# Verification report — OWR-15436-004 (Rendall's question on the Altan-Bonnet–Germain kinetic-proofreading module)

Verification date: 2026-09-30 (two referee rounds).

**Verdict.** Partial answer. The note is unrefereed.
- **Multistationarity, reaction-network sense** (mass action, independent positive rate constants): settled in both directions for the networks studied.
  - The four deficiency-one simplifications (no CD8, Lck constant, one ZAP-70 docking level; ZAP-70 constant or dynamic; Michaelis–Menten (MM) intermediates kept or collapsed): **never multistationary**. Every positive class contains exactly one steady state, positive and nondegenerate, for all rate constants.
  - Two or three docking levels with both kinases constant (four networks, deficiency 2 and 3): **multistationary**. For suitable rate constants a class contains exactly three steady states, two locally exponentially stable and one unstable.
  - The complete 57-species module (deficiency 34): **multistationary**. For suitable rate constants a class contains two locally exponentially stable steady states and at least one more.
- **Scope restriction.** All multistationary rate constants are reaction-specific; TCR–pMHC binding and unbinding constants differ by factors up to 10⁴ between receptor states. With the parameter structure of the Altan-Bonnet–Germain (ABG) model (one binding and one unbinding constant for all receptor states):
  - the MM-collapsed reductions are monostationary at every docking level (proved);
  - the MM-kept reductions showed no multistationarity in numerical searches (evidence only; in the second-round search the log-odds slope of the bound fraction stayed above 0.70);
  - the complete module is **open**.
- **Asymptotics.** Partly answered.
  - All 36 networks are persistent and have no boundary steady states in positive classes (proved).
  - In the multistationary cases the limit depends on the initial condition.
  - Global convergence in the deficiency-one case is only tested; basins and periodic orbits are open.

## Statement checked
- **Primary source.** A. D. Rendall (joint work with E. D. Sontag), "Mathematical models for T-cell activation", in *Reaction Networks and Population Dynamics*, Oberwolfach Report 28/2017, Oberwolfach Rep. 14 (2017), no. 2 (published 2018), pp. 1775–1776, doi:10.4171/OWR/2017/28. The workshop was organised by E. Baake, T. Kurtz and C. Wiuf.
  - The report PDF (fetched anonymously) was read; the relevant passage is on p. 1776. The second-round referee fetched it again independently.
  - Rendall contrasts the model of François et al. (PNAS 2013), whose kinetic-proofreading module has, by Sontag (IEEE TAC 2001), a unique globally asymptotically stable steady state, with the Altan-Bonnet–Germain model (PLoS Biol. 2005).
  - For the latter, he says things appear to be much more complicated, with even the kinetic-proofreading module having deficiency one under the strongest simplifying assumptions (a hedged statement). He writes: "It is unknown whether it admits multiple steady states."
  - Earlier in the abstract he asks what can be said about the asymptotics of solutions of the Altan-Bonnet–Germain model.
- **Corpus record.** ulamai/UnsolvedMath (version 1.6.0), OWR-15436-004 (status `open`). Its statement: does the kinetic-proofreading module, which has deficiency one under the strongest simplifying assumptions, admit multiple positive steady states, and what are the asymptotics of its solutions? This is a faithful rendering of the source.
- **The network.** Protocol S4 of Altan-Bonnet–Germain 2005 (JDesigner SBML file; open access, CC BY).
  - It has 57 species and 104 reactions; 54 kinetic laws are net rates (binding minus unbinding), so there are 158 irreversible reactions. All 104 reactions carry `reversible="false"`, although these 54 laws are net rates. The file contains the core module of Fig. 2B; SHP-1 and ERK feedbacks are not part of it.
  - The kinetic laws are of mass-action form. In 90 of the 158 mass-action terms, which lie in 66 of the 104 laws, one reactant is written as a zero-valued local parameter instead of the species (an export artefact). There are 24 such alias names, and each always stands for the same species. Read literally, these 90 terms would vanish. The paper uses the reactant and product lists (the reactions shown in Fig. 2B of ABG05) with the explicit rate constants of the laws (11 distinct nonzero values).
  - The CD8-free part has 37 SBML reactions (53 irreversible ones); the other 67 of the 104 SBML reactions (105 of the 158 irreversible reactions) involve CD8.
  - Four independent parsers (the finder's, two first-round referees' and the second-round referee's) give the same network. The author's revision check, a fifth parser, reproduces the alias and CD8 counts.
- **Ambiguity.** The source does not say which simplifications give deficiency one. We consider 36 simplifications with k = 1, 2 or 3 ZAP-70 docking levels. With no docking level (k = 0), no CD8 and Lck constant, the network is a McKeithan-type chain of deficiency 0 (weakly reversible). So the family keeps at least one docking level, and among these simplifications the minimum deficiency 1 is attained exactly by N(1, m, z). That these are the ones Rendall meant is an interpretation, stated as such in the paper.

