Generative Relational Mathematics

GRM TAXONOMY REBUILT — FROM τ

PART 0 — GRM AS GENERATIVE REBUILD PROCEDURE

0.1 GRM as a grammar for refusing premature ontology
0.2 From τ as procedure and toolkit, not inherited taxonomy
0.3 Mathematics as generated relational structure
0.4 CAN EXIST ≠ DISCOVERED ≠ REPRESENTED ≠ READOUT
0.5 Target theory as hidden comparison target, not constructor
0.6 Conventional mathematics as provenance only
0.7 Source contact before mathematical naming
0.8 Ontology stripping
0.9 Representation stripping
0.10 Target stripping
0.11 The blind-rebuild requirement
0.12 Source-seed certification
0.13 Consequence scope
0.14 Native arity preservation
0.15 Boundary preservation
0.16 Interaction preservation
0.17 Ancestry preservation
0.18 GLOBAL ≠ ΣLOCAL
0.19 OBJECT ≠ REPRESENTATION ≠ READOUT
0.20 No name before body
0.21 No carrier before operations require closure
0.22 No dual before separation and reconstruction
0.23 No correspondence before independent generation of both sides
0.24 Generated taxonomy as output rather than scaffold

PART I — BEFORE τ: FORMATION COURT

1.1 FUNDAMENTALS₀
1.2 Distinction?
1.3 Co-presence?
1.4 Constraint?
1.5 Transformation?
1.6 Recoverable persistence?
1.7 Why none is yet mathematical ontology
1.8 Source operations
1.9 Operational discriminators
1.10 Consequential versus consequence-null distinction
1.11 Consequence scope Σ_C
1.12 Expansion of probe families
1.13 Native interaction discovery
1.14 Dyadic interaction
1.15 Triadic interaction
1.16 Higher native arity
1.17 Static compatibility versus transition
1.18 Boundary events
1.19 Formation candidates
1.20 Constructor competition
1.21 Factorization attack
1.22 Minimum formation burden
1.23 Pareto minimal formation
1.24 When transition is actually forced
1.25 When transition is not yet forced
1.26 Promotion of a composable constrained transition

PART II — τ: FIRST GENERATIVE COMPRESSION

2.1 τ_K := minimal constraint-bearing composable transition
2.2 Source and target as derived interfaces
2.3 Input/output role without object ontology
2.4 Interface compatibility
2.5 Partial composability
2.6 Total composability
2.7 τ₂∘τ₁
2.8 Composition failure as information
2.9 Composition success as information
2.10 Native order
2.11 Directionality
2.12 Reversibility as a generated property
2.13 Irreversibility as surviving residue
2.14 Identity transition
2.15 Identity transition versus identity of an object
2.16 Null transition versus zero readout
2.17 Associative closure
2.18 Associativity as earned coherence
2.19 Failure of associativity
2.20 Parenthesization residue
2.21 Transition chains
2.22 Transition networks
2.23 Recursive composition
2.24 τ, τ², τ³, ...
2.25 First-return structure
2.26 Cyclic composition
2.27 Open versus closed execution

PART III — THE PRIMARY GRM GENERATOR

3.1 τ ⊗ τ → COMPOSITION
3.2 Composition audit
3.3 Preservation
3.4 Transformation
3.5 Loss
3.6 Birth
3.7 Unknown
3.8 Invariant
3.9 Covariant structure
3.10 Residue
3.11 RESIDUE := required consequence − recovered consequence
3.12 Residue is not missing ontology
3.13 Debt
3.14 Residue ownership
3.15 Residue signature
3.16 Visible residue versus root obstruction
3.17 Local residue
3.18 Interaction residue
3.19 Boundary residue
3.20 Global residue
3.21 Ancestry residue
3.22 Compression residue
3.23 Representation residue
3.24 Counterkernel
3.25 CK compilation
3.26 CK minimality
3.27 Root-cause relocation
3.28 Successor generation
3.29 Successor enumeration
3.30 Weakest lawful successor
3.31 Recomposition
3.32 Recursive execution
3.33 Liftback
3.34 Interpretation erasure
3.35 Amnesic replay
3.36 Success recompression

PART IV — GENERATED ROLE, TYPE, OBJECT AND CARRIER

4.1 Role from compositional substitutability
4.2 Type as equivalence of admissible relational role
4.3 Type failure
4.4 Stable compositional positions
4.5 Interface nodes
4.6 Derived “objects”
4.7 Object as stabilized relational interface
4.8 Object identity versus transition identity
4.9 Why object is downstream of relation
4.10 Closure pressure
4.11 Minimum carrier
4.12 Carrier adequacy
4.13 Carrier mutation
4.14 Carrier extension
4.15 Carrier reduction
4.16 Carrier quotient
4.17 Carrier genealogy
4.18 Ancestry aliasing
4.19 Carrier collapse
4.20 Carrier liftback

PART V — COMPOSITIONAL-COHERENCE MATHEMATICS

5.1 Repetition and algebraic closure
5.2 Semigroup-like structure
5.3 Monoid-like structure
5.4 Reversible transition families
5.5 Group-like structure
5.6 Partial reversibility and groupoids
5.7 Generators
5.8 Relations
5.9 Presentation versus generated body
5.10 Higher coherence
5.11 Transformations between transitions
5.12 Morphism-like structure
5.13 Categories as derived compositional bookkeeping
5.14 Functors as structure-preserving transport
5.15 Natural transformations
5.16 Higher categories
5.17 Operadic composition
5.18 Algebraic theories as closure grammars
5.19 Compositional-coherence taxonomy
5.20 Conventional algebra as projection of generated closure structures

PART VI — RECURSIVE-ARITHMETIC MATHEMATICS

6.1 Iteration before counting
6.2 Successor generated from iteration
6.3 Finite iteration classes
6.4 Counting
6.5 Addition as concatenated iteration
6.6 Repeated addition
6.7 Multiplication
6.8 Interaction of additive and multiplicative composition
6.9 Zero
6.10 Multiplicative identity
6.11 Semiring closure
6.12 Ring completion
6.13 Units
6.14 Divisibility
6.15 Irreducibility
6.16 Prime behavior
6.17 Factorization
6.18 Failure of unique factorization as residue
6.19 Congruence
6.20 Finite quotients
6.21 Finite fields
6.22 Rational extension
6.23 Ordered completion
6.24 Multiplicative order
6.25 Valuation
6.26 Archimedean valuation structure
6.27 Non-Archimedean valuation structure
6.28 Cauchy compatibility
6.29 Completion
6.30 Real completion
6.31 p-adic completion
6.32 Residue fields
6.33 Local fields
6.34 Global fields
6.35 Arithmetic locality
6.36 Arithmetic local/global incompatibility
6.37 Adelic reconstruction
6.38 Recursive-arithmetic taxonomy

PART VII — ACCESSIBILITY-DEFORMATION MATHEMATICS

7.1 Immediate composability
7.2 Relational reachability
7.3 Generated neighborhoods
7.4 Paths
7.5 Path composition
7.6 Connected components
7.7 Failed continuation
7.8 Boundary from asymmetric continuation
7.9 Local accessibility
7.10 Global accessibility
7.11 Neighborhood systems
7.12 Open structure
7.13 Closure
7.14 Connectedness
7.15 Separation
7.16 Compactness as global constraint
7.17 Topology as compositional accessibility geometry
7.18 Path deformation
7.19 Homotopy
7.20 Homotopy equivalence
7.21 Loops
7.22 Fundamental-group structure
7.23 Covering structures
7.24 Higher homotopies
7.25 Homotopy groups
7.26 Stable homotopy
7.27 Homology from persistent closed carriers
7.28 Boundary versus cycle
7.29 Accessibility-deformation taxonomy

