Distinguishability Transport and Reconstruction Theory — DTRT

PART I — FOUNDATIONAL OBJECTS

1. Probability States

1.1 Probability measures as states
1.2 Sample spaces and measurable structure
1.3 Probability law versus density representation
1.4 Support and effective support
1.5 Joint and marginal states
1.6 Conditional states
1.7 Process-valued states
1.8 Model families as sets of probability states
1.9 State equivalence
1.10 State comparison

2. Ordered Probability Pairs

2.1 (P,Q) as the primitive comparison object
2.2 Directionality of comparison
2.3 Symmetric versus asymmetric comparison
2.4 Pair equivalence
2.5 Pair orbits under transformations
2.6 Pair invariants
2.7 Pairwise distinguishability
2.8 Family-relative distinguishability
2.9 Pairwise versus multi-state comparison
2.10 Comparison without scalarization

3. Carriers

3.1 Definition of a carrier
3.2 Sample-space carriers
3.3 Observable carriers
3.4 Joint carriers
3.5 Conditional carriers
3.6 Latent-state carriers
3.7 Path-space carriers
3.8 Multiscale carriers
3.9 Model-family carriers
3.10 Interventional carriers
3.11 Carrier maps
3.12 Carrier equivalence
3.13 Carrier refinement
3.14 Carrier reduction
3.15 Carrier adequacy

PART II — DISTINCTION STRUCTURE

4. Distinctions

4.1 Observable distinctions
4.2 Probabilistic distinctions
4.3 Support distinctions
4.4 Moment distinctions
4.5 Dependence distinctions
4.6 Conditional distinctions
4.7 Temporal distinctions
4.8 Scale distinctions
4.9 Directional distinctions
4.10 Causal distinctions

5. Distinction Profiles

5.1 Scalar distinguishability versus structured distinguishability
5.2 Distinction profile Π_C(P,Q)
5.3 Profile components
5.4 Profile refinement
5.5 Profile projection
5.6 Profile equivalence
5.7 Profile dominance
5.8 Partial ordering of profiles
5.9 Hidden distinctions
5.10 Unobservable distinctions
5.11 Carrier-dependent distinctions
5.12 Task-dependent distinction profiles

6. Scalar Readouts of Distinction

6.1 KL divergence
6.2 Reverse KL
6.3 Jeffreys divergence
6.4 f-divergences
6.5 Rényi divergences
6.6 Hellinger distance
6.7 Total variation
6.8 Wasserstein distance
6.9 Integral probability metrics
6.10 Operational error exponents
6.11 Why no single scalar determines the full distinction profile

PART III — TRANSFORMATION THEORY

7. Transformation Classes

7.1 Deterministic bijections
7.2 Deterministic many-to-one maps
7.3 Markov kernels
7.4 Noise channels
7.5 Marginalization
7.6 Conditioning
7.7 Quantization
7.8 Compression
7.9 Projection
7.10 Coarse-graining
7.11 Dynamical evolution
7.12 Parameter transformations
7.13 Model transformations

8. Transformation Composition

8.1 Sequential transformations
8.2 T2∘T1
8.3 Parallel transformation paths
8.4 Commuting transformations
8.5 Noncommuting transformations
8.6 Transformation semigroups
8.7 Reversible transformation groups
8.8 Idempotent transformations
8.9 Transformation chains
8.10 Path dependence under transformation

9. Common Versus State-Dependent Transformation

9.1 Common transformation of P and Q
9.2 State-dependent maps
9.3 Adaptive transformations
9.4 Observation-conditioned maps
9.5 Learning-dependent maps
9.6 Intervention-dependent maps
9.7 Why data-processing claims require transformation typing
9.8 Transformation equivalence

PART IV — TRANSPORT OF DISTINGUISHABILITY

10. Distinguishability Transport

10.1 Distinctions before transformation
10.2 Distinctions after transformation
10.3 Transport map on distinction profiles
10.4 Exact preservation
10.5 Partial preservation
10.6 Contraction
10.7 Erasure
10.8 Concealment without destruction
10.9 Emergence of new observable distinctions
10.10 Transformation-relative distinguishability

11. Preservation Laws

11.1 Invariance under bijective transport
11.2 Representation invariance
11.3 Group invariance
11.4 Orbit invariance
11.5 Sufficient transformations
11.6 Equality under data processing
11.7 Pair-specific preservation
11.8 Family-wide preservation
11.9 Preservation under composition
11.10 Preservation without invertibility

