# NEXAH Mathematical Foundations Glossary

Status: `MAINTAINED SUBJECT GLOSSARY / NO NEW MATHEMATICS`

Maintained through: `2026-10-06`

## Purpose and scope

This glossary is the maintained mathematical-language entrance for NEXAH. It
connects local terminology to established mathematical language, keeps
different meanings of one symbol separate and records what a term does not
establish.

Established terms remain owned by their fields. A NEXAH label is not a new
mathematical object merely because it appears repeatedly in diagrams or
documents.

## Status vocabulary

| Status | Meaning |
|---|---|
| `ESTABLISHED MATHEMATICS` | standard definition, construction or result; exact hypotheses still apply |
| `LOCAL DEFINITION` | stipulated meaning inside one declared NEXAH scope |
| `WORKING NOTATION` | usable notation whose scope must accompany every use |
| `FORMAL CANDIDATE` | mathematically expressible proposal not yet proved or adopted |
| `VISUAL INTUITION` | image or metaphor that motivates a question but does not define the object |
| `UNRESOLVED PROVENANCE` | spelling, glyph, expansion or lineage is not yet source-bound |
| `EXCLUDED IDENTIFICATION` | tempting equality or attribution rejected by current evidence |

## Programme organization

| Term | Status | Working definition | Related established language | Do not infer |
|---|---|---|---|---|
| typed grammar | `LOCAL DEFINITION` | minimum roles and validity conditions required for a NEXAH description, including carrier, frame, map, failure, comparison, provenance and claim ceiling | typed schema; formal grammar; many-sorted signature | one universal ontology or executable language |
| geometry family | `LOCAL DEFINITION` | routing class for the kind of spatial, relational, combinatorial or state-space structure used by a record | metric, affine, projective, topological, graph, dynamical or information geometry as explicitly declared | that similar drawings share one geometry |
| theory family | `LOCAL DEFINITION` | routing class connecting a bounded NEXAH question to an established mathematical or disciplinary toolkit | research area; model class; formal framework | a unified NEXAH theory or transfer of results across fields |
| cross-family mapping | `FORMAL CANDIDATE` | declared relation between source and target family objects, typed as equivalence, structure-preserving map, specialization, composition, approximation, analogy or unresolved | functor or morphism only where their hypotheses are actually supplied; model correspondence | identity, shared mechanism or theorem from resemblance |
| Unified Transition Geometry / `UTG` | `LOCAL FRAMEWORK DEFINITION` | exploratory NEXAH research architecture for organizing local observations, declared structures, stability descriptions, transition candidates, navigation and reconstruction across bounded representations | hybrid-systems modelling; dynamical systems; observation and state-space models; atlas or model-family architecture, depending on each declared object | one established geometry, universal physical theory, shared mechanism across domains or validation of NEXAH as a whole |
| UTG maturity ladder | `LOCAL STATUS CLASSIFICATION` | three-level claim control: `Framework` for shared language and questions; `Formal Model` for an object-specific mathematical specification; `Validated Application` for a domain-specific controlled result | model maturity; technology/readiness classification; evidence grading | that maturity transfers between objects, media or domains, or that a framework-level visual is already a formal model |
| media constellation | `LOCAL PROVENANCE DEFINITION` | registered family of book plates, HTML instruments, GLBs, equation candidates and evidence records linked by a shared concept ID, with preserved relations and abstractions stated per medium | multimodal provenance graph; work/version lineage; multi-view representation | independent confirmation from repeated content, identity of representations or evidential promotion by media count |

