry. Viability does not generate memory and need not share predictive geometry. Removing those claims leaves a smaller but stronger paper. State is future-law sufficiency. Re- tained history is conditional future-law dependence after a matched present. Viability is a physical persistence constraint. Restoration is independent post-control validation of both. 11.3 Limitations may fail common support and leave many strata unresolved. Same-history controls only correct residual mismatch when their transport relation to the cross-history comparison is defensible. A positive G identifies predictive history dependence, not within-unit causal rewriting. Prevalence is generator-dependent by construction. Viability requires a physical realization when tangent geom- etry or reserve is discussed. The simulation calibration establishes statistical behavior on known constructions, not biological prevalence. The decisive next step is a prospective biological experiment satisfying the complete history-match-common-future design. The proposed synthetic experiment is an engineering calibration rather than evidence of natural prevalence, and its state- completion step is identifiable only where histories overlap within (M,Z). If active matching is used, controller- induced state must be treated as a possible history effect rather than silently corrected away. The 11.4 On the word “law” The word law is used in a constitutive measurement sense: it is a rule that any claimed biological state must satisfy for the declared prediction problem. Readers who reserve “law” for universal empirical regularities may substitute “biological state-sufficiency principle” without changing a def- inition, theorem, estimator, or experiment. What is empirical and genuinely open is how often conventional biological snapshots violate the rule in important systems.

Once the biological unit, admissible future family, horizon, and material response margins are fixed, a representation qualifies as state only to the extent that histories it merges induce equivalent future laws. Exact state is the zero-distance quotient of history; finite biological equivalence is a predictive ball. The empirical history effect is measured after freezing and matching the present, indexed by a named history generator, and inferred with simultaneous control of the planned maximum and an explicit bound on residual matching error. Strata that cannot be decided remain unresolved, and finite negative batteries are not extrapolated beyond their scope without a coverage theorem. This standard also sets a boundary on mechanism. Slow or hidden variables are not memory merely because they persist: to explain a retained-history defect, history must shift their conditional distribution at the matched present and a future must read that shift. A positive predictive defect can still arise from altered composition or selection rather than within-unit rewriting, so causal attribution requires additional lineage or intervention evidence. Viable continuation remains a distinct physical-state problem. Its tangent geometry, accessible dynamics, and persistence reserve belong to a physical realization, not automatically to the predic- tive quotient. Restoration therefore requires independent post-control validation of both predictive equivalence and persistence. The law is not “biology has memory. ” It is the measurement boundary on that claim: a past distinction may be discarded as state only when, within the declared horizon and biological margins, it cannot change any admissible future law. Whether conventional biology crosses that boundary

Daniel J. Murray Revised September 2026 • If Z is read only after the future challenge, preregister and pass a challenge-invariance control showing that the future does not rewrite Z. • Control the maximum jointly over the full planned battery and planned expansion stages; never interpret a raw sample supremum. • Use biological equivalence margins justified by function or decision, not by statistical signifi- tification, and restoration validation. • Label each conclusion as theorem-level, simulation-calibrated, model-derived, retrospective motivation, prospective empirical result, or untested prediction. Appendix B. Verification examples and inferential boundaries B.1 The two-sided transport bound For pseudometric 𝑑, the triangle inequality twice gives |𝑑(𝑃𝐴, 𝑃𝐵) − 𝑑(𝑃 ∗