12. Contraction Laws

12.1 Data-processing inequality
12.2 D(TP‖TQ) ≤ D(P‖Q)
12.3 Strict contraction
12.4 Pair-dependent contraction
12.5 Contraction coefficients
12.6 Strong data processing
12.7 Repeated-channel contraction
12.8 Differential contraction
12.9 Mixing-induced contraction
12.10 Information decay rates

PART V — LOSS STRUCTURE

13. Distinction Loss

13.1 Loss versus contraction
13.2 Loss versus concealment
13.3 Loss under marginalization
13.4 Loss under quantization
13.5 Loss under projection
13.6 Loss under noise
13.7 Loss under aggregation
13.8 Loss under finite resolution
13.9 Loss under model restriction
13.10 Irrecoverable loss

14. Loss Profiles

14.1 Native loss profile Λ_T(P,Q)
14.2 Scalar loss D(P‖Q)−D(TP‖TQ)
14.3 Structural loss
14.4 Directional loss
14.5 Conditional loss
14.6 Temporal loss
14.7 Scale loss
14.8 Dependence loss
14.9 Support loss
14.10 Loss-profile comparison

15. Hidden Versus Destroyed Distinction

15.1 Hidden distinction
15.2 Encoded distinction
15.3 Latent distinction
15.4 Destroyed distinction
15.5 Re-expression
15.6 Representation-induced invisibility
15.7 Distinction inaccessible to a chosen statistic
15.8 Distinction absent from the transformed law
15.9 Testing hidden versus destroyed structure
15.10 Reconstruction as the decisive test

PART VI — RECOVERABILITY

16. Reconstruction Maps

16.1 Reconstruction as a native DTRT operation
16.2 Exact reconstruction
16.3 Approximate reconstruction
16.4 Pair-specific reconstruction
16.5 Family-wide reconstruction
16.6 Deterministic decoder
16.7 Stochastic reconstruction kernel
16.8 Inverse channels
16.9 Reconstruction error
16.10 Stable versus unstable reconstruction

17. Sufficiency

17.1 Classical sufficient statistics
17.2 Minimal sufficiency
17.3 Pairwise sufficiency
17.4 Family sufficiency
17.5 Approximate sufficiency
17.6 Predictive sufficiency
17.7 Conditional sufficiency
17.8 Dynamical sufficiency
17.9 Causal sufficiency
17.10 Sufficiency as preserved distinguishability

18. Recoverability Classes

18.1 Exactly recoverable
18.2 Approximately recoverable
18.3 Recoverable only with side information
18.4 Recoverable only on a restricted family
18.5 Recoverable only probabilistically
18.6 Recoverable after carrier refinement
18.7 Locally recoverable
18.8 Globally unrecoverable
18.9 Stable recoverability
18.10 Fundamentally nonrecoverable under the declared carrier

PART VII — CARRIER ADEQUACY

19. Adequate Carriers

19.1 Carrier adequacy relative to a distinction set
19.2 Minimal adequate carrier
19.3 Redundant carrier dimensions
19.4 Underpowered carriers
19.5 Overcomplete carriers
19.6 Distinction-complete carriers
19.7 Task-relative adequacy
19.8 Family-relative adequacy
19.9 Scale-relative adequacy
19.10 Dynamical adequacy

20. Carrier Refinement

20.1 Adding observables
20.2 Restoring joint structure
20.3 Restoring latent state
20.4 Extending temporal history
20.5 Adding scale coordinates
20.6 Adding intervention labels
20.7 Adding model identity
20.8 Carrier refinement without changing probability law
20.9 Minimal refinement
20.10 Excess refinement

21. Carrier Change

21.1 marginal → joint
21.2 state → path
21.3 endpoint → trajectory
21.4 single scale → multiscale
21.5 observable → latent-augmented
21.6 observational → interventional
21.7 parameter → model family
21.8 fixed model → model-space carrier
21.9 Carrier change versus mere reparameterization
21.10 Carrier change triggered by nonrecoverability

PART VIII — LOCAL INFORMATION GEOMETRY

22. KL and Local Geometry

22.1 Nearby distributions
22.2 Second-order KL expansion
22.3 Fisher information
22.4 Fisher metric
22.5 Infinitesimal distinguishability
22.6 Local contraction
22.7 Local invariance
22.8 Geodesics
22.9 Curvature
22.10 Limits of local geometry

