Paper deep dive
A Human-AI Theorem Connecting Spontaneous and Field-Induced Mechanisms of Collective Behavior in One Dimension
Weiguo Yin
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
Last extracted: 9/2/2026, 4:41:16 AM
Summary
This paper presents a theorem derived through human-AI collaboration that establishes an exact, temperature-independent Hamiltonian-level mapping between a zero-field O(n)-vector open chain with competing nearest- and next-nearest-neighbor interactions (J1-J2 model) and a simpler O(n) open chain with nearest-neighbor interactions and an axial single-spin potential (J-h field model). The theorem applies to arbitrary inhomogeneous interaction functions, any integer n≥1, and any system size L≥1, effectively connecting spontaneous collective behavior with field-induced mechanisms. It also provides a closed-form exact solution for the J1-J2 Potts open chain.
Entities (10)
Relation Signals (6)
Theorem → discoveredby → Human-AI Collaboration
confidence 95% · The emergence of the theorems from sustained human–AI collaboration suggests that involving AI throughout a systematic research program may incubate autonomous scientific breakthroughs.
J1-J2 model → mapsto → J-h field model
confidence 95% · The homogeneous linear specialization maps the foundational frustrated J1-J2 model onto the canonical J-h field model
O(n) vector model → generalizes → Ising model
confidence 90% · n=1,2,3 being the Ising, XY, and Heisenberg classical spin models, respectively
O(n) vector model → generalizes → XY model
confidence 90% · n=1,2,3 being the Ising, XY, and Heisenberg classical spin models, respectively
O(n) vector model → generalizes → Heisenberg model
confidence 90% · n=1,2,3 being the Ising, XY, and Heisenberg classical spin models, respectively
Theorem → provenby → Householder reflection
confidence 90% · The proof is geometric... recursive Householder moving frame... construction uses reflections in the full orthogonal group O(n)
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Can an artificial intelligence (AI) generate a scientific hypothesis outside a human collaborator's active hypothesis space (AHS), and can human-AI research be organized to make such breakthroughs more likely? We document such a case while proving a theorem that connects two basic organizing mechanisms of statistical physics: collective behavior arising in zero field from competing interactions and that induced or controlled by an external field. A zero-field $O(n)$-vector open chain with arbitrary inhomogeneous nearest- and next-nearest-neighbor interaction functions $U_i(S_i\cdot{S}_{i+1})$ and $V_i(S_i\cdot{S}_{i+2})$ is microscopically, via a temperature-independent mapping at the Hamiltonian level, equivalent to a simpler $O(n)$ open chain with nearest-neighbor interaction $V_i( \sigma_i\cdot \sigma_{i+1})$ and axial single-spin potential $U_i(\sigma_i^z)$ for every integer $n\ge1$ and every system size $L\ge1$. The homogeneous linear specialization maps the foundational frustrated $J_1$-$J_2$ model onto the canonical $J$-$h$ field model---with $n=1,2,3$ being the Ising, XY, and Heisenberg classical spin models, respectively. An analogous theorem holds when the continuous $O(n)$ spins are replaced by the $q$-state Potts spins with the standard Potts interaction, implying a closed-form exact solution of the $J_1$-$J_2$ Potts open chain for every $q\ge2$ and every $L\ge1$. The emergence of the theorems from sustained human-AI collaboration suggests that involving AI throughout a systematic research program may incubate autonomous scientific breakthroughs.
Tags
Links
- Source: https://arxiv.org/abs/2609.00322v1
- Canonical: https://arxiv.org/abs/2609.00322v1
Trouble viewing inline? Open PDF directly →
Full Text
63,423 characters extracted from source content.