## Representations, maps and comparison

| Term | Status | Working definition | Related established language | Do not infer |
|---|---|---|---|---|
| carrier | `ESTABLISHED MATHEMATICS` | declared set, space, graph, manifold, sequence or other domain on which an object or map is defined | underlying set; state space; sample space | one universal NEXAH space |
| generator | `ESTABLISHED MATHEMATICS` | one named transformation from a declared generating set acting on a declared carrier | group generator; state-transition operator | that the same label denotes the same transformation on another carrier |
| generator word | `ESTABLISHED MATHEMATICS / LOCAL EXECUTION CONVENTION` | finite ordered sequence of generator symbols; this Lab's interactive convention applies symbols from left to right | word in generators; composed action | a canonical normal form or equivalence of words from equal length or similar paths |
| free involution reduction | `ESTABLISHED LOCAL REDUCTION` | repeated cancellation of adjacent equal involutive letters using `r_i r_i=I`; it is not a full Coxeter normal-form algorithm | free reduction; involutive presentation | that all Coxeter braid relations have been exhausted |
| Coxeter element | `ESTABLISHED MATHEMATICS / REGISTERED INSTANCE` | product of the simple reflections in a declared order; the registered E8 instance `C=mu_1...mu_8` has order 30 on the 240-root carrier | finite reflection group; Coxeter number | that any reordered or unrelated word has the same visible phase labelling without recomputation |
| orbit phase | `LOCAL ADDRESS` | index of a state inside a declared ordered orbit under one selected operator or word | cyclic action; orbit index | physical phase, time or an intrinsic address independent of the chosen operator and seed |
| translation instrument | `LOCAL DEFINITION` | executable inspection surface retaining carrier, typed operator, view, return and residual while translating between representations | interactive mathematical demonstrator; provenance-preserving view | physical measuring device, universal language, integrated machine or proof produced by animation |
| animation frame | `METHOD BOUNDARY` | displayed record of one computed state or transition in an execution trace | state-transition visualization | a new observation, additional sample or evidence beyond the underlying operator record |
| domain and codomain | `ESTABLISHED MATHEMATICS` | admitted inputs and declared target of a map | function signature; source and target | that every displayed arrow is defined everywhere |
| partial map | `ESTABLISHED MATHEMATICS` | map defined only on a declared subset of its proposed source | partial function | failure outside the domain is a physical boundary |
| representation map | `LOCAL DEFINITION` | declared map `F_r: D_r subseteq X -> Y_r` producing one representation or observation record | observation map; feature map; measurement map | identity between source and representation |
| representation change | `LOCAL DEFINITION` | declared map `T_(r->s)` between representation spaces | coordinate change; translation; transformation | preservation of every property |
| projection | `ESTABLISHED MATHEMATICS` | map to a lower-dimensional, quotient or selected component representation, as locally defined | linear projection; quotient map; rendering projection | that a visual projection is linear, invertible or lossless |
| registration | `ESTABLISHED MATHEMATICS` | estimation of a correspondence or transformation between representations | image registration; point-set registration; graph matching; multi-view geometry | a universal NEXAH alignment law |
| comparison defect | `LOCAL DEFINITION` | declared mismatch between two routes into a common comparison space | residual; normed error; commutation defect | that the metric is canonical |
| commutative path | `ESTABLISHED MATHEMATICS` | two composable map paths with the same result on a declared domain | commutative diagram | equality from diagrammatic resemblance alone |
| approximate commutation | `FORMAL CANDIDATE` | two routes differ by no more than a declared defect bound | stability bound; approximate intertwining | exact equivalence |
| residual | `LOCAL DEFINITION` | typed remainder after comparison, reconstruction or return | reconstruction residual; error vector; discrepancy | substance, truth or failure by itself |
| reconstruction | `ESTABLISHED MATHEMATICS` | recovery estimate of a source or latent object from declared observations | inverse problem; estimator; decoder | exact recovery unless proved |
| identifiability | `ESTABLISHED MATHEMATICS` | distinct admitted sources can be distinguished under the declared observation family and comparison rule | injectivity; statistical identifiability | provenance preservation or truth |
| non-identifiability | `ESTABLISHED MATHEMATICS` | at least two admitted sources produce equivalent observations under the declared rule | collision; aliasing; observational equivalence | that the sources are identical |
| invariant | `ESTABLISHED MATHEMATICS` | named quantity or structure unchanged under a declared transformation class | invariant theory; conserved descriptor | causality, truth or physical conservation |
| equivariant | `ESTABLISHED MATHEMATICS` | output transforms predictably when the input is transformed | group action; intertwining map | invariance or identity |
| register conversion | `ESTABLISHED MATHEMATICS / LOCAL TYPING RULE` | declared map between different quantity or representation types, including its units, domain, codomain and inverse conditions; for fixed nonzero `c`, `C_c(t)=c*t` maps time to length | dimensional analysis; unit conversion; typed isomorphism where applicable | identity of the quantity types, shared physical process or discovery from arithmetic alone |
| related not identical | `METHOD BOUNDARY` | two records may share a value, source, quotient class, abstract structure or invertible conversion while remaining distinct typed objects | equality criteria; equivalence; isomorphism; provenance | that related, equal-valued, convertible or isomorphic means literally identical in every register |

