Causal Operator Atlas — Phase II: Cross-Carrier Operator Geometry, Theorem Routing, Gram/Pythagorean Closure, and Typed Causal-Solution Integration

Causal Operator Atlas — Phase II: Cross-Carrier Operator Geometry, Theorem Routing, Gram/Pythagorean Closure, and Typed Causal-Solution Integration This release presents the complete Phase II of the Causal Operator Atlas, a cumulative research program developed within Causal Theory (CT) to test whether the local four-arm operator geometry obtained in Phase I survives transport across independent mathematical, computational, chemical, theorem-geometric, and control-system carriers without recalibrating the underlying algebra. The frozen Phase I kernel is generated by a reflection-like operator R and a quarter-turn operator J satisfying, on an authorized domain, R² = I, J² = −I, RJR = J⁻¹. Phase II does not assume that these operators, their metric realization, or the associated four classes [I], [R], [J], [RJ] must exist globally in every carrier. Instead, it develops a typed carrier protocol that distinguishes algebraic realization, metric compatibility, native gauge, projective quotient, route history, theorem occurrence, RETURN, proof authority, physical authority, and action authority. The program tests this architecture across eight major carrier families: prospective chemistry, RLC/electrical dynamics, Fourier/Hilbert/unitary geometry, Causal Differential and Integral Calculus, the Cartesian theorem carrier, HyperTriangle/B4 geometry, the AC124/DD094 Causal Solution runtime, and the Causal Reality Controller (CRC). A central result is that no non-trivial unconditional operator package containing a common native pairing, Phase-I projector/complement, reflection R, quarter-turn J, and globally canonical four-role projective quotient is authorized across all tested carriers. The strong universal-Cross hypothesis is therefore refuted within the frozen finite envelope. A maximal conditional invariant nevertheless survives: whenever a tested carrier or typed subcarrier independently supplies both an admissible R and an admissible J, the frozen signed Phase-I relations reappear on that scoped domain without refitting. The surviving object is therefore a typed sectional operator family rather than one universal global Cross. The study also substantially refines the role of Pythagorean geometry in the theorem-table architecture. Pythagoras is retained as the exact orthogonal case of the more general Gram composition law. For route vectors u and v, orthogonality permits ||u + v||² = ||u||² + ||v||², whereas general routes require the full cross term. The theorem table therefore does not force right-triangle geometry onto arbitrary theorem transitions. It uses carrier-authorized Gram geometry and activates Pythagorean simplification only when orthogonality is independently certified. This distinction is essential for the intended Periodic Cartesian Theorem Table. The project maintains strict separation between theorem identity, provider-native metric position, periodic or contextual address, route occurrence, and verified theorem use. In the current frozen theorem provider, all 118 theorem identities possess metric representations and all 6,903 unordered theorem pairs can be measured. These metric points are not automatically identified with the still-open global numerical Son owner positions. The HyperTriangle/B4 campaign provides an important higher-dimensional result. A single two-dimensional B2 Cross does not reconstruct a B4 chamber: one section compresses 384 B4 chambers to eight local states. However, the six coordinate-pair B2 sections taken together reconstruct all 384 B4 chambers exactly. The resulting interpretation is that higher-dimensional causal geometry may be represented by a bundle of sectional Cross realizations rather than by a single privileged global Cross. Runtime integration with AC124 establishes another crucial boundary. B4 state does not determine theorem role or route position. Across the 192 frozen PASS050 contexts, many theorem identities and B4 chambers occur with multiple roles. Theorem geometry can therefore measure, constrain, diagnose, and prune routes, but verified problem-local occurrence remains necessary before a theorem receives route-role or address credit. Phase II also integrates an oriented three-state equality architecture: E⁻ ↔ SIN1, E⁰ ↔ COS, E⁺ ↔ SIN2, whenever orientation authority is available. This structure records on which side of a declared closure condition a state lies. It is intentionally treated as a lower-information quotient of the four-arm operator structure, not as a replacement for [I], [R], [J], [RJ]. The Causal Reality Controller further establishes that local equality is not equivalent to complete closure. A local E⁰ condition may hold while physical-law, epistemic, protected, boundary, proof, or RETURN residuals remain open. Conversely, CRC closure may be declared under a scoped tolerance even when an underlying scalar comparison is not an exact