Expand or collapse full text
A Human–AI Theorem Connecting Spontaneous and Field-Induced Mechanisms of Collective Behavior in One Dimension Weiguo Yin Email: wyin@bnl.gov Affiliation: Condensed Matter Physics and Materials Science Division, Brookhaven National Laboratory, Upton, New York 11973, USA August 31, 2026 Abstract Can an artificial intelligence (AI) generate a scientific hypothesis outside a human collaborator’s active hypothesis space (AHS), and can human–AI research be organized to make such breakthroughs more likely? We document such a case while proving a theorem that connects two basic organizing mechanisms of statistical physics: collective behavior arising in zero field from competing interactions and that induced or controlled by an external field. A zero-field O(n)O(n)-vector open chain with arbitrary inhomogeneous nearest- and next-nearest-neighbor interaction functions Ui(i⋅i+1)U_i( S_i· S_i+1) and Vi(i⋅i+2)V_i( S_i· S_i+2) is microscopically, via a temperature-independent mapping at the Hamiltonian level, equivalent to a simpler O(n)O(n) open chain with nearest-neighbor interaction Vi(i⋅i+1)V_i( σ_i· σ_i+1) and axial single-spin potential Ui(σiz)U_i( _i^z) for every integer n≥1n≥ 1 and every system size L≥1L≥ 1. The homogeneous linear specialization maps the foundational frustrated J1J_1-J2J_2 model onto the canonical J-h field model—with n=1,2,3n=1,2,3 being the Ising, XY, and Heisenberg classical spin models, respectively. An analogous theorem holds when the continuous O(n)O(n) spins are replaced by the q-state Potts spins with the standard Potts interaction, implying a closed-form exact solution of the J1J_1-J2J_2 Potts open chain for every q≥2q≥ 2 and every L≥1L≥ 1. The emergence of the theorems from sustained human–AI collaboration suggests that involving AI throughout a systematic research program may incubate autonomous scientific breakthroughs. I Introduction Two questions about scientific breakthroughs driven by human–AI collaboration motivate this work. The first concerns scientific understanding. As general-purpose reasoning AI moves from answering well-specified questions toward generating unexpected hypotheses and solutions [1, 2, 3, 4], the practical bottleneck may invert: instead of asking only whether an AI understands a human scientist’s problem, the human scientist must also recognize and understand what the AI has produced [5, 6]. An AI response can mix hallucination, algebraic error, rediscovery, irrelevant information, useful connection, and genuinely new insight in the same channel. The scientist must distinguish among them and determine which implications survive derivation, falsification, comparison with prior work, and physical interpretation. Once the realistic results are identified and validated, can the human scientist fully understand their implications, especially those beyond the active hypothesis space (AHS), effectively leading to next breakthroughs? The second question is methodological: how can human–AI research be organized systematically to increase the chance of scientific breakthroughs, such as solving longstanding puzzles or revealing unknown unknowns beyond the human’s AHS? Recently, we developed an AI-aided research program, in which four projects [1, 7, 8, 9] were deliberately organized so that the AI’s contribution categories spanning 1d–7d in terms of a nine-dan AI contribution framework [8] could be examined as relatively clean cases. Once the designated reasoning was finished, AI did not remain embedded through every subsequent stage of the scientific project. That design was useful for attribution, but it effectively froze the AI role after the assigned contribution. Built upon these experiments, the present work was designed to take an opposite approach: AI remains involved from the beginning to the end of a research trajectory—through derivation, comparison with literature, generalization, criticism, falsification, connection, and manuscript