Triadic Semantic Cloud Theory

Adjunct Mathematics Program — Detailed Table of Contents

PART I — TSCT–ADJUNCT ARCHITECTURE

Triadic Semantic Cloud Theory

Adjunct Mathematics Program — Detailed Table of Contents

PART I — TSCT–ADJUNCT FOUNDATIONS

1. Why TSCT Requires Autonomous Adjunct Mathematics

1.1 TSCT as semantic discovery theory
1.2 Mathematical discovery versus mathematical development
1.3 Discovery object versus domain-theoretic object
1.4 Detection of boundary versus mathematics of boundaries
1.5 Detection of fracture versus mathematics of fracture
1.6 Detection of interaction versus mathematics of interaction
1.7 Carrier audit versus mathematics of carrier dependence
1.8 Classification failure versus mathematics of classification geometry
1.9 Grain change versus mathematics of scale/grain relativity
1.10 Why adjunct semantics must not enter the TSCT kernel automatically
1.11 Domain mathematics without semantic-authority transfer
1.12 Adjunct independence and replay independence

2. Generic Adjunct Contract

2.1 Source-owned input
2.2 Typed source objects
2.3 Typed operations
2.4 Distinction obligations
2.5 Residual input
2.6 Ancestry input
2.7 Grain and scale declaration
2.8 Carrier declaration
2.9 Representation declaration
2.10 Adjunct query
2.11 Branch-preserving output
2.12 Witness output
2.13 Residual output
2.14 Unknown output
2.15 Failure output
2.16 Replay receipt

3. Ontology-Neutral Adjunct Interface

3.1 No predeclared boundary requirement
3.2 No predeclared fracture requirement
3.3 No predeclared interaction class
3.4 No predeclared carrier equivalence
3.5 No predeclared classification partition
3.6 Primitive source distinctions only
3.7 Adjunct-specific derived objects
3.8 Type-safe return of newly constructed objects
3.9 Unknown object class
3.10 New-object escalation to TSCT

4. Adjunct Registry

4.1 Active adjunct set (\mathcal A_{\rm active})
4.2 Candidate adjunct set
4.3 Adjunct identity
4.4 Adjunct version
4.5 Capability signature
4.6 Input signature
4.7 Output signature
4.8 Dependency declaration
4.9 Conflict declaration
4.10 Reduction declaration
4.11 Activation
4.12 Suspension
4.13 Replacement
4.14 Splitting
4.15 Merging
4.16 Extinction

5. Adjunct Authority

5.1 Mathematical authority inside an adjunct
5.2 TSCT semantic-admission authority
5.3 Source authority
5.4 Representation authority forbidden
5.5 Readout authority forbidden
5.6 Adjunct theorem (\neq) TSCT discovery
5.7 TSCT discovery (\neq) adjunct theorem
5.8 Information backflow
5.9 Authority backflow prohibition
5.10 Cross-adjunct theorem transport

PART II — BOUNDARY GENERATION THEORY

6. Foundations of Generated Boundaries

6.1 Boundary as derived mathematical object
6.2 Imposed versus generated boundary
6.3 Intrinsic versus representational boundary
6.4 Local versus global boundary
6.5 Directional boundary
6.6 Asymmetric boundary
6.7 Grain-relative boundary
6.8 Carrier-relative boundary
6.9 Interaction-generated boundary
6.10 Temporal boundary
6.11 Ancestry-sensitive boundary
6.12 Boundary unknown state

7. Boundary-Generation Criterion

7.1 Residual as candidate generator
7.2 Failure does not imply boundary
7.3 Alternative explanations of failure
7.4 Representation-artifact elimination
7.5 Carrier-artifact elimination
7.6 Grain-artifact elimination
7.7 Operator-artifact elimination
7.8 Interaction-artifact elimination
7.9 Minimal boundary hypothesis
7.10 Boundary necessity
7.11 Boundary sufficiency
7.12 BOUNDARY_FORCED criterion

[
\boxed{
R\neq0
\not\Rightarrow
B
}
]

but

[
\boxed{
\operatorname{BOUNDARY_FORCED}(R,\Sigma)=1
\Rightarrow
B.
}
]

