This page consolidates the open conjectures of Donald Hoffman & Chetan Prakash's conscious‑agent / trace‑logic program — the unfinished mathematics and the speculative physics — into a single catalogue for researchers. It is the research‑frontier companion to the full essay, which builds the program from textbook Markov chains up to the trace logic. Here we state only what is open, and we state it so a working mathematician or physicist can begin.
The material spans three epistemic worlds — settled mathematics, theorems proven in a 2024 preprint, and frankly speculative physics — and the whole value of this catalogue is that it never conflates them. Every claim is tagged.
LEGENDHow to read this page
◆ There is no canonical "nine"
The only branded, explicitly numbered list is "The Eight Conjectures" of physics (§3) — and its home is the Trace Institute site (items 01/08–08/08), not the papers: the 2024 preprint contains no eight‑item enumeration (the formal label "Conjecture" appears exactly once, p.26; "we conjecture" twice inline; plus a six‑item list of physics identifications, p.19). The recurring "eight or nine" is a paraphrase of a spoken remark, not a printed enumeration. The genuinely tractable targets are the mathematical open problems of §1; the physics dictionary (§2) and the Eight Conjectures (§3) are research programs, each gated on that mathematics. The aim here is to state every problem so precisely that the gap — and a first step toward closing it — is visible.
MOTIVEWhy these matter — the lineage X
The program's ambition is to derive spacetime and physics from the dynamics of observation, rather than assume them. That ambition has a long lineage, which the catalogue inherits as motivation (not as proof).
◆ Observer‑first ancestors
Leibniz. The monads are windowless, perspectival perceivers; bodies, space, and time are well‑founded phenomena — appearances grounded in those perceivers, and space‑time is relational, "not an absolute being" (Leibniz–Clarke, Third Paper §5). The coordination of non‑interacting monads is the pre‑established harmony (Monadology §§78–81) — the role the global trace order plays here. Wheeler. "It from bit": every it "derives its function, its meaning, its very existence … from … bits," with spacetime itself derived in a participatory loop. Read as lineage X: monads ≈ observer‑windows, well‑founded phenomena ≈ the spacetime interface, pre‑established harmony ≈ the trace order, "it from bit" ≈ observation‑first. These are ancestors the program names — resonances, not derivations; Wheeler is an information‑first physicist, not an idealist, and Leibniz grounds the harmony in God, where the program grounds it in mathematics.
SECTION 1Mathematical open problems PC
The citable, self‑contained targets: each extends an already‑proven special case and needs no physics. The two marked ◆ student‑ready are the strongest starting points.
1 · The Combination Conjecture PC · two‑agent joins proven, general case conjectural · ◆ student‑ready
Statement. Given any pseudograph of conscious agents (any mix of directed and undirected edges), any subset of its agents — adjacent or not — can be combined into a single new conscious agent.
Proven. The two‑agent cases are theorems by explicit construction: the undirected join (Thm 1) on the product experience space \(X_1\times X_2\), and the directed join (Thm 2), with composed decision kernel \(D=D_1A_1D_2\).
Gap. The general \(n\)-agent combination is unproven; in particular, associativity / order‑independence of the combination is open.
First move. Prove the three‑agent case (chain and triangle topologies), or characterize exactly which pseudograph topologies admit an associative, order‑independent combination. Finite and combinatorial.
Toolkit. Markov kernels as block‑stochastic matrices; the join constructions of Objects of Consciousness §§5–6; basic category/pseudograph combinatorics.
Hoffman & Prakash, "Objects of Consciousness," Frontiers in Psychology 5:577 (2014), Conjecture 3; Theorems 1–2.
2 · The general join in the trace logic PC · local Booleanity proven, the general join open · ◆ student‑ready
Statement. In the trace logic, the join \(K\vee L\) and meet \(K\wedge L\) exist only between compatible (simultaneously verifiable) kernels; there is no general closed form. Verbatim: "We seek a solution for the 9 unknown matrices \((A',B',\dots,H')\) … to the 9 equations (A56)–(A58). … It is an open question whether solutions always exist and, if so, are unique."
