Parametric Identity in Models and Theories

• Parametric identity is a framework that encodes identity as a manipulable latent variable key to controlling and disentangling features across various models.
• It underpins computer vision techniques by using defined latent vectors (e.g., dimensions of 128, 512, 1344) for precise facial and body reconstruction.
• Its principles extend to language models, geometric invariants, algebraic identities, and type theory, facilitating systematic analysis and practical applications.

Parametric identity refers to the mathematical, geometric, or algorithmic instantiation of "identity" as a variable or latent representation that is explicitly encoded, manipulated, or constrained in a parametric model. Across disciplines, parametric identity enables controlled variability, disentanglement from other attributes, and systematic evaluation or alteration of the core identity object — be it a human face, body, character profile, geometric entity, algebraic solution, or syntactic construct. Key research domains include computer vision, neural modeling, language agent architectures, geometry of numbers, type theory, and mathematical identities in number theory and functional analysis.

1. Formalizations in Computer Vision and Neural Models

Parametric models in computer vision and graphics express identity as explicit latent vectors within generative or field-based representations. For example, DNPM (Details' Neural Parametric Model) and Detailed3DMM (Cao et al., 2024) encode facial identity via a coefficient vector $w_\mathsf{id} \in \mathbb{R}^{n_i}$, which, together with an expression vector $w_\mathsf{exp} \in \mathbb{R}^{n_e}$, parameterizes the bilinear model shape $V_p(w_\mathsf{id}, w_\mathsf{exp})$. Fine-grained geometric details are synthesized by mapping identity and expression coefficients through an MLP encoder $E(w_\mathsf{id}, w_\mathsf{exp})$ to a latent "detail code" $z \in W_+$ for control over StyleGAN-generated displacement maps. The full mesh is the sum $$V_h(w_\mathsf{id}, w_\mathsf{exp}) = V_p(w_\mathsf{id}, w_\mathsf{exp}) + V_r(w_\mathsf{id}, w_\mathsf{exp})$$, where $V_r$ encodes neural residuals for wrinkles and details.

Similarly, in body modeling, Neural-ABC (Chen et al., 2024) uses an identity latent variable $z_\mathsf{id} \in \mathbb{R}^{128}$ to condition signed distance fields (SDFs) for body geometry. Each identity code is auto-decoder–initialized and jointly optimized with implicit function weights. Disentanglement is ensured by data design: the same garment is worn by multiple identities, and vice versa, preventing leakage between $z_\mathsf{id}$ and garment codes $z_\mathsf{c}$.

Parametric head models such as GPHM (Xu et al., 2024) encode identity into a latent vector $z_\mathsf{id} \in \mathbb{R}^{512}$, which is mapped onto a set of 3D Gaussian ellipsoids for head shape, color, and appearance. Identity, expression ($z_\mathsf{exp}$), and non-face motion ($z_\mathsf{nf}$) are disentangled and injected via separate MLPs, allowing independent animation and reconstruction from minimal input data. The identity code controls offsets $\delta X_\mathsf{id}$ in canonical head geometry.

Learning Neural Parametric Head Models (NPHM) (Giebenhain et al., 2022) formalizes identity as a high-dimensional latent ($d_\mathsf{id} = 1344$) conditioning a canonical SDF for the neutral head shape, with expression encoded as forward deformations $\Delta(x; z_\mathsf{exp}, \hat z_\mathsf{id})$. The architecture uses blended ensembles of global and local identity fields, with latent codes optimized by backpropagation and regularized for statistical tightness.

2. Parametric Identity in LLMs and Role-Playing Agents

Parametric identity is a foundational layer for character modeling in role-playing agents (RPAs) utilizing LLMs. The construct, introduced in "Fame Fades, Nature Remains" (Jun et al., 8 Jan 2026), refers to character-specific knowledge encoded implicitly during LLM pre-training and stored internally in model parameters $\Theta$, distinct from "attributive identity" provided via persona profiles at inference. For character $c$, parametric identity is defined as $I_\mathsf{param}(c) = f_\Theta(c)$, serving as a latent embedding that co-determines generative outputs alongside explicit profile inputs $I_\mathsf{attr}(c) := p(c)$.

Controlled experiments measure the effect by contrasting "Famous" (nonzero $I_\mathsf{param}$) and "Synthetic" ($I_\mathsf{param}\approx 0$) characters within a uniform schema for attributive features. A statistically significant advantage for Famous characters is observed in single-turn scenarios ("PersonaGym"), but the effect fades in multi-turn dialogue ("CoSER"), as context accumulation overtakes parametric priors — the "Fame Fades" phenomenon. Mechanistic attention analysis reveals initial bias toward generated context for Famous characters, with later turns shifting away from profile anchors.

