Why a cheap belief field cannot see synergy: closure order, local observation, and the sign of O-information

Working title: Why a cheap belief field cannot see synergy: closure order, local observation, and the sign of O-information

Author: [withheld in the repository copy]. Contact: [withheld]. Status: DRAFT — not submitted. Luke submits by hand, and only if he chooses to. Revision: 2026-11-02 second sweep (related work, venues). Issue: none filed; slate entry P7.

Tiers: PINNED (CI gate on every change), MEASURED (run once on the owner's real data), DESIGNED (built, evaluation owed), DENIED (measured and rejected, kept with numbers), DARK (no measurement; no number typed).

Venues

Venue	Deadline (time zone)	URL	Fee / barrier	Checked
arXiv cs.IT or cond-mat.stat-mech primary; cross-list quant-ph	Rolling. Cutoff 14:00 US Eastern; announced 20:00 ET, Sunday–Thursday	https://info.arxiv.org/help/endorsement.html ; https://blog.arxiv.org/2026/01/21/attention-authors-updated-endorsement-policy	Free. cs.IT is in the cs domain (shared with P1); cond-mat.stat-mech is cond-mat; quant-ph is its own. Each new category may ask for an endorsement; moderators may reclassify. Policy since 2026-01-21: automatic endorsement needs an academic email and prior arXiv authorship in the domain; every other new author takes the personal-endorsement route, which "arXiv staff cannot waive"; no numeric endorser threshold is published	READ (policy post READ 2026-11-02)
Chaos, Special Topic "In Memory of Professor Erik Bollt" (AIP)	2027-01-31 (no time zone stated)	https://pubs.aip.org/cfp/pages/ebolt2026	Topics include information theory, causal inference, network dynamics. Subscription route free ("does not require page or color charges"); OA $3,800; AIP waivers case-by-case. Fast Track ≤ 5,750 words; non-specialist Lead Paragraph required. Re-checked 2026-11-02: still the only open Chaos special topic; five guest editors confirmed (Belykh, di Bernardo, Porfiri, Porter, Sun). No Chaos paper on O-information surfaced in three searches, so the fit rests on the call's wording, not precedent (UNVERIFIED after search 2026-11-02)	READ (re-checked 2026-11-02)
Entropy (MDPI)	Rolling	https://doaj.org/api/search/journals/issn:1099-4300	Fully OA, CC BY; APC up to 2600 CHF (DOAJ-reported); waiver policy unread; single-anonymous. Special issues UNVERIFIED (mdpi.com still 403 on 2026-11-02). Strongest-fit journal per the research note (2026-11-02): the nearest methodological negative, Gehlen et al. 2024 "Bias in O-Information Estimation", Entropy 26(10):837, sits there beside the field's PID overviews. The "Higher-Order Interactions in Physiological Networks" SI PASSED on 2026-02-25; no 2027 higher-order SI surfaced (snippets only)	DOAJ registry only; fit from verified Entropy content
Physical Review E / Physical Review Research (APS)	Rolling	https://journals.aps.org/pre/about	PRE hybrid: subscription route lists no charge; OA $2,910. PRResearch fully OA at $2,910, no waiver for a high-income independent. Single-anonymized; ORCID required; empty affiliation UNVERIFIED. PRResearch confirmed 2026-11-02 as a home for both closure admissibility (Goldobin and Dolmatova 2019, PRR 1, 033139) and O-information (Scagliarini et al. 2023, PRR 5, 013025)	READ
Frontiers in Network Physiology RT 76014 "Beyond Pairwise Interactions" (Faes, Azami, Cortes, Varley)	2026-12-01 extension (no time zone)	https://www.frontiersin.org/research-topics/76014	APC CHF 2,195 (A) / 1,750 (B); fee support "in no way influence[s]" acceptance; reviewer names published. Strongest topical match for O-information per the venues note; stays first after the 2026-11-02 re-check (nothing new for P7); the journal carries O-information gradients (Scagliarini et al. 2024)	READ (re-checked 2026-11-02)
Frontiers in Complex Systems RT 82910 "Higher order versus pairwise interactions"	2027-04-28 (no time zone)	https://www.frontiersin.org/research-topics/82910	Perspectives and Opinions encouraged; Frontiers fee schedule	READ

