---
title: "A placement criterion for catenary compartment models"
date: 2026-09-23
version: "0.1.0-candidate"
doi: 10.5281/zenodo.22909460
pdf: https://github.com/ipitchford/catenary-placement-classification/releases/download/v0.1.0-candidate/catenary-placement-classification-v0.1.0-candidate.pdf
repository: https://github.com/ipitchford/catenary-placement-classification
archive: https://zenodo.org/records/22909460
license: CC0-1.0
status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)
---

# A placement criterion for catenary compartment models

## Summary
Imagine a chain of compartments exchanging material with their neighbours. We know where material enters, where it can leak out, and which compartments we can observe. The transfer and leak rates are unknown. Does the placement of those inputs and observations leave some rates impossible to distinguish, even with ideal data?

This candidate gives a complete placement rule for **generic local structural identifiability** on a labelled bidirected chain. It applies to any chain length and any prescribed input, output and leak sets within the stated model. Local identifiability permits finitely many distant alternatives; it does not mean a unique global answer or reliable estimation from noisy data.

## Summary for specialists
Let $n\ge2$, with independently unknown positive transfers $p_i=A_{i+1,i}$ and $q_i=A_{i,i+1}$, and independently unknown positive leaks exactly at the prescribed set $L$. The diagonal is the negative total outflow. Input and output sets $I,O$ are nonempty; their coordinate selectors and gains are known. The mathematical data are the full labelled transfer matrix $C_O(sI-A)^{-1}B_I$.

Write $W=I\cup O$ and let $r(E,F)$ be the maximum number of pairwise strictly disjoint closed intervals joining vertices of $E$ to vertices of $F$. The necessary source condition is
$$r(W,L)=|L|.$$
It is sufficient when the port hulls overlap. Otherwise reflect labels, preserving input/output roles, so $b=\max I<a=\min O$. The main theorem supplies ordered counts with three thresholds:
$$
\begin{aligned}
S(I)+U(O^{\rm rev})&\ge n-1 &&(L=\varnothing),\\
A_*+B_*+c&\ge n-1 &&(\min L<a),\\
S(I)+C_*&\ge n-1 &&(\min L\ge a).
\end{aligned}
$$
The nonempty-leak cases also require the source condition. Here $c=r(I,L)+r(O,L)-r(W,L)$; the manuscript defines every recurrence, including flag timing and the augmented seed. Raw counts must not be capped prematurely. The equality $\min L=a$ belongs to the augmented case.

## Technical account
Jacobi symmetrization separates spectral changes from source-coordinate ambiguity. A Green-matrix interval-rank rule determines how many independent leak sources the ports can distinguish. Polynomial constraint spaces then supply universal upper bounds and matching constructions within each prescribed physical parameter family.

The difficult regime needs an interface constraint that changes how the output count grows. Its first contribution is one extra direction, but later updates are inclusive. With $n=7$, $I=\{1\}$, $O=\{5,6,7\}$ and $L=\{7\}$, the final augmented count is five, equal to the ordinary count rather than one larger. The threshold is six, so the model is generically locally unidentifiable.

The proof uses fixed-stiffness cofactor arguments, pinned boundary families and a scalar-lifting determinant estimate. It keeps universal upper bounds separate from boundary constructions that establish lower bounds. Jacobi, Green-function, continuant and spectral-perturbation tools are classical; the proposed contribution is their all-placement classification in the dependent-diagonal physical family. The cover depicts the seven-compartment placement schematically, not measured data.

## Evidence, assurance and limitations
The manuscript supplies a written all-length proof. The public replay compares 288,368 saved development placements with the unchanged classifier, verifies four exact reduced-system minor certificates and rejects four deliberately invalid inputs or certificates. Normal and optimized Python agree. Twenty selected physical-rank evaluations were also rerun using two supplied constructions at two primes. Larger campaigns reported in the supplied reviews were not regenerated in full.

A nonzero minor supplies a lower bound. Repeated sampled rank deficiency does not prove a generic upper bound. Model-mediated editorial roles are internal review; neither their agreement nor archive hashes establish formal verification or external journal peer review. Historical priority remains unestablished after a bounded source comparison.

The result assumes ideal full transfer information. Unknown sensor gains, inadequate excitation, measurement noise, parameter sharing, unlabelled compartments and missing edges require separate analysis. The criterion does not settle global uniqueness or practical estimation accuracy.

## Relationship to earlier work
Ahmed and collaborators identify the multiple-input/output catenary classification problem in their published Section 6. The singleton-input/output bidirected-tree criterion of Bortner and collaborators is a predecessor recovered by this candidate. Related path/cycle and free-matrix-entry results have different hypotheses; the manuscript compares them explicitly rather than treating their titles as evidence of a gap. Ovchinnikov, Pogudin and Thompson supply the transfer-coefficient interpretation for this strongly connected model class.

## Who should care, and why
| Audience | Potential use | Required caution |
|---|---|---|
| Structural-identifiability researchers | Scrutinize a proposed arbitrary-placement classification | Written all-length proof still needs unaffiliated specialist review |
| Compartment-model developers | Detect structural ambiguity before estimating rates | Check labels, gains, independence, leaks and ideal-data assumptions |
| Experimental-design researchers | Use a structural feasibility condition as one design input | Conditioning, noise and informative excitation are additional problems |
| Algebraic and spectral theorists | Study the augmented cofactor and interface mechanism | Classical tools and the new classification must remain distinguished |

## Why the problem matters
Estimation cannot recover a continuously ambiguous parameter from ideal data, however sophisticated the optimizer. A placement criterion can reveal that obstruction before numerical fitting. Conversely, passing a structural test only removes one obstacle: it does not guarantee stable or globally unique recovery. The candidate makes this first question inspectable for arbitrary chains without requiring a fresh large symbolic Jacobian for each placement.

