Proteus 2k: Even-Dimensional Unlinked Simplices

• Proteus 2k is an even-dimensional affine geometry problem that asks if every set of 2k+3 points in ℝ^(2k) contains a linked complementary pair of simplices.
• The explicit construction using the moment curve shows that all complementary pairs have even intersection parity, thereby providing an unlinked configuration.
• This result disproves the even-dimensional analogue of the classical linked-simplices theorem, highlighting fundamental distinctions in combinatorial geometry.

Proteus 2k most specifically denotes the even-dimensional Proteus-type problem in affine geometry and topological combinatorics: whether every set of $2k+3$ points in general position in $\mathbb{R}^{2k}$ must contain a linked complementary pair of simplices in the parity sense. The 2024 paper "An example of an 'unlinked' set of $2k + 3$ points in $2k$-space" proves that the answer is negative by constructing an explicit unlinked configuration on the moment curve (Starkov, 2024). In current arXiv usage, however, the label also coexists with several unrelated systems named Proteus in wearable authentication, mobile visualization adaptation, distributed ledgers, and processing-using-DRAM, so the term is context-dependent rather than globally canonical (Orzikulova et al., 2024, Liu et al., 25 Apr 2026, Mishra et al., 5 Feb 2026, Oliveira et al., 29 Jan 2025).

1. Scope and nomenclature

Within the mathematical literature represented here, Proteus 2k refers to the even-dimensional analogue of a classical linked-simplices phenomenon. The corresponding question asks whether the odd-dimensional linking theorem has a parity-analogue in dimension $d=2k$. The cited paper answers this in the negative by exhibiting a configuration in which every relevant complementary pair has even intersection parity, hence no pair is linked (Starkov, 2024).

A separate source of ambiguity is terminological. Several unrelated papers use the name Proteus for systems in distinct technical areas, but not all of them introduce a separate formal object called “Proteus 2k.” In the ledger paper, for example, the core protocol is explicitly just Proteus, and the text states that if “2k” is used elsewhere, it most plausibly denotes a particular deployment/configuration size rather than a different protocol (Mishra et al., 5 Feb 2026). This suggests a context-sensitive rather than uniform use of the label.

Usage	Domain	Core object
Proteus $2k$	Affine geometry / topological combinatorics	Even-dimensional linked-simplices problem and unlinked configuration
Proteus	Minimalist wearables	Time-bound contextual bio-IDs from sensor streams
Proteus	Visualization systems	Mobile adaptation of desktop visualizations
Proteus	Distributed ledgers	Platform-fault-tolerant append-only ledger
Proteus	Processing-using-DRAM	Runtime-adaptive PUD framework

2. The $d+3$-point framework

The mathematical Proteus $2k$ problem sits inside the standard $d+3$-point framework. The cited paper recalls the classical theorem:

Take any $d+3$ points in general position in $\mathbb{R}^{2k}$0. (a) If $\mathbb{R}^{2k}$1, then there are two linked $\mathbb{R}^{2k}$2-simplices with vertices among these points. (b) If $\mathbb{R}^{2k}$3, then there are two disjoint $\mathbb{R}^{2k}$4-tuples of these points whose convex hulls intersect (Starkov, 2024).

The same source also recalls the “opposite parity” analogue for intersections in odd dimension: for $\mathbb{R}^{2k}$5, there are two disjoint subsets of sizes $\mathbb{R}^{2k}$6 and $\mathbb{R}^{2k}$7 whose convex hulls intersect. The Proteus $\mathbb{R}^{2k}$8 question is the converse-type analogue in even dimension: must one always obtain linked simplices there as well? The answer is no (Starkov, 2024).

The relevant definitions are standard. A set of points in $\mathbb{R}^{2k}$9 is in general position if no $2k + 3$0 of them lie in a common $2k + 3$1-dimensional hyperplane. For a pair of disjoint simplices, one of dimension $2k + 3$2 and the other of dimension $2k + 3$3, the pair is linked if the $2k + 3$4-simplex intersects the boundary of the $2k + 3$5-simplex in an odd number of points. In the even-dimensional case $2k + 3$6, the complementary pair consists of a $2k + 3$7-simplex on $2k + 3$8 vertices and the simplex on the remaining $2k + 3$9 vertices; the parity question is whether the first simplex meets the boundary of the second evenly or oddly (Starkov, 2024).