## Readings
| Reading | Answer | Witness / reference |
|---|---|---|
| Deficiency-one simplifications N(1, m, z), all rate constants | no multistationarity; exactly one steady state per positive class | Thm 1.1 (Prop. 4.1, Table 2, Lemmas 4.3, 4.5) |
| N(2, m, const), N(3, m, const) (deficiency 2, 3), some rate constants | multistationary: exactly three steady states in one class, two stable, one unstable | Thm 1.2 (Table 3 and three further rate sets, stated in the proof and given in `claimant/src/pow10_*.json`) |
| Complete module, some rate constants | multistationary: two stable steady states and at least one more in one class | Thm 1.3 (exact certificate + Lyapunov + degree) |
| Reduced networks, MM collapsed, ABG sharing (one binding, one unbinding constant) | monostationary at every docking level | Prop. 7.1 |
| Reduced networks, MM kept, ABG sharing, k = 2, 3 | no decrease of φ found; smallest log-odds slope 0.7143 (k = 2) and 0.7045 (k = 3) | numerical evidence only (Sec. 7) |
| Complete module, ABG sharing or ABG parameter values | open | Problem 8.1 |
| Asymptotics | persistence for all 36 networks; bistability in the multistationary cases; global convergence for deficiency one only tested | Thm 1.4; Problem 8.2 |
| Any receptor network (Def. 3.1) with free states {T0, T1} satisfying the hypotheses of Prop. 4.1, for instance the sustained-signalling model of Bali–Rendall 2025 | φ strictly increasing, so at most one positive steady state in each positive class (existence is not claimed in this generality) | Prop. 4.1 with Lemma 3.2(a) |

## Results in the paper
- **Prop. 2.1 (deficiencies).** The 36 simplifications have the deficiencies of Table 1 (exact rank computation). The deficiency-one networks are exactly N(1, m, z). In each, all linkage classes have deficiency 0, so the Deficiency One Theorem does not apply. The case k = 0 (deficiency 0 without CD8 and with Lck constant; 2 with Lck dynamic; 4 or 6 with CD8) is discussed after Prop. 2.1. The MM collapse is applied to the steps with free Lck; collapsing the CD8-associated steps as well leaves all deficiencies unchanged.
- **Lemma 3.2 (reduction).** For the constant-kinase, no-CD8 networks (receptor networks), the positive steady states in the class (R_tot, L_tot) correspond to the roots of g(L) = L + R_tot φ(L) − L_tot. Multistationarity for some totals holds if and only if φ is not nondecreasing.
- **Lemma 3.3 (index identity).** At a positive steady state, det(−J|_S) = Δ(L) g′(L), where Δ(L) > 0 is the product of the nonzero eigenvalues of −Q(L). So a steady state with g′ < 0 has a real positive eigenvalue and is unstable.
- **Prop. 4.1 (flux balance).** With one memory state, φ/(1 − φ) is an explicit ratio with a nondecreasing numerator and a strictly decreasing denominator, so φ is strictly increasing. The proof is purely algebraic (stationary equations and the fundamental matrix of the bound block). It is equivalent to the finder's semi-Markov derivation, which the referees re-derived by hand. The sustained-signalling model analysed by Bali and Rendall (2025) is a special case (T0 = R, T1 = R*, C1 = C_N; the ligand is a conserved variable there, as in the paper).
- **Thm 4.2 (cited: Craciun–Feinberg; Feliu–Wiuf).** Injectivity criterion. Table 2 gives 208, 1472, 400 and 3100 monomials, all of sign (−1)^s = −1.
- **Lemma 4.3 and Prop. 4.4.** Siphons: every minimal siphon of each of the 36 networks contains the support of an integer conservation law (exact). So there are no boundary steady states, and by Angeli–De Leenheer–Sontag the networks are persistent (Thm 1.4).
- **Lemma 4.5 (degree).** In a positive class without boundary steady states, deg(f) = (−1)^s. So a steady state exists, and Σ sign det(−J|_S) = 1 when all steady states are nondegenerate.
- **Thm 1.2.** Rational witnesses with L1 = 6/625, L3 = 410. An exact Sturm count shows that p has exactly three simple roots in (0, L_tot) for all four networks. Exact Routh–Hurwitz tests at the two rational states. The middle state is unstable by Lemma 3.3; numerically it is a saddle with one positive eigenvalue ≈ 4.94·10⁻³. The rate sets of the three further witnesses are stated precisely: for MM kept, each Lck step of Table 3 with constant c becomes X → N_X (c), N_X → X (1), N_X → X⁺ (10³); for k = 3, the level-3 docking step C2″ → C3 gets 10⁻³ and every other new reaction gets 1, including all three reactions of each level-3 Lck step. The files `claimant/src/pow10_*.json` give the base-10 exponents.
- **Thm 1.3.** 158 positive rational rate constants and two positive rational states with f = 0 exactly, in the same class. Exact Lyapunov certificates prove both states Hurwitz (max real parts ≈ −2.77·10⁻³ and −7.34·10⁻⁵). The degree lemma gives a third steady state. The construction starts from the N(3, kept, const) rate set above; rebuilding the constants from the construction reproduces the certificate up to a relative deviation of 4.4·10⁻¹⁴.
- **Prop. 7.1.** With one binding constant a and one unbinding constant δ, φ = aL/(aL + δ) for N(k, coll, const). This holds at the ABG values a = 10⁴, δ = 8888. Uniqueness follows from Lemma 3.2(a); existence from Prop. 4.4, Lemma 4.3 and Lemma 4.5.

