The Cayley–Dickson Ladder 2: Uniformity Verification of CD Doubling—Adjointness, Witness Propagation, and Invariant Degradation

This paper examines the uniformity of Cayley–Dickson doubling at the levelof construction. Part 1 established the ladder of loss: four doubling steps successively lose order, commutativity, associativity, and alternativity. The questionof this paper is whether these four steps are four executions of one and the sameconstruction or four constructions with a similar appearance. Three readings arepreregistered according to strength: the adjoint reading, the universal-cone reading,and the inductive reading. Within the category of real algebras with involution andnondegenerate quadratic form, the adjoint reading is rejected by an initial-objectcertificate: the real field is an initial object, whereas doubling sends the real fieldto the complex field, and the complex field is not initial; hence doubling is not aleft adjoint of any endofunctor. The universal-cone reading is decomposed step bystep: the tensor reading holds only at the first step, fails at the second step atcommutativity, fails at the third step at associativity, and fails at the fourth stepat alternativity; the carrier of failure at the second step is the commutator, at thethird step the associator, and the structural proof at the fourth step is “CD doublingis alternative if and only if the underlying algebra is associative” (literature coordinate Schafer), whose explicit carrier is registered by machine check. Comparison ofobstacles shows that the two negative obstacles are incomparable under the derivedpreorder, and the obstacle-singleness claim is not supported. The inductive readingis borne by a witness-propagation format: the three-step composite of negation,embedding, and conjugation involution mechanically generates from the survivingstructure at step k a candidate witness element at step k+1; the constructive itemholds unconditionally, while well-foundedness, after explicit computational verification, does not hold and is downgraded to pending verification. On the degenerationside, three cross-generational propositions are given: the zero-divisor set is a singleton set in the first four generations and emerges by a step jump at the sedenionlevel; the trivial embedding preserves zero-divisorhood under degeneration in thefirst four steps and fails at the sedenion level; the projection sends nonzero zerodivisors of the sedenions to the set of invertible elements of the octonions, whichis disjoint from the zero-divisor set of the preceding generation. Equational losshas an endpoint, and the appearance of zero divisors has a starting point; both arestep events. The negative theorem is conditional on the completeness of the candidate space and on a bridging premise; when no completeness certificate is obtained,the conclusion is limited to the statement that, within the preregistered candidatespace, the existing certificates do not support a single universal property bearingall four losses