L0 - Gauge Selection and the Fourier-Mellin Simplex

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 \(r=\pm\infty\) — 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 \(\beta\) × cumulant/tensor order \(n\). For unstructured labels, the apex is the moment-generating function \(K_y(\beta)\) of the centered log-ledger. When labels carry \(\mathbb Z/N\) structure, the apex upgrades to the Fourier–Mellin transform \[ \mathcal M_x(\beta,c)=\tfrac{1}{N}\textstyle\sum_{s\in\mathbb Z/N}e^{\beta y_s}\overline{\chi_c(s)}, \] whose \(c=0\) mode is \(e^{K_y}\), whose Taylor jets at \(\beta=0\) 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: 1. Gauge Selection Principle. Sections of \(\mathbb R_{>0}^N/\mathbb R_{>0}^\times\) form the \(L^r\) power-mean family (\(r\in\mathbb R\cup\{\pm\infty\}\)); choosing \(r\) chooses the affine substance. 2. Apex Generators. \(K_y(\beta)=\log\tfrac1N\sum_i e^{\beta y_i}\) is the entire cumulant generator of the centered log-ledger; under \(\mathbb Z/N\) structure it upgrades to \(\mathcal M_x(\beta,c)\) with \(\mathcal M_x(\beta,0)=e^{K_y(\beta)}\). 3. Two Sampling Diagonals. Taylor jets at fixed \(c\) give the moment/cumulant tower; integer \(\beta\) samples give the power-closure tower; both obey \(a_1+\cdots+a_n\equiv c\pmod N\) (negative samples via reciprocal-ledger substitution). 4. \(\beta\)-Flow. The escort family \(P_\beta\propto e^{\beta y}\) satisfies the constant-fitness replicator equation; \(K_y\) is its free energy (\(K_y'=\mathbb E_{P_\beta}[y]\), \(K_y''=\mathrm{Var}_{P_\beta}(y)\ge 0\)). 5. Prime Decomposition. For \(N=p^a\), 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 \(\mathcal M_x\) factorizes exactly when the ledger is separable. 6. Radical Arithmetic & Anchor. Helmert normalizers \(\sqrt{k(k+1)}\) accumulate into \(K_N=\mathbb Q(\sqrt p:p\le N)\) with field growth at prime \(N\); \(N=6=2\cdot 3\) is the smallest integer with non-trivial cross-prime interaction in \(\mathcal M_x\), the canonical multi-prime worked example. Harvest. AJLS's \(S^r(\Omega)\) 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–L12 form a coherent forward program on the same surface. The opening sentence, restated: A statistical theory begins by answering: what is additive?