PART VIII — REFINEMENT-COMPLETION MATHEMATICS

8.1 Refining a transition
8.2 Compatible subdivision
8.3 Directed refinement systems
8.4 Refinement invariants
8.5 Limit consistency
8.6 Continuous transition families
8.7 Continuity without coordinates
8.8 Continuity without metric
8.9 Local perturbation
8.10 Stable variation
8.11 Convergence
8.12 Cauchy structure
8.13 Completion residue
8.14 Completion as required relational closure
8.15 Compactness under refinement
8.16 Uniformity
8.17 Regularity
8.18 Approximation
8.19 Error transport
8.20 Differentiability as stable local first-order refinement
8.21 Differential
8.22 Higher variation
8.23 Distributional extension
8.24 Functional closure
8.25 Functional analysis
8.26 Refinement-completion taxonomy

PART IX — TRANSPORT-GEOMETRIC MATHEMATICS

9.1 When local comparison requires transport
9.2 Transportable relational structure
9.3 Path transport
9.4 Composition of transport
9.5 Transport independence
9.6 Path dependence
9.7 Closed positional execution
9.8 Relational nonclosure
9.9 Holonomy
9.10 Holonomy as surviving transport ancestry
9.11 Small-loop fracture
9.12 Curvature
9.13 Curvature as infinitesimal holonomy density
9.14 Local transport rule
9.15 Connection
9.16 Parallel transport
9.17 Bundle-like carrier structure
9.18 Local triviality
9.19 Transition functions
9.20 Gauge-equivalent local descriptions
9.21 Forced compensating transport
9.22 Gauge structure
9.23 Abelian transport
9.24 Non-Abelian transport
9.25 Self-coupled transport residue
9.26 Torsion-like closure fracture
9.27 Geodesic-like self-consistent continuation
9.28 Metric emergence
9.29 Metric as independently earned comparison law
9.30 Differential versus gradient
9.31 Dual-lift requirement for gradient
9.32 Riemannian-type geometry
9.33 Symplectic-type geometry
9.34 Complex-type geometry
9.35 Transport-geometric taxonomy

PART X — LOCAL↔GLOBAL RECONSTRUCTION MATHEMATICS

10.1 Local descriptions
10.2 Restriction
10.3 Pairwise overlap
10.4 Higher overlap
10.5 Interface transport
10.6 Local compatibility
10.7 Local closure
10.8 Why local closure gives no global authority
10.9 Globalization constructor
10.10 INTERIOR ⊗ INTERACTION ⊗ BOUNDARY
10.11 Higher compatibility
10.12 Gluing
10.13 Descent
10.14 Failure to descend
10.15 Cocycle-like residue
10.16 Repairable discrepancy
10.17 Coboundary-like repair
10.18 Persistent discrepancy
10.19 Cohomology
10.20 Higher cohomological residue
10.21 Sheaf-like organization
10.22 Sheaves
10.23 Derived local/global structure
10.24 Obstruction classes
10.25 Characteristic classes
10.26 Local systems
10.27 Global sections
10.28 Descent data
10.29 Stack-like reconstruction
10.30 Local↔global taxonomy

PART XI — VARIATION-DEFORMATION-MODULI MATHEMATICS

11.1 Perturb a generated structure
11.2 Allowed infinitesimal variation
11.3 Forbidden variation
11.4 Linearized deformation
11.5 Integrability of infinitesimal deformation
11.6 Deformation obstruction
11.7 Rigidity
11.8 Flexible structure
11.9 Families of stable reconstructions
11.10 Representational equivalence
11.11 Quotient by redundancy
11.12 Genuine inequivalence
11.13 Moduli carrier
11.14 Singular points of moduli
11.15 Automorphisms and stabilizers
11.16 Deformation complexes
11.17 Formal deformation
11.18 Global deformation
11.19 Bifurcation
11.20 Singularity theory
11.21 Variation-deformation-moduli taxonomy

PART XII — ACTION-REPRESENTATION MATHEMATICS

12.1 Generated transformation systems
12.2 Action
12.3 Action carrier
12.4 Faithful versus nonfaithful action
12.5 Representation emergence
12.6 SOURCE ACTION ≠ REPRESENTATION
12.7 Linearization
12.8 Linear representation
12.9 Modules
12.10 Invariant subcarriers
12.11 Irreducibility
12.12 Decomposition
12.13 Intertwiners
12.14 Equivalent representations
12.15 Characters
12.16 Trace as compressed representation data
12.17 Tensor products
12.18 Induced representations
12.19 Restriction
12.20 Representation categories
12.21 Action-representation taxonomy

PART XIII — PROBE-DUALITY MATHEMATICS

13.1 Probe families
13.2 Separating probes
13.3 Failure of separation
13.4 Pairings
13.5 Annihilators
13.6 Dual carrier
13.7 Double dual
13.8 Reconstruction
13.9 DUALITY := SEPARATION ⊗ RECONSTRUCTION
13.10 Weak duality
13.11 Separating duality
13.12 Reconstructive duality
13.13 Bidirectional duality
13.14 Fourier-type duality
13.15 Pontryagin-type duality
13.16 Geometric dualities
13.17 Physical dualities
13.18 Duality versus mere correspondence
13.19 Probe-duality taxonomy

PART XIV — SPECTRAL-HARMONIC MATHEMATICS

14.1 Stable action modes
14.2 Eigenstructure
14.3 Spectral decomposition
14.4 Spectrum as downstream action readout
14.5 Character decomposition
14.6 Invariant measure
14.7 Measure from symmetry
14.8 Harmonic analysis
14.9 Fourier transform
14.10 Convolution
14.11 Plancherel-type reconstruction
14.12 Operator spectrum
14.13 Resolvent
14.14 Spectral measure
14.15 Local harmonic analysis
14.16 Global harmonic analysis
14.17 Adelic harmonic analysis
14.18 Automorphic functions
14.19 Automorphic representations
14.20 Hecke operators
14.21 Hecke eigenstructure
14.22 L-functions as compressed local/global spectral data
14.23 Spectral-harmonic taxonomy

PART XV — SYMMETRY-EXTENSION MATHEMATICS

15.1 Extension of a generated arithmetic carrier
15.2 Base-preserving transformations
15.3 Extension symmetry
15.4 Fixed substructure
15.5 Galois correspondence
15.6 Finite Galois extensions
15.7 Compatible restriction
15.8 Inverse systems
15.9 Absolute Galois structure
15.10 Profinite topology
15.11 Local Galois structure
15.12 Decomposition groups
15.13 Inertia
15.14 Wild inertia
15.15 Residue-field action
15.16 Frobenius
15.17 Local arithmetic signature
15.18 Global arithmetic symmetry
15.19 Étale reconstruction
15.20 Arithmetic geometry
15.21 Symmetry-extension taxonomy

PART XVI — PROBABILISTIC-UNCERTAINTY MATHEMATICS

16.1 Multiple admissible successors
16.2 Unresolved branching
16.3 Ontic branching versus projection-created ignorance
16.4 Stable weighting
16.5 Additivity
16.6 Normalization
16.7 Probability measure
16.8 Conditional structure
16.9 Independence
16.10 Dependence
16.11 Random variables as readouts
16.12 Stochastic process
16.13 Filtration as ancestry carrier
16.14 Martingale structure
16.15 Markov compression
16.16 Memory beyond Markov compression
16.17 Stopping structures
16.18 Concentration
16.19 Large deviations
16.20 Random geometry
16.21 Random graphs
16.22 Probabilistic-uncertainty taxonomy

PART XVII — RECURSIVE-DYNAMICAL MATHEMATICS

17.1 State succession
17.2 Orbit
17.3 Invariant set
17.4 Fixed point
17.5 Periodic orbit
17.6 Recurrence
17.7 Recurrence without forgetting
17.8 Regeneration as ancestry-null interface
17.9 Stability
17.10 Instability
17.11 Attractor
17.12 Mixing
17.13 Ergodicity
17.14 Sensitivity
17.15 Chaos
17.16 Conserved structure
17.17 Integrability
17.18 Integrability as stable invariant compression
17.19 Lax-type representation
17.20 Spectral curve
17.21 Bifurcation
17.22 Dynamical taxonomy