## Relations, graphs and order

| Term | Status | Working definition | Related established language | Do not infer |
|---|---|---|---|---|
| relation | `ESTABLISHED MATHEMATICS` | subset of a Cartesian product, generalized to a declared arity where needed | binary or n-ary relation | function, causality or mechanism |
| typed relation | `LOCAL DEFINITION` | relation record whose roles, carrier, arity and admissible values are explicit | many-sorted relation; relational schema | universal ontology |
| equivalence relation | `ESTABLISHED MATHEMATICS` | reflexive, symmetric and transitive relation | partition; quotient relation | similarity without verified axioms |
| reachability `R*` | `ESTABLISHED MATHEMATICS` | reflexive-transitive closure of a directed relation | path reachability | direct reciprocity |
| strongly connected component | `ESTABLISHED MATHEMATICS` | maximal class of mutually reachable vertices | SCC | direct two-way edge only |
| quotient | `ESTABLISHED MATHEMATICS` | space or set of equivalence classes under a declared equivalence relation | quotient set; quotient graph | preservation of all source information |
| partial order | `ESTABLISHED MATHEMATICS` | reflexive, antisymmetric and transitive relation; a strict version is irreflexive and transitive | poset; strict order | cycles such as `Q < P < Q` |
| return to type | `LOCAL DEFINITION` | arrival at a record of the same declared type or role | typed transition; re-entry | return to the identical state |
| `Q_0 -> P -> Q_1` | `FORMAL CANDIDATE` | corrected process reading of the historical `Q < P < Q` visual, with `Q_0` and `Q_1` distinct unless identity is proved | state transition; typed path | a strict partial-order cycle |
| typed C4 family | `EXACT FINITE BRIDGE / NAMESPACE CONTROL` | separates `C4_group=Z/4Z`, a four-element fiber orbit, `C4_graph`, a chosen `K2 square K2` coordinate realization, observer-bin `C^(epsilon)_4`, musical pitch `C_4` and pitch class `[C]`; only the declared group/Cayley/graph/product bridge is exact | cyclic group; Cayley graph; Cartesian graph product; torsor; observer quantization; pitch-class quotient | identity across the namespaces, intrinsic phase zero/product axes, historical OIL/RIS semantics or one physical mechanism |