development. This longitudinal setting allows a different possibility to be tested: can the AI’s role itself escalate to 8d/9d as the shared scientific landscape accumulates? Here, we show that addressing the two questions (Sec. IV.1) has driven remarkably new AI-led insight, far beyond the human’s AHS, into longstanding problems in statistical mechanics and magnetic materials. The two kinds of physics models at the center of this history are themselves foundational. The first kind is the so-called n-vector spin model with nearest-neighbor (N) exchange interaction J [10] (n=1,2,3n=1,2,3 are the classical Ising, XY, and Heisenberg models, respectively [11, 12, 13]), which is the canonical setting for studying collective behavior arising in zero field and magnetic response to an external magnetic field h, where exchange-driven collective correlations compete with field-driven polarization and determine magnetization and field-dependent thermodynamics. For spins forming a one-dimensional (1D) lattice, it is described by the J-h Hamiltonian ℋJ−h=−J∑i⋅i+1−h∑iiz,H_J-h=-J _i S_i· S_i+1-h _i S_i^z, (1) where i S_i is a unit vector in the n-dimensional spin space at lattice site i. The q-state clock model is a discrete n=2n=2 model with ℤqZ_q symmetry, while the simplest case (q=2q=2) is the Ising model. In addition, the q-state standard Potts model is a simpler generalization of the Ising model (q=2q=2) to permit full permutation qS_q symmetry. The second kind is the so-called J1J_1-J2J_2 spin model, which is a minimal archetype of frustrated magnetism: interactions at two competing ranges—N and next-nearest-neighbor (N)—can favor incompatible local arrangements and generate nontrivial correlations, modulated or helical ground states, dimerization, and degeneracy across discrete and continuous spin settings [14, 15, 16]. For a 1D chain, the J1J_1-J2J_2 model is described by ℋJ1−J2=−J1∑i⋅i+1−J2∑i⋅i+2.H_J_1-J_2=-J_1 _i S_i· S_i+1-J_2 _i S_i· S_i+2. (2) The corresponding transfer-matrix or transfer-integral problems can be formulated exactly [17] and evaluated numerically [18, 19, 20]. Of particular interest for these textbook 1D systems is whether we can achieve exact analytical results that provide classroom-level deep understanding. Table 1 (J−hJ-h and J1−J2J_1-J_2 lines) summarizes a representative status of solving these chain models in closed form, showing that most of them are longstanding unsolved problems. Table 1: Representative status of exact results and J1J_1-J2J_2 ↔ J-h mapping for the spin chains. T-map and H-map stand for thermodynamic and Hamiltonian-level mappings respectively. X means an unsolved problem. I, I, I, and IV stand for the discovery types of this work (see Table 2). Model Ising Potts clock/XY Heisenberg O(n)O(n) (n=1n=1) (n=2n=2) (n=3n=3) J 1925 [21] 1952 [22] 1967 [23] 1964 [24] 1969 [10] J-h 1941 [17] 1994 [25] X X X J1J_1-J2J_2 1969 [26] 2025 [1] X X X T-map 1969 [26] 2025 [1] 1984 [27] 1990 [28] I → I H-map 1969 [26] IV 1984 [27] I/I I → IV In the transfer matrix formulation, the range-two J1J_1-J2J_2 chain appears much more complicated than the range-one J-h chain: For a q-state Potts or clock realization, the order of the transfer matrix of the J1J_1-J2J_2 chain is q2q^2, compared with q for the J-h chain; for continuous O(n)O(n) spins, the transfer problems for the J1J_1-J2J_2 and J-h chains are represented in pair-spin and single-spin configuration spaces, respectively. However, an exact thermodynamic mapping between the J1J_1-J2J_2 and J-h chains has been established for the n≤3n≤ 3 and Potts cases, in which J=J2J=J_2 and h=J1h=J_1, as shown in Table 1 (T-map line). The implication of these results is remarkable: the exact bridge between the two canonical models not only reduces a seemingly much more complicated problem to a simpler one, but also connects two basic organizing mechanisms of statistical physics—collective behavior arising spontaneously [29] from competing interactions and collective