23. Higher-Order Distinction Geometry

23.1 Third-order KL structure
23.2 Amari–Chentsov tensor
23.3 Alpha-connections
23.4 Exponential connection
23.5 Mixture connection
23.6 Duality
23.7 Directional structure beyond Fisher
23.8 Same Fisher metric, different global distinguishability
23.9 Local equivalence versus global fracture
23.10 Higher-order transport behavior

PART IX — GLOBAL PAIR GEOMETRY

24. Pair-Orbit Geometry

24.1 Transformation group acting on probability pairs
24.2 Pair orbit
24.3 Stabilizer
24.4 Quotient space
24.5 Complete pair invariants
24.6 Symmetric invariants
24.7 Directional invariants
24.8 Orbit dimension
24.9 Divergence complexity versus orbit dimension
24.10 Distinguishability readouts on orbit space

25. Cauchy Laboratory

25.1 Location-scale Cauchy family
25.2 Möbius action
25.3 Pair invariant q
25.4 KL symmetry
25.5 Hyperbolic pair geometry
25.6 Fisher readout
25.7 f-divergence collapse
25.8 One-dimensional pair-orbit structure
25.9 What survives common transformation
25.10 Global distinguishability collapse

26. Gaussian Laboratory

26.1 Location-scale Gaussian family
26.2 Ordered affine invariants
26.3 Directional KL
26.4 Fisher geometry
26.5 Symmetric quotient
26.6 Lost directional information
26.7 Multivariate covariance structure
26.8 Relative orientation
26.9 Source symmetry versus metric symmetry
26.10 Multi-coordinate distinction profile

PART X — INFERENCE AS DISTINGUISHABILITY TRANSFORMATION

27. Bayesian Updating

27.1 Prior state
27.2 Likelihood
27.3 Posterior state
27.4 New evidence as carrier augmentation
27.5 Information gain
27.6 Posterior distinguishability
27.7 Sequential updating
27.8 Filtering
27.9 Distinguishing state change from channel contraction
27.10 Bayesian information transport

28. Variational Approximation

28.1 Target law
28.2 Restricted model family
28.3 Forward KL
28.4 Reverse KL
28.5 Projection-induced loss
28.6 Mode loss
28.7 Tail loss
28.8 Dependency loss
28.9 Approximation-family inadequacy
28.10 Carrier refinement through richer variational families

29. Learning Dynamics

29.1 Log-loss
29.2 Cross-entropy
29.3 Gradient dynamics
29.4 Natural gradient
29.5 Mirror descent
29.6 KL-proximal updates
29.7 Online learning
29.8 Representation learning
29.9 Distinction preservation under learned embeddings
29.10 Representation collapse

PART XI — COARSE-GRAINING

30. Coarse-Graining Maps

30.1 Many-to-one observation
30.2 Aggregation
30.3 Binning
30.4 Marginalization
30.5 Partial observation
30.6 Block variables
30.7 Effective observables
30.8 Repeated coarse-graining
30.9 Coarse-graining semigroups
30.10 Scale-indexed transformations

31. Distinction Survival Across Scale

31.1 Microscopic distinctions
31.2 Mesoscopic distinctions
31.3 Macroscopic distinctions
31.4 Relevant distinctions
31.5 Irrelevant distinctions
31.6 Persistent distinctions
31.7 Scale-dependent distinguishability
31.8 Loss of dependence structure
31.9 Memory generated by coarse-graining
31.10 Scale-relative recoverability

32. Renormalization

32.1 Coarse-grain
32.2 Rescale
32.3 Iterate
32.4 Distributional fixed points
32.5 Relevant directions
32.6 Irrelevant directions
32.7 Universality classes
32.8 Universality as distinction quotient
32.9 RG trajectory distinguishability
32.10 Multiscale carrier requirements

PART XII — LARGE DEVIATIONS

33. Empirical-Measure Distinguishability

33.1 Empirical law
33.2 Typical states
33.3 Atypical states
33.4 Sanov structure
33.5 Pr(L_n≈Q) ~ exp(-nD(Q‖P))
33.6 KL as rarity cost
33.7 Constraint sets
33.8 Most probable atypical state
33.9 Information projection
33.10 Distinction under empirical sampling

34. Large-Deviation Transport

34.1 Transformation of empirical measures
34.2 Contraction principle
34.3 Transformation of rate functions
34.4 Hidden variables and contracted rate functions
34.5 Coarse-grained rarity
34.6 Recovering microscopic fluctuation structure
34.7 Multiple rate-function carriers
34.8 Rare-event distinguishability
34.9 Static versus dynamical rarity
34.10 When empirical-measure carriers cease to be adequate

