GRM — New Mathematics Discovery Program

New math topic	Core object / question	Discovery target
Residual Geometry	\(F(x)\mapsto R_F(x)\)	Geometry encoded by failure, defect, obstruction, mismatch
Constructor Geometry	constraints \(\to\) minimal constructors	Which operators/objects are forced by constraints
Semantic Curvature	transport around semantic loops	Nontrivial return defect under representation/presentation transport
Fracture Mathematics	decomposition \(\to\) recomposition loss	Information existing only in broken interactions
Interaction Cohomology	irreducible \(n\)-ary interaction	Structure invisible to all lower-order marginals
Carrier-Relative Mathematics	law \(\times\) carrier	When changing carrier changes mathematical meaning
Local Model Mutation	\(M_x\to M'_x\) under residual pressure	Mathematics whose local grammar changes across a domain
Boundary Generation Theory	compatibility failure \(\to\) boundary object	When interfaces/defects/boundaries must emerge
Zero-Defect Mathematics	\(R=0\)	Distinguish exactness, flatness, splitting, degeneracy, hidden quotient
Semantic Phase Transitions	continuous source change \(\to\) grammar change	Thresholds where the correct mathematical ontology changes
Ancestry Geometry	object + construction history	When ancestry is constitutive rather than eliminable
Compression Residue Theory	\(X\to C(X)\to \operatorname{Expand}(C(X))\)	Mathematical structure destroyed by semantic compression
Discovery Topology	space of possible mathematical structures	Neighborhoods defined by minimal semantic mutation
Grammar Cohomology	obstruction outside \(\operatorname{closure}(\Gamma)\)	Detect when a language requires new primitives/laws
Representation Friction	operation transport between representations	Structural cost or defect despite semantic equivalence
Irreducible Interaction Geometry	whole not generated by proper subsystems	Geometry of genuinely higher-order interaction
Semantic Renormalization	fine language \(\to\) coarse language	Laws preserved, lost, or emergent under semantic coarse-graining
Constraint-Generated Ontology	constraints \(\to\) distinctions \(\to\) operators \(\to\) objects	Derive ontology instead of assuming it
Mathematics of Nonclosure	\(\Gamma_0\to\Gamma_1\to\Gamma_2\to\cdots\)	Invariants of unavoidable successor-language growth
Constructor Dynamics	\(S_t\xrightarrow{R_t}S_{t+1}\)	Attractors, cycles, bifurcations and invariants of discovery itself

Detailed Table of Contents

Part I — Foundations of Constructed Mathematics

GRM as a Domain-Neutral Mathematical Constructor
1.1 Mathematics as constructed structure rather than inherited vocabulary
1.2 Source, carrier, representation, operator, residual, and readout
1.3 Constructor semantics versus subject-matter semantics
1.4 Why GRM must not absorb the mathematics it constructs
1.5 Domain-neutral primitives
1.6 Typed operators and executable obligations
1.7 Semantic branching and noncanonical alternatives
1.8 Residual-first discovery
1.9 Constructor-language mutation
1.10 Replay, ancestry, and rehydration
1.11 Local closure versus global closure
1.12 Discovery without authority backflow
Constraint-Generated Ontology
2.1 Constraints before objects
2.2 Distinctions as primitive discovery pressure
2.3 From distinctions to required operators
2.4 From operators to emergent carriers
2.5 Minimal ontology under semantic sufficiency
2.6 Competing ontologies satisfying the same constraints
2.7 Ontology equivalence and ontology residue
2.8 When new primitives are unavoidable
2.9 Ontology mutation under unresolved residuals
2.10 Constructor criteria for genuinely new mathematical objects
Constructor Geometry
3.1 Constructor spaces
3.2 Minimal generating systems
3.3 Constructor adjacency
3.4 Constructor deformation
3.5 Constructor equivalence
3.6 Irreducible constructor directions
3.7 Local versus global constructor sufficiency
3.8 Constructor singularities
3.9 Constructor moduli
3.10 Geometry induced by repair trajectories