8. Boundary Objects

8.1 Boundary points
8.2 Boundary loci
8.3 Boundary regions
8.4 Boundary strata
8.5 Boundary complexes
8.6 Nested boundaries
8.7 Higher-codimension boundaries
8.8 Boundary intersections
8.9 Junctions
8.10 Networks
8.11 Boundary fields
8.12 Dynamic boundaries

9. Typed Boundary Algebra

9.1 Boundary domains
9.2 Boundary codomains
9.3 Composition conditions
9.4 Partial composition
9.5 Union compatibility
9.6 Intersection compatibility
9.7 Boundary fusion
9.8 Boundary splitting
9.9 Boundary cancellation
9.10 Boundary inversion
9.11 Identity boundaries
9.12 Noncommuting operations
9.13 Associativity conditions
9.14 Closure failures

10. Boundary Invariants and Load

10.1 Boundary support
10.2 Boundary rank
10.3 Boundary orientation
10.4 Boundary multiplicity
10.5 Boundary persistence
10.6 Consequence load
10.7 Distributed load
10.8 Joint load
10.9 Boundary ablation
10.10 Load transfer
10.11 Boundary equivalence
10.12 Boundary stability

11. Boundary Transport

11.1 Transport along boundaries
11.2 Transport across boundaries
11.3 Directional crossing
11.4 Reverse crossing
11.5 Boundary-preserving morphisms
11.6 Boundary-changing morphisms
11.7 Carrier-changing transport
11.8 Grain-changing transport
11.9 Transport defect
11.10 Boundary holonomy

12. Boundary Reconstruction

12.1 Reconstruction from either side
12.2 Reconstruction from boundary data
12.3 Identifiability
12.4 Reconstruction ambiguity
12.5 Minimal sufficient boundary data
12.6 Hidden boundary structure
12.7 Boundary information loss
12.8 Exact reconstruction
12.9 Reconstruction residual

13. Boundary Dynamics and TSCT Binding

13.1 Boundary birth
13.2 Motion
13.3 Stabilization
13.4 Bifurcation
13.5 Merger
13.6 Dissolution
13.7 Hysteresis
13.8 Recurrence
13.9 Boundary lineage
13.10 TSCT boundary query
13.11 BGT return state
13.12 TSCT semantic-admission test

PART III — FRACTURE MATHEMATICS

14. Foundations of Fracture

14.1 Fracture versus decomposition
14.2 Fracture versus partition
14.3 Fracture versus information loss
14.4 Fracture versus boundary
14.5 Fracture versus singularity
14.6 Intrinsic fracture
14.7 Representation-induced fracture
14.8 Carrier-induced fracture
14.9 Grain-induced fracture
14.10 Interaction-induced fracture

15. Admissible Fracture Criterion

15.1 Decomposition family
15.2 Reconstruction family
15.3 Admissible reconstruction
15.4 Reconstruction equivalence
15.5 Algorithm-dependent false fracture
15.6 Carrier-dependent false fracture
15.7 Grain-dependent false fracture
15.8 Representation-dependent false fracture
15.9 Persistent reconstruction failure
15.10 Irreducible fracture

[
\boxed{
X\neq\operatorname{Recompose}_D(X_i)
}
]

is insufficient.

The stronger object is:

[
\boxed{
\forall D\in\mathcal D_{\rm admissible},
\quad
\operatorname{Recompose}_D(X_i)\not\equiv X.
}
]

16. Fracture Operators

16.1 Decomposition
16.2 Projection
16.3 Ablation
16.4 Localization
16.5 Marginalization
16.6 Coarse-graining
16.7 Participant removal
16.8 Representation change
16.9 Carrier change
16.10 Grain change
16.11 Composite fracture operators

17. Fracture Residues

17.1 Recompositional residue
17.2 Interaction residue
17.3 Boundary residue
17.4 Orientation residue
17.5 Carrier residue
17.6 Grain residue
17.7 Ancestry residue
17.8 Information residue
17.9 Structural versus procedural residue
17.10 Residual factorization

18. Fracture Invariants

18.1 Fracture rank
18.2 Depth
18.3 Support
18.4 Multiplicity
18.5 Persistence
18.6 Orientation
18.7 Stability
18.8 Complexity
18.9 Equivalence class
18.10 Minimal fracture representation