## Boundaries, transitions and dynamics

| Term | Status | Working definition | Related established language | Do not infer |
|---|---|---|---|---|
| geometric boundary | `ESTABLISHED MATHEMATICS` | boundary of a declared set or region relative to a topology | topological boundary; manifold boundary | an observer-dependent relation unless one is defined |
| boundary record | `LOCAL DEFINITION` | typed record of structural, computational, representational, censoring or epistemic limits | failure record; interface state; censored observation | one universal boundary operator |
| boundary relation | `FORMAL CANDIDATE` | classification of a state relative to a declared boundary, frame, time and rule | side predicate; interface relation; event classification | universal ontology or physical wall |
| boundary status | `LOCAL DEFINITION` | provisional status `IN`, `OUT`, `CROSSING` or `UNRESOLVED` in the relation-record line | point-in-region classification | that unknown equals zero or outside |
| event function | `ESTABLISHED MATHEMATICS / LOCAL TYPING RULE` | declared scalar function `g: X × T -> R` used to identify an event surface or threshold on a declared state/time carrier | event detection; guard function; level-set function | one universal NEXAH function or physical cause of the event |
| event boundary / event surface | `ESTABLISHED MATHEMATICS / FORMAL CANDIDATE INSTANCE` | level set `B_t={x in X | g(x,t)=0}` for one declared event function; its meaning depends on `X`, `T`, `g` and regularity assumptions | level set; switching surface; guard surface | observation aperture, material wall, regime change or crossing without an evaluation rule |
| gate | `LOCAL DEFINITION` | declared subset of a boundary together with an explicit admissibility or transition rule | guard condition; switching surface; interface condition | automatic physical aperture, causal portal or permission merely because `g=0` |
| crossing | `LOCAL DEFINITION / ONE OBJECT-SPECIFIC CONTROL PASS` | observed or computed change of declared side status under an explicit sampling and classification rule; `UTG-FORMAL-01` exercises this on Lorenz-63 | boundary crossing; event detection | direction, continuity or transversality without trajectory and resolution assumptions |
| transversal crossing | `ESTABLISHED MATHEMATICS / ONE OBJECT-SPECIFIC CONTROL PASS` | for differentiable `g` and trajectory `x(t)`, an event time `t*` with `g(x(t*),t*)=0` and nonzero total derivative along the trajectory; the Lorenz control uses `g=x` and `dg/dt=sigma*y` | transversality; simple zero; event detection | that every discrete sign change resolves a continuous crossing, or that tangency and near-miss are crossings |
| transition map / reset map | `ESTABLISHED MATHEMATICS / LOCAL TYPING RULE` | declared map applied after an admitted event to produce a target state or record | hybrid-system reset map; state-transition map | that every gate changes state, that the map is invertible, or that observation causes transition |
| escape | `FORMAL CANDIDATE` | selected exit from a declared local domain, basin, chart, regime or representation window | first-exit event; basin escape; threshold crossing; switching event | nonlocality, discontinuity or irreversibility by default |
| first-exit time | `ESTABLISHED MATHEMATICS` | first time a declared trajectory leaves a declared domain | stopping time; exit time | applicability without a trajectory and time model |
| return map | `ESTABLISHED MATHEMATICS` | map from a declared section or state class back to a comparison section or class | Poincare map; recurrence map | identity of returned and initial states |
| return without identity | `LOCAL DEFINITION` | return to a related type, frame or comparison class with retained mismatch or unresolved identity | reinjection; recurrence comparison; residual return | sameness merely from repeated labels |
| cut | `LOCAL DEFINITION` | declared operation that makes a distinction or selection operational | restriction; selection predicate; mask operation; sampling rule | creation or destruction of the source |
| aperture / observation aperture | `LOCAL DEFINITION` | admitted window, section or subset `W` through which a cut or sampling operation produces a record | observation window; support; field of view; section; mask | transition permission, event boundary, physical opening, gate or cut operation by default |
| UTG Aperture / Transition | `FORMAL CANDIDATE / ONE OBJECT-SPECIFIC CONTROL PASS` | framework label whose formal use must separate observation aperture `W`, event boundary `B`, admissible gate `G` and optional transition map `R`; `UTG-FORMAL-01` passes this separation on one Lorenz-63 carrier | observation operator plus hybrid-system event/guard/reset structure | equivalence of aperture and gate, validity of the Series XIV plate equation, a nontrivial reset or a universal transition operator |
| closure rule | `LOCAL DEFINITION` | declared rule by which an open aperture or selected record is treated as closed for one comparison | closure operation; boundary completion; record-finalization rule | topological closure unless a topology is declared |
| bubble record | `LOCAL DEFINITION` | admitted inside together with its declared boundary, closure rule and provenance | bounded record; closed region; cell or chart domain where specified | physical substance, created universe or topologically closed set by image alone |
| regime | `ESTABLISHED MATHEMATICS` | parameter or state region with declared qualitative behavior | dynamical regime; phase region | objective universal partition without criteria |