Part II — Mathematics of Residuals and Failure

Residual Geometry
4.1 Residuals as mathematical objects
4.2 Residual carriers
4.3 Residual factorization
4.4 Residual support
4.5 Residual locality
4.6 Residual symmetry
4.7 Residual rank and multiplicity
4.8 Residual interactions
4.9 Residual transport
4.10 Residual composition
4.11 Residual equivalence
4.12 Residual stratification
4.13 Residual singularities
4.14 Residual-generated operators
4.15 Residual-generated ontology
Zero-Defect Mathematics
5.1 Why zero is not a single semantic state
5.2 Operator-indexed zero semantics
5.3 Exact zero
5.4 Integrable zero
5.5 Flat zero
5.6 Split zero
5.7 Symmetry-generated zero
5.8 Degenerate zero
5.9 Quotient-induced zero
5.10 Information-destroying zero
5.11 False zero under representation collapse
5.12 Zero-class transitions
5.13 Zero as a discovery trigger
Mathematics of Nonclosure
6.1 Closure as a local property
6.2 Stable nonclosure
6.3 Successor-language chains
6.4 Nonclosure depth
6.5 Nonclosure rank
6.6 Nonclosure recurrence
6.7 Irreducible closure gaps
6.8 Finite versus transfinite successor growth
6.9 Nonclosure invariants
6.10 When incompleteness is structural rather than temporary

Part III — Fracture, Interaction, and Higher Arity

Fracture Mathematics
7.1 Decomposition as an information-destroying operation
7.2 Fracture operators
7.3 Recomposability
7.4 Fracture residuals
7.5 Pairwise-preserving but globally destructive fractures
7.6 Local fracture versus global fracture
7.7 Carrier fracture
7.8 Representation fracture
7.9 Scale fracture
7.10 Ancestry fracture
7.11 Fracture invariants
7.12 Fracture-induced new primitives
Irreducible Interaction Geometry
8.1 Whole structure not generated by proper subsystems
8.2 Interaction arity
8.3 Higher-order interaction carriers
8.4 Pairwise shadows
8.5 Joint-only invariants
8.6 Interaction support
8.7 Interaction boundaries
8.8 Interaction transport
8.9 Interaction curvature
8.10 Irreducible interaction classes
8.11 Interaction geometry under coarse-graining
Interaction Cohomology
9.1 Cochains indexed by interaction arity
9.2 Lower-order projections
9.3 Hidden higher-order classes
9.4 Interaction coboundaries
9.5 Failure of reconstruction from marginals
9.6 Arity spectral sequences
9.7 Obstruction classes for interaction collapse
9.8 Local-to-global interaction assembly
9.9 Interaction persistence
9.10 Constructor discovery from nontrivial classes

Part IV — Representation, Carrier, and Semantic Transport

Carrier-Relative Mathematics
10.1 Mathematical law versus carrier
10.2 Carrier ownership
10.3 Carrier substitution
10.4 Carrier-preserving transport
10.5 Carrier mutation
10.6 Wrong-carrier validity
10.7 Carrier-local truths
10.8 Carrier-global claims
10.9 Carrier equivalence
10.10 Carrier residue
10.11 When the same formula defines different mathematics
Representation Friction
11.1 Representation equivalence is not transport triviality
11.2 Friction of operator transport
11.3 Friction of normalization
11.4 Friction of localization
11.5 Friction of quotienting
11.6 Friction under coordinate change
11.7 Computational versus semantic friction
11.8 Friction tensor/cocycle candidates
11.9 Friction accumulation along paths
11.10 Friction singularities
11.11 Friction as a discovery signal
Semantic Curvature
12.1 Semantic transport paths
12.2 Loops in representation space
12.3 Failure of semantic return
12.4 Semantic holonomy
12.5 Semantic curvature operators
12.6 Flat semantic regions
12.7 Curvature concentrated on representation transitions
12.8 Curvature generated by quotient or compression
12.9 Curvature under model mutation
12.10 Semantic Bianchi-type constraints
12.11 Discovery from nonzero semantic holonomy

Part V — Compression, Coarse-Graining, and Renormalization

Theory of Mathematical Compression Residues
13.1 Semantic compression versus textual compression
13.2 Compression operators
13.3 Exact expansion
13.4 Lost distinctions
13.5 Lost interactions
13.6 Lost ancestry
13.7 Compression residual
13.8 Compression equivalence
13.9 Minimal sufficient representations
13.10 Irreversible compression
13.11 Compression-induced false invariants
13.12 Compression residue as a new mathematical object
Semantic Renormalization
14.1 Fine and coarse mathematical languages
14.2 Semantic coarse-graining operators
14.3 Scale-dependent primitives
14.4 Stable laws under coarse-graining
14.5 Emergent laws
14.6 Disappearing laws
14.7 Semantic fixed points
14.8 Relevant and irrelevant distinctions
14.9 Renormalization of operator vocabularies
14.10 Renormalization of residuals
14.11 Universality classes of mathematical descriptions