19. Fracture Topology

19.1 Stable regions
19.2 Fracture loci
19.3 Fracture adjacency
19.4 Connected fracture sets
19.5 Fracture networks
19.6 Intersections
19.7 Nested fracture structures
19.8 Basin formation
19.9 Fracture topology transitions

20. Fracture, Identity, and Ancestry

20.1 Identity interiors
20.2 Identity boundaries
20.3 Identity from fracture complements
20.4 Identity splitting
20.5 Identity merging
20.6 Partial identity
20.7 Carrier-relative identity
20.8 Grain-relative identity
20.9 Ancestry-relative fracture
20.10 Extensional equality versus generative equality

21. Fracture Dynamics and TSCT Binding

21.1 Fracture creation
21.2 Propagation
21.3 Healing
21.4 Stabilization
21.5 Cascading fracture
21.6 Higher-order fracture
21.7 Critical fracture
21.8 TSCT fracture query
21.9 FM return state
21.10 Boundary escalation
21.11 Interaction escalation
21.12 Carrier escalation

PART IV — INTERACTION COHOMOLOGY

22. Foundations of Interaction Structure

22.1 Participant set
22.2 Interaction carrier
22.3 Interaction arity
22.4 Proper-subset projection
22.5 Lower-order reconstruction
22.6 Irreducible joint information
22.7 Structural versus causal interaction
22.8 Carrier-relative interaction
22.9 Grain-relative interaction

23. The Arity Complex

23.1 Unary level
23.2 Dyadic level
23.3 Triadic level
23.4 General (n)-ary level
23.5 Inclusion maps
23.6 Projection maps
23.7 Arity boundary maps
23.8 Arity consistency
23.9 Role-sensitive arity
23.10 Carrier-sensitive arity
23.11 Grain-sensitive arity

24. Cochain Construction

24.1 Interaction cochains (C^n)
24.2 Coefficient objects
24.3 Coboundary maps

[
d^n^n\rightarrow C^{n+1}
]

24.4 Nilpotence condition

[
d^{n+1}d^n=0
]

24.5 Cocycles
24.6 Coboundaries
24.7 Relative complexes
24.8 Filtered complexes
24.9 Functoriality

25. Interaction Cohomology Classes

25.1 (H^1_{\rm int})
25.2 (H^2_{\rm int})
25.3 (H^3_{\rm int})
25.4 (H^n_{\rm int})
25.5 Nonzero class interpretation
25.6 Vanishing class interpretation
25.7 Persistent classes
25.8 Relative interaction cohomology
25.9 Interaction-class transport

26. Reconstruction Obstructions

26.1 Proper-marginal reconstruction
26.2 Equal-access reconstruction
26.3 Enlarged-carrier reconstruction
26.4 Conditional reconstruction
26.5 Participant removal
26.6 Intervention reconstruction
26.7 Hidden synergy
26.8 Masked higher-order structure
26.9 Reconstruction obstruction class

27. Causal Interaction Classes

27.1 Structural arity
27.2 Causal arity
27.3 Necessary participant sets
27.4 Joint necessity
27.5 Redundant participants
27.6 Context dependence
27.7 Causal cocycles
27.8 Restoration tests
27.9 Arity disagreement

28. Interaction Transport and Boundaries

28.1 Carrier transport
28.2 Grain transport
28.3 Representation transport
28.4 Role-preserving transport
28.5 Role-changing transport
28.6 Interaction birth/death loci
28.7 Arity-transition boundaries
28.8 Boundary-supported interactions
28.9 Interaction-induced fracture

29. IC → TSCT Binding

29.1 TSCT joint-structure query
29.2 Proper-subset ablation
29.3 Cohomology computation
29.4 Irreducibility result
29.5 Reconstruction challenge
29.6 Causal-arity return
29.7 Boundary escalation
29.8 Fracture escalation
29.9 Carrier escalation
29.10 Semantic admission

PART V — CARRIER-RELATIVE MATHEMATICS

30. Foundations of Carrier Relativity

30.1 Carrier as mathematical structure
30.2 Formula versus carrier
30.3 Law versus carrier
30.4 Operation versus carrier
30.5 Carrier ownership
30.6 Carrier adequacy
30.7 Carrier dependence
30.8 Hidden carrier assumptions
30.9 Carrier-local validity
30.10 Carrier-global overreach

