Introduce the conditional result that an isotropic additive completion cost is proportional to trace. Explain why proportionality leaves an overall calibration unresolved.
The complexification manuscript starts with source histories and a selected ordinary complex structure, then asks what remains after a specified completion protocol. Packets are represented in a two-dimensional complex carrier. The retained tests are quadratic in each packet, add responses from separate writes and do not retain their relative phases. Complete tests then identify a history with a positive Hermitian two-by-two matrix. This is a precise operational quotient of supplied histories. It is not a claim that every physical detector follows this protocol, or that source unity alone forces the carrier, tests and retention rule.
Let T(X) assign a nonnegative cost to each positive record. The draft assumes continuity, additivity under appending records, strict positivity for nonzero records, and invariance under all SU(2) rotations of the carrier. These properties represent a particular isotropic apparatus or completion frame. The theorem asks what functional is compatible with them. It does not begin with an identified laboratory clock, and its result should not silently be read as proper time.
Every Hermitian matrix is a difference of positive matrices, so additivity extends the cost consistently to the Hermitian difference space. Continuity makes the extension real linear, and every such functional has the form tr(HX) for a Hermitian H. Rotation invariance forces H to commute with the irreducible SU(2) action, leaving H=cI. Strict positivity fixes c>0. Thus trace is the unique isotropic additive cost direction up to one multiplicative calibration.
The record diag(3,1) and the isotropic record diag(2,2) have the same trace four, hence the same additive isotropic cost. Their determinants differ, so the cost does not exhaust their geometric information. A rank-one packet diag(a,0) costs ca even though its determinant is zero. Appending two such packets adds costs, regardless of whether their carrier directions coincide. This illustrates why trace cost and determinant interval answer different questions about a history.
A determinant-preserving congruence B=diag(e^(s/2),e^(−s/2)) sends diag(1,0) to diag(e^s,0), changing its trace. A nonzero trace cost therefore cannot remain invariant under all such boosts. It belongs to an isotropic completion frame, while determinant-based intervals have a different transformation property. A physical DTT clock must specify how its apparatus, gain and record law transform between observers. The mathematical uniqueness theorem helps by narrowing the allowed additive cost, but it leaves the overall scale and its operational relation to outer time unresolved.
The trace theorem assigns positive cost to every nonzero packet even though a rank-one packet has determinant zero. This is not inconsistent: the two functionals answer different operational questions. Trace counts an additive amount of completed norm in one isotropic frame; determinant measures a quadratic relation in the difference carrier. A source clock built from the first may tick during a record that has vanishing determinant interval. Therefore a physical identification must explain how the apparatus distinguishes accumulation cost from elapsed proper time. This is a sharper target than declaring every scalar extracted from a record to be the outer-time reading.
In SMM, a shared source motivates the demand that the cost of a completed record be defined consistently across its different history decompositions. Here that motivation becomes an explicit additivity and symmetry problem. The trace functional is the selected cost only after those mathematical assumptions and the chosen record carrier are supplied.
DTT distinguishes inner recurrence from outer completion; this model makes the distinction concrete by showing which source information survives in records. The six nested inner stages describe generative organization, while outer completion is the public interface. Their interpretation does not add six independent clock coordinates to the matrix construction. To turn the derived record carrier into physical spacetime, one must additionally identify accessible events, spatial differences, clock readings and compatible future dynamics. Lorentz-cone algebra is a valuable intermediate result, but it does not by itself supply observer-equivalent apparatus, matter propagation or Einstein’s equations. The article keeps those interfaces explicit.
Exact cone reconstruction assumes arbitrary nonnegative packet weights and sufficiently rich directional controls. With only finitely many packet directions, the attainable unit-cost section is their convex hull, generally a polyhedron rather than a round ball. For example six axial directions give an octahedron, whose diagonal reach differs from its axial reach. Thus the quaternionic or complex carrier alone does not guarantee isotropic attainable geometry. A physically complete completion apparatus must explain its directional resolution, allowed amplitudes and refinement law, in addition to the algebraic dimension of its possible records.
KEEP THE CONNECTION
Scope of the argument. Conditional uniqueness of a completion cost; no identification with calibrated proper time.
Draft file: complexification_from_temporal_completion_v9.pdf
Working manuscript. Article-level references identify the consulted sections and the scope of the interpretation. No public URL was supplied for this exact manuscript version.
Source register →