Abstract (draft)

A belief field cheap enough to run live pays for its price in what it cannot see. We report three measured limits of such a field, a density matrix over clusters of two-level systems, under three measures: total correlation (TC, how far the joint sits from the product of its parts), O-information (Ω, whose sign separates redundancy from synergy) and variation of information (VI). First, local single-system observations factorize the joint state: from a product start, 60–120 one-hot observations leave TC ≈ 0.0000, and exchange coupling that builds TC ≈ 1.9 in isolation is crushed to ≈ 0.0001 under live observation. One crack: a latent, never-observed concept system coupled to observed beliefs sustains a correlation of ≈ 0.08, 800 times the floor, while Ω's sign stays at the floor. Second, a second-order cumulant closure zeros the three-body cumulant that carries Ω's sign: on pure XOR the closure reads TC ≈ 0 and flat where the exact state reads TC = 1, Ω = −1, so a cheap cluster feels binding but cannot name its character. Third, VI built from quantum conditional entropies goes negative on entangled states (Bell pair VI = −2), so it is a metric only in the separable regime. The positive result: a linear readout over self-discovered product features forecasts a held-out pure-parity target (order 2 yes, order 1 no). We assemble the closure limit from Marcinkiewicz, Schneidman and Amari, state Barrett's Gaussian caveat, and report a runtime null for these measures; we found no published runtime null for them (search scope: DARK; the two owed phrase searches are still to run).

Contributions

1. Three limits as theorems-with-numbers. Each is a pinned test with its figures: the architectural null under local observation, the order-2 closure's sign blindness, and the separable-only validity of VI (DENIED, measured and kept).
2. The corollary route for sign blindness. Order-2 closures retain connected information of order 2 and discard order ≥ 3 (Schneidman et al. 2003; Amari 2001); truncating above order 2 is non-positive (Marcinkiewicz; Rajagopal and Sudarshan 1974); hence a cheap closure is blind to purely third-order Ω, with Barrett's (2015) Gaussian regime as the stated exception.
3. Canonical pins. Product 0 / 0; classical GHZ TC = 2, Ω = +1; XOR TC = 1, Ω = −1 with zero pairwise correlation; pure GHZ TC = 3, Ω = 0; TC proxy within 1% of exact in the weak regime; VI triangle inequality with 0 violations on pinned classical states (PINNED).
4. The pulse-and-skeleton reading of one real tape. On 138 sampled ticks of a game's pantry, grain flat 100%, TC > 1e-4 on 0%, max ‖C‖ 1e-5, while learned zz couplings persist at |J| ≤ 0.22: higher-order is a pulse riding live correlation; zz is the skeleton (MEASURED NULL).
5. Synergy found by the forecaster, not the field. On a pure-parity world, order-1 context scores svp ≈ −0.06 against persistence and order-2 scores ≈ +0.20; an engine-free ridge search beats brute-force second order (0.355 vs 0.415) by discovering a sparse product set (PINNED).
6. A proposed law of scale, stated as interpretation. Small dense clusters name grain; composed clusters, cumulant inside and mean-field bridges between, are grain-blind. A reading of rows 1–4, not a measurement.

Outline (about 6,500 words; Fast Track at Chaos needs ≤ 5,750)