## Number theory, combinatorics and attribution controls

| Term | Status | Working definition | Related established language | Do not infer |
|---|---|---|---|---|
| Catalan number | `ESTABLISHED MATHEMATICS` | `C_n = (1/(n+1)) binom(2n,n)`, counting many recursively structured families | Dyck paths; balanced parentheses; polygon triangulations; binary trees | that any branching visual has Catalan structure |
| Catalan core | `FORMAL CANDIDATE` | local name for a grammar hypothesized to persist across different geometric views | combinatorial class; generating grammar; incidence structure | invariance until an encoding and isomorphism criterion are supplied |
| Rogers-Ramanujan identities | `ESTABLISHED MATHEMATICS` | specific identities connecting q-series and infinite products with partition interpretations | q-series; partitions; modular forms | a generic modulo-five escape mechanism |
| Ramanujan summation | `ESTABLISHED MATHEMATICS` | one regularization framework assigning finite values in contexts where ordinary sums diverge | summability; analytic continuation; regularization | equality with ordinary convergence |
| `1+2+3+... = -1/12` | `EXCLUDED IDENTIFICATION` | shorthand sometimes used for a regularized value, not an ordinary convergent sum | zeta regularization; Ramanujan summation with conventions | an ordinary arithmetic identity or physical boundary |
| Euler equations | `ESTABLISHED MATHEMATICS` | equations for inviscid fluid motion under declared assumptions | PDE; continuum mechanics | Euler polynomial or Euler-Mascheroni constant |
| Euler prime-generating polynomial | `ESTABLISHED MATHEMATICS` | historically notable polynomial such as `n^2+n+41` producing primes for a finite integer range | polynomial values; prime-generating heuristics | a universal prime generator |
| Euler-Mascheroni constant | `ESTABLISHED MATHEMATICS` | `gamma = lim_(n->infinity)(H_n - log n)` | asymptotic analysis; harmonic numbers | Euler flow, Euler polynomial or an arbitrary gamma glyph |
| Riemann zeta function | `ESTABLISHED MATHEMATICS` | analytic function obtained from `sum n^(-s)` where convergent and by continuation elsewhere | analytic number theory | that every boundary labelled Riemann is a zeta result |
| mathematician role label | `LOCAL DEFINITION` | mnemonic assignment such as Euler-flow, Riemann-boundary or Ramanujan-escape inside a visual | conceptual crosswalk | historical attribution, theorem or endorsement |
| `2501` Keystone | `LOCAL DEFINITION` | local role name for the midpoint/path hinge in `2101 + 2901 = 2*2501 = 5002`; the same integer also occurs as a historical Scarab bridge label | midpoint; commutative path diagram; factorization `2501=41*61` | universal constant, prime, physical keystone or proven Scarab operator |
| Scarab geometry | `VISUAL INTUITION` | historical layered architecture using field, axis, bridge, gate, mirror and compression/expansion roles | typed graph; carrier-map-constraint-filter representation | one established transition operator or physical mechanism |
| Quaternion Gate | `EXCLUDED IDENTIFICATION` | historical name for a four-class center-reflection quotient in the audited `P_7` construction | four-state quotient; finite partition | quaternion algebra or quaternion group without defined operations and identities |
| Root-40 quaternion record | `LOCAL DEFINITION` | registered coordinate embedding `q=2+2i+4j+4k` with standard norm `N(q)=40` | quaternion coordinates; normed division algebra | that the Prime Lens or four-state gate generated the quaternion operations |