PART XVIII — COLLECTIVE-STATISTICAL MATHEMATICS

18.1 Many-body composition
18.2 Interaction-generated joint carrier
18.3 Effective arity beyond pairwise graph arity
18.4 Correlation
18.5 Collective modes
18.6 Macro-compression
18.7 Equilibrium
18.8 Nonequilibrium
18.9 Ensemble
18.10 Entropy
18.11 Free-energy-type compression
18.12 Stable phase
18.13 Order parameter
18.14 Multiple stable closures
18.15 Symmetry realization
18.16 Spontaneous symmetry breaking
18.17 Phase transition
18.18 Criticality
18.19 Long-range correlation
18.20 Topological order
18.21 Collective-statistical taxonomy

PART XIX — SCALE-RECOMPRESSION MATHEMATICS

19.1 Resolution as generated distinction
19.2 Coarse-graining
19.3 Lost-distinction ledger
19.4 Scale transport
19.5 Effective carrier
19.6 Effective interaction
19.7 Recursive recompression
19.8 Renormalization
19.9 Relevant structure
19.10 Irrelevant structure
19.11 Marginal structure
19.12 Running parameters
19.13 Fixed point
19.14 Limit cycle
19.15 Runaway flow
19.16 Universality
19.17 Universality class
19.18 Critical scaling
19.19 Effective field theory
19.20 Multiscale analysis
19.21 Scale-recompression taxonomy

PART XX — OPERATOR-EVOLUTION MATHEMATICS

20.1 Operation as carrier
20.2 Composition algebra of operations
20.3 Linear operator emergence
20.4 Kernel
20.5 Image
20.6 Cokernel
20.7 Spectrum
20.8 Resolvent
20.9 Semigroup
20.10 Generator
20.11 Evolution operator
20.12 Functional calculus
20.13 Compactness
20.14 Fredholm structure
20.15 Differential operators
20.16 PDE evolution
20.17 Nonlinear operators
20.18 Noncommutative operator structures
20.19 Operator-evolution taxonomy

PART XXI — FIELD / LOCAL-INTERACTION MATHEMATICS

21.1 State distributed across a locality carrier
21.2 Local interaction
21.3 Propagation
21.4 Finite versus infinite propagation structure
21.5 Constraint fields
21.6 Classical fields
21.7 Gauge fields
21.8 Gauge transport versus gauge dynamics
21.9 Matter response
21.10 Charge as representation response
21.11 Non-Abelian self-coupling
21.12 Yang–Mills-type dynamics
21.13 Symmetry-stabilizer structure
21.14 Higgs-type branch
21.15 Quantized field representation
21.16 Excitation carriers
21.17 Quantum field theory
21.18 Topological field theory
21.19 Emergent gauge structure
21.20 Field/local-interaction taxonomy

PART XXII — OBSTRUCTION-INDEX MATHEMATICS

22.1 Failed extension
22.2 Failed lifting
22.3 Stable obstruction
22.4 Obstruction hierarchy
22.5 Local obstruction
22.6 Global obstruction
22.7 Cohomological obstruction
22.8 Analytic residue
22.9 Topological residue
22.10 Common residue carrier
22.11 Index
22.12 Analytic index
22.13 Topological index
22.14 Index theorem as common-ancestry reconstruction
22.15 Characteristic classes
22.16 Anomalies
22.17 Anomaly as obstruction to structure-preserving lift
22.18 Obstruction-index taxonomy

PART XXIII — CORRESPONDENCE MATHEMATICS

23.1 Why correspondence is higher-order structure
23.2 Independent generation requirement
23.3 Side A formation
23.4 Side B formation
23.5 Common invariant search
23.6 Common parameter carrier
23.7 Transport into common grammar
23.8 Liftback to side A
23.9 Liftback to side B
23.10 Correspondence versus identity
23.11 Correspondence versus duality
23.12 Functorial correspondence
23.13 Trace-formula comparison
23.14 Geometric/spectral correspondence
23.15 Index correspondences
23.16 Mirror-symmetry-type correspondences
23.17 Arithmetic/geometric correspondences
23.18 Correspondence taxonomy

PART XXIV — FROM τ TO LANGLANDS

24.1 Arithmetic branch regenerated from τ
24.2 Recursive arithmetic
24.3 Field closure
24.4 Valuation
24.5 Completion
24.6 Local arithmetic
24.7 Global arithmetic
24.8 Extension symmetry
24.9 Galois structure
24.10 Frobenius structure
24.11 Representation branch regenerated independently
24.12 Action
24.13 Linearization
24.14 Harmonic analysis
24.15 Spectral decomposition
24.16 Automorphic structure
24.17 Hecke structure
24.18 Local spectral parameters
24.19 Local factors
24.20 Global coherence
24.21 Independent maturity of Galois branch
24.22 Independent maturity of automorphic branch
24.23 Search for common parameter grammar
24.24 Dual-group structure
24.25 L-parameters
24.26 Local correspondence
24.27 Global correspondence
24.28 Functorial transport
24.29 Geometric Langlands branch
24.30 Langlands as stable higher-order correspondence ecology
24.31 Why τ ↛ Langlands by necessity
24.32 Langlands as one generated branch of mathematics, not terminal ontology

PART XXV — CROSS-FAMILY MATHEMATICAL COMPOUNDS

25.1 Why mature mathematics lives at intersections
25.2 Theory family versus generated axis
25.3 Gauge theory as transport × action × local/global × field interaction
25.4 Cohomology as local/global × obstruction × accessibility
25.5 Differential geometry as refinement × transport × comparison
25.6 Algebraic geometry as arithmetic × accessibility × local/global × moduli
25.7 Arithmetic geometry as arithmetic × symmetry-extension × geometry
25.8 QFT as field interaction × probability/amplitude × operator evolution × scale
25.9 Statistical mechanics as collective × probability × dynamics × scale
25.10 Integrable systems as dynamics × spectral × duality × geometry
25.11 Topological QFT as topology × local/global × field interaction × correspondence
25.12 Mirror symmetry as geometry × moduli × duality × correspondence
25.13 Index theory as operator × topology × obstruction × correspondence
25.14 Langlands as arithmetic × symmetry × representation × spectral × duality × local/global × correspondence
25.15 Compound-theory decomposition
25.16 Shared generated basis
25.17 Theory overlap
25.18 Theory collapse
25.19 Theory splitting
25.20 New theory-family discovery

PART XXVI — GRM CONCEPT FORMATION

26.1 Structure versus mechanism
26.2 Mechanism versus mature concept
26.3 Mature concept versus theory family
26.4 Theory family versus correspondence ecology
26.5 Abstraction scale Λ
26.6 Level 0: source distinction
26.7 Level 1: operation
26.8 Level 2: relation
26.9 Level 3: carrier/structure
26.10 Level 4: mechanism
26.11 Level 5: mature concept
26.12 Level 6: theory family
26.13 Level 7: correspondence ecology
26.14 Same-body/different-readout merge
26.15 Different-body/same-name split
26.16 Concept persistence under representation erasure
26.17 Concept recurrence across realizations
26.18 Concept minimality
26.19 Concept genealogy
26.20 Concept formation certificate

PART XXVII — GRM TAXONOMY GENERATION

27.1 Why conventional taxonomy is not imported
27.2 Generated concept population
27.3 Independent perturbation of generated properties
27.4 Consequence matrix
27.5 Axis extraction
27.6 Orthogonal versus coupled axes
27.7 Hierarchical axes
27.8 Cross-cutting axes
27.9 Classification by generator
27.10 Classification by residue
27.11 Classification by carrier
27.12 Classification by interaction arity
27.13 Classification by locality/globality
27.14 Classification by ancestry
27.15 Classification by scale
27.16 Classification by reconstruction strength
27.17 Classification by representation dependence
27.18 Taxonomic merge
27.19 Taxonomic split
27.20 Taxonomic demotion to representation
27.21 Taxonomic promotion to mature concept
27.22 New generated category
27.23 New generated axis
27.24 Taxonomy stability under replay
27.25 GRM_TAXONOMYΩ