behavior induced or controlled by an external field. Moreover, a microscopic mapping at the Hamiltonian level was found for n=1n=1 (Ising) [26] and n=2n=2 (XY or planar) [27] open chains. This is a much stronger mapping because it holds for arbitrary finite L, rather than only in the thermodynamic limit L→∞L→∞. For n=3n=3, Harada and Mikeska found an exact mapping for every L≥1L≥ 1 via matching the transfer integrals of the J1J_1-J2J_2 and J-h open chains; however, they explicitly argued that this duality could not be derived at the Hamiltonian level because rotations in three dimensions do not commute [28]. Can we overcome this 1990 stated obstruction and derive a Hamiltonian-level mapping for n=3n=3 and, more generally, every n≥1n≥ 1? If successful, what implications might this cleanest mapping carry? In addition, a knowledge gap is uncovered in Table 1 (H-mapping line): a Hamiltonian-level mapping has not been published for the standard q-state Potts chain yet, to the best of our knowledge. Can we close this knowledge gap? The fact that the J-h Potts open chain has been solved in closed form [25, 30] makes the question particularly interesting: If found, such a microscopic mapping would yield a closed-form exact solution to the J1J_1-J2J_2 Potts open chain for every q≥2q≥ 2 and every L≥1L≥ 1, a clear leap from the 2025 L→∞L→∞ result [1]. Figure 1: Illustration of a human–AI research developed from a single point (big blue dot [1]) to a four-axis AHS: The green axis stands for model choice, red for inverse system size 1/L1/L, and orange for homogeneous arbitrary interactions U(x),V(x)U(x),V(x), and black for inhomogeneous Ui(x),Vi(x)U_i(x),V_i(x). The main driver of different subspaces is indicated as Human or AI. Human–AI means that they drove the n−Ln-L subspace together. The initial AHS is represented by the green arrow and marked as Type I. The 1969 and 1984 red dashed lines stand for the microscopic proofs for the Ising and XY models, respectively [26, 27]. X denotes the 1990 obstruction [28]. In this article, we present a human–AI collaboration in which the human-designed initial AHS for the AI’s progressive development consisted of the thermodynamic mapping problems between the J1J_1-J2J_2 and J-h chains, with challenge progression through the Potts, clock, n=2n=2, n=3n=3, and O(n)O(n) models [31]. The human–AI trajectory then continued instead of terminating after a successful reasoning. It demonstrates a helix of the AI autonomous hypothesis generation and the human understanding of the AI results, as documented in Sec. IV.2 and Sec. IV.3. The final result is a theorem far beyond the initial AHS: the exact Hamiltonian-level mapping between the range-two and range-one open chains with arbitrary inhomogeneous interaction functions—not merely the homogeneous J1J_1-J2J_2 vs J-h interactions—for every n≥1n≥ 1 (Sec. I) or every q≥2q≥ 2 (Sec. I), and every L≥1L≥ 1, as illustrated in Fig. 1. I Theorems and Microscopic proof Consider an open chain of L classical unit vectors i∈Sn−1,n≥1, S_i∈ S^n-1, n≥ 1, (3) with the range-two Hamiltonian ℋrange 2OBC(L)=∑i=1L−1Ui(i⋅i+1)+∑i=1L−2Vi(i⋅i+2),H OBC_ range\,2(L)= _i=1^L-1U_i( S_i· S_i+1)+ _i=1^L-2V_i( S_i· S_i+2), (4) where Ui(x)U_i(x) and Vi(x)V_i(x) are arbitrary real functions on x∈[−1,1]x∈[-1,1] and may vary arbitrarily with the site index i, allowing the interactions to be independently inhomogeneous along the chain. OBC means the open boundary condition. Empty sums are understood to vanish. The corresponding range-one open chain with another set of L−1L-1 classical unit vectors i∈Sn−1 σ_i∈ S^n-1 is ℋrange 1OBC(L−1)=∑i=1L−1Ui(^⋅i)+∑i=1L−2Vi(i⋅i+1).H OBC_ range\,1(L-1)= _i=1^L-1U_i( z· σ_i)+ _i=1^L-2V_i( σ_i· σ_i+1). (5) The N interaction UiU_i of the range-two model becomes an axial single-spin potential, whereas the N interaction ViV_i becomes the N interaction. Theorem 1. For every positive integer n, every chain size L, and arbitrary real functions Ui(x),Vi(x)U_i(x),V_i(x) on