Cosine Problem: Multidisciplinary Perspectives

• Cosine Problem is a context-dependent term referring to diverse challenges in geometry, harmonic analysis, machine learning, and quantum gravity.
• It addresses issues such as Euclidean law deformations on curved surfaces, ambiguity in cosine similarity, and extremal behavior of cosine sums.
• The topic integrates geometric formulations, computational decision problems, and semiclassical analyses to understand cosine-related phenomena.

Searching arXiv for recent and foundational uses of “cosine problem” across domains. In arXiv literature, “Cosine Problem” does not denote a single canonical question. The expression is used for several technically distinct problems: the generalization of the law of cosines and Pythagoras to infinitesimal geodesic triangles on curved surfaces; the ambiguity of cosine similarity on embeddings trained with dot-product objectives; extremal questions for cosine sums, cosine polynomials, and their zeros; the decision problem $\exists t\in\mathbb{N}$ with $\cos(t\theta)=c$; the zero-two law for Banach-algebra cosine functions; and the appearance or disappearance of $\cos S_{\mathrm{Regge}}$ in spin-foam semiclassics (Zachos, 2020, Bouhsine, 23 Feb 2026, Mercer, 2012, Jha, 2021, Esterle, 2015, Jercher et al., 2024). This suggests that the term functions primarily as a context-dependent label for problems in which cosine, cosine similarity, or cosine-type asymptotics is structurally central.

1. Context-dependent uses of the term

In differential geometry, the phrase refers to the problem of how Euclidean length identities deform on a curved $C^2$-regular surface when a triangle is geodesic and infinitesimal. In representation learning, it refers to whether cosine similarity is geometrically meaningful for learned embeddings, especially under diagonal “gauge” transformations that preserve dot-product objectives but alter angles (Zachos, 2020, Bouhsine, 23 Feb 2026).

In harmonic analysis and additive combinatorics, the term designates several extremal problems. One is Chowla’s cosine problem, which asks how negative $\sum_{a\in A}\cos(at)$ must become for a finite $A\subset\mathbf{N}$; another is Littlewood’s problem on how many zeros a cosine polynomial $\sum_{n\in A}\cos(nt)$ must have; a third is the problem of sharp upper bounds for weighted sums $\sum a_i\cos\alpha_i$ under angle-sum constraints (Bedert, 5 Sep 2025, Bedert, 2024, Barbara, 2017). In operator theory it appears in the zero-two law for Banach-algebra cosine functions, while in computation it appears as the “$\cos t\theta$ problem” linked to Skolem-type reachability questions (Esterle, 2015, Jha, 2021).

The phrase is also used in quantum gravity, where the “cosine problem” is the appearance of two semiclassical saddles $e^{\pm iS_{\mathrm{Regge}}}$ rather than a single Feynman-like factor, and in few-shot learning, where cosine-normalized attention is proposed as a remedy for instability in scaled dot-product cross-attention (Jercher et al., 2024, Nguyen et al., 2022).

2. Geometric law-of-cosines formulations

A differential-geometric version is given by the study of infinitesimal geodesic triangles $\cos(t\theta)=c$0 on a $\cos(t\theta)=c$1-regular surface $\cos(t\theta)=c$2. For such triangles, one obtains a generalized cosine theorem

$\cos(t\theta)=c$3

where $\cos(t\theta)=c$4 is a rational function in $\cos(t\theta)=c$5. An equivalent form rewrites the error in terms of the angle excess,

$\cos(t\theta)=c$6

When $\cos(t\theta)=c$7, this yields a generalized Pythagorean theorem on a surface (Zachos, 2020).

The same paper uses a local Gauss–Bonnet relation

$\cos(t\theta)=c$8

for the region $\cos(t\theta)=c$9 bounded by the geodesic triangle, together with Toponogov’s sine theorem for infinitesimal triangles. The leading-order behavior is Euclidean, while curvature enters through higher-order terms encoded by the angle excess. A plausible implication is that, in this local regime, the “cosine problem” is the problem of identifying precisely how curvature modifies Euclidean trigonometric identities.

A classical Euclidean counterpart is the geometric representation of the law of cosines associated with Al Cuoco and discussed by Claudio Bernardi. In that construction, the law of cosines is derived by dissecting the three squares built externally on the sides of a triangle into rectangles cut by the altitudes. The cosine term appears as a projection factor: for example, the area of a key rectangle is $\cos S_{\mathrm{Regge}}$0, yielding

$\cos S_{\mathrm{Regge}}$1

The same configuration reduces to the Pythagorean theorem when $\cos S_{\mathrm{Regge}}$2, and for obtuse triangles the argument is interpreted using signed areas (Bernardi, 2016).