Part VI — Grammar, Language, and Successor Mathematics

Grammar Cohomology
15.1 Mathematical language as an executable grammar
15.2 Expressible versus inexpressible distinctions
15.3 Grammar defects
15.4 Grammar cocycles
15.5 Primitive-extension classes
15.6 Law-extension classes
15.7 Arity-extension classes
15.8 Transport failures as grammar obstructions
15.9 Grammar cohomology and successor languages
15.10 Vanishing classes and sufficient languages
15.11 Persistent classes and irreducible mathematical novelty
Discovery Topology
16.1 Space of candidate mathematical structures
16.2 Semantic neighborhoods
16.3 Minimal mutations as adjacency
16.4 Residual-directed neighborhoods
16.5 Discovery paths
16.6 Discovery loops
16.7 Discovery barriers
16.8 Discovery singularities
16.9 Connected components of hypothesis space
16.10 Topological obstruction to local repair
16.11 Successor-language transitions as topology change
Local Model Mutation
17.1 One global grammar as an unjustified prior
17.2 Local mathematical models
17.3 Model transition maps
17.4 Mutation triggers
17.5 Residual-localized mutation
17.6 Compatibility across model boundaries
17.7 Model atlases
17.8 Model holonomy
17.9 Mutation propagation
17.10 Stable heterogeneous mathematics
Semantic Phase Transitions
18.1 Continuous source change versus discrete ontology change
18.2 Constructor instability
18.3 Primitive birth and death
18.4 Law mutation
18.5 Representation bifurcation
18.6 Residual criticality
18.7 Semantic order parameters
18.8 Critical constructor regimes
18.9 Hysteresis in mathematical languages
18.10 Phase diagrams of mathematical descriptions

Part VII — Boundary and Interface Generation

Boundary Generation Theory
19.1 Boundaries as emergent rather than assumed objects
19.2 Compatibility failure
19.3 Interface residues
19.4 Boundary constructors
19.5 Boundary support
19.6 Boundary transport
19.7 Boundary-local interactions
19.8 Boundary ancestry
19.9 Boundary equivalence
19.10 Nested and higher-codimension boundaries
19.11 When bulk mathematics forces new boundary mathematics
Interface and Defect Algebra
20.1 Defect composition
20.2 Defect fusion
20.3 Defect cancellation
20.4 Defect transport
20.5 Interface branching
20.6 Defect invariants
20.7 Defects as constructors
20.8 Defects that mutate the surrounding language

Part VIII — Ancestry and Generative Identity

Ancestry Geometry
21.1 Objects with construction histories
21.2 Extensional equality versus generative equality
21.3 Ancestry paths
21.4 Ancestry branching
21.5 Ancestry cycles
21.6 Minimal ancestry
21.7 Ancestry-preserving transformations
21.8 Ancestry curvature
21.9 Ancestry collapse
21.10 When provenance becomes mathematically constitutive
Generative Identity Theory
22.1 Identity through recursion
22.2 Equivalent outputs from inequivalent constructions
22.3 Constructor-sensitive identity
22.4 Identity under compression
22.5 Identity under transport
22.6 Identity under mutation
22.7 Identity under branching
22.8 Fractured identity
22.9 Rehydrated identity

Part IX — Dynamics of Mathematical Discovery

Constructor Dynamics
23.1 Construction states
23.2 Residual-driven transitions
23.3 Repair trajectories
23.4 Constructor attractors
23.5 Constructor cycles
23.6 Dead-end grammars
23.7 Bifurcation of constructor languages
23.8 Conserved semantic quantities
23.9 Discovery entropy
23.10 Irreversible discovery events
23.11 Long-term dynamics of self-expanding mathematics
Repair Dynamics
24.1 Residual extraction
24.2 Capability requirements
24.3 Constructor synthesis
24.4 Failed repair as secondary residual
24.5 Repair depth
24.6 Repair branching
24.7 Repair convergence
24.8 Repair cycles
24.9 Repair-induced language growth
24.10 Stable repair versus superficial relabeling
Mathematical Search Geometry
25.1 Search spaces generated by constraints
25.2 Search directions induced by residuals
25.3 Blind spots and streetlights
25.4 Search anisotropy
25.5 Search barriers
25.6 Search curvature
25.7 Local optimum versus semantic closure
25.8 Search-space expansion by primitive creation