1. Lead paragraph and introduction (600 words). Plain statement: a system that keeps its beliefs cheap can tell that three things move together but not whether they agree or conspire. Then the coined words: "chorus" is redundancy (Ω > 0), "conspiracy" is synergy (Ω < 0), "grain" is the sign of Ω. Rosas et al. (2019) give the measure; Rosas et al. (2022) the vocabulary of high-order behaviour against mechanism.
2. The measures on a cluster (700 words). Per dense cluster, TC = Σ S(ρ_i) − S(ρ_X); Ω = (n−2) S(X) + Σ_i [S({i}) − S(−i)]; VI(i, j) = H_i + H_j − 2·MI; all from the cumulants the mutual-information sensor already touches. Javarone et al. (2024) define the same Ω on density operators; unitary evolution can flip its sign.
3. Limit one: local observation factorizes the state (1,000 words). The live field takes evidence one qubit at a time. From a product start, 60–120 one-hot observations leave TC ≈ 0.0000; a diagonal ZZ Hamiltonian commutes with a diagonal state (0.5 and 2.0 byte-identical); exchange builds TC ≈ 1.9 unobserved and ≈ 0.0001 observed; an installed one-hot mixture reads TC = 1.32, Ω = −1.27, so the sensor is correct. The crack: a latent concept qubit sustains C ≈ 0.08 and TC ≈ 0.01–0.05 with grain at the floor. This is the filtering-side corollary of Girolami et al. (2022) and Brandão, Piani and Horodecki (2015): redundant local records leave the classical part. Classical shadows (Huang, Kueng and Preskill 2020) are the counterpoint: many copies with random local bases recover global properties; a live belief has one copy, one basis, sequential evidence.
4. Limit two: order-2 closure is blind to the sign (1,000 words). Second-order O-information is identically zero; |Ω|/TC falls 5.0e-2 → 6.6e-3 as ε shrinks 0.08 → 0.01; on pure XOR the cumulant reads TC ≈ 0 and flat while dense reads TC = 1, Ω = −1. The corollary: order-2 retains I_C^(2) and discards I_C^(≥3) (Schneidman et al. 2003; Amari 2001); the Marcinkiewicz theorem (Rajagopal and Sudarshan 1974) makes order 2 the last positive truncation; Walck and Lyons (2008) and Zhou (2008) show GHZ-type correlation is what lower-order reduced states miss. Barrett (2015): for Gaussian variables Ω follows from covariance alone, so the blindness is to purely third-order dependence, not universal.
5. Limit three: VI is not a metric on entangled states (500 words). VI = 2S(AB) − S(A) − S(B) uses quantum conditional entropies that go negative when entangled (Cerf and Adami 1997; Horodecki, Oppenheim and Winter 2005). A Bell pair gives H = 1, MI = 2, VI = −2. Meilă (2007) proves VI is a metric on partitions; the quantum version is one only in the separable regime. Cross-cluster higher-order is not computed; bridges are mean-field.
6. The real tape (500 words). 138 sampled ticks; learned zz web |J| ≤ 0.22; grain flat 100%; TC > 1e-4 on 0%; max ‖C‖ 1e-5, ten times below the floor. The pulse-and-skeleton reading, and the fog-law discipline: a null is a measured zero with its age, not an absence.
7. The positive result (700 words). The synergy world, C = A × B, no first-order signal, held out on disjoint seeds: order-1 svp ≈ −0.06, order-2 svp ≈ +0.20; the engine-free batch-ridge search scores 0.355 against brute-force second order 0.415 by finding a sparse product set. XOR is the degree-2 monomial, a leap-2 function in Abbe, Boix-Adserà and Misiakiewicz (2023); Barak et al. (2022) show SGD learns sparse parities with hidden progress. Synergy the field cannot hold, the readout can still forecast.
8. Related work and the negative space (900 words). As below.
9. Threats, DARK, what this paper is not (600 words). As below.

