Integrated-Gamma Geometry Framework

• Integrated-Gamma Geometry is a mathematical framework that employs gamma distributions and their integrals to characterize cumulative observable profiles in spatial, probabilistic, and geometric contexts.
• Empirical validations in financial order-book analysis show that integrated-gamma models consistently achieve high fits (R² between 0.78–0.92) compared to alternative approaches.
• The framework bridges analytic techniques with categorical structures in quantum cohomology via the Gamma-conjecture, supporting applications in mirror symmetry and functional inequalities.

Integrated-Gamma Geometry is a mathematical framework and empirical paradigm for describing spatial, probabilistic, or functional structures in diverse physical, financial, and geometric contexts using gamma-like forms and their integrated (cumulative) realizations. It manifests in order-book liquidity analysis, statistical mechanics, mirror symmetry, and functional inequalities, where gamma distributions and their integrals encode both underlying geometric constraints and emergent observables.

1. Foundational Principles and Mathematical Formulation

Integrated-gamma geometry arises when observable densities or profiles are constrained to gamma-like functional forms by underlying geometric, dynamical, or combinatorial hypotheses. In its canonical instantiation, for $x>0$, the density takes the shape

$q(x) = C \cdot x^{\gamma} e^{-\lambda x}$

and, in the standard gamma parameterization (shape $k$, scale $\theta$),

$$f(x; k, \theta) = \frac{x^{k-1} e^{-x/\theta}}{\theta^k \Gamma(k)}$$

with cumulative distribution

$$F(x; k, \theta) = \int_0^x f(u; k, \theta) du = \frac{\gamma(k, x/\theta)}{\Gamma(k)}$$

where $\gamma(k, z)$ denotes the lower incomplete gamma function. In discrete or empirical settings, "integrated-gamma" profiles refer to cumulative observables modeled as scaled gamma integrals, often fit as $S(x) = A \cdot \gamma(k, x/\theta)$ (Cruz, 24 Jan 2026).

2. Inflationary Relational Systems and Spectral Projection

The integrated-gamma geometry framework was rigorously developed in financial order-book analysis, abstracting the market as a combinatorial graph $G=(V,E)$ without primitive metric, price, or time. Edges (possibly vertices in extensions) evolve via inflationary (multiplicative, preferential-attachment-style) dynamics, yielding broad, hub-dominated non-equilibrium degree distributions. Observable coordinates such as “price” are only emergent: the first nontrivial eigenvector $\phi_1$ of the Laplacian $L = D-A$ is used to project entities to a 1D embedding $p_i = \phi_1(i)$, interpreted as synthetic price. Distances from the mid (defined as $p^* = (p_\text{best-bid}+p_\text{best-ask})/2$) become the only intrinsic length scale in subsequent analysis (Cruz, 24 Jan 2026).

This projection-induced regularity implies that, under the minimal single-scale log-slope hypothesis (where only distance to the mid and a global decay $\lambda$ enter), the empirical liquidity densities on each side (bid, ask) must be gamma-like. The discrete data observed in high-frequency Level II order books naturally yield integrated-gamma cumulative profiles.

3. Empirical Validation and Comparative Model Fit

Empirical studies employ per-second Level II quote snapshots for select US equities, binning liquidity by tick distance and averaging over intraday windows. Integrated-gamma cumulative models are fit via log-normal noise likelihoods or least squares to observed volume profiles, then compared to alternative forms such as cumulative log-normal or truncated power-law using AIC/BIC criteria.

Empirical outcomes demonstrate:

• Integrated-gamma fits consistently outperform alternatives, with median $R^2$ in the 0.78–0.92 range and $\Delta$AIC improvements of $-5$ to $-15$ relative to rivals.
• Typical parameter values: shape $k \approx 2.5$ (bid), $k \approx 2.0$ (ask); scale $\theta \approx 5$–$15$ ticks.
• Log-residuals collapse around zero for $x \gtrsim 3$ ticks, indicating systematic fit quality; deviations at small $x$ reflect discrete matching and order-size granularity.
• For sparse books such as GS, discrete support replaces smooth gamma, consistent with projection limits.

Simulations of inflationary relational dynamics (edge growth via degree-weighted attachment) and Laplacian projection reproduce the same integrated-gamma structures, without explicit agent behavior or price formation rules (Cruz, 24 Jan 2026).