Part X — Cross-Theory Synthesis

Residual ↔ Fracture Duality
26.1 Residual as failed reconstruction
26.2 Fracture as induced residual
26.3 Shared invariants
26.4 Interaction loss and residual support
Curvature ↔ Friction Correspondence
27.1 Local friction
27.2 Loop accumulation
27.3 Holonomy from representation cost
27.4 Flatness conditions
Grammar ↔ Ontology Correspondence
28.1 Grammar determines constructible objects
28.2 Ontology pressures grammar
28.3 New primitive versus new law
28.4 Grammar-ontology fixed points
Compression ↔ Renormalization Correspondence
29.1 Semantic loss versus scale transformation
29.2 Stable coarse descriptions
29.3 Emergent invariants
29.4 Irreducible compression residue
Interaction ↔ Boundary Correspondence
30.1 Higher-order interaction as interface generator
30.2 Boundary emergence from failed decomposition
30.3 Interaction localization
30.4 Boundary-supported residuals
Ancestry ↔ Identity Correspondence
31.1 When extensional equality is insufficient
31.2 History-dependent mathematical identity
31.3 Canonical ancestry
31.4 Ancestry-preserving compression

Part XI — GRM Discovery Tests

Domain-Neutrality Test
32.1 Can GRM instantiate the topic without kernel changes?
32.2 Topic primitives versus constructor primitives
32.3 Detecting semantic contamination
32.4 Forced kernel growth as failure signal
Novelty Test
33.1 Renaming existing mathematics versus new structure
33.2 New invariant criterion
33.3 New operator criterion
33.4 New obstruction criterion
33.5 New equivalence criterion
33.6 New constructor criterion
Irreducibility Test
34.1 Reduction to known theory
34.2 Partial reduction
34.3 Residual after reduction
34.4 Genuine primitive requirement
Executable Discovery Test
35.1 Explicit domain and codomain
35.2 Constructible operator
35.3 Structured residual
35.4 Falsifiable transport law
35.5 Replayable discovery path
35.6 Rehydratable theory state
Streetlight Test
36.1 Is the topic defined by available tools rather than source structure?
36.2 Are hard-to-measure interactions being silently deleted?
36.3 Does canonicalization suppress alternatives?
36.4 Is representation convenience becoming ontology?
36.5 Is finite search being mistaken for impossibility?

Part XII — Priority Discovery Programs

Program A — Residual Geometry
37.1 Minimal axioms
37.2 Residual composition
37.3 Residual transport
37.4 Zero classification
37.5 Residual-generated geometry
37.6 First nontrivial examples
Program B — Fracture Mathematics
38.1 Formal fracture operator
38.2 Recomposability
38.3 Fracture residue
38.4 Interaction recovery
38.5 Higher-order fracture invariants
Program C — Grammar Cohomology
39.1 Grammar complex
39.2 Obstruction classes
39.3 Primitive extension
39.4 Law extension
39.5 Successor-language criterion
Program D — Semantic Curvature
40.1 Semantic connections
40.2 Transport loops
40.3 Holonomy
40.4 Curvature
40.5 Flat semantic structures
Program E — Interaction Cohomology
41.1 Arity complex
41.2 Marginal projections
41.3 Hidden joint classes
41.4 Interaction obstruction
41.5 Higher-order reconstruction
Program F — Constraint-Generated Ontology
42.1 Distinction generation
42.2 Operator necessity
42.3 Primitive synthesis
42.4 Ontology branch space
42.5 Minimal generative ontology