PART XXVIII — BLIND REBUILD AND TARGET REVEAL

28.1 Target hiding
28.2 Vocabulary erasure
28.3 Conventional theorem erasure
28.4 Conventional dependency erasure
28.5 Blind generation
28.6 Nonaliased route generation
28.7 Branch ledger
28.8 Common-mode ancestry audit
28.9 Unexplored branch debt
28.10 Generative-equivalence court
28.11 Target reveal
28.12 Generated-to-conventional mapping
28.13 Preserved conventional concepts
28.14 Split conventional concepts
28.15 Merged conventional concepts
28.16 Representation-only conventional concepts
28.17 Missing expected concepts
28.18 Newly generated concepts
28.19 Newly generated taxonomy axes
28.20 Blind replay after target reveal

PART XXIX — MATHEMATICAL QUANTIFICATION GATE

29.1 Structural discovery is not sufficient
29.2 Quantification trigger
29.3 Exact identity
29.4 Inequality
29.5 Bound
29.6 Constant
29.7 Rank
29.8 Dimension
29.9 Measure
29.10 Rate
29.11 Spectrum
29.12 Asymptotic
29.13 Finite witness
29.14 Counterexample
29.15 Minimal numerical discriminator
29.16 Quantitative liftback
29.17 Provisional concept without quantitative discriminator
29.18 Mathematical body certification

PART XXX — GENERATIVE EQUIVALENCE AND MINIMAL BASIS

30.1 Replay stability versus uniqueness
30.2 Alternative generative bases
30.3 Generator deletion
30.4 Primitive deletion
30.5 Relation deletion
30.6 Carrier deletion
30.7 Concept deletion
30.8 Taxonomy-axis deletion
30.9 Reconstruction test
30.10 Pareto-minimal basis
30.11 Multiple incomparable minimal bases
30.12 Generative equivalence
30.13 Generative multiplicity
30.14 Basis compression
30.15 Success-driven ontology reduction
30.16 Minimum generative mathematics

PART XXXI — FAILURE, FRONTIER AND NEW PRIMITIVE FORMATION

31.1 Failure is executable information
31.2 Failure witness
31.3 Failure localization
31.4 Common-mode failure
31.5 Nonlocal failure
31.6 Boundary failure
31.7 Globalization failure
31.8 Compression failure
31.9 Formation failure
31.10 Counterkernel saturation
31.11 Constructibility closure
31.12 Exhaustion criterion
31.13 New primitive pressure
31.14 Provisional primitive birth
31.15 Primitive factorization attack
31.16 Primitive independence
31.17 Primitive replay
31.18 New primitive candidate
31.19 Frontier payload
31.20 Why frontier is not “I do not know”
31.21 Recovery closure relative to current primitive/operation closure

PART XXXII — GRM NOVELTY AND DISCOVERY

32.1 Re-description
32.2 Recompression
32.3 New dependency ordering
32.4 Conventional category collapse
32.5 Conventional category split
32.6 New invariant
32.7 New obstruction
32.8 New counterkernel
32.9 New construction
32.10 New theorem under weaker assumptions
32.11 Stronger theorem
32.12 New exact bound
32.13 New counterexample
32.14 New correspondence
32.15 New transferable generator
32.16 Novel taxonomy axis
32.17 Novelty gate
32.18 RECOMPRESSION_ONLY
32.19 NEW_GENERATED_MATHEMATICS

PART XXXIII — THE REBUILT GRM MAP OF MATHEMATICS

33.1 Mathematics as intersecting generated grammars
33.2 Compositional-coherence
33.3 Recursive-arithmetic
33.4 Accessibility-deformation
33.5 Refinement-completion
33.6 Transport-geometric
33.7 Local/global reconstruction
33.8 Variation-deformation-moduli
33.9 Action-representation
33.10 Probe-duality
33.11 Spectral-harmonic
33.12 Symmetry-extension
33.13 Probabilistic-uncertainty
33.14 Recursive-dynamical
33.15 Collective-statistical
33.16 Scale-recompression
33.17 Operator-evolution
33.18 Field/local-interaction
33.19 Obstruction-index
33.20 Correspondence
33.21 Why these are generated axes rather than conventional disciplines
33.22 Why mature theories occupy intersections
33.23 Why taxonomic boundaries remain defeasible
33.24 Why new mathematics can generate new taxonomy

PART XXXIV — MASTER GRM FIELD

34.1 Full generative dependency graph
34.2 FUNDAMENTALS₀ → formation
34.3 formation → τ_K where earned
34.4 τ_K → composition
34.5 composition → invariant ⊕ residue
34.6 invariant → compression
34.7 residue → CK
34.8 CK → successor
34.9 successor → recomposition
34.10 recomposition → recursive mathematical ecology
34.11 Ecology → mature concept
34.12 Mature concepts → theory families
34.13 Theory families → independent branches
34.14 Independent branches → common parameter grammar
34.15 Common grammar → correspondence
34.16 Correspondence → functorial transport
34.17 Liftback
34.18 Erasure
34.19 Replay
34.20 Recompression
34.21 Taxonomy regeneration

The compressed dependency of the entire book is:

𝔽₀
→ FORMATION
→ τ_K
→ COMPOSITION
→ INVARIANT ⊕ RESIDUE
→ COMPRESSION ⊕ CK/SUCCESSOR
→ RECOMPOSITION ↺
→ generated mathematical grammars
→ mature concepts
→ intersecting theory families
→ dualities / obstructions / correspondences
→ GRM_TAXONOMYΩ.

GRM TAXONOMY REBUILT — `FROM τ`

Generative Glossary

This glossary is ordered by generative dependency, not alphabetically. The original From τ to Langlands glossary uses the same rule: GRM → FUNDAMENTALS₀ ⊗ DISCOVERY₀ → MATHEMATICS₀ → τ_K → composition → invariant/residue → symmetry/arithmetic/local-global → representation/duality → Galois⊗automorphic → parameter → Langlands. The glossary below extends that pattern to the rebuilt GRM taxonomy of mathematics.

I. GRM / EPISTEMIC GRAMMAR

GRM — Generative Relational Mathematics.
Minimal grammar governing when mathematical structure has actually been earned. GRM is not itself an ontology; it prevents candidate structures from being promoted merely because they are familiar or mathematically available.

? — Ontological suspension.
Marks a candidate distinction, carrier, relation or structure as something to discriminate rather than assume.

SOURCE.
The currently admitted source of consequential distinctions and executable relations before downstream mathematical representation.

SOURCE CONTACT.
First executable encounter with source structure before a mathematical object or theory has been imposed.

SOURCE SEED.
A provisional source-owned distinction or operation admitted to the rebuild.

SOURCE-SEED CERTIFICATE.
Evidence that a seed is operationally discriminable, source-owned, target-independent and not imported from a downstream representation.

TARGET.
The mature concept, theorem or domain against which the completed reconstruction may later be compared; never a constructor.

TARGET HIDE.
Removal of mature target names, conventional taxonomy and expected endpoints during generative reconstruction.

ONTOLOGY ERASE.
Removal of object names and category assumptions that have not yet been generated.

REPRESENTATION ERASE.
Removal of coordinates, matrices, formulas, spaces or other encodings to determine which source structure actually survives.

BLIND REBUILD.
Generation from source constraints without allowing knowledge of the desired mature taxonomy to select the construction path.

PROVENANCE.
Where a concept, theorem, representation or observation came from. Provenance can guide retrieval but does not supply generative authority.

BODY.
Executable relational structure that must exist before a mathematical name may be promoted.

NAME ↛ BODY.
Naming a structure does not construct it.

GENERATED.
Constructed from earlier source-owned operations through admissible formation, interaction, transport and replay.

PROVISIONAL.
Currently useful but still subject to factorization, deletion, retyping or stronger reconstruction.

II. `FUNDAMENTALS₀`

FUNDAMENTALS₀.
Ontologically suspended candidate basis:
DISTINCTION? ⊗ CO-PRESENCE? ⊗ CONSTRAINT? ⊗ TRANSFORMATION? ⊗ RECOVERABLE-PERSISTENCE?.

DISTINCTION.
Operationally discriminable difference. A distinction does not yet imply objects possessing properties.

CONSEQUENTIAL DISTINCTION.
A distinction whose alteration changes at least one admissible execution, probe, continuation or reconstruction within the declared consequence scope.

