# Three-Carrier Admissibility Taxonomy v1 **Date:** 24 October 2026 **Status:** exact names for registered finite objects and operational contracts; cross-carrier identities remain proposals ## Purpose The science programme, Claimipedia and Sourcery are related, but they do not contain the same things. | Carrier | Contains | Primary question | |---|---|---| | Science programme | mathematical structures, physical models, reductions, observables and experiments | What survives reduction, transforms lawfully or remains physically testable? | | Claimipedia | claims, versions, evidence, challenges, provenance and authored relationships | What has been said, with what qualification, dependency, support or objection? | | Sourcery | typed objects, declared transformations, assumptions, losses, evidence references and receipts | Which declared routes are structurally admissible, and why are others refused? | The governing rule is: > A number, relation or algebraic shape crosses carriers only through an > explicit typed map. Shared dimensions and shared words do not transfer > identity or authority. The machine-readable companion is the [three-carrier register](../registers/three_carrier_admissibility_taxonomy_v1.json). The [programme checkpoint](../programmes/frontier/wave-7/frontier_wave7_45_three_carrier_programme_checkpoint.md) states which parts of this finite architecture are exhausted, which physical gates remain open and why F05 is the shortest useful execution route. ## I. Science programme ### Quotient-Scar Admissibility Quiver The **Quotient-Scar Admissibility Quiver** \(Q_{\rm QS}^{(1)}\) is a six-coloured directed quiver with 16 vertices and 27 arrows, defined by the [quotient-scar register](../programmes/quotient-scars/quotient_scar_register_v1.json). Its arrow multiplicities are \((7,5,5,5,3,2)\). ### Constructive Twelve The **Constructive Twelve** \(K_{12}\) are twelve actual arrows. | Label | Edge | Type | Name | |---|---|---|---| | K01 | QR-006 | extends | projector-to-holonomy extension | | K02 | QR-008 | extends | projection-to-reflexive-closure extension | | K03 | QR-011 | extends | memory-to-wall extension | | K04 | QR-014 | extends | wall-to-response extension | | K05 | QR-019 | extends | invariant-to-detector extension | | K06 | QR-022 | extends | orientation-to-selector extension | | K07 | QR-025 | extends | selector-to-provenance extension | | K08 | QR-001 | instantiates | Markov detector realization | | K09 | QR-015 | instantiates | anchored-closure realization | | K10 | QR-017 | instantiates | projective-overlap realization | | K11 | QR-020 | instantiates | label-folding realization | | K12 | QR-023 | instantiates | wall-detector realization | ### Guardrail Fifteen The **Guardrail Fifteen** \(G_{15}\) are the remaining fifteen arrows. | Label | Edge | Type | Name | |---|---|---|---| | G01 | QR-005 | requires enrichment | triality-to-Freudenthal carrier gate | | G02 | QR-009 | requires enrichment | memory-to-oriented-holonomy gate | | G03 | QR-012 | requires enrichment | wall-to-coherent-holonomy gate | | G04 | QR-018 | requires enrichment | overlap-to-oriented-lift gate | | G05 | QR-021 | requires enrichment | population-to-coherence gate | | G06 | QR-007 | separates | holonomy from hidden-state memory | | G07 | QR-010 | separates | retained memory from pure-BF forgetting | | G08 | QR-016 | separates | determinant selection from coherent holonomy | | G09 | QR-024 | separates | equilibrium lift selection from accumulated holonomy | | G10 | QR-027 | separates | measured coefficient from universality and provenance | | G11 | QR-002 | shared action, not carrier | triality-root versus Markov-state seam | | G12 | QR-004 | shared action, not carrier | BF remnant versus triality-lattice seam | | G13 | QR-013 | shared action, not carrier | wall topology versus singularity topology seam | | G14 | QR-003 | rules out identification | selector index is not the G2 triple bond | | G15 | QR-026 | rules out identification | current Peirce line is not the required orientation carrier | Together they define the **constructive–guardrail polarization** \[ C_1(Q_{\rm QS}^{(1)})=K_{12}\oplus G_{15}. \] ### Fundamental Return Twelve and Boundary Fifteen The underlying graph independently has cycle rank 12 and incidence rank 15. These are dimensions, not finite sets. To name basis