## Computations (exact unless stated; scripts and outputs in reproducibility/)
- **Finder** (`claimant/`).
  - It parses the SBML file and computes the deficiency table.
  - It builds and exactly verifies the four witnesses of Thm 1.2, including Routh–Hurwitz.
  - It computes three of the four injectivity determinants symbolically (fraction-free over ℤ[x, κ]).
  - It checks Prop. 4.1 on random rational instances and the 1150 nonnegative coefficients for N(1, coll, const).
  - It constructs the certificate of Thm 1.3 and exact Lyapunov certificates (about 50 min).
  - Exploratory GTH scans (floating point) are included; no proof rests on them. The audit recovered their seeds, and they reproduce exactly.
- **Author, audit** (`lead/`).
  - Exact polynomials w_B, w_F, p for all four witnesses, own Sturm sequences (exactly three simple roots each), and an exact check of Lemma 3.3 at the rational states.
  - The fourth injectivity determinant (1472 monomials, all negative), computed with the finder's determinant code.
  - Prop. 4.4 for all 36 networks, with exact integer conservation laws.
  - Prop. 7.1 exactly for k = 1, 2, 3; the identity fails with MM intermediates kept (control).
  - The shared-parameter search for k = 2 (numerical), rerunning the scripts of a first-round referee (`referee/referee2/`).
  - Revision (second round): `revision2_checks.py`, a separate parser, reproduces the alias counts (66 of 104 laws, 90 of 158 terms, 24 consistent aliases), the CD8 counts (67 of 104, 105 of 158) and the k = 0 deficiencies (0 for no CD8 and Lck constant, weakly reversible); `br25_sustained_check.py` (sympy) confirms that the sustained-signalling model as described in the paper reproduces the response function in Table 2 of Bali–Rendall, and does not if rebinding led to C0; a rerun of the second-round referee's log-odds search reproduces its output exactly (minima 0.714308 and 0.704495).
- **First-round referee 1** (independent; `referee/referee1/`). It reran all finder scripts (logs in `rerun_logs/`) and wrote its own code:
  - a regex SBML parser;
  - an independent certificate check with its own left kernel and its own stability proof (reaction-vector basis, rigorous rounding bound, exact LDLᵀ);
  - the witnesses with Hessenberg characteristic polynomials and Hurwitz determinants;
  - a Sturm count for the level-2 witness;
  - injectivity coefficients by Cauchy–Binet enumeration (all four variants; a control network fails);
  - siphons by branch and prune (the full module, the networks above and a failing control);
  - an exact lumpability check and a numerical shared-parameter probe.
- **First-round referee 2** (independent; `referee/referee2/`).
  - An ElementTree re-parse and certificate check, and 50-digit eigenvalues.
  - The level-2 witness.
  - A numerical shared-parameter search for k = 3 (1500 random vectors, 12 Nelder–Mead optimisations).
  - The OpenCitations literature record.
- **Second-round referee** (independent; `referee/round2_referee2/`; code written from the text of the paper before any finder script was opened; the finder's certificate and rate files were read as data only; exact parts use Python `fractions` or integers).
  - Own ElementTree parser and network construction: Table 1 and Prop. 2.1 for both ZAP-70 variants, the complete-module invariants, the k = 0 case, the alias and CD8 counts.
  - Table 2 by Laplace expansion over column subsets (exact integer polynomials), with det M = det(J|_S) checked; control N(2, coll, const) has mixed signs (28 positive, 3188 negative).
  - Prop. 4.4 by SAT and complete minimal-siphon enumeration (2–5 minimal siphons per network).
  - Prop. 4.1 on 60 random rational instances; the 1150 coefficients (Bareiss/DP, no interpolation); Prop. 7.1 with the kept-intermediate control.
  - All four witnesses of Thm 1.2 (w_X by Bareiss over ℚ[L], own Sturm, Hessenberg characteristic polynomials, Routh), both readings of the k = 3 rule.
  - Thm 1.3: own reaction matching and conservation laws; f(x¹) = f(x³) = 0 and x³ − x¹ ∈ S exactly; a separate rigorous stability proof (dyadic rounding of J|_S, exact balancing by powers of 2, an exact perturbation bound, integer Bareiss). Rebuilding the constants from the construction gives a maximal relative deviation of 4.392·10⁻¹⁴ with the rule stated in the paper (constant 1 for the level-3 Lck steps), and 0.999 with the alternative reading (10³ for their catalytic steps).
  - Numerical evidence: the log-odds search of Sec. 7 (3000 random vectors and 40 Nelder–Mead runs per k); the saturating log-slope objective (minima of order 10⁻¹⁵ at the ends of the L range); 600 random instances of N(1, coll/kept, const), all Hurwitz. The dynamic-ZAP-70 half of the finder's 600-sample scan was not rerun.