equality. Computation, proof, physical validity, and action authority therefore remain separate layers. The terminal theorem changes the practical target of the theorem table. The goal is no longer to force every theorem or problem into one universal metric or one universal Cross. Instead, the theorem-table interface is made total by typed classification. For each admitted ROOT problem, the system must use the strongest structure actually authorized by the carrier: a full metric four-arm realization, a local or sectional realization, Gram-only geometry, history/RETURN structure, or diagnostic-only information. If the required structure is absent, the correct output is an explicit state such as PARTIAL, NON_IDENTIFIABLE, NO_GO, REFUTED, OPEN_EMPIRICAL, or FAIL_CLOSED_MISSING_PROVIDER_OR_BRIDGE. The resulting Causal Solution routing architecture is: SAME ROOT → CURRENT STATE → FULL_DELTA → typed carrier classification → authorized geometry/history → theorem-route diagnosis → verified occurrence and proof gate → CRC authority gate → RETURN → SAME ROOT → REMEASURE. The terminal Phase II status is: CLOSED_FINITE_TYPED_OPERATOR_FAMILY_WITH_MAJOR_NO_GO_PARTITION. The release includes the complete cumulative Phase II archive, cold-replay and integrity receipts, cross-carrier invariant matrices, the final No-Go Atlas, the terminal theorem, the total typed routing contract for theorem-table/Causal-Solution integration, and complete English and French scientific syntheses. Several important application and validation debts remain explicit rather than being absorbed into the closure claim: numerical owner Son positions for the 118 theorems, prospective prediction of useful theorem or lemma steps on unseen problems, verified multi-event theorem-use routes, owner-bound HyperTriangle transport receipts, empirical CRC/RETURN-C tests, and independent external validation. Phase II therefore closes the finite internal operator-atlas problem while leaving the next research stage well defined: materialize and prospectively test a dynamic theorem table capable of guiding the Causal Solution on previously unseen problems. Causal Operator Atlas — Phase II : géométrie opératorielle inter-carriers, routage théorématique, fermeture Gram/Pythagore et intégration typée à la Solution causale Cette publication présente la Phase II complète du Causal Operator Atlas, programme de recherche cumulatif développé dans le cadre de la Théorie causale (CT) afin de déterminer si la géométrie opératorielle locale à quatre bras obtenue en Phase I survit lorsqu’elle est transportée vers des carriers mathématiques, computationnels, chimiques, théorématiques et de contrôle indépendants, sans recalibrer l’algèbre sous-jacente après observation des résultats. Le noyau gelé de Phase I est engendré par un opérateur de type réflexion R et un opérateur de quart-de-tour J satisfaisant, sur un domaine autorisé, R² = I, J² = −I, RJR = J⁻¹. La Phase II ne suppose pas que ces opérateurs, leur réalisation métrique ou les quatre classes associées [I], [R], [J], [RJ] doivent exister globalement dans chaque carrier. Elle développe au contraire un protocole typé qui sépare réalisation algébrique, compatibilité métrique, gauge native, quotient projectif, histoire de route, occurrence théorématique, RETURN, autorité de preuve, autorité physique et autorité d’action. Le programme teste cette architecture sur huit grandes familles de carriers : chimie prospective, dynamique RLC/électrique, géométrie Fourier/Hilbert/unitaire, Calcul différentiel et intégral causal, carrier cartésien des théorèmes, HyperTriangle/B4, runtime de la Solution causale AC124/DD094 et Causal Reality Controller (CRC). Un résultat central est qu’aucun package opératoriel non trivial et inconditionnel comprenant un pairing natif commun, le projecteur/complément de Phase I, un miroir R, un quart-de-tour J et un quotient projectif global canonique à quatre rôles n’est autorisé sur l’ensemble des carriers testés. L’hypothèse forte d’une Cross universelle est donc réfutée dans l’enveloppe finie gelée. Un invariant conditionnel maximal survit néanmoins : chaque fois qu’un carrier ou sous-carrier testé fournit indépendamment un R admissible et un J admissible, les relations signées gelées de Phase I réapparaissent sur ce domaine sans refit. L’objet survivant est donc une famille opératorielle sectionnelle typée plutôt qu’une Cross globale unique. La Phase II précise également profondément le rôle de la géométrie pythagoricienne dans le tableau des théorèmes. Pythagore demeure le cas orthogonal exact de la loi de composition générale de Gram. Pour des vecteurs de route u et v, l’orthogonalité permet ||u + v||² = ||u||² + ||v||², alors qu’une route générale doit conserver le terme croisé. Le tableau ne force donc plus une géométrie de triangle rectangle sur des transitions théorématiques quelconques. Il utilise la géométrie de Gram autorisée par le carrier et active la simplification pythagoricienne seulement lorsque l’orthogonalité est certifiée indépendamm