directions, v1 declares the **QR-ordered fundamental basis**: process `QR-001` through `QR-027` in identifier order and retain an edge exactly when it joins two previously disconnected components. The twelve non-tree chords label the **Fundamental Return Twelve** \(F_{12}^{\rm QR}\): | Label | Chord | Name | |---|---|---| | F01 | QR-009 | selection–memory–holonomy return | | F02 | QR-010 | memory–BF–triality return | | F03 | QR-012 | wall–holonomy enrichment return | | F04 | QR-013 | wall–triality carrier-separation return | | F05 | QR-015 | wall–response feedback return | | F06 | QR-016 | selection–holonomy separation return | | F07 | QR-018 | overlap–orientation-lift return | | F08 | QR-020 | detector-folding closure return | | F09 | QR-021 | population–coherence boundary return | | F10 | QR-023 | wall–orientation detector return | | F11 | QR-024 | equilibrium-lift–holonomy return | | F12 | QR-027 | measurement–provenance return | The fifteen retained tree edges label the dual fundamental-cut basis, the **Boundary Fifteen** \(B_{15}^{\rm QR}\): | Label | Edge | Name | |---|---|---| | B01 | QR-001 | detector-folding bridge | | B02 | QR-002 | triality–Markov carrier seam | | B03 | QR-003 | selector-index prohibition | | B04 | QR-004 | BF–triality carrier seam | | B05 | QR-005 | triality–Freudenthal enrichment cut | | B06 | QR-006 | projector–holonomy trunk | | B07 | QR-007 | holonomy–memory separation cut | | B08 | QR-008 | projector–reflexive-closure trunk | | B09 | QR-011 | memory–wall cut | | B10 | QR-014 | wall–response cut | | B11 | QR-017 | response–overlap cut | | B12 | QR-019 | overlap–detector cut | | B13 | QR-022 | overlap–wall-orientation cut | | B14 | QR-025 | selector–provenance cut | | B15 | QR-026 | Peirce-carrier prohibition | Thus \(\dim K_{12}=\dim F_{12}^{\rm QR}=12\) and \(\dim G_{15}=\dim B_{15}^{\rm QR}=15\), but the paired spaces are not equal. ### Mixed interface, cocircuit and backbone The **Mixed Admissibility Interface** is \[ I_8=\operatorname{im}\partial_K\cap\operatorname{im}\partial_G, \qquad\dim I_8=8. \] It records closed paths that exist only because constructive moves and critical boundaries compensate one another. The exact sequence has dimensions \(0\to(2+2)\to12\to8\to0\). The interface is now resolved integrally. Seven vertices are forced to zero by being isolated in one of the two typed subgraphs. The remaining **Mixed Core Nine** is `QS-002`, `QS-009`, `QS-010`, `QS-011`, `QS-012`, `QS-013`, `QS-014`, `QS-015`, `QS-016`; the only constraint is zero coordinate sum. Therefore \[ I_{\rm mix}(Q_{\rm QS}^{(1)};\mathbb Z)\cong A_8. \] In source-node order its eight simple transfers are detector–holonomy, holonomy–memory, memory–wall, wall–response, response–overlap, overlap–detector, detector–selector and selector–provenance. The equal-weight vertex pairing has Cartan determinant nine and discriminant group \(\mathbb Z/9\mathbb Z\). Its coordinate cubic is real but its projected product is not Jordan. See the [resolution](../programmes/frontier/wave-7/frontier_wave7_39_mixed_interface_resolution.md). The three `shares_action_not_carrier` arrows form the **Carrier-Separation Cocircuit** \(S_3\). Deleting all three disconnects the graph; any one reconnects it. Their deletion leaves the **Relation Backbone** \(R_{24}\), which carries: - the **Quotient-Scar Incidence Lattice** \(L_{\rm QS}=(\mathbb Z^{24},2I+B^TB)\), even, positive definite, rootless and minimum-norm four, but not unimodular; and - the **Seam-Projected Cycle Code** \(C_{\rm QS}\), a binary \([24,12,2]\) code, not the extended Golay code. Leech, Golay, exceptional Jordan and E8 remain comparison or frontier labels, not names for these exact objects. ## II. Claimipedia Claimipedia contains authored claims and relationships, not the sixteen science structures themselves. ### Claim Carrier and Relation Quartet The pinned capture defines the **Claim Carrier** \(V_{38}\), the rational span of 38 claim or crystal nodes appearing in 43 immutable relationship records. Two repeated triples leave 41 distinct typed edges. The four relation types form the **Claimipedia Relation Quartet** \[ C=A_{\rm Challenges},\quad D=A_{\rm DependsOn},\quad Q=A_{\rm Qualifies},\quad S=A_{\rm Supports}, \] with record counts \((2,7,27,7)\). Their exact closure is the **Captured Claim Relation Algebra** \(\mathcal