Related work and the negative space

Classical higher-order information. Watanabe (1960), IBM J. Res. Dev.: total correlation. Rosas, Mediano, Gastpar and Jensen (2019), PRE: O-information, sign > 0 redundancy, < 0 synergy. Meilă (2007), J. Multivariate Analysis: VI as a metric. James and Crutchfield (2017), Entropy: "dyadic camouflage". Schneidman, Still, Berry and Bialek (2003), PRL: connected information of order k, the entropy drop against the order-(k−1) maximum-entropy model. Amari (2001), IEEE Trans. Inf. Theory: orthogonal decomposition of interactions by order. Liardi, Rosas, Marinazzo, Varley, Gastpar and Mediano (2026), arXiv:2610.03764 (v2 6 Oct 2026; no journal reference), "Statistical Dark Matter: Synergy is everywhere but is hard to capture": pairwise maximum-entropy Ising and Gaussian fits, factor-analytic reconstructions and tree-structured Bayesian networks fitted to higher-order generators give "a systematic underestimation of synergistic structure", "commonly used probabilistic graphical models cannot express such synergistic structures", yet pairwise maxent models "can still exhibit some degree of synergy"; larger random systems are increasingly synergy-dominated; analytical and Monte Carlo, offline; the nearest prior for pairwise models underestimating synergy. Gehlen, Li, Hourican, Tassi, Mishra, Lehtimäki, Kähönen, Raitakari, Bosch and Quax (2024), Entropy 26(10):837, DOI 10.3390/e26100837, "Bias in O-Information Estimation": the finite-sample Ω of independent systems "is severely biased towards synergy if the sample size is smaller than the number of jointly possible observations". Varley et al. (2023), PNAS 120(30):e2300888120: partial entropy decomposition; synergies "largely invisible to standard functional connectivity analyses". Antonacci, Barà, Sparacino, Mijatovic, Minati and Faes (2025), arXiv:2503.12421: a time-varying VAR fitted by recursive least squares with a forgetting factor that tracks synergy and redundancy over time and frequency, an online-capable estimator applied offline to recorded physiological data. Toolboxes, neither with a streaming mode: Neri et al. (2024), J. Open Source Software 9(103):7360, DOI 10.21105/joss.07360, HOI (Oinfo, TC, DTC, Sinfo, PID atoms, gradients; Gaussian-copula, kNN, binning and kernel estimators; JAX); Belloli, Mediano, Cofré, Fernandez Slezak and Herzog (2025), arXiv:2501.03381, THOI (Gaussian-copula joint entropy, batched on CPU/GPU/TPU). Pérez-Reche (2026), arXiv:2609.20142 (single-author review, 30 pp.): higher-order interactions by mechanism, behaviour, representation and reducibility, the last formulated as comparing a higher-order model with a class of lower-order models against a target and a tolerance. Deployment side: Liang et al. (2023), NeurIPS 2023, arXiv:2302.12247, PID statistics of multimodal datasets and trained models with a robotic-perception case study, offline; Lv, Yu, Wen and Principe (2020/21), arXiv:2007.10692 (J. Franklin Inst.), pairwise mutual-information matrices over windows on Tennessee Eastman, the measure process monitoring uses in place of TC, DTC or Ω.

Partial information decomposition and syntheses. Williams and Beer (2010), arXiv:1004.2515: the redundancy lattice. Lizier, Bertschinger, Jost and Wibral (2018), Entropy: the field overview. Barrett (2015), PRE: the Gaussian PID; net synergy from covariance alone. Rosas et al. (2022), Nature Physics correspondence: behaviour against mechanism. Robiglio et al. (2025), PRL: a synergistic signature of group mechanisms invisible to pairwise measures. Stramaglia, Scagliarini, Daniels and Marinazzo (2021), Frontiers in Physiology: O-information on recorded spiking, offline.

Quantum correlation measures. Cerf and Adami (1997), PRL: negative conditional entropy. Horodecki, Oppenheim and Winter (2005), Nature: its operational meaning. Javarone, Rosas, Facchi, Pascazio and Stramaglia (2024), PRA: Q-information; sign flips under pairwise and three-body Hamiltonians, invariant under single-qubit ones. van Enk (2023), arXiv:2308.04499: a measurement-free quantum PID; the quantum conditional entropy "is negative only when ρ_TA is entangled". Ericson, Wang and van Enk (2026), arXiv:2606.07880: capacity-based quantum PID; redundant information "characterizes the classical world".

Closures. Kubo (1962), J. Phys. Soc. Jpn.: the generalized cumulant expansion. Rajagopal and Sudarshan (1974), PRA: Marcinkiewicz generalized; truncation above order 2 yields non-positivity. Goldobin and Dolmatova (2019), Phys. Rev. Research 1, 033139, DOI 10.1103/PhysRevResearch.1.033139: "Truncation of circular cumulant expansions, except for the Ott-Antonsen case, is forbidden", against linear variables where the Gaussian second-cumulant truncation is admissible; a second published "only admissible truncation" statement beside Marcinkiewicz for the step "why the ladder stops at order 2". Plankensteiner, Hotter and Ritsch (2022), Quantum: order-n truncation as neglected joint cumulants.

