Bounds, Decompositions and Null Behaviour of KRATOS: A Mathematical Specification of a Recognition-Comparability Diagnostic

Standard citation metrics treat all citations as equal, obscuring the social and structural pathways through which scholarly influence propagates. I introduce Citation-Constellation, a freely available no-code tool for citation network analysis with two complementary bibliometric scores that decompose a researcher's citation profile by network proximity between citing and cited authors. BARON (Boundary-Anchored Research Outreach Network score) is a strict binary metric counting only citations from outside the detected collaborative network. HEROCON (Holistic Equilibrated Research Outreach CONstellation score) applies graduated weights assigning partial credit to in-group citations based on relationship proximity. The gap between scores serves as a diagnostic of inner-circle dependence. An extended abstract with full details appears in the paper. The tool implements this through a phased architecture: (1) self-citation analysis, (2) co-authorship graph traversal, (3) temporal institutional affiliation matching via ROR, and (4) AI-agent-driven venue governance extraction using a local LLM. Phases 1-3 are fully operational; Phase 4 is under development. Key design choices include ORCID-validated author identity resolution, an UNKNOWN classification for citations with insufficient metadata, and comprehensive audit trails documenting every classification decision. A no-code web interface enables researchers to compute scores without programming, installation, or registration. I present these scores as structural diagnostics, not quality indicators. BARON and HEROCON describe where in the social graph citations originate. They should not be used for hiring, promotion, or funding decisions. HEROCON weights are experimental and require empirical calibration.

Matrix-valued memories make rank the natural budget of a learned representation: the number of independent directions a state spans bounds what it can bind, compose, and track. We report causal evidence, on a group-composition testbed trained under a hard single-state bottleneck with a fixed decoder that cannot launder rank, that gradient descent recruits precisely the rank the task's algebra demands. A companion paper [Larson, 2026a] establishes the analogous recruitment and causal necessity pattern on a $K$-pair associative-binding testbed, where exact recovery provably requires state rank at least $K$; this paper inherits that instrument and extends the rank law from a scalar capacity bound to a representation-theoretic one. We train toward chosen minimal faithful reference representations embedded in larger matrices. On group-composition state tracking over five finite groups spanning the solvable/non-solvable divide, the recruited rank equals the group's minimal faithful real representation dimension $d_{\min}$ (Spearman $ρ= 0.9747$, the design's tie-capped maximum), the dimension-matched solvable/non-solvable pair $S_4$/$A_5$ is statistically equivalent under a pre-registered test, and a pre-registered force-rank test separates a guaranteed similarity ceiling from empirical recovery at the target dimension: one rank below $d_{\min}$, cosine similarity is capped by the target's tied unit spectrum at $\sqrt{(d_{\min}{-}1)/d_{\min}} \le 0.894$, below the $0.9$ threshold in every group by construction, with observed cells at 86-95% (mean 91%) of that ceiling; at $d_{\min}$, not guaranteed a priori, recovery clears the pre-registered anchor-relative bar at four seeds per group in all five groups. Within this testbed, measured effective rank tracks representation dimension; the matched-dimension $S_4$/$A_5$ comparison establishes equivalence within the pre-registered tolerance.

Leaderboards rank long-horizon agents by their final outputs. Yet a higher score alone does not establish whether two systems are comparable or which stage accounts for the difference. Unequal evidence, inputs, or budgets can affect scores, and statistical corrections do not remove this mismatch. We introduce compositional controllability to address these questions. A comparison window covers one stage, several stages, or the whole agent. Our central result bounds the gap between observed and controlled score differences using only nuisance outside the window. This yields an admissibility test applied before scores are inspected. Inadmissible comparisons are refused. For admissible pairs, an ordering is certified only when the score gap exceeds the combined sampling and nuisance radii; otherwise, it remains undecided. These decisions give each system a rank interval. We introduce BioLitBench, a benchmark of 2,042 biomedical articles represented as structured claim graphs. Among seven published pipelines, a conventional statistical analysis declares a winner in 14 of 21 pairwise comparisons. Yet the top-ranked system alone received the target review's bibliography. To isolate pipeline performance, our test requires matched inputs and a fixed backbone model. It refuses 11 of the 21 comparisons, including every comparison involving the top-ranked system. Seven of the 14 conventional conclusions fall within these refused pairs. The same comparison windows support stage-level training. We train SCRIBE on Qwen3.8-27B using rewards measured at each stage's exit. Under matched evidence, SCRIBE achieves a certified rank interval of [1,2], with certified advantages over all evaluated published pipelines and the evaluated Claude and OpenAI agents. Under same pool, SCRIBE matches the strongest published retriever and is certified above three published pipelines.