CONSEQUENCE SCOPE Σ_C.
The active family of probes, interventions, resolutions, outcomes and scopes relative to which distinctions are judged consequential.

CONSEQUENCE-NULL DISTINCTION.
A difference that no admitted consequential discriminator can detect at current scope.

CO-PRESENCE.
Multiple distinctions participating in one relational situation without yet assuming set membership, space, simultaneity or locality.

CONSTRAINT K.
Structure discriminating admissible from inadmissible relational possibilities: possible ≠ admissible.

TRANSFORMATION.
Comparable relational change. Initially carries no automatic assumption of time, motion, causality, function or geometry.

RECOVERABLE PERSISTENCE.
Structure reconstructible across transformation.

RELATION.
Structured dependence among distinctions. Relation is not an underlying substance.

NATIVE ARITY.
Minimum number of jointly participating distinctions required by a relation or interaction.

NATIVE ORDER.
Source-required ordering of participating distinctions or operations.

INTERACTION.
Joint execution whose consequences cannot be reconstructed from independent treatment of its components.

BOUNDARY.
A load-bearing carrier/interface where continuation, compatibility, transport or closure changes. Boundary is not automatically an edge of a pre-existing space.

ANCESTRY.
Execution history that remains consequential for current structure.

III. `DISCOVERY₀`

DISCOVERY₀.
TYPE? ⊗ CARRIER? ⊗ TRANSPORT? ⊗ DEBT? ⊗ RESIDUE? ⊗ CK? ⊗ SUCCESSOR? ⊗ LIFTBACK? ⊗ REPLAY?.

TYPE.
Equivalence of admissible relational role generated by substitutability in execution.

TYPE FAILURE.
Failure caused by substituting structures that appeared similar but do not occupy the same consequential role.

CARRIER.
Minimum support on which an earned operation/relation closes. The original glossary compresses carrier as the “minimum support on which the extension closes.”

CARRIER ADEQUACY.
Whether a proposed carrier can actually support every required operation, relation, boundary and transport.

CARRIER GENEALOGY.
Constructed ancestry recording how a carrier arose, what it inherited, what it lost and how it lifts back.

TRANSPORT.
Lawful movement of relational structure between carriers/interfaces while explicitly tracking preservation, transformation and loss.

DEBT.
Explicit unsatisfied obligation generated by an attempted construction.

RESIDUE.
Consequential structure not recovered by the current composition, transport or reconstruction. The source defines it as required consequence minus recovered consequence and explicitly warns RESIDUE ≠ NEW ONTOLOGY.

RESIDUE OWNER.
The source, relation, carrier, interaction, boundary, transport, representation or globalization step responsible for retaining the unresolved structure.

RESIDUE SIGNATURE.
Typed description of the failure: owner, location, arity, preserved/lost/born structure, boundary dependence, ancestry dependence, scale dependence and replay stability.

VISIBLE RESIDUE.
Where failure becomes observable.

ROOT RESIDUE.
Earliest causally responsible dependency whose repair mechanically discharges the downstream failure.

COUNTERKERNEL CK.
Minimum source-realizable intervention that discriminates or disables a failure while preserving the maximal valid prefix. In the original glossary: “minimal repair-discriminating intervention.”

SUCCESSOR.
Smallest source-lawful extension forced by residue plus CK; not the familiar theory that happens to resemble the failure.

LIFTBACK.
Reconstruction of native source structure from a downstream carrier, representation or solution.

REPLAY.
Independent re-execution of the construction from stored source conditions rather than from its interpretation or desired result.

AMNESIC REPLAY.
Replay after mature names, interpretation and target information have been erased.

RECOMPRESSION.
Deletion of structures shown by successful replay to be unnecessary, followed by regeneration from a smaller basis.

IV. FORMATION AND `τ`

FORMATION COURT.
Competition among candidate minimal relational formations before privileging any one constructor.

FORMATION.
First source-owned executable relational body satisfying the current consequential distinctions.

τ_K.
Minimum stable constraint-bearing composable transition. This is the original GRM-PHOTON.

GRM-PHOTON.
Name given to τ_K as the first highly compressed mathematical structure in the original From τ reconstruction; not a claim that every source is fundamentally particle-like.

COMPOSABILITY.
Ability of transitions to join through compatible interfaces.

COMPOSITION ∘.
Lawful joining of generated transitions.

τ_K ∘ τ_K.
First relational interaction in the source glossary.

COMPOSITIONAL INTERFACE.
Derived source/output position through which transitions can lawfully compose.

IDENTITY TRANSITION 𝟙.
Operational neutral transition satisfying composition neutrality when such a transition is forced.

OBJECT IDENTITY.
Persistent identity attached to a stabilized interface; later than operational identity.

ASSOCIATIVITY.
Parenthesization-insensitive composition when independently earned.

REVERSIBILITY.
Existence of a transition that lawfully undoes another.

IRREVERSIBLE RESIDUE.
Consequential structure not reconstructible after execution.

ITERATION.
Repeated execution τ, τ², τ³,… before numerical counting is assumed.

RECURSION.
Reuse of generated compositional structure as input to further generation.

CYCLE.
Finite compositional return.

CLOSED EXECUTION.
Execution whose positional/interface condition returns to its starting equivalence class.

V. PRIMARY GENERATIVE SPLIT

INVARIANT.
What survives an admissible transformation/composition. This is the source glossary's compressed definition.

COVARIANT.
Structure that changes lawfully while retaining reconstructible correspondence.

LOST.
Previously consequential distinction no longer recoverable.

BORN.
Consequential distinction generated by interaction that was not present independently in its inputs.

UNKNOWN.
Current execution cannot classify a distinction as preserved, transformed, lost or born.

PRIMARY GRM GENERATOR.
COMPOSITION → INVARIANT ⊕ RESIDUE. The original construction explicitly identifies this as the first major generator.

COMPRESSION.
Replacement of a larger relational description by a smaller sufficient structure preserving all required consequences.

MINIMALITY ORDER.
Declared partial order over burden—assumptions, arity, carrier capacity, retained information, intervention strength, dependencies and resources—used to justify claims of “minimum.”

PARETO MINIMUM.
A candidate not dominated across the active burden dimensions; several incomparable minima may remain.

VI. COMPOSITIONAL-COHERENCE FAMILY

COMPOSITIONAL COHERENCE.
Stable preservation of interface and relational compatibility under repeated composition.

SEMIGROUP-LIKE STRUCTURE.
Closed associative transition composition without requiring identity or inverses.

MONOID-LIKE STRUCTURE.
Associative closed composition with an earned identity.

GROUP-LIKE STRUCTURE.
Reversible compositional transition system with earned identity and inverses.

GROUPOID-LIKE STRUCTURE.
Partially composable reversible transitions.

GENERATOR.
Smaller transition family from which a larger compositional structure can be reconstructed.

RELATION/PRESENTATION.
Constraint compressing how generators compose.

CATEGORY-LIKE STRUCTURE.
Derived bookkeeping of stabilized interfaces and composable transitions; the source explicitly treats category as derived from τ, not primitive ontology.

FUNCTOR.
Higher-order transport preserving relevant compositional structure between generated categories.

NATURAL TRANSFORMATION.
Coherent transformation between functorial transports.

HIGHER CATEGORY.
Recursive promotion of transformations themselves to carriers of further transformation.

COMPOSITIONAL-COHERENCE MATHEMATICS.
GRM taxonomy family generated when lawful composition and its higher coherence are the primary retained structure.

VII. RECURSIVE-ARITHMETIC FAMILY

SUCCESSOR.
Here, specifically, the next distinguishable iteration class generated by recursion; distinct from GRM's general repair-successor.

COUNT.
Stable indexing of distinguishable finite iteration.

ADDITION.
Composition/concatenation of iteration counts.

MULTIPLICATION.
Repeated additive composition once a second law is forced.

ARITHMETIC.
Stable recursive compositional structure supporting counting, multiplication and factorization.

DIVISIBILITY.
Whether one generated multiplicative transition factors through another.