Local observation and redundancy. Linden, Popescu and Wootters (2002), PRL: almost every three-qubit pure state is fixed by its two-party reduced states; GHZ is the exception. Walck and Lyons (2008), PRL: only GHZ-type states are undetermined by their reduced states. Zhou (2008), PRL: irreducible k-party correlations by maximum entropy. Zurek (2009), Nature Physics: quantum Darwinism, redundancy as the prerequisite for objectivity. Brandão, Piani and Horodecki (2015), Nature Communications: indirect observers access at most classical information about one measurement. Girolami, Touil, Yan, Deffner and Zurek (2022), PRL: redundantly amplified information suppresses quantum correlations. Huang, Kueng and Preskill (2020), Nature Physics: classical shadows, the many-copy counterpoint. Wood and Spekkens (2015), New J. Phys.: latent common causes and fine-tuning; the frame for the concept-qubit crack.

Parity learning. Barak, Edelman, Goel, Kakade, Malach and Zhang (2022), NeurIPS: hidden progress on sparse parities. Abbe, Boix-Adserà and Misiakiewicz (2023), COLT: leap complexity.

The negative space. The nearest neighbours are Javarone et al. (2024), who compute Ω on exact four-qubit states under exact unitaries and never under a cumulant-truncated or filtered state; Plankensteiner et al. (2022), who automate order-n truncation and never ask what information measures it destroys; Schneidman et al. (2003) and Amari (2001), who decompose dependence by order and never write "second-order closure" and "O-information" in one sentence; Girolami et al. (2022) and Brandão et al. (2015), who show redundant local records leave only the classical part and never say so for a filter fed sequential local evidence; Cerf and Adami (1997) and Horodecki et al. (2005), who make conditional entropy negative and never note what that does to VI; Wood and Spekkens (2015), who treat latent common causes for Bell correlations and never put a latent qubit in a classical filter; and Stramaglia et al. (2021) with every other O-information application found, all offline, now including Antonacci et al. (2025), whose recursive estimator is online-capable but applied to recorded data, and the HOI and THOI toolboxes, neither with a streaming mode. The nearest prior for the closure limit is now Liardi et al. (2026), who fit pairwise-maximum-entropy, factor-analytic and tree-structured models to higher-order generators and record a systematic underestimation of synergy, offline and never for a cumulant-truncated or filtered state; a referee will expect it cited before the sign-blindness corollary, and Pérez-Reche (2026) supplies the "reducibility to a lower-order model class" vocabulary. No paper found computes Ω under cumulant order n against exact ρ; none states the sign-blindness corollary in its exact form, that an order-2 cumulant closure of a belief state zeros the third cumulant and hence the sign of a purely third-order Ω; none studies synergy decay under sequential local conditioning; none defines a quantum VI or names its metric failure (the nearest structural neighbour is Matthias, Makkeh, Wibral and Gutknecht 2025/26, arXiv:2512.16662, where PID cannot keep non-negativity, the chain rule and invariance together, a measure losing a property when extended, but not about VI or entanglement); none reports O-information or TC as a live runtime signal with a published null. The research note marks these "not found" under a search budget exhausted twice. The two phrase searches owed by the first draft ran on 2026-11-02 and returned nothing: "total correlation" with process monitoring finds only pairwise correlation-structure monitoring (canonical-variate and partial-correlation statistics; the pairwise MI matrix of Lv et al. 2021), and "synergy" with sensor fusion finds PID theory only (one "fusion" hit, BioSync, arXiv:2609.04504, never mentions PID or synergy: a false lead). Two searches were NOT RUN for lack of budget and must run before submission: the recommender / closed-loop-control / robotics flank of the live-diagnostic question, and the explicit published-null search (a higher-order measure with "negative result", "no improvement" or "did not improve" in deployed anomaly detection). Until they run, the runtime-null sentence stands with two caveats printed beside it: Gehlen et al. (2024) give a mechanism by which a live Ω reads as decorative or misleading (finite-sample Ω of independent variables is biased toward synergy), and Liardi et al. (2026) show synergy is "hard to capture" from pairwise summaries.