A_{38}\): \[ \dim\mathcal A_{38}=66, \quad\dim Z(\mathcal A_{38})=14, \quad\dim[\mathcal A_{38},\mathcal A_{38}]=53. \] ### Typed two-rail carrier and mirror Before adjacency, path quotient or Jordanization, the same capture has a canonical **Typed Two-Rail Relation Carrier** \[ \mathcal H_{\rm rel} =V_{38}\otimes\mathbb R^{\{S,T\}}\otimes\mathbb R^{\{C,D,Q,S\}}, \qquad \dim\mathcal H_{\rm rel}=304. \] Each record \(e=(u\to v,t)\) contributes the **Typed Relation Ray** \[ d_e d_e^{\mathsf T}, \qquad d_e=|v,T,t\rangle-|u,S,t\rangle. \] Their sum is the **Relation Gram Kernel** \(K_{\rm rel}\succeq0\). Its source--target blocks reconstruct all 41 distinct directed typed records, and its rank is 41. The **Rail-Swap Mirror** \(J\) exchanges source and target rails and satisfies \[ J K_{\rm rel}(G)J^{\mathsf T}=K_{\rm rel}(G^{\rm rev}), \qquad J^2=I. \] This is an exact pre-quotient positive order and arrow-reversal covariance. It is not an authored reverse relation, a path-algebra dagger, a contragredient map, or the symplectic quarter-turn whose square is \(-I\). Identifying rails and relation types produces the **Scalar Support Quotient**, the ordinary rank-33 Laplacian. The rank loss \(41\to33\) is the **Eight-Direction Relation Quotient Scar**. It records exactly how much independent edge information the scalar support face forgets, without claiming that rank alone identifies the lost semantics. See the [relation--face compiler](../programmes/admissibility/admissibility_124_relation_face_compiler_from_adjacency_to_jordan_holonomy.md), [live support-order audit](../programmes/admissibility/admissibility_125_claimipedia_support_native_order_and_semantic_boundary.md) and [typed two-rail lift](../programmes/admissibility/admissibility_126_typed_two_rail_positive_lift_preserves_the_mirror_face.md). ### Source-native Clifford rail cell The rail carrier also has an exact **Source-Native Clifford Cell**. Authored role grading (G=\operatorname{diag}(-1,+1)) anticommutes with rail swap (J), so (I=GJ) squares to minus one and the null mixtures ((J\pm I)/2) are the two directed within-fibre rail transfers. This cell does not survive the scalar support quotient. More importantly, (I) exchanges the signed Relation Gram Kernel with an unsigned **Co-incidence Kernel** rather than preserving either one separately. It is exact pre-quotient role geometry; path dynamics and a semantic relation-type adjoint remain open. See [AZ-131](../programmes/admissibility/admissibility_131_the_authored_rail_grading_closes_the_local_clifford_cell_but_opens_a_coincidence_channel.md) and the [cross-corpus atlas](../programmes/admissibility/admissibility_132_cross_corpus_clifford_cell_atlas_and_exhaustion.md). ### Scalar topology shadow and adjoint boundary The doubled rail response has an exact scalar topology shadow. Signed incidence descends to the rank-33 Laplacian; unsigned co-incidence descends to the rank-35 signless Laplacian. Their two-rank difference counts the two non-bipartite connected components. However, exhausting every involution of the four authored relation types leaves only the reciprocal Spin--Temple Supports/Qualifies pair and fails on 41 records. The whole-source semantic adjoint is therefore absent; the free mirror must not be relabelled as an authored path star. See [AZ-133](../programmes/admissibility/admissibility_133_the_incidence_coincidence_doublet_detects_odd_cycle_topology.md) and [AZ-134](../programmes/admissibility/admissibility_134_no_global_semantic_adjoint_exists_on_the_current_claimipedia_relation_source.md). ### Carrier--selector--detector physical cell The minimal physical reading is now executable rather than metaphorical: - **carrier = thing:** the three resonator amplitudes that can move; - **dual face = selector:** the equal/opposite parity chosen on each coupling; - **triality relation = detector:** the gauge-invariant product obtained only after the third edge closes the loop. For a weighted triangle, switching the loop product leaves the trace and quadratic spectral moment unchanged but shifts the determinant by $4\kappa_{12}\kappa_{23}\kappa_{31}$. Thus the detector is not a third object placed beside the carrier and selector. It is the closure record that exists only when the three pairwise relations compose. See [AZ-142](../programmes/admissibility/admissibility_142_a_three_resonator_parity_switch_turns_the_dual_face_into_a_detector.md). ### Spin–Temple cell and context The opposite `Supports` and `Qualifies` arrows between Spin and Temple define the **Spin–Temple Reversible Cell** \(W_{\rm ST}\), whose associative closure is \(M_2(\mathbb Q)\) and commutator closure is \(\mathfrak{sl}_2(\mathbb Q)\). The incoming Christianity–kenosis qualification is the **Kenosis Descent Scar**: it proves that the two-node cell is not an autonomous quotient. Predecessor closure gives the **Spin–Temple Context Closure** \(P_{15}^{\rm ST}\), with 15 nodes and 17 edges. Its algebra fits into \[ 0\longrightarrow J_{30}\longrightarrow\mathcal A_{15}^{\rm ST} \longrightarrow\mathbb Q\oplus M_2(\mathbb Q)\longrightarrow0. \] The Jacobson radical \(J_{30}\) is the **Directed History Radical**. The quotient is the **Stable Reading Skeleton**: one scalar channel plus the reversible cell. No second root cell is present, so this direct route does not close to \(C_2\) or \(Spin(2,3)\). ### Full operator exhaustion and programme core The full 38-node relationship-path algebra sharpens that local result. Its 58-dimensional nilpotent radical leaves the semisimple quotient \[ \mathbb Q\oplus M_2(\mathbb Q)\oplus\mathbb Q[C_3], \] so its only nonabelian semisimple commutator factor is \(\mathfrak{sl}_2(\mathbb Q)\). Separately, the reciprocal nine-node Program–Version–Example core is a genuine double-centralizer \(M_3(\mathbb Q)\otimes I_3\) with commutant \(I_3\otimes M_3(\mathbb Q)\). Its root system is exactly \(A_2\), hence its commutator algebra is \(\mathfrak{sl}_3(\mathbb Q)\). Evidence incidence is square-zero and the authored provenance-stage algebra is scalar plus strictly triangular. Neither contributes a semisimple Lie factor; the provenance operators preserve no nondegenerate alternating form. The full captured semisimple Lie content is therefore \[ \mathfrak{sl}_2(\mathbb Q)\oplus\mathfrak{sl}_3(\mathbb Q), \] not \(\mathfrak{sp}_4\). Reopening requires a new evidence-bound operator on a shared carrier that supplies a second root pair, a nondegenerate preserved alternating form, the asymmetric \((-1,-2)\) Cartan coupling and the \(C_2\) Serre relations. See the [programme-core calculation](../programmes/frontier/wave-7/frontier_wave7_39_programme_a2_double_centralizer_and_c2_no_go.md) and [operator exhaustion](../programmes/frontier/wave-7/frontier_wave7_40_claimipedia_operator_exhaustion_and_sp4_gate.md). ### Kenotic junction topology The pinned relationship one-complex has 38 vertices, 40 simple undirected edges, five components and first cycle rank seven. The Christianity–kenosis node is the **Kenotic Cycle-Bearing Articulation**: its deletion adds two components and removes three cycle classes, while its authored orientation has five incoming arrows, one outgoing arrow and no directed return. The distinct Wick-Rotation kenosis node is the **Kenotic Acyclic Bridge**: its deletion adds one component but removes no cycle class. These names classify exact graph roles, not theological or physical objects. Neither node is an absent pole or puncture. The next admissible object would be a three-sector cover with declared local sections and overlap maps, followed by a globalization test. See the [topology audit](../programmes/frontier/wave-7/frontier_wave7_45_kenotic_pole_topology_audit.md). The natural cover is now resolved. Its branches have sizes 15, 1 and 7 before the junction is restored, and covers of sizes 16, 2 and 8 afterward. All pairwise and triple overlaps are the same singleton. The **Kenotic Filled-Nerve Cover** is therefore a full two-simplex with constant-coefficient \(H^1=0\); scalar sections that agree at the junction glue uniquely. This closes the adjacency-only absent-pole route. A richer test requires distinct or higher-dimensional overlaps with authored restriction maps. See the [cover-gluing no-go](../programmes/frontier/wave-7/frontier_wave7_46_kenotic_cover_gluing_no_go.md). The smallest exact enrichment is the **Kenotic Sign-Holonomy Loop**: three distinct pairwise overlaps, no triple overlap and transition signs \(g_{12},g_{23},g_{31}\). Its gauge-invariant classifier is \(h=g_{31}g_{23}g_{12}\). The \(h=+1\) control admits a nonzero global marker; the \(h=-1\) control returns the marker reversed and admits none. Over