UNIT.
Multiplicative operational identity.

IRREDUCIBLE.
Element not decomposable under the generated factorization law except by units.

PRIME.
Irreducible structure satisfying the stronger generated divisibility behavior of the relevant multiplicative carrier.

FACTORIZATION.
Decomposition into multiplicative constituents.

CONGRUENCE.
Arithmetic equivalence after quotienting a specified difference.

VALUATION.
Generated structured measure of multiplicative divisibility/order.

COMPLETION.
Adjoining relational limits required by an earned compatibility/Cauchy structure; not ontological enlargement by default.

LOCAL FIELD.
Complete valued arithmetic carrier possessing the additional earned field structure.

GLOBAL FIELD.
Arithmetic carrier whose coherent structure spans multiple local completions.

ADELIC RECONSTRUCTION.
Global arithmetic organization formed from local carriers together with their compatibility and restriction structure.

RECURSIVE-ARITHMETIC MATHEMATICS.
Taxonomy family generated when repeated compositional structure forces counting, multiple laws, factorization and arithmetic local/global structure.

VIII. ACCESSIBILITY / REFINEMENT FAMILY

ACCESSIBILITY.
Which generated continuations can be reached by admissible compositions.

NEIGHBORHOOD.
Local family generated by immediate admissible continuation.

PATH.
Composable chain of transitions.

CONNECTEDNESS.
Generated mutual accessibility within the relevant carrier.

TOPOLOGY.
Stable accessibility/neighborhood structure generated from admissible continuation; the source describes it as “compositional accessibility geometry.”

DEFORMATION.
Admissible continuous alteration of generated structure while preserving declared constraints.

HOMOTOPY.
Equivalence of paths/structures generated by admissible deformation.

FUNDAMENTAL-GROUP STRUCTURE.
Compositional classes of closed paths under admissible deformation.

HOMOLOGY.
Classification of persistent closed carriers modulo those generated as higher boundaries.

REFINEMENT.
Replacement of a transition/description by a more discriminating compatible one.

CONTINUITY.
Stability of consequential structure under arbitrarily fine compatible refinement.

CONVERGENCE.
Stable approach under an earned comparison/refinement structure.

COMPLETENESS.
Absence of unresolved missing-limit residue relative to the chosen compatibility structure.

REGULARITY.
Degree to which increasingly refined local variation remains coherently controlled.

ACCESSIBILITY-DEFORMATION MATHEMATICS.
Family generated primarily by reachability and deformation.

REFINEMENT-COMPLETION MATHEMATICS.
Family generated primarily by stable limiting, approximation and regularity structure.

IX. TRANSPORT-GEOMETRIC FAMILY

LOCAL COMPARISON.
Comparison among structures within one generated local carrier.

CROSS-LOCAL COMPARISON.
Comparison requiring relational transport between distinct local carriers.

PARALLEL TRANSPORT.
Coherent continuation of relational structure along a generated path.

CONNECTION.
Generated local rule enabling coherent transport composition.

CLOSED TRANSPORT.
Transport around a closed path.

HOLONOMY.
Relational transport ancestry surviving positional closure:
closed path ∧ nonclosed transported relation.
In GRM compression: nonclosure of transport under closure of path.

CURVATURE.
Stable infinitesimal density of closed-transport fracture when the relevant limiting structure has been earned.

MONODROMY.
Global path-class transport residue, especially when local transport is flat.

GAUGE STRUCTURE.
Local descriptive equivalence together with consequence-bearing cross-local comparison that forces compensating transport.

ABELIAN TRANSPORT.
Generated transport composition whose relevant transformations commute.

NON-ABELIAN TRANSPORT.
Generated transport composition whose order remains consequential.

GAUGE THEORY.
Mature theory family formed from local descriptive redundancy, compensating relational transport, interaction and dynamical/global extensions.

TRANSPORT-GEOMETRIC MATHEMATICS.
Family generated when cross-local comparison forces path-sensitive transport structure.

X. LOCAL↔GLOBAL RECONSTRUCTION FAMILY

LOCALITY.
Structure relative to one generated local carrier/completion. The original glossary specializes this to one completion/place in the arithmetic branch.

GLOBALITY.
Coherence that survives transport across local carriers; not the sum of local closures.

GLOBAL ≠ ΣLOCAL.
Local closure provides no authority for global closure.

RESTRICTION.
Transport from broader structure to a local/subcarrier while recording what is retained.

OVERLAP.
Carrier on which independently formed local structures interact and their compatibility becomes consequential.

GLUING.
Construction of a larger carrier from compatible local bodies and interfaces.

DESCENT.
Executable local→global reconstruction together with the compatibility conditions required for liftback.

COCYCLE-LIKE RESIDUE.
Compatibility data surviving pairwise/local reconstruction and appearing at a higher overlap level.

COBOUNDARY-LIKE REPAIR.
Discrepancy removable by a lower-order admissible redefinition.

COHOMOLOGY.
Organized irreducible failure-to-glue after lower-order repair redundancy has been factored out. The source explicitly characterizes cohomology as organized failure-to-glue.

OBSTRUCTION.
Stable residue preventing extension, lift or globalization.

CHARACTERISTIC CLASS.
Global invariant/residue carried by locally simpler or locally trivialized structure.

SHEAF-LIKE STRUCTURE.
Organization of local data, restrictions and gluing constraints.

STACK-LIKE STRUCTURE.
Higher local/global organization required when objects and their equivalences must both descend.

GLOBALIZATION CONSTRUCTOR.
INTERIORS ⊗ INTERACTIONS ⊗ BOUNDARIES ⊗ OVERLAPS ⊗ HIGHER OVERLAPS ⊗ TRANSPORT ⊗ ANCESTRY.

LOCAL↔GLOBAL RECONSTRUCTION MATHEMATICS.
Family generated when the decisive problem is reconstruction of a coherent whole from locally valid structures.

XI. VARIATION / MODULI FAMILY

VARIATION.
Controlled perturbation of a generated structure.

INFINITESIMAL DEFORMATION.
First-order admissible variation after an appropriate local comparison structure has been earned.

DEFORMATION.
Generated family of admissible variations preserving specified source constraints.

DEFORMATION OBSTRUCTION.
Residue preventing an infinitesimal/partial deformation from extending to a full one.

RIGIDITY.
Absence of consequential nontrivial admissible deformations after equivalence has been removed.

FLEXIBILITY.
Persistence of non-equivalent admissible deformation directions.

MODULI.
Carrier of genuinely inequivalent stable reconstructions after representational/gauge redundancy has been quotiented away.

SINGULAR MODULUS.
Point/sector where the local deformation grammar changes type, rank, stabilizer or reconstruction behavior.

VARIATION-DEFORMATION-MODULI MATHEMATICS.
Family generated by nonuniqueness of stable reconstruction and the structure of its admissible variations.

XII. ACTION / REPRESENTATION / DUALITY FAMILY

ACTION.
One independently generated transition system acting consequentially on another carrier.

REPRESENTATION.
Downstream encoding of source transition structure on another carrier. The source explicitly insists SOURCE TRANSITION SYSTEM ≠ REPRESENTATION.

LINEARIZATION.
Representation of generated transitions as linear operations after additive/scalar structure has independently been earned.

IRREDUCIBLE REPRESENTATION.
Representation containing no proper nontrivial invariant subcarrier of the specified type.

INTERTWINER.
Map transporting one representation action into another while preserving the represented action relation.

CHARACTER.
Compressed representation readout, commonly through multiplicative/trace information; not the representation itself.

PROBE.
Operation interrogating a carrier for a consequential distinction.

SEPARATING FAMILY.
Probe family capable of distinguishing every relevant pair of source states.

DUAL CARRIER.
Carrier generated from a separating family of probes.

DUALITY.
Recoverable alternative carrier generated through separation plus reconstruction.

WEAK DUALITY.
Transport between two carriers without full reconstruction.

RECONSTRUCTIVE DUALITY.
Dual relation admitting recovery of the relevant source structure.

BIDIRECTIONAL DUALITY.
Each side reconstructs the other within declared scope.