A pedagogical reconstruction appears in the “Primary Gasing Triangle,” a right triangle with hypotenuse $\cos S_{\mathrm{Regge}}$3, where

$\cos S_{\mathrm{Regge}}$4

Here cosine is literally the adjacent side length, or, in coordinate form, the $\cos S_{\mathrm{Regge}}$5-coordinate of the point $\cos S_{\mathrm{Regge}}$6. This framework is used to derive the standard cosine rule

$\cos S_{\mathrm{Regge}}$7

by decomposing a non-right triangle into scaled right triangles (Surya et al., 3 Mar 2025).

3. Cosine similarity, normalization, and retrieval geometry

In machine learning and information retrieval, the best-known modern “cosine problem” concerns learned embeddings optimized with dot-product objectives and then queried using cosine similarity. Steck, Ekanadham, and Kallus showed that matrix factorization models admit a diagonal gauge freedom: $\cos S_{\mathrm{Regge}}$8 with $\cos S_{\mathrm{Regge}}$9 invertible and diagonal. This transformation preserves $C^2$0 and therefore the training objective, but it does not preserve cosine similarity between item vectors. Bouhsine’s response argues that the pathology is not cosine similarity itself but the use of cosine on unnormalized embeddings trained with an incompatible dot-product objective (Bouhsine, 23 Feb 2026).

On the unit sphere $C^2$1, the ambiguity disappears. For normalized vectors $C^2$2,

$C^2$3

so

$C^2$4

The paper states that this monotonic equivalence implies identical nearest-neighbor rankings under cosine distance and Euclidean distance on normalized embeddings, and that the $C^2$5-matrix ambiguity “vanishes identically” under sphere constraints (Bouhsine, 23 Feb 2026).

A related but distinct problem arises in exact similarity search. Cosine similarity is not a metric, and the obvious dissimilarity $C^2$6 fails the standard triangle inequality. Schubert develops a triangle inequality for cosine similarity by passing through the angular metric $C^2$7 on the unit sphere. The resulting lower and upper bounds are

$C^2$8

and

$C^2$9

These bounds are described as tight and suitable for exact similarity search with structures such as the VP-tree, Cover-tree, and M-tree (Schubert, 2021).

A further machine-learning usage appears in few-shot classification. The “Few-shot Cosine Transformer” replaces scaled dot-product cross-attention by “Cosine Attention,” computed from cosine-normalized query–key interactions without softmax. The paper attributes instability in standard cross-attention to sensitivity to feature magnitudes across support and query sets, and reports that cosine attention improves accuracy “from 5% to over 20%” relative to the default scaled dot-product mechanism (Nguyen et al., 2022).

4. Extremal cosine sums, inequalities, and zero sets

One major harmonic-analysis usage is Chowla’s cosine problem. For a finite $\sum_{a\in A}\cos(at)$0, let

$\sum_{a\in A}\cos(at)$1

Chowla asked for the asymptotic behavior of how negative $\sum_{a\in A}\cos(at)$2 must be. Bedert proves that there exists an absolute constant $\sum_{a\in A}\cos(at)$3 such that

$\sum_{a\in A}\cos(at)$4

thereby establishing polynomial bounds for the problem (Bedert, 5 Sep 2025). Independently, Jin, Milojević, Tomon, and Zhang prove that there exists an absolute constant $\sum_{a\in A}\cos(at)$5 such that for every finite $\sum_{a\in A}\cos(at)$6,

$\sum_{a\in A}\cos(at)$7

and identify this as the first polynomial bound obtained through spectral graph methods (Jin et al., 3 Sep 2025).

A finite-$\sum_{a\in A}\cos(at)$8 extremal version was analyzed earlier through the quantities

$\sum_{a\in A}\cos(at)$9

For $A\subset\mathbf{N}$0, the exact value is

$A\subset\mathbf{N}$1

attained by $A\subset\mathbf{N}$2, since $A\subset\mathbf{N}$3 has minimum $A\subset\mathbf{N}$4. For $A\subset\mathbf{N}$5,

$A\subset\mathbf{N}$6

attained by $A\subset\mathbf{N}$7, since $A\subset\mathbf{N}$8 has minimum

$A\subset\mathbf{N}$9

This formulation is explicitly described as Chowla’s cosine problem in the sense of the “largest minimum” of a cosine sum (Mercer, 2012).

Another extremal formulation studies weighted sums under angle constraints. For positive coefficients $\sum_{n\in A}\cos(nt)$0 and positive angles $\sum_{n\in A}\cos(nt)$1 with $\sum_{n\in A}\cos(nt)$2, the pentagonal inequality gives a sharp upper bound for

$\sum_{n\in A}\cos(nt)$3

by a positive real fraction in the coefficients alone. The normal form is

$\sum_{n\in A}\cos(nt)$4

with equality characterized by

$\sum_{n\in A}\cos(nt)$5

A corresponding heptagonal inequality is given for seven cosine terms (Barbara, 2017).