Second sweep (2026-11-02). Sixteen web queries ran on the higher-order note before the shared budget closed a second time; nine queued searches did not execute, and verification continued by direct fetch (arXiv abstract pages, Crossref, Semantic Scholar, Open Library, NeurIPS proceedings, PMC, ORA). Verdicts: higher-order information on a live, cumulant-truncated belief state, UNCHANGED, nearest Liardi et al. 2026; the sign-blindness corollary as a stated theorem, NARROWED, the "systematic underestimation" half now in print (Liardi 2026; Varley 2023) and the Barrett-2015 caveat confirmed by the same source, with the exact order-2-zeros-the-third-cumulant statement still unfound, and Goldobin–Dolmatova 2019 added beside Marcinkiewicz; synergy decay under sequential local observation, UNCHANGED (Antonacci 2025 is estimation from data, not a statement about what a filter can hold); quantum VI and its metric failure, UNCHANGED, Bell-pair VI = −2 unwritten in anything retrieved; the latent qubit as carrier of correlation, UNCHANGED (not searched this pass; Blume-Kohout and Zurek 2006 now verified as PRA 73, 062310); O-information or TC as a deployed runtime diagnostic with a published null, NARROWED on the estimator flank (Antonacci 2025; HOI; THOI) and UNCHANGED on the deployment and null flanks, with the recommender/control and explicit-null searches NOT RUN; the PID–parity bridge, UNCHANGED, with Daniely and Malach (NeurIPS 2020) and Blum, Kalai and Wasserman (2003), JACM 50(4):506–519, now verified anchors and the two literatures still not citing each other; quantum PID on filtered states, UNCHANGED, Ericson, Wang and van Enk 2026 still without a journal reference. Venue gap: Journal of Physics: Complexity, Chaos and PRX Quantum remain UNVERIFIED after search 2026-11-02 as homes for any cited work; Entropy, Physical Review Research, Frontiers in Network Physiology and NeurIPS gained verified entries, and the note names Entropy the strongest fit for a methods-and-negative-result paper. Still UNVERIFIED after search 2026-11-02 and not cited above as verified: Rosas et al., "Characterising high-order interdependence via entropic conjugation", Communications Physics 2025 (snippet only; Spiral 429), which would claim O-information as the unique skew-symmetric measure with linear estimation cost; Ji, Wang and Sun's sliding-window MI-matrix monitoring (ADCHEM 2021); the 2012 AIChE multidimensional-MI dissimilarity; Jia et al. 2022 IoT subsystem correlation; Mijatovic et al. 2024, arXiv:2408.15617, still without a journal reference. Owner numbers in this file are unchanged by the sweep.

Evidence each claim rests on

Claim	Pointer	Tier
Per dense cluster TC, Ω, grain = sign(Ω), VI and MI-graph constellations from the cumulants the MI sensor touches	DARK: needs public evidence (a public repro or measurement) before it can be claimed	DARK
Canonical pins: product 0 / 0; classical GHZ TC = 2, Ω = +1; XOR TC = 1, Ω = −1, zero pairwise; pure GHZ TC = 3, Ω = 0; TC proxy vs exact < 1% weak regime (−0.04%); VI triangle 0 violations	same	PINNED
Clusters run dense (2ⁿ) or second-order cumulant closure (e₁, e₂; O(n²))	substrate/cumulant_cluster.py, substrate/cluster.py	PINNED (code)
Architectural null: 60–120 one-hot observations → TC ≈ 0.0000; diagonal ZZ 0.5 vs 2.0 byte-identical; exchange TC → 1.9 unobserved, ≈ 0.0001 observed; installed mixture TC = 1.32, Ω = −1.27; latent concept-qubit C ≈ 0.08 (800× floor), TC ≈ 0.01–0.05, grain at floor	DARK: needs public evidence (a public repro or measurement) before it can be claimed	DARK
Dense-only: second-order Ω identically zero;	Ω	/TC 5.0e-2 → 6.6e-3 over ε 0.08 → 0.01; pure XOR: cumulant TC ≈ 0 flat, dense TC = 1, Ω = −1
Separable-regime-only: Bell pair H = 1, MI = 2, VI = −2; cross-cluster higher-order not computed (bridges mean-field)	DARK: needs public evidence (a public repro or measurement) before it can be claimed	DARK
Real tape (manifold_game pantry): 138 ticks,	J	≤ 0.22; grain flat 100%; TC > 1e-4 on 0%; max ‖C‖ 1e-5
Synergy world: order-1 svp ≈ −0.06; order-2 svp ≈ +0.20; ridge feature search 0.355 vs brute-force 0.415	DARK: needs public evidence (a public repro or measurement) before it can be claimed	DARK
A null renders DARK, never 0; a measured zero is 0 with its age	the-broadcast docs/papers/01-truth-tracking.md §5	PUBLISHED

DARK (do not claim a number)

• Live O-information or TC as a decision signal: never promoted.
• Ω under cumulant order n along Javarone's H₂ / H₃ evolutions against exact ρ: proposed, not computed.
• Cross-cluster higher-order: not computed; bridges are mean-field.
• Coherence advantage on any real data: owed.
• Synergy on any real tape beyond the 138-tick pantry: unmeasured.
• Any neural, physiological or human data: none.