31. Typed Carrier Morphisms

31.1 Embedding
31.2 Quotient
31.3 Extension
31.4 Completion
31.5 Localization
31.6 Refinement
31.7 Coarse-graining
31.8 Partial carrier map
31.9 Noninvertible carrier transformation
31.10 Morphism-class typing

[
\tau(m)
\in
{
embed,quotient,extend,complete,
localize,refine,coarsen,\ldots
}.
]

32. Variance-Aware Carrier Transport

32.1 Covariant transport
32.2 Contravariant transport
32.3 Mixed variance
32.4 Partial transport
32.5 Forward transport
32.6 Reverse transport
32.7 Liftback
32.8 Transport residual
32.9 Information loss
32.10 Transport impossibility

33. Carrier Equivalence

33.1 Behavioral equivalence
33.2 Structural equivalence
33.3 Consequence equivalence
33.4 Interaction equivalence
33.5 Boundary equivalence
33.6 Fracture equivalence
33.7 Grain-relative equivalence
33.8 Partial equivalence
33.9 Failed equivalence
33.10 Ancestry-sensitive equivalence

34. Carrier Invariants

34.1 Stable laws
34.2 Stable identities
34.3 Stable interactions
34.4 Stable boundaries
34.5 Stable fractures
34.6 Stable residuals
34.7 Carrier-relative invariants
34.8 Carrier-sensitive invariants

35. Carrier Mutation

35.1 Carrier inadequacy
35.2 Residual-driven mutation
35.3 Enlargement
35.4 Restriction
35.5 Splitting
35.6 Merging
35.7 Successor carrier
35.8 Minimal adequate carrier
35.9 Carrier mutation residue

36. Carrier–Operator Interaction

36.1 Fixed operator / varying carrier
36.2 Fixed carrier / varying operator
36.3 Joint perturbation
36.4 Nonseparable carrier/operator effect
36.5 Carrier-conditioned operator validity
36.6 Operator-conditioned carrier adequacy
36.7 Joint residual
36.8 Joint invariance

37. CRM → TSCT Binding

37.1 Carrier audit
37.2 Cross-carrier comparison
37.3 Forward transport
37.4 Reverse transport
37.5 Liftback
37.6 Invariance test
37.7 Boundary escalation
37.8 Fracture escalation
37.9 Classification escalation
37.10 Successor-carrier pressure

PART VI — MATHEMATICAL CLASSIFICATION GEOMETRY

38. Classification as a Mathematical Object

38.1 Latent mathematical structure space (\mathcal M)
38.2 Classification map

[
\kappa:\mathcal M\to\mathcal C
]

38.3 Category partitions
38.4 Category adjacency
38.5 Category distance
38.6 Classification ancestry
38.7 Classification stability
38.8 Classification residual

39. Representation-Induced Equivalence

39.1 Representation family (\rho)
39.2 Observational equivalence

[
x\sim_\rho y
]

39.3 Representation-induced quotient

[
\mathcal M/!\sim_\rho
]

39.4 False fusion
39.5 False separation
39.6 Hidden distinction
39.7 Artificial distinction
39.8 Representation-dependent naturalness

41. Classification Rotation and Lifting

41.1 Alternative classification axes
41.2 Repartitioning
41.3 Classification rotation
41.4 Higher-dimensional latent structure
41.5 Classification projection
41.6 Classification lifting
41.7 Lost dimensions
41.8 Reconstructed dimensions

42. Classification Geometry

42.1 Category neighborhoods
42.2 Structural adjacency
42.3 Historical adjacency
42.4 Classification curvature
42.5 Classification singularity
42.6 Classification boundary
42.7 Category transition path
42.8 Classification topology

43. Taxonomy Reconstruction

43.1 Label erasure
43.2 Representation erasure
43.3 Behavior preservation
43.4 Operator comparison
43.5 Residual comparison
43.6 Transport comparison
43.7 Interaction comparison
43.8 Boundary comparison
43.9 Carrier comparison
43.10 Latent taxonomy recovery

44. MCG → TSCT Binding

44.1 TSCT category query
44.2 Representation-artifact audit
44.3 Fusion candidate
44.4 Fission candidate
44.5 Rotation candidate
44.6 Lifting candidate
44.7 Taxonomic residual
44.8 Semantic-cloud repartition proposal
44.9 TSCT independent admission