| emergent quaternion frame | `FORMAL CANDIDATE` | proposed stage in which three typed directed operations emerge from a `4+1` seam/binder and generational relation process | oriented frame; endomorphism composition; Hamilton units | quaternion identity until squares, ordered products and sign reversal reproduce |
| `root i` | `TYPED AMBIGUITY` | either the complex quarter-turn `i=sqrt(-1)` or a generative provenance root in a dependency graph; the two require an explicit bridge map | complex structure; provenance graph | identity by shared word, or automatic quaternion unit |
| inner `q` / outer `Q` Hopf typing | `FORMAL CANDIDATE WITH TESTED OPERATOR` | lowercase `q_hat` is a normalized representative in `S3 subset H`; uppercase `Q=[q_hat]` is its phase-equivalence class/base point in `S3/U(1) ~= CP1 ~= S2`, computed by the repository's tested Hopf map | principal circle fibration; quotient; gauge/global-phase equivalence | that every historical uppercase `Q` has this meaning, that a stacked grid is a Hopf fibration, or that Hamilton multiplication follows |
| Root/sign-to-Hopf observation bridge | `EXACT FINITE CANDIDATE / HISTORICAL USE UNVERIFIED` | under declared coefficientwise axioms, `F40(s)=2s0+2s1 i+4s2 j+4s3 k`; its sixteen normalized states map to four Hopf base points with four-state `Z/4Z` phase fibers, while `(base,phase)` is injective | finite group action; unit quaternions; Hopf fibration; quotient and reconstruction fibers | source-forced basis assignment, P7-class identity, global phase transport, historical PCA implementation or shared physical mechanism |
| Hopf chart zipper | `ESTABLISHED MATHEMATICS / TESTED FINITE RESTRICTION` | transition function `g_NS(x,y,z)=(x+i y)/sqrt(x^2+y^2)` between declared north/south local sections of the Hopf bundle; local phases obey `lambda_S=conj(g_NS) lambda_N`; on the four `F40` base points its image is exactly `{1,i,-1,-i}=C4` | transition function; bundle atlas; local trivialization; gauge change | absolute fiber phase, path-dependent parallel transport, historical use of the word `zipper`, physical seam or P7 identity |
| Hopf connection and holonomy candidate | `ESTABLISHED MATHEMATICS / TESTED DECLARED PATH` | on the declared F40 latitude `gamma(phi)=(4/5 cos phi,4/5 sin phi,-3/5)`, the chosen Hopf connection has `Gamma_N=(4/5)dphi`, `Gamma_S=-(1/5)dphi` and gauge-invariant loop phase `exp(i*8pi/5)=exp(-i*2pi/5)` | principal connection; horizontal lift; Berry phase; curvature; holonomy | historical OIL/RIS transport, C4-valued closure, zero-curvature mechanism, relativity result or closure of the finite F40 alphabet under transport |
| visible/hidden paired observation | `EXACT LOCAL CHANGE OF BASIS` | paired record `M=(p+q)/2`, `S=(p-q)/2` with inverse `p=M+S`, `q=M-S`; `M` may be selected as shared/visible and `S` as antisymmetric/hidden only under a declared observation contract | sum/difference coordinates; mid/side transform; shear decomposition; scaled Hadamard transform | that hidden means physically inaccessible, that one channel alone is invertible, or that every forward/return pair has this form |
| clamp/residual observation | `EXACT FINITE TESTED PATTERN` | ACR-38 split `C_A(x)=min(x,c)`, `C_B(x)=x-C_A(x)`, with exact reconstruction `x=C_A+C_B`; the clamped A-side alone is lossy on the tested outer domain | saturation; residual coding; sufficient paired statistic | universal buffer, energy reservoir or physical side channel |
| affine cyclic zipper | `FORMAL CANDIDATE` | matching of two ordered cyclic boundaries by `sigma(k)=epsilon k+delta mod n`, where `epsilon=+1` preserves orientation, `epsilon=-1` reverses it and `delta` retains an address shift | cyclic/dihedral action; boundary matching; permutation | recovered OIL/RIS implementation, Möbius topology, Hopf transition function or one physical mechanism until a source and carrier map are supplied |