PART XIII — PATH-SPACE DTRT

35. Probability Measures on Trajectories

35.1 State law versus path law
35.2 Path-space carrier
35.3 Trajectory likelihood ratio
35.4 Path-space KL
35.5 Relative entropy rate
35.6 Endpoint projection
35.7 Information lost by endpoint-only description
35.8 Temporal ordering
35.9 Memory
35.10 Path distinguishability

36. Dynamical Distinction

36.1 Same stationary distribution, different dynamics
36.2 Same endpoint law, different trajectories
36.3 Transition-law distinctions
36.4 Current distinctions
36.5 Waiting-time distinctions
36.6 Temporal-correlation distinctions
36.7 Markov versus non-Markov structure
36.8 Path-space recoverability
36.9 Minimal temporal carrier
36.10 Dynamical carrier adequacy

37. Irreversibility

37.1 Forward process
37.2 Reverse process
37.3 Forward/reverse path distinguishability
37.4 Entropy production
37.5 Detailed balance
37.6 Broken detailed balance
37.7 Probability currents
37.8 Temporal asymmetry
37.9 Hidden irreversibility
37.10 Time-direction distinction requires path carrier

PART XIV — OPEN SYSTEMS

38. Hidden Environment Structure

38.1 System-environment joint state
38.2 Marginal system state
38.3 Environment elimination
38.4 Memory generation
38.5 Effective non-Markovian dynamics
38.6 Hidden correlation structure
38.7 Information backflow
38.8 Environment-assisted recoverability
38.9 Joint carrier restoration
38.10 Distinguishability redistribution

39. Information Flow

39.1 Mutual information dynamics
39.2 Conditional mutual information
39.3 Directed information
39.4 Transfer entropy
39.5 Predictive information
39.6 Information currents
39.7 Redistribution versus destruction
39.8 Local contraction with global preservation
39.9 Subsystem distinction profiles
39.10 Networked distinguishability transport

PART XV — MODEL-FAMILY CARRIERS

40. Model Families

40.1 Model as set of distributions
40.2 Parametric family
40.3 Nonparametric family
40.4 Nested families
40.5 Non-nested families
40.6 Singular families
40.7 Latent-variable families
40.8 Model-family inclusion
40.9 Family equivalence
40.10 Model carrier

41. Model-to-Model Distinguishability

41.1 Point-to-family comparison
41.2 Family-to-family comparison
41.3 Best achievable divergence
41.4 Worst-case distinguishability
41.5 Approximation regions
41.6 Family overlap
41.7 Family separation
41.8 Structural versus parametric difference
41.9 Model-space distance is not ordinary distribution distance
41.10 Geometry of model classes

42. Model Change

42.1 Parameter update versus carrier change
42.2 Family enlargement
42.3 Family restriction
42.4 Latent-variable addition
42.5 Dependency-structure change
42.6 Symmetry-class change
42.7 Dimensional change
42.8 Observation-model change
42.9 Model-space trajectories
42.10 Distinguishability before and after model change

PART XVI — CAUSAL AND INTERVENTIONAL CARRIERS

43. Observational Distinguishability

43.1 Observational distributions
43.2 Conditional independence
43.3 Markov equivalence
43.4 Observationally indistinguishable models
43.5 Hidden causal distinctions
43.6 Confounding
43.7 Observational carrier limits
43.8 Environment-indexed observation
43.9 Mechanism invariance
43.10 Nonidentifiability

44. Interventional Distinguishability

44.1 Intervention-labelled distributions
44.2 Interventional carrier
44.3 Breaking observational equivalence
44.4 Mechanism comparison
44.5 Intervention selection
44.6 Experimental distinguishability
44.7 Causal recoverability
44.8 Counterfactual extension
44.9 Minimal intervention set
44.10 When observational probability space is inadequate

PART XVII — STATISTICAL EXPERIMENTS AND TRANSFORMATION ORDER

45. Statistical Experiments

45.1 Experiment as a family of probability laws
45.2 Comparing experiments
45.3 Garbling
45.4 Blackwell order
45.5 Sufficiency
45.6 Deficiency
45.7 Decision-theoretic informativeness
45.8 Transformation preorder
45.9 Equivalence of experiments
45.10 Incomparable experiments