PART VII — SCALE–GRAIN RELATIVITY

45. Grain as Mathematical Structure

45.1 Grain versus parameter
45.2 Grain ownership
45.3 Grain refinement
45.4 Grain coarsening
45.5 Local grain
45.6 Global grain
45.7 Mixed-grain systems
45.8 Grain ancestry

46. Grain Transport

46.1 Fine-to-coarse transport
46.2 Coarse-to-fine reconstruction
46.3 Grain-preserving transport
46.4 Grain-changing morphisms
46.5 Grain transport residual
46.6 Irreversible grain loss
46.7 Grain liftback

47. Grain-Relative Objects

47.1 Grain-relative boundary
47.2 Grain-relative fracture
47.3 Grain-relative interaction
47.4 Grain-relative carrier
47.5 Grain-relative identity
47.6 Grain-relative invariant

48. Grain Transitions

48.1 Boundary birth under refinement
48.2 Boundary disappearance under coarsening
48.3 Fracture birth/death
48.4 Arity transition
48.5 Carrier-equivalence transition
48.6 Classification transition
48.7 Grain criticality

49. Grain Invariance

49.1 Stable structure across grains
49.2 Grain-equivalent descriptions
49.3 Grain-sensitive laws
49.4 Grain-sensitive identity
49.5 Grain-sensitive interaction
49.6 Grain-sensitive classification
49.7 Multi-grain invariants

50. SGR → TSCT Binding

50.1 Grain audit
50.2 Cross-grain comparison
50.3 Grain-refinement request
50.4 Grain-coarsening request
50.5 Boundary escalation
50.6 Fracture escalation
50.7 Interaction escalation
50.8 Classification escalation

PART VIII — COUPLED ADJUNCT SEMANTICS

51. Adjunct Dependency Graph

[
\mathcal G_A=(V_A,E_A)
]

51.1 Adjunct dependency
51.2 Conditional dependency
51.3 Strong dependency
51.4 Weak dependency
51.5 Cyclic dependency
51.6 Dependency ancestry
51.7 Dependency mutation

52. Non-Symmetric Adjunct Relations

52.1 CRM (\to) IC
52.2 CRM (\to) FM
52.3 CRM (\to) BGT
52.4 IC (\to) FM
52.5 IC (\to) BGT
52.6 FM (\to) BGT
52.7 SGR (\to) all scale-sensitive adjuncts
52.8 MCG (\to) taxonomy-sensitive outputs
52.9 Reverse feedback relations

53. Adjunct Refinement Dynamics

53.1 Initial adjunct states
53.2 Local update
53.3 Dependency-triggered recomputation
53.4 Residual propagation
53.5 Branch propagation
53.6 State refinement
53.7 Contraction
53.8 Stabilization

54. Fixed-Point Semantics

\Phi_{\mathcal G_A}(\mathbf A_t)
]

54.1 Monotone refinement
54.2 Nonmonotone refinement
54.3 Stable fixed point
54.4 Multiple fixed points
54.5 Oscillation
54.6 Divergence
54.7 Unresolved cycle
54.8 Fixed-point witness

55. Nonconvergence Semantics

55.1 Oscillation receipt
55.2 Divergence receipt
55.3 Dependency-cycle residue
55.4 Insufficient adjunct set
55.5 Missing mathematical theory
55.6 Successor-adjunct pressure

56. Branch-Preserving Joint State

Replace

[
\langle B,F,H,C\rangle
]

with

{
\mathcal S_A
\mid
A\in\mathcal A_{\rm active}
}
}
]

where every (\mathcal S_A) retains:

56.1 Branches
56.2 Witnesses
56.3 Residuals
56.4 Unknowns
56.5 Ancestry
56.6 Carrier context
56.7 Grain context
56.8 Representation context

PART IX — CROSS-ADJUNCT RESIDUAL ALGEBRA

57. Residual Types

57.1 Boundary residual
57.2 Fracture residual
57.3 Interaction residual
57.4 Carrier residual
57.5 Classification residual
57.6 Grain residual
57.7 Cross-adjunct residual

