Paper 13: The Complex Structure of Scale: From Configurational Deficit to the Conjecture that the Scale Coordinate is a Single Complex Value

Papers 10 and 11 of the scale-space series established that the corrected 5D metric requires gtt =−(1+2/L)c^2, and that the Einstein tensor of this metric reveals a configurational deficit of 3c^2/L^3 in the tt sector. Paper 11 identified this deficit as pointing to either an additional geometric dimension (the σ conjecture) or something structurally unaccounted for in the current model, and left the fork open. This paper takes the second branch. The argument proceeds in five stages, building a bridge from the familiar language of real and imaginary parts to a conjecture that requires going beyond it. Stage 1: The real framework is incomplete. The correction terms of Papers 9–11 are quantitative signatures that the real-valued treatment of the scale coordinate s is missing something. Stage 2: Transitional language — s= sR + isI. In conventional complex-number language, the missing piece can be named: the framework has been using only the real projection of s, and the correction terms are the fingerprint of what is discarded. This language is introduced explicitly as a bridge — familiar notation connecting the reader to the deficit — not as the paper’s final position. Stage 3: The projection warning. Separating a complex value into real and imaginary parts is mathematically valid, but it is projection onto two lower-dimensional subspaces. Carrying both projections as a pair (sR,sI) does not reconstitute the original complex value: information about its intrinsic complex structure is lost in the decomposition. The pair is an ordered pair of reals; the single complex value has multiplicative structure, argument, and rotation that the pair carries only by external convention. Stage 4: The conjecture. The scale coordinate s is conjectured to be a single complex value zs ∈C whose complexity is intrinsic and not reducible to two separable components without loss. The current framework’s ln(R/R0) is not Re(zs): it is what you obtain when zs is forced through a numeric system that can only output real numbers. A system in which zs is a primitive single value — not decomposed into projections — is what the CNRS programme [14,15] conjectures. Stage 5: Correction terms as projection proxies. Without the full zs, the correction terms of Papers 9–11 are the best the real framework can achieve: quantitative proxies for the information lost in projecting zs onto the real axis. With a system that carries zs natively, the correction would not need to be patched in — it would emerge from the full complex treatment. The paper does not carry out the full complex calculation. It establishes the bridge, states the conjecture precisely, and identifies the programme required to verify it.