Part XIII — Unified New-Mathematics Field

Common Constructor Spine
\[ \Sigma \to DIST \to REQ \to OP \to R \to CLASS \to NEWREQ \to SYNTH \to \Delta\Gamma \to \Sigma' \]
Common Geometric Spine
\[ \text{local structure} \to \text{transport} \to \text{interaction} \to \text{residual} \to \text{curvature/fracture} \to \text{new geometry} \]
Common Language Spine
\[ \Gamma \to GAP \to H^\bullet_{\Gamma} \to NEWPRIM/NEWLAW \to \Gamma' \]
Common Discovery Spine
\[ \text{failure} \not\to \text{rejection} \]\[ \text{failure} \to \text{structured information} \to \text{constructor pressure} \to \text{new mathematics} \]
Unified Research Objective
47.1 Discover structures not visible inside inherited mathematical vocabularies
47.2 Preserve higher-order interaction and local curvature
47.3 Prevent representation from becoming ontology
47.4 Turn failure into mathematical signal
47.5 Permit mathematics to change its own language
47.6 Separate constructor machinery from constructed subject matter
47.7 Build genuinely new mathematical domains rather than renamed old ones

Below is a working glossary for the GRM New Mathematics Discovery Program, using the terminology established in the program rather than importing standard meanings where the terms are intentionally new.

New Mathematics Discovery Program — Glossary

A

Adjacency
A relation specifying which mathematical states, constructors, models, representations, or semantic regions are reachable from one another by an admissible minimal transformation.

Ancestry
The construction history of a mathematical object, including the operators, transformations, dependencies, representations, and prior states from which it was generated.

Ancestry Geometry
The study of mathematical structure carried by construction histories themselves: branching, merging, transport, cycles, minimal ancestry, and history-dependent identity.

Ancestry-Preserving Transformation
A transformation that changes a mathematical object without destroying construction information required for its semantic interpretation.

Arity
The number of simultaneously participating inputs or components required for an interaction. Higher arity can contain structure invisible to all lower-arity projections.

Arity Extension
A successor-language repair in which the current grammar must admit higher-order interactions because existing unary, pairwise, or lower-order constructions cannot represent the observed residual.

B

Behavioral Equivalence
Equivalence determined by preservation of admissible operations, consequences, and tests rather than syntactic identity or representation form.

Boundary Generation
The emergence of a new interface, defect, boundary object, or edge structure because bulk structures fail to compose, glue, transport, or interact coherently.

Boundary Generation Theory
The proposed mathematics of when and how boundaries are forced into existence by compatibility residuals rather than assumed in advance.

Branch
A semantically distinct valid construction that cannot legitimately be collapsed into another candidate by representation choice, byte order, or arbitrary canonical selection.

Branch Residue
The mathematical consequences unique to one semantic branch after consequences common to all branches have been removed.

C

Canonicality, Mathematical
Uniqueness or preferred status forced by mathematics—for example through a universal property, behavioral equivalence class, or necessary ancestry.

Canonicality, Serialization
A deterministic choice of textual or computational representation. It has no authority to decide mathematical identity.

Carrier
The structure or domain on which mathematical laws, operations, or representations live. Changing the carrier may change the semantics even when formulas remain identical.

Carrier Mutation
Replacement or modification of the carrier in response to evidence that the existing carrier cannot support the required structure.

Carrier-Relative Mathematics
The proposed study of how mathematical truth, structure, and validity depend on the carrier supporting them.

Compression Residue
The semantic structure lost when a mathematical object is compressed and subsequently reconstructed.

Constraint-Generated Ontology
A mathematical program in which objects are not assumed first. Constraints generate required distinctions; distinctions generate operators; operators force the ontology.

Constructor
An executable mathematical operation that produces a new object, relation, operator, representation, law, or mathematical state.

Constructor Dynamics
The proposed study of mathematical construction as a dynamical system whose states evolve through residuals, repairs, bifurcations, cycles, and successor-language events.

Constructor Geometry
The proposed study of the geometry of constructor spaces, including adjacency, deformation, equivalence, singularity, branching, and minimal generating systems.

Constructor Space
The set or structured space of admissible mathematical constructors available under a given grammar and source constraint system.

Coverage Gap
A source distinction, behavior, interaction, or residual that the current mathematical language cannot explain, distinguish, or construct.

D

Debt
A compressed classification of an unresolved mathematical requirement. In the discovery program, debt is downstream of the full residual and must never replace it.

Discovery Dynamics
The recursive process
[
RUN\to RESIDUAL\to REQUIREMENT\to SYNTHESIS\to REPAIR\to RUN.
]

Discovery Topology
The proposed topology on mathematical hypothesis or constructor space, with neighborhoods defined by minimal semantic transformations rather than numerical distance.

Distinction
A source-grounded difference that must remain detectable by any admissible mathematical reconstruction.

