Situation. Information geometry — Rao, Chentsov, Amari, and the modern functional-analytic synthesis of Ay–Jost–Lê–Schwachhöfer (AJLS) — parameterizes statistical models by tensor order \ (n\) along a single track, treating the choice of representative on each ray of measures as a technical normalization. Gap. That choice is *modeling content*: arithmetic-mean center vs log-density affine vs squared-amplitude additive selects a section of the positive projective cone, and the choice determines which arithmetic, cumulant projection, and classical literature applies. The known cells — Aitchison centered log-ratio (clr) transformation at \ (r=0\), Fisher–Rao at \ (r=2\), softmax at \ (r=1\), harmonic at \ (r=-1\), tropical at \ (r=\) — are samples of a single corridor. Action. We name the gauge selection principle, fix the apex generator that encodes the full corridor, and structure the program as a two-axis grid: gauge exponent \ (\) × cumulant/tensor order \ (n\). For unstructured labels, the apex is the moment-generating function \ (Kᵧ () \) of the centered log-ledger. When labels carry \ (Z/N\) structure, the apex upgrades to the Fourier–Mellin transform \ Mₓ (, c) =1Nₒ ₙ/₍e^ yₛc (s), \ whose \ (c=0\) mode is \ (e^Kᵧ\), whose Taylor jets at \ (=0\) are the character-twisted moment/cumulant tensors of L2–L3, whose integer samples are the L6 (https: //doi. org/10. 5281/zenodo. 20392428) power closures, whose Fisher gradient flow is the constant-fitness replicator equation, and whose Chinese Remainder Theorem (CRT) factorization over the primes dividing \ (N\) stratifies structure into prime-local depth and cross-prime interaction. Result. Six core claims: Gauge Selection Principle. Sections of \ (R>₀N/ R>₀^\) form the \ (Lʳ\) power-mean family (\ (r R\\\) ) ; choosing \ (r\) chooses the affine substance. Apex Generators. \ (Kᵧ () =1Nᵢ e^ yᵢ\) is the entire cumulant generator of the centered log-ledger; under \ (Z/N\) structure it upgrades to \ (Mₓ (, c) \) with \ (Mₓ (, 0) =e^Kᵧ () \). Two Sampling Diagonals. Taylor jets at fixed \ (c\) give the moment/cumulant tower; integer \ (\) samples give the power-closure tower; both obey \ (a₁++aₙ c N\) (negative samples via reciprocal-ledger substitution). \ (\) -Flow. The escort family \ (P_ e^ y\) satisfies the constant-fitness replicator equation; \ (Kᵧ\) is its free energy (\ (Kᵧ'= E_y\), \ (Kᵧ''=Var_ (y) 0\) ). Prime Decomposition. For \ (N=pᵃ\), the conductor lattice is the \ (p\) -adic depth filtration; for \ (N\) with multiple distinct prime factors, CRT + analysis-of-variance (ANOVA) /Möbius separate single-axis from cross-axis strata, and \ (Mₓ\) factorizes exactly when the ledger is separable. Radical Arithmetic \ (N=6=2 3\) is the smallest integer with non-trivial cross-prime interaction in \ (Mₓ\), the canonical multi-prime worked example. Harvest. AJLS's \ (Sʳ () \) measure powers are the analytic shadow of the corridor; the canonical \ (n\) -tensor on \ (S^1/n\) is the moment-root diagonal; L6's \ (m\) -power conductor mixing is the closure diagonal — visible only when the corridor is named. L1–L6 are worked cells; L7–L10 (https: //doi. org/10. 5281/zenodo. 20437847) form a coherent forward program on the same surface. The opening sentence, restated: A statistical theory begins by answering: what is additive?
Leonardo Murillo Montero (Thu,) studied this question.