| OIL/RIS | `SOURCE-BOUND VISUAL KINEMATIC GRAMMAR` | generator metadata expands OIL as outer orbit, inner orbit and link/transfer orbit, and RIS as rotation, inversion and shift; the state grammar ends in a shifted return | orbit diagram; transformation vocabulary; finite-state grammar | Euler replacement, orbital physics, mechanical coupling, source-defined operator composition or a new physical mechanism |
| PCA phase-space view | `BOUND HISTORICAL REPRESENTATION` | a time-ordered record in selected coordinates of a PCA basis fitted from a declared centered/scaled source matrix | covariance eigenspace; linear projection; reduced trajectory | attractor, physical mode, Hopf base, quaternion space or invariant cross-system axes by title or shape |
| regime landscape / navigation path | `FORMAL CANDIDATE` | a separately declared scalar or partition model over a typed state domain plus an ordered path evaluated against dynamics, constraints and endpoint criteria | optimization; reachability; control; basin analysis | stabilization, optimality or control capability from a contour plot and drawn curve |
| regime boundary / separatrix | `TYPED DISTINCTION` | a regime boundary belongs to a declared partition or classifier; a separatrix is an invariant manifold or basin boundary under declared dynamics | dynamical systems; classification; invariant manifolds | identity from a colored edge or shared visual location |
| FTLE | `ESTABLISHED MATHEMATICS / LOCAL COMPUTATION UNVERIFIED` | `FTLE_T(x0)=(1/abs(T))*log(sqrt(lambda_max((D Phi_T)^T D Phi_T)))` under a declared flow, interval, metric and derivative/integration method | finite-time stretching; Lagrangian coherent-structure analysis | gradient plot, rough surface, colorbar label or asymptotic Lyapunov exponent without the required operator |
| high-dimensional projection | `BOUND HISTORICAL REPRESENTATION` | a declared map from an n-coordinate feature record to a visible 2D/3D screen | dimensionality reduction; projection; reconstruction tests | n-dimensional attractor, injectivity or recovered surface from the screen alone |
| `4+1` seam architecture | `LOCAL DEFINITION` | four oriented zones plus one selective seam/binder that preserves separation and relation | N+1 architecture; interface; relation binder | fifth quaternion component, privileged centre or hidden physical dimension |
| `5+2` trace schema | `LOCAL DEFINITION` | five local measures `(P,D,C,Phi,S)` plus the two primary sides `(IN,OUT)` | typed record schema; boundary side | identity with decimal `5002`, seven physical dimensions or a natural law |
| GLB geometry | `LOCAL DEFINITION` | scene graph, mesh, node-transform and animation data extracted from a declared GLB/glTF artifact | computer graphics scene; affine transforms; spatial model | physical mechanism, invariant or semantic truth from node names or shape similarity |
| Triple-Six gate-hit test | `FORMAL CANDIDATE` | historical angle-gate and Monte Carlo comparison requiring generator, population, tolerance, seed and null-model replay | preregistered randomization test; multiple-comparison control | resonance mechanism from a plotted extreme alone |
| resonance pentagon | `FORMAL CANDIDATE` | five-node incidence record shown under decimal, binary, ternary and graph encodings | labelled graph; encoding/decoding map; codeword representation | preservation across encodings until decoders and incidence criteria are explicit |