Proven. On the downset of any fixed kernel \(N\) (its traces), the logic is a Boolean algebra with well‑defined \(\wedge,\vee,\neg\) (Thm 4.12, local Booleanity).
Gap. Whether the least upper bound \(K\vee L\) exists, and is unique, for incomparable kernels globally; no general notation exists for it.
First move. Solve the A56–A58 matrix system for small state spaces (2‑ and 3‑state kernels sharing two states) and extract an explicit compatibility criterion — characterize precisely which pairs of Markov kernels admit a join.
Toolkit. Schur complements / censored chains; the trace‑chain formula \(p_A=P_{AA}+P_{AA'}(I-P_{A'A'})^{-1}P_{A'A}\); lattice theory (least upper bounds, compatibility).
Hoffman, Prakash & Chattopadhyay, "Traces of Consciousness," Preprints.org doi:10.20944/preprints202410.1305.v1 (17 Oct 2024), Remark 4.10, Appendix A.4 (printed page "43 of 46"), eqs. A53–A58.
3 · The commute‑time periodicity conjecture C · best "could become a theorem"
Statement. An ergodic Markov chain on \(n\) states has minimal total expected commute time between its states if and only if it is periodic with period \(n\). (The total commute time is \(\mathcal{T}=\sum_{i\lt j}T_{ij}\), with \(T_{ij}\) the expected round‑trip time between states \(i\) and \(j\).)
Proven. Verified by the authors only for \(n=2\) (their Figure 9). Standard tool: for a reversible chain, \(C(a,b)=2m\,R_{\mathrm{eff}}(a,b)\) (Chandra et al.), tying the quantity to effective resistance / Dirichlet energy.
Gap. The iff for all \(n\): that total commute time over the Markov polytope \(\mathcal{M}_n\) is minimized exactly by the period‑\(n\) (cyclic) kernels.
First move. Prove the iff for \(n=3,4\) by direct optimization over \(\mathcal{M}_n\) using resistance identities; then attempt the general bound. This underwrites the physics claim that the fastest chains — deterministic cycles, hence zero entropy rate — move at the limiting speed \(c\) (§2).
Toolkit. Effective resistance / electrical‑network theory (Doyle–Snell; Chandra et al.); mean first‑passage and commute times; convex optimization over the Birkhoff/Markov polytope.
"Traces of Consciousness" (2024), p.26 (Figure 9; proposal p.25).
4 · The Conscious‑Agent Thesis C · a falsifiable empirical thesis (Church–Turing‑style), not a theorem
Statement. "Every property of consciousness can be represented by some property of a dynamical system of conscious agents" (2014). Operationalized as a Church–Turing‑style empirical claim: a conscious process (recognition, inference, choice) not representable by the action of any Markov kernel would falsify it.
Status. A falsifiable thesis. Corroborated by representing further cognitive operations as kernels (CA‑Networks 2018: memory, planning, learning); falsified by a single counterexample.
First move. Pin down the representability class: which cognitive operations are Markov‑kernel‑representable, and at what state‑space cardinality — a theory‑of‑computation question.
Toolkit. Markov kernels / measurable dynamics; Church–Turing and representability arguments; the CA‑Networks 2018 memory/planning constructions as templates.
Hoffman & Prakash 2014, p.10 ("Hypothesis 2"); operational reframing in Fields, Hoffman, Prakash & Singh, "Conscious agent networks," Cognitive Systems Research 47 (2018) 186–213, p.190 (eScholarship qt2d34n6zf).
5 · The agent–particle correspondence & the Markov→decorated‑permutation completion C · ◆◆ hard (math half)
Statement. Conjecture: "a particle (in spacetime) is an aspect of a physical projection of the dynamics of a communicating class of conscious agents to a face of an amplituhedron." The supporting object is a one‑way construction (Fusions 2023, Definition 2) sending a Markov chain to a decorated permutation via its communicating classes — a definition, not a proven bijection. A communicating class of size \(\ell\) is made to code an \((\ell-1)\)-dimensional cell of the positive Grassmannian \(\mathrm{Gr}_{\ge 0}(k,n)\).