rational vector-space coefficients the twisted local system has \(H^0=H^1=0\); the surviving datum is the nontrivial \(\mathbb Z_2\) line-bundle class, not a nonzero rational \(H^1\). This is a proposed reopening template, not a fourth structure extracted from the current graph. See the [enriched-loop template](../programmes/frontier/wave-7/frontier_wave7_47_kenotic_enriched_loop_reopening_template.md). Across carriers, the **Central-Return Character** is the exact map \(\chi:\{\pm I_2\}\to\{\pm1\}\). It sends the science carrier's coherent \(-I_2\) frame return to the sign-holonomy toy's \(-1\) class while the projector returns unchanged. Determinant is not this map: \(\det(-I_2)=+1\). The detector must therefore retain a coherent endpoint reference and compare one traversal with two. See the [physical-detector bridge](../programmes/frontier/wave-7/frontier_wave7_48_sign_holonomy_physical_detector_bridge.md). ## III. Sourcery Sourcery accepts caller-supplied graphs, so it has no single fixed node count. Its definite object is a parameterized gated path system. ### Signature carrier and transformations A **Scientific Object Signature** has ten context axes \[ (\text{domain, rank, variance, basis, regime, order, scheme, symmetry, CP, support}) \] plus a seven-coordinate SI-base dimension vector. Their collection is the **Scientific Signature Carrier**. An **Evidence-Bearing Transformation** is a directed edge with exact endpoint signatures, a named operation, declared changes, assumptions, irreversible losses, evidence, dimensional delta, analogy flag and bounded authority. Passing edges form the **Admissible Transformation Graph** \(G_{\rm adm}\). Failing edges form the **Rejection Boundary** and remain visible in the receipt. ### Bounded paths and ALE valuation For source \(s\), target \(t\) and depth \(d\), Sourcery searches the **Bounded Derivation Path Set** \[ P_{\le d}(s,t;G_{\rm adm}). \] Each accepted path remains a **Derived Claim Proposal**. It carries the **Assumption–Loss–Evidence valuation** \[ \nu(p)=(A(p),L(p),E(p)), \] with componentwise distinct set union, so for composable paths \(\nu(pq)=\nu(p)\cup\nu(q)\). This is the exact sense in which a route remembers its cost and warrant. More precisely, an admitted derivation is an **Ordered Path with ALE Join Valuation**. The path word composes by noncommutative concatenation while the ALE ledger is a commutative idempotent monoid (a finite join-semilattice for a finite token universe). Loss is monotone, exact duplicates require canonical scope-and-version keys, and rejected rules remain outside the path algebra in a separate receipt. See the [ALE algebra](../programmes/frontier/wave-7/frontier_wave7_43_sourcery_ale_algebra.md). ### Verification envelope The **Derivation Verification Envelope** contains four surfaces: 1. Query; 2. Candidate Paths; 3. ALE Ledger; 4. Rejection and Authority Receipt. The bounded unexplored remainder is the **Search Frontier**, not evidence of absence. Sourcery's `scientific-derivation-composition/common-algebra-2` is a declarative process of assertions, filters, maps, folds, union and hashing. It is not the vector algebra \(\mathcal A_{38}\), the science cycle space or a Lie algebra. The safe mathematical name is the **Gated Provenance Path System**. ## IV. Cross-carrier passage The governed cycle is \[ \text{science structure} \to\text{Claimipedia claim graph} \to\text{Sourcery route receipt} \to\text{human review, proof or experiment} \to\text{revised science structure or boundary}. \] Each arrow requires a distinct act: - science to Claimipedia requires authorship, evidence and provenance; - Claimipedia to Sourcery requires exact signatures, declared changes, assumptions, losses, evidence and authority; - Sourcery to science requires the proof, computation, review or experiment appropriate to the claim. ## V. Rule for new names Every new taxonomic name must declare: 1. its carrier; 2. its object kind—arrow set, space, basis, quotient, algebra, path set, receipt or frontier; 3. its status—intrinsic, convention-fixed, exact no-go, operational or speculative; and 4. its source version, capture hash or admitted generation. This allows the taxonomy to grow without allowing a memorable number or shared diagram to erase the difference between physics, authored knowledge and checked reasoning.