The number $2k$0 is therefore not incidental. When $2k$1, one has $2k$2, so the natural combinatorial configuration is exactly a set of $2k$3 points. One complementary subset has size $2k$4, the other has size $2k$5, and the geometric problem becomes a parity statement about boundary intersections (Starkov, 2024).

3. Main theorem: existence of an unlinked set

The central statement is Assertion 2 of the paper:

There are $2k$6 points in general position in $2k$7 such that any $2k$8-simplex formed by $2k$9 of them intersects the boundary of the simplex formed by the remaining $d=2k$0 points at an even number of points (Starkov, 2024).

Equivalently, the number of linked pairs of simplices is zero for that configuration. The result is described as disproving the natural even-dimensional analogue of the classical linked-simplices theorem, and it does so in the strongest possible way: not merely by showing that a linked pair need not exist, but by producing a configuration in which all complementary pairs are unlinked (Starkov, 2024).

The explicit construction is

$d=2k$1

where $d=2k$2 is the moment curve map

$d=2k$3

and

$d=2k$4

The paper identifies this as a cyclic-polytope configuration and shows that the rigidity of convex-hull intersections along the moment curve makes the parity analysis tractable (Starkov, 2024).

This construction yields an unlinked set of $d=2k$5 points in $d=2k$6-space. In the paper’s terminology, “unlinked” means that every complementary $d=2k$7-simplex / $d=2k$8-simplex pair has even intersection parity. Since linkedness is defined by odd parity, the configuration contains no linked pairs at all (Starkov, 2024).

4. Moment-curve mechanism and parity reduction

The proof is geometric in formulation but combinatorial in execution. A preliminary fact, Lemma 4, states that for points in general position, the convex hulls of two disjoint simplices can intersect in at most one point. As the paper notes, this allows parity of intersections to be identified with whether an intersection occurs at all, because every contributing face meets at most once (Starkov, 2024).

General position of the chosen configuration is obtained from Lemma 5: any $d=2k$9-hyperplane intersects the moment curve in at most $2k$0 points. Applied to $2k$1, this ensures that the selected points on the moment curve are in general position (Starkov, 2024).

The decisive structural input is Lemma 6. For disjoint $2k$2 with $2k$3,

$2k$4

Moreover, if they intersect, they intersect in exactly one point. Here “alternate” means that the elements of the two $2k$5-sets can be ordered so that the numbers alternate along $2k$6, for example

$2k$7

or the same with the roles swapped (Starkov, 2024).

For a fixed $2k$8, the paper studies the number $2k$9 of intersection points between

$d+3$0

where $d+3$1. It then identifies $d+3$2 in several equivalent ways: as the number of boundary intersections, as the number of $d+3$3-faces of $d+3$4 meeting $d+3$5, and as the number of $d+3$6-subsets $d+3$7 such that $d+3$8. By Lemma 6, this is exactly the number of $d+3$9-subsets $2k$0 alternating with $2k$1 (Starkov, 2024).

The geometric claim is thereby reduced to Lemma 7: for every $2k$2-subset $2k$3, the number of $2k$4-subsets $2k$5 alternating with $2k$6 is even. The proof is a case analysis. If $2k$7 contains adjacent integers, then no $2k$8 alternates with $2k$9. If $d+3$0 has no adjacent integers, one distinguishes whether $d+3$1 and/or $d+3$2 lie in $d+3$3. In the most symmetric case,

$d+3$4

the only alternating complements are

$d+3$5

so the count is exactly $d+3$6. In every case the total is $d+3$7 or $d+3$8, hence even (Starkov, 2024).

5. Conceptual significance

The principal conceptual point is the separation between intersection theorems and linking theorems. In $d+3$9, the $d+3$0-point theorem guarantees linked $d+3$1-simplices. In $d+3$2, one instead has a theorem about intersecting convex hulls of disjoint $d+3$3-subsets. The paper shows that the naive parity-analogue of the odd-dimensional linking theorem does not extend to even dimensions (Starkov, 2024).

A common misconception is that the odd-dimensional linked-simplices theorem should have a formally parallel even-dimensional statement. The result rules this out by producing a configuration in which all complementary pairs have even intersection parity. The failure is therefore not marginal but absolute for the constructed example: every potentially relevant pair is unlinked (Starkov, 2024).

This also clarifies the role of cyclic-polytope combinatorics. The moment curve is not used merely as a convenient generic-position source; it provides an explicit combinatorial criterion for convex-hull intersections via alternation. A plausible implication is that the even-dimensional obstruction is deeply tied to the rigid oriented-matroid structure of cyclic-polytopal configurations, rather than to an accident of a particular coordinate choice.