Proven (adjacent). On the pure‑math side, the Postnikov–Williams bijection between decorated permutations and positroid cells of \(\mathrm{Gr}_{\ge 0}\) is established; the amplituhedron's tree‑level BCFW triangulation is now proven.
Gap. The Markov side: upgrade Definition 2 from a one‑way map to a characterized (ideally bijective) correspondence between Markov communicating classes / polytope cells and decorated permutations / positroid cells.
First move. Characterize the image of Definition 2 and ask when it is injective/surjective onto positroid cells for small \((k,n)\). (The physical half — locating a gluon‑scattering decorated permutation inside an actual agent's dynamics — is a separate, much harder program.)
Toolkit. Positroid combinatorics and decorated permutations (Postnikov; Williams); the positive Grassmannian \(\mathrm{Gr}_{\ge 0}(k,n)\); communicating‑class structure of finite Markov chains.
Hoffman, Prakash & Prentner, "Fusions of Consciousness," Entropy 25(1):129 (2023), §7 and Definition 2 (PMC9858210); Williams, ICM 2021, arXiv:2110.10856.
SECTION 2The physics‑projection dictionary C
The authors' proposed identifications of Markov / recurrent‑communicating‑class (RCC) quantities with physical observables. All are conjectural; the recurring obstacle for every entry is the same — an undetermined dimensional bridge constant, since the projected quantity is dimensionless and the physical one is not.
| Observable | Proposed projection | Where | Note |
|---|---|---|---|
| Mass | entropy rate of the RCC, \(h(Q)=-\sum_i\pi_i\sum_j Q_{ij}\log Q_{ij}\); a deterministic cycle (zero entropy rate) \(\Rightarrow\) massless | eq. 47 | the cleanest entry |
| Spin | determinant of \(Q\) collapsed via a geometric‑algebra "C‑spin" to \(S\in\{0,\tfrac12,1\}\) | eq. 50 | ⚠ thin sketch — verify eq. 50 in the PDF |
| Speed | total commute time of the RCC; periodic \(\Rightarrow\) minimal commute \(\Rightarrow\) maximal speed \(c\) | eqs. 51–57 | tied to §1 problem 3 |
| Distance | commute time as the squared distance, \(T_{ab}=\lVert a-b\rVert^2\) — the metric is \(\sqrt{T_{ab}}\); reversible case \(C(a,b)=2m\,R_{\mathrm{eff}}\) — a resistance / Dirichlet‑energy quantity | p. 24 | geometry is rigorous outside Hoffman (Doyle–Steiner) |
| Momentum / energy | \(p=h/d,\ E=hc/d\), with \(d\) the number of asymptotic states (\(h\)=Planck, not the entropy rate) | eq. 44 | de Broglie form; \(d\leftrightarrow\) wavelength asserted, not derived |
Contested · harmonic functions = free‑particle wavefunction
The authors assert (eqs. 36–39) that the harmonic functions of the enhanced chain are "identical in form" to a free‑particle wavefunction. The crux is a real‑vs‑complex gap: the harmonic functions are a real‑valued discrete cis‑sum over asymptotic events, while the wavefunction is a continuum complex plane wave. Open: establish a genuine identity of functional form (same eigenvalue/PDE structure) or demonstrate it is only an analogy. Until resolved, not citable as a result.