3. Parametric Identity in Geometry and Physical Theories

Parametric identities appear as invariant differential relations in geometric and physical models. In Parameterized Absolute Parallelism (PAP) geometry (Wanas et al., 2016), the parametric identity is a generalization of Bianchi-type identities governing conservation laws. The canonical form is

$$E^\mu{}_{\nu\|\mu} = b(1-b) E^{\mu\alpha} \gamma_{\alpha\mu\nu} \equiv 0,$$

where $b$ interpolates between Riemannian ($b=0$) and teleparallel ($b=1$) limits, and $\gamma$ is the contortion tensor. This identity encodes conservation of the stress–energy tensor $T_{\mu\nu}$ and underlies all PAP-based gravitational theories, with no need for an action principle. The framework supports continuous variation in geometric structure and coupling to physical fields.

4. Algebraic and Number-Theoretic Parametric Identities

Parametric identity also refers to families of algebraic or combinatorial identities that admit full parametric solutions or classification in terms of underlying parameters.

In the cubic identity studied in (Fejzić, 20 Aug 2025),

$$\sum_{j=1}^{n} j^3 + x^3 - k^3 = \left( \sum_{j=1}^{n} j + x - k \right)^2,$$

the set of integer solutions $(k,x,n)$ is characterized parametricly via the equation $a^2 + ab + b^2 = n^2 + n + 1$, with $(a, b)$ related to $(k, x)$. Nontrivial solutions exist precisely when $n^2+n+1$ has at least two prime factors congruent to $1 \bmod 3$, due to the factorization properties of the norm in Eisenstein integers $\mathbb{Z}[\omega]$, $\omega = (1 + \sqrt{-3})/2$.

In parametric geometry of numbers, identities such as Jarník's (Summerer, 2019) connect Diophantine exponents via weighted inequalities. The parametric version arises in the study of successive minima functions $L_i(q)$ within convex lattice bodies parameterized by a scalar $q$, yielding relations such as

$(2-\nu_1)\,_1+(1+\nu_1)\,_3 \leq -3\,_1\,_3 \leq (2-\nu_2)\,_1+(1+\nu_2)\,_3,$

with explicit dependence on the weight vector $(\nu_1, \nu_2)$ and the parameter $q$.

In functional analysis, Samart (Samart, 2020) establishes a parametric functional identity for Mahler measures,

$$m\bigl(P_{1,k}\bigr) = 2(m^+(P_{a,c})-m^-(P_{a,c})) + \frac{1}{2}\log\frac{k-4}{k+4},$$

where $(a, c)$ are algebraic functions of $k$.

5. Parametric Identity in Type Theory and Syntax

Parametric identity has deep implications in type theory, especially in the formalization of parametricity and uniqueness principles. The notion that a function of type $\forall A.\,A \to A$ must be the identity is enforced in internal parametricity models (Altenkirch et al., 2023) via strict definitional equations and span-based modal operations in extended Martin-Löf type theory. The unique inhabitant follows by the algebraic propagation of span-legs, degeneracies, and isomorphisms, ensuring that for any $f: \forall A.\,A \to A$,

$f \equiv \lambda A.\,\mathsf{id}_A$

holds strictly, validated by the presheaf–span model over the BCH cube category. This result is foundational for computational canonicity and higher observational type theory.

6. Disentanglement and Applications

Parametric identity is typically disentangled from other latent variables by architectural, data, and loss design. In neural models, careful conditioning and regularization prevent leakage of garment/style/pose variation into identity codes (Chen et al., 2024, Giebenhain et al., 2022). In LLM RPAs, the disambiguation from attributive identity enables controlled experiments and precise benchmarking (Jun et al., 8 Jan 2026). In physical geometry, parametric identities enable the modulation or constraint of field equations across a family of geometric settings (Wanas et al., 2016).

Practical applications span facial detail synthesis, head/avatar reconstruction, privacy-preserving face obfuscation, fully controllable body models, interactive RPAs, automated geometric analysis, and algebraic characterization across number fields. Evaluation frequently targets identity-specific effects in reconstruction error, perceptual studies, and quantitative benchmarks.

7. Summary Table: Parametric Identity in Selected Domains

Domain	Representation/Definition	Distinctive Features
Neural face/body	Latent code $z_\mathsf{id}$, coefficient vector $w_\mathsf{id}$	Style/geometry control
LLM character agent	$I_\mathsf{param}(c) := f_\Theta(c)$ in model weights	Context-independent
Differential geometry	Tensorial identity parameterized by $b$	Conservation laws
Number theory	Solutions to parametric algebraic/Diophantine equation	Factorization criteria
Type theory	Uniqueness theorem for $\forall A.\,A\to A$	Strict internal laws

Parametric identity thus provides a precise, manipulable, and often disentangled key to controlled analysis, synthesis, and classification across mathematical, computational, and algorithmic frameworks.