## How to inspect or reproduce the recorded checks
Download and extract the versioned evidence archive. With Python 3.10 or later:
```sh
python3 replay.py > /tmp/catenary-replay.json
python3 -O replay.py > /tmp/catenary-replay-optimized.json
cmp replay-results.json /tmp/catenary-replay.json
cmp replay-results.json /tmp/catenary-replay-optimized.json
shasum -a 256 -c MANIFEST.sha256
```
Read Theorem 1 and the proof dependency map before following the appendices. Review the source-to-physical-model argument separately from the executable checks. The package records the exact original review coverage and the subsequent editorial repairs.

## The most valuable next projects
Independently scrutinize the family-preserving cofactor and perturbation arguments. Determine global identifiability degrees for placements that pass the local test. Study robust design with noise and uncertain gains. Extend the placement analysis to graphs beyond paths while retaining the true physical parameter dependencies.

## What is in the evidence package
PDF, TeX and Markdown manuscripts; the unchanged placement classifier; saved finite tables and exact certificates; a portable replay and expected results; original review responses and frozen internal editorial records; claim, source, environment and licence records; and a complete file manifest. Original prose and data are CC0, original code MIT. Supplied third-party review text and code are not redistributed or relicensed.




## Open directions for follow-up research

- Obtain unaffiliated specialist scrutiny and reproduction of the arbitrary-placement proof.
- Determine global identifiability degrees and characterize the remaining finite ambiguity.
- Study robust experiment and sensor design under noise, limited excitation and uncertain gains.
- Extend beyond labelled paths while preserving the prescribed physical parameter dependencies.

## Research process, metrics and reusable methods

This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work.

- Work ID: ep-work:catenary-placement-classification
- Attempt and metric receipts: ep-attempt:catenary-placement-classification-publication: published / positive; scope assurance-through-publication; target Internal editorial approval, immutable GitHub and Zenodo candidate assets, and complete Evidence Press page/media/canonical readback.; active forecast 180 minutes (120-300); Fermi components Proof exposition and mathematical typesetting: 1 x 60/90/160 minutes low/central/high (Procedural prior; no comparative acceleration claim.); Reproducibility and internal editorial gate: 1 x 25/40/65 minutes low/central/high (Procedural prior; no comparative acceleration claim.); Archive, media and guarded deployment: 1 x 35/50/75 minutes low/central/high (Procedural prior; no comparative acceleration claim.); positive-signal/closure probabilities 0.95/0.85 within 300 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 54/unknown/unknown/2/0/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 4; architectures tested/rejected 1/0; result target-closed; target reached true; forecast error -126 minutes; ratio 0.3; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: Human effort not instrumented.; computeMinutes: No separately instrumented total; ordinary validation excluded.; deduplicatedModelTokens: Fork-aware runtime counter unavailable.; uncachedInputTokens: Runtime token/cache accounting unavailable.. Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json
- Intended aims: science
- Artifact roles: research-output, evidence-assessment, communication
- Decision object: other — Generic local structural identifiability from port and leak placement in labelled bidirected paths. Scope: Independently unknown positive physical rates, prescribed leak set, full labelled transfer response and known gains; finite ambiguity permitted.
- Reusable methods: Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json
- Targeted clocks: assurance, publication
- Semantic bridge: explicit — A physical source-coordinate parametrization converts response-preserving variations into polynomial and source quotients. Universal bounds and physical attainment yield the placement criterion; finite replay checks only stated finite consequences. Remaining risks: All-length analytic and cofactor arguments require specialist scrutiny.; Finite sampled deficiencies do not prove generic upper bounds.; Global uniqueness and noisy-data estimation are separate questions..
- Human judgement gates: Assess the source-to-claim correspondence and written proof.; Preserve rights, status and priority boundaries.; Publication is authorised; external review is a separate dimension.
- Next assurance action: Inspect and independently reproduce the bounded result; explore extensions separately. External review is not a publication prerequisite.
- Claim ceiling: Unrefereed candidate. No formal verification, unaffiliated journal peer review, certified historical priority or practical parameter-estimation benefit is established.
- Aim-scoped impact evidence:
  - science: NO_IMPACT_EVIDENCE — Inspectable structural identifiability classification in Producer-coordinated mathematical publication; design none; comparator None.; estimand No acceleration or impact effect estimated.; no real-world effect evidence asserted



## Verification status

Unrefereed candidate. No formal verification, unaffiliated journal peer review, certified historical priority or practical parameter-estimation benefit is established.

## References

1. Ahmed et al. (2026), Identifiability of directed-cycle and catenary linear compartmental models: source classification problem. <https://doi.org/10.1137/25M1728636>
2. Bortner et al. (2023), Identifiability of linear compartmental tree models and a general formula for input-output equations: singleton-port predecessor. <https://doi.org/10.1016/j.aam.2023.102490>
3. Bortner and Meshkat (2022), Identifiable paths and cycles in linear compartmental models: related path/cycle machinery. <https://doi.org/10.1007/s11538-022-01007-5>
4. Ovchinnikov, Pogudin and Thompson (2023), Input-output equations and identifiability of linear ODE models: transfer-data interpretation. <https://doi.org/10.1109/TAC.2022.3145571>
5. Boukhobza, Hamelin and Simon (2014), A graph theoretical approach to the parameters identifiability characterisation: free-entry model comparison. <https://doi.org/10.1080/00207179.2013.856519>