Sketch · special relativity, and the missing SR → GR passage
Tracing a period‑\(n\) kernel onto \(m\lt n\) states (keeping the first and \(n\)th) yields a factor \(\beta\sim n/m\) (so \(m\ll n\Rightarrow v\to c\)) — where the paper defines \(\beta:=1/\sqrt{1-v^2/c^2}\), i.e. the quantity physics standardly calls the Lorentz factor \(\gamma\), not \(\beta=v/c\); under that definition the sketch is internally consistent (\(\beta=n/m\ge 1\), and \(m=n\Rightarrow\beta=1\Rightarrow v=0\)). This is explicitly a sketch ("we expect to explore further," p.29); "time dilation" never appears. The corpus reaches only special‑relativistic / conformal / twistor geometry (\(G(2,4)\cong\) Minkowski, rotor group \(SU(2,2)\)); there is no worked passage to general relativity (despite Conjecture 02/08). First move (SR only): upgrade \(\beta\sim n/m\) from a ratio to a derived full Lorentz transformation — boosts compose, the \((1-v^2/c^2)^{-1/2}\) structure appears, Lorentz covariance of the trace dynamics holds. Do not attempt the GR step.
SECTION 3The Eight Conjectures of physics C
The program's one branded, numbered list — the aspirational derivation targets, each gated on the mathematics of §1–§2. Source: Trace Institute, "Research → The Eight Conjectures," items 01/08–08/08 (a pre‑publication source; the trace‑logic anchor is itself a non‑peer‑reviewed preprint). The 2024 preprint contains no such list — cite it only for the individual correspondences, not for the branding.
| # | Target | Conjectured claim |
|---|---|---|
| 01 | Special relativity | Minkowski space emerges as the limiting behavior of Markov chains representing \(n\)-cycles as \(n\to\infty\). |
| 02 | General relativity | Curved spacetime emerges from special classes of non‑cyclic chains. (Entirely unrealized in the papers — only the SR sketch exists.) |
| 03 | Cosmology | Cosmology and cosmic evolution arise from long samples of certain trace chains. |
| 04 | Planck‑scale failure | The breakdown of spacetime at the Planck scale follows from energy growing with the number of states in a trace. |
| 05 | Wavefunction & Born rule | Free‑particle wavefunctions and the Born rule are recovered from the asymptotics of enhanced chains. |
| 06 | Elementary particles | Each Standard‑Model particle is identified with a particular class of Markov chains. |
| 07 | Scattering amplitudes | Amplitudes arise from chain properties, with ABHY associahedra as subpolytopes of the Markov polytope. |
| 08 | Entanglement | Disjoint traces of an ergodic chain create spacelike‑separated observers with hidden interactions, yielding entanglement. |
SECTION 4Outside extrapolations — not the authors' own X
Three appealing problems that are not Hoffman–Prakash's published claims. A researcher must not cite these as the authors' conjectures.
- The lattice identity. Whether the global trace logic is exactly a partial Boolean algebra (Kochen–Specker) or something weaker. The papers say only that it is not an "orthocomplemented modular lattice" (the paper's exact phrase, §3 p.9; its Summary, p.32, garbles this as "orthomodular complemented lattices," and the standard term "orthomodular lattice" never appears in the PDF). Outside extrapolation.
- Meta‑policy recursion. Whether a Markov chain whose states are observer‑windows (policies of policies) follows formally from trace transitivity (Thm 2.4) plus the strategy‑kernel result, or needs new axioms. Appears as pre‑publication "recursive trace logic" (Trace Institute 2026), not a printed theorem.
- The Dirichlet metric. Whether the commute‑time "distance" genuinely yields a Dirichlet‑form / diffusion metric on observers. Rigorous outside Hoffman (Doyle–Steiner; Fukushima–Oshima–Takeda), but the machinery is absent from the corpus — a clean extrapolation to pose, not the authors' stated problem.
One item not promoted above: an entropy‑rate monotonicity conjecture (if \(P\preceq_t Q\) then the entropy rate of \(P\) is at most that of \(Q\)) appears in a direct read of the 2024 preprint but could not be independently re‑confirmed in verification; treat it as needs‑verification before citing.
CONTRIBUTEHow to engage
If you want to work on one of these, the two ◆ student‑ready problems (the Combination Conjecture and the general join) are the best entry points: both extend an already‑proven special case and need no physics. The commute‑time periodicity iff is the strongest "could become a theorem." The physics dictionary and the Eight Conjectures are full research programs — use them as motivation, and do the provable work in §1.