58. Residual Relations

58.1 Explains
58.2 Refines
58.3 Dominates
58.4 Contradicts
58.5 Is independent of
58.6 Causes
58.7 Masks
58.8 Cancels
58.9 Transports to

59. Residual Composition

59.1 Sequential composition
59.2 Parallel composition
59.3 Conditional composition
59.4 Residual pullback
59.5 Residual pushforward
59.6 Residual factorization
59.7 Residual localization
59.8 Residual persistence

60. Logical Conflict Semantics

Replace pairwise disagreement by:

[
\boxed{
\operatorname{INCOMPATIBLE}
(a_i,a_j,\Sigma)
}
]

60.1 Genuine contradiction
60.2 Merely different outputs
60.3 Scope mismatch
60.4 Grain mismatch
60.5 Carrier mismatch
60.6 Representation mismatch
60.7 Ancestry mismatch
60.8 Incomparable branches

61. Conflict Court

61.1 Conflict preservation
61.2 Conflict localization
61.3 Stronger source intervention
61.4 Carrier audit
61.5 Grain audit
61.6 Representation audit
61.7 Ancestry audit
61.8 Adjunct-set expansion
61.9 Unresolved-conflict terminal

PART X — TSCT DISCOVERY INTEGRATION

62. Adjunct Invocation

62.1 Residual-driven invocation
62.2 Capability-driven invocation
62.3 Dependency-driven invocation
62.4 Multi-adjunct invocation
62.5 Deferred invocation
62.6 Unknown-theory invocation

63. Adjunct-Assisted Discovery

[
\Sigma
\to
TSCT
\to
\mathcal A_{\rm active}
\to
\mathfrak A^*
\to
TSCT
\to
\delta_{\rm semantic}
]

63.1 Candidate generation
63.2 Adjunct query generation
63.3 Dependency execution
63.4 Fixed-point refinement
63.5 Return of structured mathematics
63.6 Consequence test
63.7 Counterfactual test
63.8 Replay test
63.9 Semantic mutation test
63.10 Commit/reject/unknown

64. Authority Firewall

64.1 Adjunct theorem validity
64.2 TSCT admission validity
64.3 No automatic discovery from adjunct theorem
64.4 No TSCT override of valid adjunct mathematics
64.5 No readout authority
64.6 No representation authority
64.7 No target authority

65. Ancestry-Sensitive Admission

65.1 Extensional equality
65.2 Generative equality
65.3 Ancestry equivalence
65.4 Distinct ancestry with identical output
65.5 Ancestry-sensitive boundaries
65.6 Ancestry-sensitive fractures
65.7 Ancestry-sensitive classifications
65.8 Admission under ancestry divergence

66. Semantic-State Mutation

66.1 New identity
66.2 New relation
66.3 New carrier
66.4 New boundary
66.5 New interaction class
66.6 New grain regime
66.7 New classification partition
66.8 New operator
66.9 New adjunct requirement

PART XI — ADVERSARIAL VALIDATION

67. Boundary False Positives

67.1 Representation artifact
67.2 Sampling gap
67.3 Grain artifact
67.4 Carrier artifact
67.5 Operator discontinuity
67.6 Interaction artifact

68. Fracture False Positives

68.1 Arbitrary decomposition
68.2 Weak reconstruction
68.3 Wrong carrier
68.4 Wrong grain
68.5 Missing operator
68.6 Artificial information loss

69. Interaction False Positives

69.1 Pairwise structure mislabeled higher-order
69.2 Weak reconstruction masquerading as synergy
69.3 Role-label artifact
69.4 Carrier-induced arity inflation
69.5 Representation-induced interaction

70. Carrier False Positives

70.1 Representation difference mistaken for carrier difference
70.2 Incomplete transport
70.3 Missing liftback
70.4 Grain mismatch
70.5 Operator mismatch
70.6 Loss-ledger omission

71. Classification and Grain False Positives

71.1 Historical difference mistaken for structural difference
71.2 Shared notation mistaken for structural identity
71.3 Representation collapse
71.4 Classification overfitting
71.5 False category fusion
71.6 False category fission
71.7 Grain-induced false distinction
71.8 Grain-induced false equivalence

PART XII — SUCCESSOR-ADJUNCT DISCOVERY

72. Detecting Adjunct Insufficiency

72.1 Persistent unexplained residual
72.2 Cross-adjunct contradiction
72.3 Unresolved object class
72.4 Missing invariant
72.5 Missing operator family
72.6 Missing equivalence relation
72.7 Persistent nonconvergence

73. Adjunct Reduction Test

Define explicit reduction:

[
A\preceq B
]

iff every licensed (A)-construction can be translated into (B) with declared semantic preservation.

