# 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 | DARK: needs public evidence (a public repro or measurement) before it can be claimed | DARK |
| 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 | DARK: needs public evidence (a public repro or measurement) before it can be claimed | DARK |
| 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.