Threats to validity

The pins run on synthetic states and one 138-tick real tape with small learned couplings (|J| ≤ 0.22), so its null may reflect weak coupling rather than the mechanism; the proof file separates the two on synthetic data only. The closure is the engine's own order-2 implementation, not a reference package, so it is not reproducible without the private engine. The Gaussian regime (Barrett 2015), where order 2 does carry Ω, is stated but not tested. The VI result is a two-line corollary of known negative conditional entropy. The "law of scale" is an interpretation of four rows and must be labelled so in print. The TC proxy is within 1% only in the weak regime; the MI floor behind "800×" is the engine's own constant.

What this paper is not

• Not a claim that synergy is absent in the world; only that this field cannot hold it live.
• Not a neuroscience or physiology paper; no brain or body data is used.
• Not a new partial information decomposition, quantum or classical.
• Not a quantum-advantage claim; the field is simulated classically.
• Not a challenge to offline O-information studies.
• Not a report on any earlier home deployment.

Luke's steps, if he wants to submit

1. Register at arxiv.org ("Independent" as organization; ORCID linked). Choose the primary by method: cond-mat.stat-mech if the closure argument leads, cs.IT if the measures lead; cross-list quant-ph. Each category may ask for an endorsement; cs is shared with P1.
2. Find endorsers through "Which authors of this paper are endorsers?" on recent neighbours: Javarone et al., arXiv:2310.03681; Ericson, Wang and van Enk, arXiv:2606.07880; Robiglio et al., arXiv:2401.11588; Liardi et al., arXiv:2610.03764. One email per endorser with the six-character code and the PDF. Since 2026-01-21 this personal route is the only one for an author without an academic email and a prior arXiv paper in the domain, and "arXiv staff cannot waive endorsement requirements" (READ 2026-11-02).
3. Pick the arXiv perpetual non-exclusive license if the journal is undecided; CC BY 4.0 if Entropy or Frontiers is the target. Submit before 14:00 ET.
4. Chaos Special Topic: write the non-specialist Lead Paragraph, keep the Fast Track under 5,750 words, submit by 2027-01-31 (no time zone; finish a day early) at https://pubs.aip.org/cfp/pages/ebolt2026; take the subscription route unless OA is wanted at $3,800.
5. Frontiers RT 76014: submit by 2026-12-01 (no time zone) at https://www.frontiersin.org/research-topics/76014; request fee support at submission (CHF 2,195 A / 1,750 B); link the ORCID in place of an institutional email.
6. Entropy or PRE: read mdpi.com's special-issue and waiver pages by hand (Entropy is the research note's strongest-fit journal as of 2026-11-02, with Gehlen et al. 2024 as the nearest methodological negative already there; cite it and Liardi et al. 2026 before claiming the first runtime null); at APS take the subscription route, supply the ORCID and try "Independent researcher" as affiliation (handling UNVERIFIED).
7. Before any submission re-run [private, withheld], both test_higher_order pins, test_vi_breaks_on_bell and [private, withheld]; record the dates. The first draft's two phrase searches ("total correlation" + "process monitoring"; "synergy" + "sensor fusion") ran on 2026-11-02 and found nothing; run the two that did not for lack of budget: "partial information decomposition" OR "O-information" with recommender system OR "closed-loop control" OR robotics online, and a higher-order measure with "negative result" OR "no improvement" OR "did not improve" in deployed anomaly detection. Spell out "partial information decomposition" in any control-systems query: "PID" collides with proportional–integral–derivative control and returns controller tuning. (DARK: needs public evidence (a public repro or measurement) before it can be claimed.)
8. Log the submission in the authority record when there is a receipt.