73.1 Definitional reduction
73.2 Behavioral reduction
73.3 Lossless compilation
73.4 Functorial embedding
73.5 Consequence equivalence
73.6 Partial reduction
73.7 Irreducible residue

74. New Adjunct Eligibility

74.1 Independent object class
74.2 New invariant
74.3 New operator family
74.4 New obstruction type
74.5 Nonreducibility
74.6 Replayability
74.7 Domain independence
74.8 TSCT relevance
74.9 Semantic gain

75. Adjunct Evolution

75.1 Specialization
75.2 Generalization
75.3 Splitting
75.4 Merging
75.5 Replacement
75.6 Extinction
75.7 Version compatibility
75.8 Cross-version ancestry

PART XIII — MINIMAL EXECUTABLE ARCHITECTURE

76. Minimal Adjunct Interface

[
Q_A:
\Sigma\times R\times Context
\rightarrow
\mathcal S_A
]

where

[
Context=
\langle
carrier,grain,representation,ancestry
\rangle.
]

77. Minimal Dependency Engine

[
\mathcal G_A
\to
SCHEDULE
\to
REFINE
\to
FIXPOINT
]

77.1 Dependency resolution
77.2 Fair scheduling
77.3 Recompute trigger
77.4 Branch preservation
77.5 Cycle detection
77.6 Nonconvergence terminal

78. Minimal TSCT Return Contract

[
\mathcal S_A
\to
{
THEOREM,
RESIDUAL,
UNKNOWN,
BRANCH,
NEWOBJECT,
NEWADJUNCT
}.
]

79. Minimal Authority Contract

[
\boxed{
A\text{-truth}
\neq
TSCT\text{-discovery}
}
]

[
\boxed{
TSCT\text{-admission}
\neq
A\text{-truth modification}
}
]

PART XIV — UNIFIED TSCT ADJUNCT FIELD

80. Open Adjunct Field

\left(
\mathcal A_{\rm active},
\mathcal G_A,
{\mathcal S_A},
\mathcal R,
\mathcal W,
\mathcal H
\right)
}
]

where

[
\begin{aligned}
\mathcal A_{\rm active}&=\text{active adjunct theories},\
\mathcal G_A&=\text{dependency/refinement graph},\
\mathcal S_A&=\text{branch-preserving adjunct states},\
\mathcal R&=\text{cross-adjunct residual algebra},\
\mathcal W&=\text{witness/receipt system},\
\mathcal H&=\text{ancestry graph}.
\end{aligned}
]

81. Unified Discovery Dynamics

[
\Sigma_t
\to
R_t
\to
\mathcal A_t
\to
\mathfrak A_t^*
\to
\Delta_t
\to
\Sigma_{t+1}
]

with

[
\mathcal A_t
\neq
\mathcal A_{t+1}
]

permitted.

Thus discovery can modify not only semantic structure but also the mathematical decomposition used to analyze that structure.

82. Final Architectural Law

[
\boxed{
\text{TSCT must never assume that its current adjunct partition is the true partition of mathematics.}
}
]

Therefore:

[
\boxed{
\text{BOUNDARY},
\text{FRACTURE},
\text{INTERACTION},
\text{CARRIER},
\text{CLASSIFICATION},
\text{GRAIN}
}
]

are not eternal TSCT primitives.

They are currently successful mathematical decompositions subject to:

[
\boxed{
\text{fusion}
;\lor;
\text{fission}
;\lor;
\text{replacement}
;\lor;
\text{lifting}
;\lor;
\text{extinction}.
}
]

The mature TSCT–Adjunct architecture is therefore:

[
\boxed{
\text{SOURCE}
\to
TSCT
\to
\text{open mathematical decomposition}
\to
\text{coupled adjunct refinement}
\to
\text{semantic mutation}
}
]

rather than:

[
\boxed{
TSCT+BGT+FM+IC+CRM.
}
]