## Independent adversarial audit
**First round** (2026-09-30; two independent verifiers):

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (the choice of simplification is an interpretation) |
| Proofs | CONFIRMED (no mathematical error found by either verifier) |
| Computations | CONFIRMED (all scripts reproduce; independent code agrees) |
| Answer as posed | PARTIAL (reaction-network sense settled; ABG-parametrised question open for the full module; asymptotics partly answered) |
| Novelty | CONFIRMED as far as can be checked (no priority claim) |
| Presentation | CONFIRMED_WITH_FIXES |

All required fixes of both first-round verifiers were applied (the pre-audit write-up is preserved in the problem folder):
1. Every "multistationary" claim is qualified as "for some choice of independent (reaction-specific) rate constants". The witnesses' non-uniform TCR–pMHC constants are stated explicitly, and the contrast with the ABG parametrisation and with kinetic proofreading is explained.
2. The lumpability converse (Prop. 7.1) was added, with the numerical evidence for the MM-kept networks. The ABG-parametrised question for the complete module is stated as open (Problem 8.1).
3. "Rendall's deficiency-one simplification" was renamed "the deficiency-one simplifications identified here".
4. The limitation on ABG parameters was corrected: the parameter values are explicit local parameters, only species aliases are artefacts, and the mapping is not ambiguous.
5. The corpus note is now partial and names the specific family of deficiency-one networks.
6. The middle steady state: "saddle" is now labelled numerical, and its instability is proved (Sturm count and Lemma 3.3; the verifier's degree-theory route is also given, as Lemma 4.5).
7. The reproduction list is complete: `check_semimarkov.py`, `verify_trial.py 0 381`, `scan_levels.py` with seeds 1–4, `scan_phi2.py 0 400`, and the search, simplify and embed steps. The search seed was not recorded.
8. The certificate re-check by `verify_full_certificate.py` is marked as not independent of the finder's parser; the independent parses are cited.
9. Persistence was extended from five networks to all 36.
10. The injectivity result for N(1, kept, const) (1472 monomials) is cited as a second proof.
11. The literature record was completed (OpenCitations citers, rate-limited services).

Beyond the required fixes, the first audit added: the index identity (Lemma 3.3); exact Sturm counts for all four witnesses ("exactly three" steady states); the degree lemma and the third steady state of the complete module; and the shared-parameter search for k = 2.