## Current NEXAH notation and source-local terms

| Term | Status | Working definition | Established neighbors | Do not infer |
|---|---|---|---|---|
| `Q°_N(t)` | `WORKING NOTATION` | preferred current notation for the declared local observation state at time `t`, including observer position, orientation, horizon, time, instrument and projection rule | observation configuration; reference frame; measurement context | source, truth, universal origin, physical centre, universal operator or fifth quaternion component |
| `Q°_r` | `WORKING NOTATION` | projection-relative orientation marker inside a declared view/frame record | view marker; projection reference; derived record | object coordinate, invariant centre or observer entity |
| `Q°_t` | `WORKING NOTATION` | experiment-local historical label for a weighted state estimate in one compatible comparison space | weighted mean; estimator; Frechet mean only where defined | identity with `Q°_N(t)`, universal aggregation or estimator accuracy |
| `\mathcal{Q}^\circ` | `EXCLUDED IDENTIFICATION` | historical relational-control, binder or joystick template retained as genealogy but not adopted as the current universal Q-degree definition | context-dependent control map after a complete signature is supplied | OLS primitive, unique update law, fifth quaternion component or validated controller |
| `Q_1` | `WORKING NOTATION` | target or returned Q-type record in the repaired process `Q_0 -> P -> Q_1`; distinct from `Q_0` unless identity is proved | indexed state; re-entry record; target object | universal operator, first Q-mode or identity with `Q_0` |
| Janus | `VISUAL INTUITION` | local metaphor for directional selection at a gate | selector; switching rule; two-sided interface | one universal operator |
| Transversum | `LOCAL DEFINITION` | documentary architecture of typed paths across bounded non-identical representations, including cuts, frames, translations, provenance, loss and return | typed diagram; atlas and transition maps; provenance graph; alignment workflow | universal physical space, replacement for the universe or one operator |
| transversal relation | `FORMAL CANDIDATE` | declared cross-representation relation between two bounded records inside a Transversum trace | correspondence; transport relation; representation bridge | differential-topological transversality without manifolds, tangent spaces and its defining condition |
| transversal residual potential | `FORMAL CANDIDATE` | family or envelope of typed residual outcomes possible over a declared class of cross-representation paths | attainable-set analysis; family of reconstruction errors; uncertainty set, depending on formalization | scalar potential, energy, universal residue or universal operator |
| Ramanujan Escape | `VISUAL INTUITION` | historical NEXAH label for branch, discrete continuation or escape from a local representation | exit event; branch; continuation; reinjection | Ramanujan theorem or validated mechanism |
| The Room Looks Back | `VISUAL INTUITION` | mnemonic for frame-, aperture- and boundary-conditioned observation and return | contextual measurement; observer-system coupling; inverse-problem residual | agency or consciousness of space |
| `ALGO` | `UNRESOLVED PROVENANCE` | in the located `LANG-ALGO` line, an even-residue / square-root-five rail paired with `OLGO` | routing layer; residue-class channel | generic algorithm, Algol modulus or BoundaryRelation grammar |
| Algol modulus | `VISUAL INTUITION` | report-local name for the proposed carrier `11357 = 41 * 277` | composite modulus; Chinese Remainder Theorem after exact specification | cross-domain physical unification |
| `ALG°` | `UNRESOLVED PROVENANCE` | recalled glyph with no independent exact historical occurrence yet located in readable repository sources | unknown until source-bound | normalization to `ALGO`, Algol or algorithm |
| ILAU | `LOCAL DEFINITION` | ledger of retained/invariant, lost, added/introduced and unresolved content | change ledger; provenance-aware comparison | causal explanation or four substances |

## Maintenance rule

A new mathematical or math-adjacent term is entered here when it changes one
of the following:

- the objects or carrier;
- a domain, codomain or mapping rule;
- identity, equality or equivalence semantics;
- a boundary, failure or uncertainty class;
- a claim status;
- attribution to established mathematics;
- the meaning of an existing symbol.

Every new local term must include its bounded definition, nearest established
language, explicit non-equivalence and owning source. Open source or notation
questions belong in the [maintenance register](13_MATHEMATICAL_GLOSSARY_MAINTENANCE_REGISTER.md).

The current UTG source and first formal-candidate audit is maintained in
[`UTG_SOURCE_AND_FORMAL_CONCORDANCE_01.md`](../SCIENCE_LAB/NAVIGATION/EVIDENCE/UNIFIED_TRANSITION_GEOMETRY_2026-10-06/UTG_SOURCE_AND_FORMAL_CONCORDANCE_01.md).
It is an owning candidate record with one object-specific control pass, not authority to promote UTG or its plate
equations.

The current source-to-thesis ownership and discriminating tests are maintained
in the [NEXAH Thesis Evidence Matrix](14_NEXAH_THESIS_EVIDENCE_MATRIX.md).