46. Distinguishability Monotones

46.1 KL as monotone
46.2 f-divergence families
46.3 Complete monotone sets
46.4 Partial order versus scalar ranking
46.5 Conversion relations
46.6 Monotone insufficiency
46.7 Asymptotic conversion
46.8 Catalytic transformations
46.9 Resource-like interpretation
46.10 Distinction profiles as richer monotone objects

PART XVIII — FAILURE MODES INTERNAL TO DTRT

47. Scalarization Failure

47.1 Equal KL, different distinction profiles
47.2 Equal Fisher distance, different global distinction
47.3 Equal marginals, different joints
47.4 Equal endpoints, different paths
47.5 Equal macrostates, different microstructure
47.6 Equal predictive loss, different mechanisms
47.7 Scalar agreement without structural equivalence
47.8 Divergence choice dependence
47.9 Metric substitution failure
47.10 Comparison collapse

48. Carrier Failure

48.1 Carrier cannot encode demanded distinction
48.2 Carrier aliases distinct states
48.3 Temporal compression failure
48.4 Scale compression failure
48.5 Marginal carrier failure
48.6 Observational carrier failure
48.7 Fixed-model carrier failure
48.8 Hidden-state carrier failure
48.9 False adequacy from coarse metrics
48.10 Detecting inadequate carriers

49. Reconstruction Failure

49.1 No inverse map
49.2 Multiple valid reconstructions
49.3 Unstable reconstruction
49.4 Reconstruction requiring side information
49.5 Reconstruction outside the declared family
49.6 Support ambiguity
49.7 Latent-state ambiguity
49.8 Temporal ambiguity
49.9 Scale ambiguity
49.10 Exact versus approximate nonrecoverability

PART XIX — FORMAL DTRT CORE

50. Primitive DTRT Object

50.1 Carrier C
50.2 Ordered pair (P,Q)
50.3 Distinction profile Π_C(P,Q)
50.4 Transformation T:C→C'
50.5 Transformed pair (TP,TQ)
50.6 Survival operator S_T
50.7 Loss profile Λ_T
50.8 Recoverability class R_T
50.9 Carrier adequacy relation A(C,Π)
50.10 Carrier refinement relation C≼C'

51. Native DTRT Relations

51.1 PRESERVE_T(δ)
51.2 CONTRACT_T(δ)
51.3 ERASE_T(δ)
51.4 HIDE_T(δ)
51.5 RECOVER_T(δ)
51.6 APPROX_RECOVER_T(δ)
51.7 INADEQUATE_CARRIER(C,δ)
51.8 REFINE(C→C',δ)
51.9 EQUIV_T(P,Q)
51.10 DOMINATE(Π1,Π2)

52. Core DTRT Questions

52.1 What distinction is being compared?
52.2 On what carrier is it represented?
52.3 What transformation acts on that carrier?
52.4 Which distinctions survive?
52.5 Which contract?
52.6 Which disappear?
52.7 Which remain reconstructible?
52.8 Which require additional carrier structure?
52.9 What is the minimal adequate carrier?
52.10 Does the scalar divergence reflect the full distinction profile?

PART XX — SYNTHESIS

53. DTRT as a General Theory of Transformation-Relative Distinguishability

53.1 Distinguishability is carrier-relative
53.2 Transformation determines what can remain visible
53.3 Invertibility preserves more than scalar divergence
53.4 Data processing is one survival law, not the whole theory
53.5 Recoverability distinguishes concealment from destruction
53.6 Carrier refinement distinguishes missing representation from missing information
53.7 Path-space structure reveals distinctions invisible to state distributions
53.8 Multiscale carriers reveal distinctions erased by coarse-graining
53.9 Interventional carriers reveal distinctions invisible observationally
53.10 Model-family carriers reveal distinctions invisible within fixed parameterizations

54. Final Architecture

(P,Q)
→ carrier C
→ distinction profile Π_C(P,Q)
→ transformation T:C→C'
→ transformed profile Π_C'(TP,TQ)
→ {preserved ∥ contracted ∥ hidden ∥ erased}
→ recoverability class
→ {current carrier adequate ∥ carrier refinement required}
→ minimal adequate carrier
→ new distinction profile.

The strongest DTRT question is therefore not “how much information was lost?” It is what distinction changed status under transformation, whether that distinction remains reconstructible, and whether the current probability carrier is expressive enough to represent the comparison at all.