XIII. SPECTRAL-HARMONIC FAMILY

INVARIANT MEASURE.
Measure earned from an action/invariance requirement rather than assumed generically.

HARMONIC ANALYSIS.
Decomposition of generated action/translation structure using separating/invariant modes.

SPECTRUM.
Downstream decomposition/readout of stable action modes.

EIGENSTRUCTURE.
Structure transformed by an action through a simplified stable response.

FOURIER TRANSFORM.
Dual transport between a carrier and a generated character/probe carrier when inversion/reconstruction conditions hold.

CONVOLUTION.
Composition law for functions/readouts induced by an underlying translation/action structure.

AUTOMORPHIC STRUCTURE.
Global spectral structure generated from arithmetic quotient/symmetry carriers.

AUTOMORPHIC REPRESENTATION.
Irreducible representation-level organization of global automorphic spectral structure.

HECKE STRUCTURE.
Commuting/localized correspondence operations whose eigenstructure compresses arithmetic-spectral data.

L-FUNCTION.
Generated local/global spectral-arithmetic readout; never fundamental ontology in the source reconstruction.

SPECTRAL-HARMONIC MATHEMATICS.
Family generated by decomposition of actions into stable modes and their local/global reconstruction.

XIV. SYMMETRY-EXTENSION / GALOIS FAMILY

EXTENSION.
Carrier containing a previously generated arithmetic/algebraic carrier while preserving a specified embedding/base structure.

BASE-PRESERVING SYMMETRY.
Transformation of an extension that leaves the base relational grammar fixed.

GALOIS STRUCTURE.
Arithmetic symmetry branch generated from extension-preserving transformations. In the original glossary: “arithmetic symmetry branch.”

LOCAL GALOIS STRUCTURE.
Galois structure relative to a local completion/place.

DECOMPOSITION STRUCTURE.
Subsymmetry preserving a selected local place/valuation.

INERTIA.
Part of local symmetry acting trivially on the residue-level quotient while retaining finer local structure.

FROBENIUS STRUCTURE.
Compressed local arithmetic transition arising from finite residue behavior.

FROBENIUS PARAMETER.
Local arithmetic/Galois information compressed into conjugacy/eigenvalue data.

SYMMETRY-EXTENSION MATHEMATICS.
Family generated when extension structure is reconstructible through its preserving symmetries.

XV. PROBABILISTIC / DYNAMICAL / COLLECTIVE FAMILY

UNRESOLVED SUCCESSOR MULTIPLICITY.
Multiple admissible continuations remain after current constraints have been conditioned upon.

PROBABILITY.
Stable normalized weighting grammar generated over unresolved alternatives once aggregation/additivity requirements are earned.

RANDOMNESS.
Unresolved successor distinction relative to the current admitted constraint/probe carrier; distinct from ignorance created by projection.

STOCHASTIC PROCESS.
Repeated probabilistically weighted transition ecology.

FILTRATION.
Increasing ancestry/information carrier available during sequential execution.

MARKOV COMPRESSION.
Current carrier sufficient to predict admitted future transition probabilities without additional retained ancestry.

RECURRENCE.
Persistent return/coverage under recursive execution.

REGENERATION.
Ancestry-null interface after which the relevant future can be reconstructed independently of the pre-interface history.

ERGODICITY.
Long-execution statistical structure collapsing appropriate initial distinctions into a common invariant readout.

MIXING.
Progressive loss of specified dependence between transported portions of the system.

CHAOS.
Recursive dynamics in which small consequential distinctions are amplified so that the chosen finite compression cannot retain predictive equivalence.

INTEGRABILITY.
Dynamics whose apparent complexity recompresses into a stable invariant grammar permitting reconstruction.

COLLECTIVE CARRIER.
Joint state required when many interacting components cannot be represented by independent marginals.

PHASE.
Stable equivalence class of collective closure structure.

PHASE TRANSITION.
Retyping of stable collective closure as a control parameter/constraint changes.

CRITICALITY.
Regime in which scale-local compression ceases to isolate a characteristic stable scale and structure persists across refinement scales.

TOPOLOGICAL ORDER.
Robust global relational distinctions not reconstructible from bounded local order data.

XVI. SCALE-RECOMPRESSION FAMILY

RESOLUTION.
Level at which distinctions are admitted as consequential.

COARSE-GRAINING.
Controlled quotient/removal of fine distinctions with an explicit lost-information ledger.

EFFECTIVE CARRIER.
Compressed carrier sufficient for a declared scale/scope.

RENORMALIZATION.
Recursive scale transformation consisting of controlled distinction erasure, transport of surviving residue and minimal recompression.

RELEVANT STRUCTURE.
Structure amplified or increasingly consequential under scale recompression.

IRRELEVANT STRUCTURE.
Structure consistently erased without altering the declared large-scale consequences.

MARGINAL STRUCTURE.
Structure whose scale burden is neither clearly amplified nor erased.

RENORMALIZATION FIXED POINT.
Generative grammar reproduced, within equivalence, after another admissible scale-recompression step.

UNIVERSALITY.
Different microscopic constructors becoming indistinguishable because the same irreducible generator/carrier survives refinement.

UNIVERSALITY CLASS.
Equivalence class under surviving scale grammar rather than microscopic ancestry.

SCALE-RECOMPRESSION MATHEMATICS.
Family generated when change of resolution itself becomes the consequential transformation.

XVII. OPERATOR / FIELD / OBSTRUCTION FAMILY

OPERATOR CARRIER.
Carrier whose elements are transformations that themselves compose and interact.

OPERATOR THEORY.
Mathematics generated when transformation bodies become the primary compositional carrier.

KERNEL.
Source distinctions erased by an operator relative to the specified zero/neutral readout.

COKERNEL.
Residual target structure not attained by the operator image.

RESOLVENT.
Derived response representation of an operator where an inverse of a shifted/scaled operator is earned.

SEMIGROUP EVOLUTION.
One-parameter/ordered compositional family describing repeated evolution after that parameter structure has been generated.

FIELD.
Local assignment of relational state over a generated locality carrier with compatibility/interaction structure.

FIELD THEORY.
Mature ecology governing local assignments, interactions, propagation and global reconstruction.

GAUGE FIELD THEORY.
Field theory whose local descriptive equivalences force comparison transport and whose transport carrier is dynamical.

QUANTUM FIELD THEORY.
Field/local-interaction ecology represented through quantum amplitude/operator structure once those structures have independently been earned.

TOPOLOGICAL FIELD THEORY.
Field-theoretic structure whose admitted readouts survive removal of constitutive local metric detail.

OBSTRUCTION THEORY.
Systematic classification of residues preventing extension/lift/reconstruction.

INDEX.
Compressed imbalance/residue invariant associated with a generated transformation problem.

INDEX THEORY.
Ecology in which independently generated analytic and geometric/topological residue descriptions are shown to encode a common source structure.

ANOMALY.
Obstruction to transporting a symmetry/constraint consistently into a richer representation or global carrier.

XVIII. CORRESPONDENCE ECOLOGY

INDEPENDENT FORMATION.
Each side of a proposed correspondence must be generated without using the other as its constructor.

COMMON PARAMETER CARRIER.
Third carrier into which independently generated structures can be transported while retaining the required invariants.

CORRESPONDENCE.
Structured relation between independently generated sides mediated by preserved structure; correspondence is not identity.

FUNCTORIAL TRANSPORT.
Higher-order lawful transport between parameter/representation carriers preserving the required compositional structure. The source uses this reading explicitly.

TRACE FORMULA.
Global comparison architecture relating geometric/conjugacy and spectral/automorphic decompositions of one global system.

ENDOSCOPY.
Additional relational carrier required when naive one-group comparison fails to account for stable multiplicity/conjugacy structure.

TRANSFER.
Transport of orbital/spectral structures between distinct symmetry carriers while preserving the required comparison data.

MIRROR-SYMMETRY-TYPE CORRESPONDENCE.
Bidirectional reconstruction between independently generated geometric ecologies through deeper invariant/dual structures.