4. Gamma Geometry in Probability, Functional Inequalities, and Analytic Interpolation

The geometric significance of the gamma law is codified in the Bakry–Émery CD$(\rho, \infty)$ curvature-dimension framework. The natural operator on $L^2(\gamma_{\alpha, \lambda})$ is the Laguerre generator $L f(x) = x f''(x) + (\alpha - \lambda x) f'(x)$, whose iterated carré du champ obeys a curvature lower bound $Γ_2(f) \geq \frac{\lambda}{2} Γ(f)$ for $\alpha \geq 1/2$ (Arras et al., 2015).

Interpolations (“smart path”) between arbitrary measures and gamma distributions admit explicit constructions, leading to De Bruijn-type identities: $$\partial_t D(\mu_t \,||\, \gamma_{\alpha,\lambda}) = - I_\gamma(\mu_t)$$

$$D(\mu \,||\, \gamma_{\alpha,\lambda}) = \int_{0}^{1} I_\gamma(\mu_t) \, dt$$

where $I_\gamma$ is a relative Fisher information respecting gamma geometry. These identities underpin functional inequalities:

• Logarithmic Sobolev: $D(\mu || \gamma) \leq \frac{1}{2} I_\gamma(\mu)$
• HSI-type: $$D(\mu || \gamma) \leq S(\mu)^2 \log(1 + \frac{I_\gamma(\mu)}{S(\mu)^2})$$ Both capture geometric regularization and entropic contraction along Otto–Wasserstein geodesics in probability space (Arras et al., 2015).

5. Gamma Integral Structures in Mirror Symmetry and Quantum Cohomology

The Gamma-conjecture connects the topological K-group and the Gamma-class (a characteristic class containing transcendental Riemann $\zeta$-value coefficients) to integral structures in quantum cohomology and their mirror duals. For a Calabi–Yau manifold $X$,

$$\widehat\Gamma_X = \prod_{i=1}^{n} \Gamma(1+\delta_i) \in H^*(X; \mathbb{R})$$

with $\delta_i$ the Chern roots of $TX$. The expansion involves

$$\widehat\Gamma_X = \exp\left(-\gamma\,c_1(X) + \sum_{k=2}^\infty (-1)^k \frac{\zeta(k)}{(k-1)!} \mathrm{ch}_k(TX) \right)$$

The Gamma-integral structure comprises flat sections of the Dubrovin (quantum) connection, constructed as images of K-theory classes twisted by $\widehat\Gamma_X$ (Iritani, 2023).

Tropicalization of mirror periods yields error terms matching precisely the $\zeta$-value corrections of the Gamma-class, as demonstrated for both two and three-dimensional cases. This provides explicit, concrete evidence for the mirror-symmetric identification of integral structures posited by the Gamma-conjecture.

6. Integrated-Gamma Geometry for Quadrics and Exceptional Collections

In the context of Fano manifolds, specifically smooth quadrics $Q \subset \mathbb{P}^n$, Gamma Conjecture II predicts that asymptotic behaviors of flat sections of the Dubrovin connection are encoded via the $\widehat\Gamma$-integral structure associated to full exceptional collections in $\mathcal{D}^b(Q)$. The proof involves:

• Explicit construction of the Gamma-class $\widehat\Gamma_Q$, and K-group framed flat sections $s_E(z)$ for $E$ in Kapranov's collection.
• Direct computation of Chern characters for spinor bundles.
• Verification that the ordered asymptotics of these flat sections yield a basis of asymptotically exponential fundamental solutions (AEFS) per Dubrovin's criterion.
• Reduction, via deformation invariance, of primitive GW invariants to ambient invariants, guaranteeing convergence of quantum cohomology.

Consequently, the family $\{ (\widehat\Gamma_Q, ch(E_i)) \}$, with corresponding Laplace-type expansions $s_{E_i}(z)$, realizes an integrated Gamma structure that bridges categorical data with analytic quantum connection properties, confirming the mirror-symmetric correspondence for quadrics (Hu et al., 2021).

7. Summary and Theoretical Implications

Integrated-gamma geometry elucidates multiple phenomena where projection, symmetry, or curvature constraints enforce gamma-like and cumulative gamma forms as robust models for observables—whether liquidity densities, functional interpolants, or period asymptotics. Its presence across finance, probability theory, quantum cohomology, and mirror symmetry stems from intrinsic geometric principles rather than specific microdynamic or agent-based rules. A plausible implication is that wherever data-generating processes are symmetric, inflationary, or projection-induced over a single scale, integrated-gamma geometry provides the correct functional envelope. This suggests that future analytic, geometric, and empirical models in high-dimensional and categorical settings may benefit from incorporating projected gamma-class constraints and their integrated realizations.