**Second round** (2026-09-30; one further independent referee; report in the problem folder, `audit/REFEREE_REPORT_2.md`):

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED WITH FIXES (quotation and paraphrases match the Oberwolfach report; the zero-docking case had to be stated) |
| Proofs | CORRECT (every lemma, proposition and theorem checked line by line; no mathematical error; one missing citation in the proof of Prop. 7.1) |
| Computations | ALL REPRODUCED by independent code, including a separate rigorous stability proof for Thm 1.3; the dynamic-ZAP-70 half of the 600-sample Hurwitz scan was not rerun |
| Novelty | no prior or concurrent publication of the result found; not fatal |
| Presentation / house style | CONFIRMED WITH FIXES |
| Fatal | No |

All twelve required fixes of the second round were applied (the pre-revision source is preserved as `paper/main_v1_prereferee_2026-09-30.tex` in the problem folder):
1. The zero-docking case: k = 0 gives a McKeithan-type network of deficiency 0 (no CD8, Lck constant); the family keeps at least one docking level, and the minimum deficiency 1 is attained exactly by N(1, m, z). Stated after Prop. 2.1 and in Scope and priority.
2. The alias artefact is quantified (66 of 104 laws, 90 of 158 terms, 24 consistent alias names; read literally these terms vanish; the reactant and product lists of Fig. 2B with the explicit constants are used), and the `reversible="false"` flags are mentioned. Also here.
3. The rate sets of the other three witnesses of Thm 1.2 are unambiguous: for N(3, kept, const) the level-3 Lck steps have constant 1 in all three reactions (the rate set of the Thm 1.3 construction). The proof points to `claimant/src/pow10_*.json` and explains the base-10-exponent encoding.
4. The CD8 count reads "67 of the 104 SBML reactions (105 of the 158 irreversible reactions)".
5. Verification paragraph: the N(1, kept, const) determinant used the finder's code, and the k = 2 shared-constant search reran a first-round referee's scripts; the second round is described (new item), here and in the paper. The referee's code is released in `reproducibility/referee/round2_referee2/`, with README rows.
6. Sec. 7 uses the log-odds slope d log(φ/(1 − φ))/d log L (≡ 1 for the collapsed networks) and reports the minima 0.7143 (k = 2) and 0.7045 (k = 3) from 3000 random vectors and 40 Nelder–Mead runs each; the saturation of the earlier objective is explained; "evidence, not proof" is kept.
7. Franco–Velázquez, "Critical lack of equilibrium in stochastic kinetic proofreading", is cited as published: Eur. J. Appl. Math. (2026) 1–47, doi:10.1017/S0956792526100400, with the arXiv id as a note.
8. BibTeX: `{De Leenheer}, Patrick` (label ADS07); `{Rey Barreiro}, Xabier` (label RFV26; ORCID lists the family name as Rey Barreiro, while the Crossref record splits it as "Barreiro, Xabier Rey"); "Version 1.6.0" in the UnsolvedMath note.
9. Bali–Rendall is credited explicitly: their sustained-signalling model (a single memory state with rebinding into the last complex; ligand a conserved variable; unique steady state stated via the Deficiency Zero Theorem) is the closest known analogue of Thm 1.1 and a special case of Prop. 4.1; the phrase "cites [ABG05] in its introduction only" was removed.
10. The "Readings" row on other single-memory lumpings was made precise (at most one positive steady state per class, by Prop. 4.1 with Lemma 3.2(a)).
11. The release was regenerated (tectonic build without warnings, all 14 pages rendered and checked, new `source.zip`, zenodo copies and checksums).
12. The proof of Prop. 7.1 cites Lemma 4.3 together with Prop. 4.4 before applying Lemma 4.5.

The optional recommendations were also applied: Lipniacki et al. 2008 and Owens et al. 2010 are cited in Scope and priority; Feinberg's Deficiency One Algorithm (1995) is mentioned as a possible alternative; the search description names the Europe PMC screen of the 381 citers of ABG05 and the OpenAlex rate limits; the abstract and introduction follow Rendall's hedging; the related works get short descriptions (as "other related work", since most of them are not among the OpenCitations citers of Rendall–Sontag); "No proof rests on" replaces "No claim rests on"; the MM collapse is specified for the CD8-associated steps; the Oberwolfach report is noted as published 2018.