◆ Honest source status
The trace‑logic results (the partial order, local Booleanity, the homomorphism) rest on a non‑peer‑reviewed preprint ("Traces of Consciousness," 2024). "The Eight Conjectures" live on the pre‑publication Trace Institute site. The peer‑reviewed anchors are Objects of Consciousness (2014), Conscious Agent Networks (2018), Fitness Beats Truth (2020), and Fusions of Consciousness (2023). Citation precision matters: the Church–Turing passage is on p.190 of CA‑Networks; the general‑join remark is on printed page "43 of 46"; and eq. 50 (spin) should be checked directly in the PDF.
To cite this page: "Open Conjectures of the Conscious‑Agent Program," scottn66.github.io/conscious-agents-conjectures.html (2026). To discuss or contribute: open an issue or discussion on the project repository. A fuller working brief — every statement with its citation, status, and a tractable sub‑question — lives in the repo as HMM-hoffman-open-problems.md.
SOURCESReferences
Adversarially verified where possible (3‑vote, ≥2/3 to refute). The trace‑logic theorems rest on a full read of a non‑peer‑reviewed preprint; the physics dictionary and Eight Conjectures are the authors' acknowledged conjectures. Tiers as in the legend.
- Traces of Consciousness — Hoffman, Prakash & Chattopadhyay, 2024, Preprints.org, doi 10.20944/preprints202410.1305.v1 · ResearchGate. Trace order, trace logic, the join (Rmk 4.10), the commute‑time conjecture, the physics dictionary.
- Objects of Consciousness — Hoffman & Prakash, 2014, Frontiers in Psychology 5:577 · full text. The Combination Conjecture (Conj. 3); join Theorems 1–2; the Conscious‑Agent Thesis.
- Conscious Agent Networks — Fields, Hoffman, Prakash & Singh, 2018, Cognitive Systems Research 47:186–213 · eScholarship PDF. Church–Turing falsifiability (p.190); kernel models of memory/planning.
- Fusions of Consciousness — Hoffman, Prakash & Prentner, 2023, Entropy 25(1):129 · MDPI. PMC. Markov polytope; the Markov→decorated‑permutation map (Def. 2); the agent–particle correspondence.
- Fitness Beats Truth — Prakash, Stephens, Hoffman, Singh & Fields, 2020, Acta Biotheoretica · Springer. The evolutionary motivation.
- Decorated permutations / positive Grassmannian — Williams, 2021 (ICM) · arXiv:2110.10856. The proven positroid‑cell bijection (the math anchor for problem 5).
- The Amplituhedron — Arkani‑Hamed & Trnka, 2013 · arXiv:1312.2007.
- Commute‑time geometry of Markov chains — Doyle & Steiner, 2011 · arXiv:1107.2612. commute time as squared distance — the metric is \(\sqrt{T_{ab}}\); reversible case \(C(a,b)=2m\,R_{\mathrm{eff}}\) (problems 3, §2, §4).
- The electrical resistance of a graph — Chandra, Raghavan, Ruzzo, Smolensky & Tiwari, 1996, Comput. Complexity 6 · doi 10.1007/BF01270385. commute time = \(2m\,R_{\mathrm{eff}}\).
- Dirichlet Forms & Symmetric Markov Processes — Fukushima, Oshima & Takeda, 2011, de Gruyter. The Dirichlet‑form machinery absent from the corpus (§4).
- Monadology — G. W. Leibniz, 1714 (Latta trans., 1898) · Wikisource. Windowless monads (§7); the "mill" (§17); pre‑established harmony (§§78–81).
- Information, Physics, Quantum — J. A. Wheeler, 1989/1990, in Zurek (ed.), Complexity, Entropy, and the Physics of Information (Addison‑Wesley), pp. 309–336 · archive PDF. "It from bit" (p. 311); the participatory loop.
- Trace Institute — Research · traceinstitute.org. "The Eight Conjectures" (01/08–08/08); the 2026 Recursive Trace Logic pre‑publication materials.
- Skeptical voices — P. A. Murphy, "Lost in Maths" · QRI, "Reflections on Fusions".