Littlewood’s cosine problem concerns zeros rather than minima. For

$\sum_{n\in A}\cos(nt)$6

Sahasrabudhe proved that the number of distinct roots in $\sum_{n\in A}\cos(nt)$7 is at least

$\sum_{n\in A}\cos(nt)$8

for every fixed $\sum_{n\in A}\cos(nt)$9 and sufficiently large $\sum a_i\cos\alpha_i$0 (Sahasrabudhe, 2016). Bedert later improved the lower bound to

$\sum a_i\cos\alpha_i$1

where $\sum a_i\cos\alpha_i$2 is the minimum number of zeros among all such cosine polynomials with $\sum a_i\cos\alpha_i$3 (Bedert, 2024). A plausible implication is that the harmonic-analysis “cosine problem” has split into at least two mature branches: extremal negativity and zero-count growth.

5. Analytic, computational, and spectral formulations

In Banach algebra theory, a cosine function is a family $\sum a_i\cos\alpha_i$4 satisfying the d’Alembert equation

$\sum a_i\cos\alpha_i$5

Esterle proves a zero-two law: if $\sum a_i\cos\alpha_i$6 is a cosine function in a unital Banach algebra and

$\sum a_i\cos\alpha_i$7

then

$\sum a_i\cos\alpha_i$8

The paper also proves a spectral-radius version and derives the norm result through a square-root functional calculus argument (Esterle, 2015).

In computation theory, the “$\sum a_i\cos\alpha_i$9 problem” is the decision problem

$\cos t\theta$0

where $\cos t\theta$1 and $\cos t\theta$2 are real algebraic numbers. The paper shows that $\cos t\theta$3 is a linear recurrence sequence over $\cos t\theta$4, relates the problem to the Skolem problem, and gives a polynomial-time algorithm by reducing it to the Orbit problem. The formal theorem states that for algebraic $\cos t\theta$5 and $\cos t\theta$6 with $\cos t\theta$7 and $\cos t\theta$8, there is a polynomial-time algorithm deciding whether there exists $\cos t\theta$9 such that $e^{\pm iS_{\mathrm{Regge}}}$0 (Jha, 2021).

In quasiperiodic spectral theory, a different “cosine problem” is the dry Ten Martini problem for $e^{\pm iS_{\mathrm{Regge}}}$1 cosine-type quasiperiodic Schrödinger operators

$e^{\pm iS_{\mathrm{Regge}}}$2

For $e^{\pm iS_{\mathrm{Regge}}}$3, $e^{\pm iS_{\mathrm{Regge}}}$4 $e^{\pm iS_{\mathrm{Regge}}}$5 cosine-type, and $e^{\pm iS_{\mathrm{Regge}}}$6 sufficiently large, the paper proves that the dry Ten Martini problem holds, and also proves homogeneity of the spectrum and absolute continuity of the integrated density of states. The authors emphasize that analyticity can be replaced by a geometric cosine-type condition in this low-regularity setting (Ge et al., 10 Mar 2025).

6. Semiclassical and causal formulations in quantum gravity

In spin-foam quantum gravity, the “cosine problem” refers to the fact that the semiclassical asymptotics of a vertex amplitude often yields

$e^{\pm iS_{\mathrm{Regge}}}$7

rather than a single Feynman-like factor. In a coherent spin-foam model for $e^{\pm iS_{\mathrm{Regge}}}$8-dimensional Lorentzian quantum gravity, the stationary phase approximation leads to gluing equations whose solutions determine whether one obtains one or two semiclassical saddles (Jercher et al., 2024).

The specific result is more nuanced than the standard formulation. When all triangles are either spacelike or timelike, two solutions exist, and the usual cosine behavior survives. In any other case, only a single solution is obtained, thus yielding a single Regge exponential rather than a cosine of the Regge action. The paper therefore describes a “partial absence of cosine problem” in 3D Lorentzian spin foams (Jercher et al., 2024).

This use of the term differs sharply from the analytic and geometric usages, but it preserves a common structural motif: the issue is whether a mathematically natural cosine contribution is fundamental or whether it can be reduced to a more primitive oriented object. In harmonic analysis the question is how negative or how oscillatory cosine sums must be; in embedding geometry it is whether cosine similarity is well-founded; in spin foams it is whether $e^{\pm iS_{\mathrm{Regge}}}$9 is an unavoidable semiclassical output. Across these domains, “Cosine Problem” names a technical obstruction or extremal question centered on the role of cosine itself.