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.

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. The programme checkpoint 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. Its arrow multiplicities are \((7,5,5,5,3,2)\).

Constructive Twelve

The Constructive Twelve \(K_{12}\) are twelve actual arrows.

Guardrail Fifteen

The Guardrail Fifteen \(G_{15}\) are the remaining fifteen arrows.

Together they define the constructive–guardrail polarization

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 \(F_{12}^{\rm QR}\):

The fifteen retained tree edges label the dual fundamental-cut basis, the Boundary Fifteen \(B_{15}^{\rm QR}\):

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

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

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.

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

with record counts \((2,7,27,7)\). Their exact closure is the Captured Claim Relation Algebra \(\mathcal A_{38}\):

Typed two-rail carrier and mirror

Before adjacency, path quotient or Jordanization, the same capture has a canonical Typed Two-Rail Relation Carrier

Each record \(e=(u\to v,t)\) contributes the Typed Relation Ray

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

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, live support-order audit and typed two-rail lift.

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 and the cross-corpus atlas.

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 and AZ-134.

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.

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

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

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

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 and operator exhaustion.

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.

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.

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.

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.

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

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

Each accepted path remains a Derived Claim Proposal. It carries the Assumption–Loss–Evidence valuation

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.

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

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.