## Relation to the literature, novelty and scope
- **Searches (September 2026).**
  - arXiv API: kinetic proofreading with multistationarity, steady states, bistability, deficiency or reaction networks; Altan-Bonnet with Germain or steady states; all fields "Altan-Bonnet"; Rendall with T cell; rebinding with multistationarity; T cell receptor with multiple steady states or reaction network; multistationarity with T cell; "deficiency one" with receptor.
  - Crossref (bibliographic queries and DOI lookups of all cited works); zbMATH ("kinetic proofreading", 26 records); Europe PMC, including a keyword screen of all 381 records citing ABG05 and the 9 records citing Rendall–Sontag.
  - One web search by the finder and one by each verifier.
- **Citers checked (OpenCitations).** All were checked; none addresses multistationarity of the ABG module.
  - Of Rendall–Sontag, R. Soc. Open Sci. 4 (2017) 170821, there are 13 records:
    - Kreusser–Rendall, Bull. Math. Biol. 2021 (Lck autophosphorylation);
    - Shevyrev–Tereshchenko–Sennikov, Int. J. Mol. Sci. 2022 (two records);
    - Li et al., Immunology 2025;
    - Bali–Rendall, Sci. Rep. 2025, and its bioRxiv preprint;
    - Nikolaev–Zloza–Sontag, Front. Immunol. 2019, and its bioRxiv preprint;
    - Vassena, SIAM J. Appl. Dyn. Syst. 2023 and Math. Methods Appl. Sci. 2024;
    - Proulx-Giraldeau–Rademaker–François, Biophys. J. 2017;
    - Gálvez–Gálvez–García-Peñarrubia, Front. Immunol. 2019;
    - Franco–Velázquez, SIAM J. Appl. Math. 2025.
  - Of the Oberwolfach report there is 1 record: Anderson–Higham–Leite–Williams, Multiscale Model. Simul. 2019.
  - OpenAlex list and filter queries and Semantic Scholar cited-by lookups were rate-limited and not used; single OpenAlex work lookups worked.
- **Related work credited in the paper:**
  - Sontag 2001 (McKeithan's model, deficiency zero);
  - Rendall–Sontag 2017 (François et al. model, three steady states for N = 3);
  - Kreusser–Rendall 2021;
  - Bali–Rendall 2025 (phenotypic models; their "kinetic proofreading with sustained signalling", reviewed by Lever et al. 2014, is a single memory state with rebinding into the last complex, and they state a unique steady state via the Deficiency Zero Theorem; the closest known analogue of Thm 1.1 and a special case of Prop. 4.1);
  - Lever–Maini–van der Merwe–Dushek 2014 (review of phenotypic models);
  - Lipniacki–Hat–Faeder–Hlavacek 2008 (bistability from the competition of SHP-1 and ERK feedbacks in a TCR signalling model) and Owens–Timmis–Greensted–Tyrrell 2010 (analysis of a detailed model of this kind); the paper's results concern the kinetic-proofreading module without these feedbacks;
  - Bali–Rendall–Quapp 2026;
  - Rey Barreiro–Faro–Villaverde 2026;
  - Yu–Sontag 2024;
  - Franco–Velázquez 2025 (SIAM J. Appl. Math.) and 2026 (Eur. J. Appl. Math.);
  - Vassena 2023 and 2024;
  - the injectivity, persistence and degree theory of Craciun–Feinberg, Feliu–Wiuf, Wiuf–Feliu, Angeli–De Leenheer–Sontag, Craciun–Helton–Williams and Conradi–Feliu–Mincheva–Wiuf;
  - Feinberg 1995 (Deficiency One Algorithm; mentioned as a possible alternative, not used) and Banaji–Pantea (inheritance; not used).
- **Novelty.** No treatment of the multistationarity of the ABG module was found in either referee round. This negative search is not a proof of priority.
- **Scope.**
  - The note answers Rendall's multistationarity question in the reaction-network sense for the 36 simplifications and the complete module, with the certificates listed above.
  - The following are **not** settled:
    - multistationarity under the ABG parameter structure for the complete module (and the MM-kept reductions);
    - global asymptotic stability in the deficiency-one case;
    - basins and periodic orbits in the multistationary cases;
    - the intermediate networks with dynamic kinases or CD8 (no separate certificates);
    - the coupled model with SHP-1 and ERK feedbacks.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/