The bottom line of the paper is: $d+3$4 An explicit example is

$d+3$5

and the proof proceeds by counting alternating complementary subsets and showing that the count is always even (Starkov, 2024).

6. Other Proteus systems in current arXiv usage

Outside combinatorial geometry, Proteus names several unrelated systems. In wearable computing, "Proteus" (Orzikulova et al., 2024) is a framework for generating time-bound contextual bio-IDs from live sensor data so that devices can authenticate users and collaborate without relying on static pairing alone. It uses IMU and PPG data from devices such as earbuds, wristband, and headband; trains a global bio-ID model using contrastive learning inspired by SimCLR; segments data into 20-second windows for IMU-only data and 30-second windows for PPG-only or IMU/PPG data; resamples to 100 Hz; and uses a 1D convolutional embedding network with three convolutional blocks, one max-pooling layer, and a projection head of three fully connected layers. Experiments on FatigueSet with 12 participants report, among other outcomes, TPR = 84%, FPR = 9%, and FNR = 16% in the left/right earbud IMU+PPG setting, with TPR stays above 80%, ranging from 83% to 89% across activity types (Orzikulova et al., 2024).

In visualization systems, "Proteus" (Liu et al., 25 Apr 2026) is an automated framework for adapting desktop-authored visualizations to mobile screens. It formalizes a three-level design space—Global Topology, Reference Frame, and Visual Elements—and implements an LLM-driven five-agent workflow consisting of Semantic Parser, Data Extractor, Design Planner, Frontend Engineer, and Visual Critic. The framework is organized around R1 Perceptual Scalability, R2 Semantic Equivalence, R3 Exchange Space for Time, and R4 Respect Cross-Level Dependencies. In a within-subject study with 12 participants over 67 real desktop visualizations, Proteus reports a render success rate: 91.8% versus 87.8% for a stronger multi-agent baseline, with $d+3$6 for data fidelity and readability and $d+3$7 for aesthetics and interaction reasonableness (Liu et al., 25 Apr 2026).

In distributed systems, "Proteus" (Mishra et al., 5 Feb 2026) is a platform-fault-tolerant append-only ledger that embeds a Byzantine fault-tolerant audit protocol inside a crash-fault-tolerant protocol with no additional messages. The formal replica bound is

$d+3$8

with $d+3$9. Each batch has the form

$\mathbb{R}^{2k}$00

A batch is committed if it is replicated at a majority of nodes in the same view, while auditQCs require a supermajority signed set of Resp messages. The protocol uses hash chaining, pipelining, incremental quorums, pendingQCs, and a stabilization-based branch-selection rule that prefers the highest auditQC, then fast-path candidates, then highest view, then highest sequence number. In the default LAN setup, it reaches 345k txn/s at peak, about 87% of the throughput of signed Raft with seven nodes, and is about 1.9× faster than Engraft (Mishra et al., 5 Feb 2026).

In computer architecture, "Proteus" (Oliveira et al., 29 Jan 2025) is a processing-using-DRAM framework that dynamically chooses bit-precision, data representation, and arithmetic implementation at runtime. Its three principal mechanisms are dynamic bit-precision reduction for narrow values, concurrent execution of independent in-DRAM primitives across multiple DRAM arrays via One-Bit-Per-Subarray (OBPS) mapping, and adaptive selection among bit-serial, bit-parallel, and redundant binary representation (RBR) algorithms. The dynamic precision engine tracks the maximum value of each registered PUD object and can compute required precision as

$\mathbb{R}^{2k}$01

The paper evaluates the design on 12 real applications and reports, using a single DRAM bank, average performance per mm$\mathbb{R}^{2k}$02 of 17× over CPU, 7.3× over GPU, and 10.2× over SIMDRAM in the latency-optimized configuration, together with 90.3× lower energy than CPU, 21× lower than GPU, and 8.1× lower than SIMDRAM in the energy-optimized dynamic configuration (Oliveira et al., 29 Jan 2025).

Taken together, these usages show that “Proteus 2k” has one precise mathematical meaning tied to the even-dimensional unlinked-set problem, while “Proteus” functions more broadly as a reused system name across several technical domains. The mathematically specific sense is the one anchored by the explicit $\mathbb{R}^{2k}$03-notation and by the theorem on unlinked sets in $\mathbb{R}^{2k}$04 (Starkov, 2024).