Domain Neutrality
The requirement that GRM construct mathematics without embedding the semantics of a particular mathematical topic into its own kernel.

E

Emergent Law
A law that becomes valid only after transformation, compression, coarse-graining, interaction, or constructor evolution and is not primitive at the finer level.

Executable Mathematics
Mathematics expressed with sufficient typing, domains, codomains, operators, and application rules that its consequences and failures can be mechanically evaluated.

Extensional Equivalence
Equality determined only by observable behavior or output. It may be weaker than generative identity when ancestry matters.

F

Failure Signature
A normalized description of a failed construction. It is useful for indexing failures but is weaker than the complete structured residual that produced it.

Fracture
A transformation that decomposes, projects, localizes, averages, partitions, or otherwise separates a mathematical system in a way that may destroy interaction structure.

Fracture Invariant
A quantity or structure characterizing what is preserved or destroyed under fracture.

Fracture Mathematics
The proposed study of the information and structure lost when mathematical wholes are decomposed and later recomposed.

Fracture Residue
The difference between an original mathematical structure and what can be reconstructed from its fractured components.

G

Generative Identity
Identity determined not only by final behavior but also by the construction process that produced the object.

Generative Identity Theory
The proposed mathematics of when two extensionally equivalent outputs should remain distinct because their construction histories differ.

Grammar
The currently available mathematical primitives, operators, typing rules, composition rules, and laws from which constructions may be formed.

Grammar Cohomology
The proposed study of obstructions showing that a mathematical phenomenon lies outside the expressive closure of the current grammar.

Grammar Gap
A residual that cannot be represented, classified, or repaired using the current mathematical language.

H

Higher-Order Interaction
Structure arising only when multiple components participate jointly and which cannot be reconstructed from all lower-order interactions.

Holonomy, Semantic
The residual semantic change obtained after transporting a mathematical object or operation around a closed loop of representations or models.

I

Information Backflow
The permitted use of downstream results or readouts to generate hypotheses, search directions, or probes for upstream reconstruction.

Interaction Cohomology
The proposed theory of cohomological structures indexed by interaction arity, designed to detect information invisible to lower-order marginals.

Interaction Geometry
Geometry generated by relations among interacting components rather than by the components considered independently.

Irreducible Interaction
A joint structure that cannot be reconstructed from any collection of proper lower-order subsystems.

Irreducible Interaction Geometry
The proposed geometry of genuinely higher-order interaction structure.

L

Language Mutation
Modification of the mathematical grammar through a new primitive, operator, law, arity, carrier, or composition rule.

Local Model
A mathematical language or structure valid within a restricted semantic region rather than globally across the entire source domain.

Local Model Mutation
Replacement or modification of the mathematical model in a specific region because the current local grammar ceases to represent the source adequately.

Local Closure
Successful closure within a declared local domain, scale, representation, branch, or finite scope. It carries no automatic global authority.

M

Mathematical Compression
Replacement of a mathematical structure by a smaller representation intended to preserve specified semantic consequences.

Mathematical Compression Residue
The structure not recoverable after compression followed by expansion.

Mathematics of Nonclosure
The proposed study of systems whose correct mathematical description necessarily generates successor languages rather than reaching a final closed grammar.

Model Atlas
A family of local mathematical models together with transition rules specifying how they interact across overlapping semantic regions.

N

New Primitive
A mathematical object or operator that cannot be constructed from the current primitive closure and therefore requires successor-language expansion.

Nonclosure Depth
A measure of how many successive language extensions are required before a specified mathematical obligation can be represented or resolved.

Novelty Test
A procedure for determining whether a proposed new field contributes genuinely new invariants, operators, equivalence relations, obstructions, or constructors rather than renaming existing mathematics.

O

Ontology
The set of mathematical object-types the current theory treats as constructible or primitive.

Ontology Mutation
A change in the admissible mathematical object-types forced by unresolved constraints or residuals.

Operator
A typed transformation with explicit domain, codomain, application behavior, residual semantics, effects, and ancestry.

Operator-Indexed Zero
The principle that the mathematical meaning of a zero residual is determined by the operator producing it rather than by a universal notion of “nothing happened.”

P

Phase Transition, Semantic
A transition in which a continuous change in the source forces a discontinuous change in the mathematical language, ontology, constructor family, or primitive structure.

Primitive Closure
Everything constructible from the current primitive set using licensed operations and composition laws.