CORRESPONDENCE ECOLOGY.
Highest rebuilt taxonomy level where mature mathematical families themselves become independently generated sides of a larger reconstruction.

XIX. LANGLANDS BRANCH

GALOIS BRANCH.
Arithmetic/symmetry reconstruction producing Galois/parameter data.

AUTOMORPHIC BRANCH.
Harmonic/spectral reconstruction producing automorphic data.

Ĝ.
Generated dual symmetry carrier.

SATAKE PARAMETER.
Local automorphic structure compressed into dual-group conjugacy data.

FROBENIUS PARAMETER.
Local arithmetic/Galois structure compressed into corresponding conjugacy/eigenvalue data.

^LG.
Arithmetic-aware dual parameter carrier in the reconstructed Langlands branch.

LANGlands PARAMETER.
Common third carrier mediating independently generated arithmetic/Galois and automorphic/spectral structures.

LOCAL CORRESPONDENCE.
Structure-preserving matching through the common parameter grammar at a local arithmetic/spectral carrier.

GLOBAL COHERENCE.
Compatibility surviving assembly of local correspondences across the relevant global carrier.

LANGlands.
Stable local/global transport architecture relating independently generated arithmetic-symmetry and automorphic-spectral structures through a common dual parameter grammar.

τ ↛ LANGLANDS.
Langlands is not logically forced by τ alone; it is one deep branch generated after additional structures become independently necessary. The original construction explicitly lists associative closure, reversibility, discrete recursion, multiplicative structure, field closure, valuations, completion, local/global ancestry, linearization, invariant integration, reductive symmetry, duality, spectral decomposition and global coherence among structures that had to be added.

XX. REBUILT TAXONOMY TERMS

GENERATIVE FAMILY.
Class of mature mathematics sharing the same primary irreducible generative problem/body.

MATHEMATICAL ECOLOGY.
Interacting collection of generated carriers, operations, invariants, residues, representations and reconstruction rules stable under replay.

MATURE CONCEPT.
Replay-stable generated body recurring across non-identical realizations and surviving representation erasure.

THEORY FAMILY.
Large coherent collection of mature concepts generated by a shared relational grammar.

CORRESPONDENCE ECOLOGY.
Structure formed when entire independently generated theory families become sides of a higher reconstruction.

ABSTRACTION SCALE Λ.
Generated-level classifier:

Λ0 source distinction
Λ1 operation
Λ2 relation
Λ3 carrier/structure
Λ4 mechanism
Λ5 mature concept
Λ6 theory family
Λ7 correspondence ecology.

SIBLING RULE.
Taxonomic siblings should inhabit the same abstraction level unless an explicit cross-level relation is being expressed.

CONCEPT FORMATION.
Promotion of a recurring irreducible generated body to mature-concept status.

CONCEPT MERGE.
Collapse of differently named structures when their differences vanish under representation erasure and they share the same irreducible generator.

CONCEPT SPLIT.
Division of one conventional concept when it contains multiple independently generated, non-interderivable bodies.

TAXONOMY AXIS.
Independently consequential generated distinction capable of separating mature concepts/families.

AXIS DISCOVERY.
Perturbation of generated properties followed by causal discrimination to determine the independent dimensions of the taxonomy.

GRM TAXONOMY.
Taxonomy emitted after generative reconstruction, concept formation, level normalization and axis discovery—not a rearrangement of conventional subject names.

CROSS-FAMILY COMPOUND.
Mature theory formed at the intersection of several independently generated families.

Examples from the rebuilt taxonomy:

GAUGE THEORY
≈ transport-geometric ⊗ local/global ⊗ action-representation ⊗ field-interaction.

RENORMALIZATION
≈ scale-recompression ⊗ dynamics ⊗ collective/statistical structure.

COHOMOLOGY
≈ local/global reconstruction ⊗ obstruction ⊗ accessibility/deformation.

LANGLANDS
≈ recursive-arithmetic ⊗ symmetry-extension ⊗ representation ⊗ spectral-harmonic ⊗ duality ⊗ local/global ⊗ correspondence.

XXI. VALIDATION / DISCOVERY TERMS

NONALIASED ROUTES.
Generative routes whose apparent independence does not arise merely from different surface representations over shared assumptions/carriers/normalizations.

COMMON-MODE FAILURE.
Failure shared because several apparently independent routes inherit the same hidden constructor.

GENERATIVE EQUIVALENCE.
Two different construction bases reproduce all required mathematics with bidirectional source liftback.

GENERATIVE MULTIPLICITY.
Persistence of multiple irreducible, non-equivalent generative bases after replay and recompression.

SUCCESS RECOMPRESSION.
Attack on successful mathematics by deleting generated primitives/carriers/concepts one at a time and testing whether the complete structure can still be regenerated.

QUANTIFICATION TRIGGER.
Requirement to move from qualitative structure to an exact mathematical discriminator once type, carrier and relation stabilize.

QUANTITATIVE DISCRIMINATOR.
Exact identity, inequality, bound, constant, rank, dimension, measure, rate, spectrum, asymptotic, finite witness or counterexample capable of separating competing generated bodies.

NOVELTY GATE.
Criterion distinguishing genuine new mathematics from reinterpretation.

RECOMPRESSION ONLY.
Status when a GRM reconstruction supplies a cleaner basis or dependency graph but no new mathematical consequence.

NEW GENERATED MATHEMATICS.
Status requiring at least one substantive gain such as weaker assumptions, stronger result, new invariant, new obstruction, new construction, new counterexample, new correspondence or validated taxonomy collapse/split.

RECOVERY CLOSURE.
Closure relative to a declared primitive, operation and consequence scope—not absolute mathematical completeness.

NEW PRIMITIVE CANDIDATE.
Provisional primitive admitted only after current constructibility closure is exhausted and irreducibility, transport and replay survive.

FRONTIER PAYLOAD.
Exact surviving residue plus its source ancestry, attempted CKs and smallest currently executable next edge.

XXII. FINAL GRM COMPRESSION

The original glossary compressed From τ to Langlands as:

GRM
→ τ_K
→ τ_K∘τ_K
→ INVARIANT ⊕ RESIDUE
→ SYMMETRY ⊕ ARITHMETIC
→ LOCAL ⊗ GLOBAL
→ REPRESENTATION ⊗ DUALITY
→ GALOIS ⊗ AUTOMORPHIC
→ ^LG / L-PARAMETERS
→ LANGLANDS.

The rebuilt taxonomy expands that into:

GRM
→ FUNDAMENTALS₀ ⊗ DISCOVERY₀
→ FORMATION COURT
→ τ_K where earned
→ COMPOSITION / INTERACTION
→ INVARIANT ⊕ RESIDUE
→ COMPRESSION ⊕ CK/SUCCESSOR
→ RECOMPOSITION ↺
→ {
COMPOSITIONAL-COHERENCE
⊗ RECURSIVE-ARITHMETIC
⊗ ACCESSIBILITY-DEFORMATION
⊗ REFINEMENT-COMPLETION
⊗ TRANSPORT-GEOMETRIC
⊗ LOCAL↔GLOBAL
⊗ VARIATION-MODULI
⊗ ACTION-REPRESENTATION
⊗ PROBE-DUALITY
⊗ SPECTRAL-HARMONIC
⊗ SYMMETRY-EXTENSION
⊗ PROBABILISTIC
⊗ DYNAMICAL
⊗ COLLECTIVE
⊗ SCALE-RECOMPRESSION
⊗ OPERATOR-EVOLUTION
⊗ FIELD-INTERACTION
⊗ OBSTRUCTION-INDEX
}
→ mature concepts
→ intersecting theory families
→ independently generated sides
→ common parameter carriers
→ correspondence ecologies
→ GRM_TAXONOMYΩ.

The non-collapse laws remain:

τ_K ≠ object
relation ≠ carrier
carrier ≠ representation
representation ≠ source
local ≠ global
invariant ≠ ontology
residue ≠ missing-object name
parameter ≠ either mediated side
duality ≠ identity
correspondence ≠ identity
replay stability ≠ unique ontology
generated taxonomy ≠ conventional taxonomy.