Probe
A hypothesis or experimental construction generated from downstream information but carrying no semantic authority until reconstructed independently.

R

Readout
A downstream representation, measurement, invariant, zero-set, statistic, or other output used to inspect a mathematical construction.

Representation
A particular encoding or coordinate realization of a mathematical object. A representation is not identical to the represented object.

Representation Friction
The proposed measure of structural cost, defect, or nontriviality encountered when transporting operations between semantically equivalent representations.

Representation Naturality
The requirement that mathematical operations behave compatibly across an explicitly declared family of admissible representations.

Residual
The structured output describing how a candidate construction fails, differs, or remains incomplete under an operator.

Residual Algebra
A system of operations for composing, comparing, transporting, factoring, localizing, and interpreting residuals.

Residual Geometry
The proposed mathematical field treating residuals as geometric objects capable of carrying support, interaction, symmetry, locality, transport, and higher structure.

Residual-First Discovery
The rule that the full residual must be retained and analyzed before it is compressed into a failure category or debt label.

Residual Support
The source region, component set, interaction domain, or semantic locus on which a residual is nontrivial.

S

Semantic Canonicality
Canonicality established by mathematical behavior, universal structure, or forced ancestry rather than serialization order.

Semantic Compression
Compression that preserves a declared set of mathematical consequences and admits exact or explicitly loss-accounted reconstruction.

Semantic Curvature
The proposed measure of failure to return semantically unchanged after transport around a closed loop of representations, models, or mathematical descriptions.

Semantic Friction
Local resistance encountered when transporting mathematical behavior across representations or model boundaries.

Semantic Phase Transition
See Phase Transition, Semantic.

Semantic Renormalization
The proposed theory of how mathematical languages, operators, laws, and distinctions transform under semantic coarse-graining.

Semantic Spine
The dependency structure that determines mathematical meaning independently of the discovery process used to find it.

Source
The domain of distinctions, constraints, operations, and behavior from which mathematical structure must be reconstructed.

Source Sovereignty
The principle that downstream representations, targets, readouts, or desired results cannot acquire authority over source semantics.

Stable Nonclosure
A regime in which continued successor-language generation is itself persistent structure rather than temporary failure.

Streetlight Effect
Bias introduced when mathematical search is restricted to structures that are easy to represent, compute, measure, or formalize while harder but source-relevant structures are ignored.

Successor Language
A new mathematical grammar produced when the current language cannot represent a source-required distinction, residual, operator, or law.

T

Target Blindness
The requirement that candidate mathematical structures be constructed without allowing desired answers or downstream outputs to determine upstream semantics.

Transport
Movement of mathematical structure across a morphism, representation, carrier, model, scale, or context together with explicit accounting of what is preserved, transformed, lost, or created.

Typed Residual
A residual whose meaning is explicitly determined by the operator, input, output, carrier, and residual schema that produced it.

Z

Zero Class
A semantic classification of a zero residual such as exactness, flatness, integrability, splitting, degeneracy, symmetry, quotient collapse, or information loss.

Zero-Defect Mathematics
The proposed mathematical theory classifying structurally different meanings of vanishing residuals rather than treating every zero as identical.

Core Discovery Relations

[
\boxed{
SOURCE
\neq
REPRESENTATION
\neq
READOUT
\neq
VALIDATION
}
]

[
\boxed{
CONSTRAINT
\to
DISTINCTION
\to
OPERATOR
\to
RESIDUAL
\to
REQUIREMENT
\to
CONSTRUCTOR
}
]

[
\boxed{
FAILURE
\not\Rightarrow
IMPOSSIBILITY
}
]

[
\boxed{
ZERO
\not\Rightarrow
ABSENCE
}
]

[
\boxed{
LOCAL\ CLOSURE
\not\Rightarrow
GLOBAL\ CLOSURE
}
]

[
\boxed{
INFORMATION\ BACKFLOW
\neq
AUTHORITY\ BACKFLOW
}
]

[
\boxed{
SEMANTIC\ CANONICALITY
\neq
SERIALIZATION\ CANONICALITY
}
]

[
\boxed{
CURRENT\ GRAMMAR
\not\Rightarrow
COMPLETE\ MATHEMATICS
}
]

[
\boxed{
RESIDUAL
\to
LANGUAGE\ PRESSURE
\to
SUCCESSOR\ MATHEMATICS
}
]