x∈[−1,1]x∈[-1,1], Eqs. (4) and (5) can be mapped exactly onto each other under the change of variables, ℋrange 2OBC(L)=ℋrange 1OBC(L−1).H_ range\,2 OBC(L)=H_ range\,1 OBC(L-1). (6) Equivalently, the transformation is a bijection after retaining the decoupled global-orientation variable 1 S_1. The Hamiltonian mapping is temperature independent, and at every finite temperature for which the corresponding Boltzmann integrals exist, the partition functions obey the exact identity Zrange 2OBC(L)=Zrange 2OBC(1)⋅Zrange 1OBC(L−1),Z_range\,2 OBC(L)=Z_range\,2 OBC(1)· Z_range\,1 OBC(L-1), (7) with Zrange 1OBC(0)=1Z_range\,1 OBC(0)=1. The proof is geometric and particularly transparent. We first demonstrate it using the familiar O(3)O(3) geometry. I.1 n=3n=3 Define the bond angle ϑi _i by cosϑi=i⋅i+1, _i= S_i· S_i+1, (8) and let φi _i be the signed dihedral angle between the planes (i,i+1)( S_i, S_i+1) and (i+1,i+2)( S_i+1, S_i+2). Then i⋅i+2=cosϑicosϑi+1+sinϑisinϑi+1cosφi. S_i· S_i+2= _i _i+1+ _i _i+1 _i. (9) Now introduce azimuth angles αi _i recursively by αi+1−αi=φi, _i+1- _i= _i, (10) with arbitrary α1 _1 (for example, α1=0 _1=0) and construct an auxiliary Heisenberg spin i=(sinϑicosαi,sinϑisinαi,cosϑi). σ_i=( _i _i, _i _i, _i). (11) Then σiz=i⋅i+1,i⋅i+1=i⋅i+2. _i^z= S_i· S_i+1, σ_i· σ_i+1= S_i· S_i+2. (12) Substitution into Eq. (4) gives Eq. (5). The dihedral angle φi _i is undefined for degenerate collinear configurations, when sinϑisinϑi+1=0 _i _i+1=0. This coordinate singularity is absent from the general Householder construction below. I.2 n≥1n≥ 1: a recursive Householder moving frame Householder reflections are standard symmetric orthogonal transformations in numerical linear algebra [32, 33], while moving-frame constructions have a long geometric history [34]. Hyperplane reflections also appear as symmetry operations in Monte Carlo algorithms for classical O(n)O(n) spins [35, 36]. We use the phrase recursive Householder moving frame descriptively for the construction below: the frame is propagated site by site through a product of local Householder reflections. To our knowledge, the recursive use of Householder reflections as an exact change of spin variables that reduces a 1D range-two Hamiltonian to a range-one Hamiltonian has not previously been formulated. First, fix once and for all a reference unit vector ^∈Sn−1 e∈ S^n-1. Then, for every ∈Sn−1 x∈ S^n-1 define a Householder reflector that sends e to x, ()=−(^−)(^−)1−^⋅,≠^,,=^. H( x)= cases I- ( e- x)( e- x) T1- e· x,& x≠ e,\\[8.53581pt] I,& x= e. cases (13) Since ‖^−‖2=2(1−^⋅)\| e- x\|^2=2(1- e· x), Eq. (13) is the standard Householder form −2/() I-2 v v T/( v T v) with =^− v= e- x. It obeys ()=(),()2=,()^=, H( x) T= H( x), H( x)^2= I, H( x) e= x, (14) so ()∈O(n) H( x)∈ O(n). The special definition at = x= e is sufficient; no globally smooth choice of frame is required for the change of variables or the partition-function identity. Choose 1=(1),1^=1, G_1= H( S_1), G_1 e= S_1, (15) and define recursively, for i=1,…,L−1i=1,…,L-1, i=ii+1,i+1=i(i). σ_i= G_i T S_i+1, G_i+1= G_i H( σ_i). (16) Explicitly, i=(1)(1)H(2)⋯(i−1). G_i= H( S_1) H( σ_1)H( σ_2)·s H( σ_i-1). Because ii=ii= G_i T G_i= G_i G_i T= I, i G_i is orthogonal. Thus, ii=i+1i+1=1 σ_i T σ_i= S_i+1 T S_i+1=1; each i σ_i is again a unit n-vector. Moreover, i+1^=i(i)^=ii=i+1, G_i+1 e= G_i H( σ_i) e= G_i σ_i= S_i+1, (17) so induction gives i=i^(i=1,…,L). S_i= G_i e (i=1,…,L). (18) The transformation is invertible for an open chain: given 1 S_1 and ii=1L−1\ σ_i\_i=1^L-1, Eqs. (16) and (18) reconstruct all original spins uniquely. The two identities responsible for the mapping are now immediate. First, i⋅i+1 S_i· S_i+1 =^(i)^=^⋅i. = e T H( σ_i) e= e· σ_i. (19) Second, using i+2=i(i)(i+1) S_i+2= G_i H( σ_i) H( σ_i+1) e, i⋅i+2 S_i· S_i+2 =^(i)(i+1) = e T H( σ_i) H( σ_i+1) e =[(i)^][(i+1)^] = [ H( σ_i) e ] T [ H( σ_i+1) e ] =i⋅i+1. = σ_i· σ_i+1. (20) Importantly, Eq. (20) uses the symmetry = H T= H, not commutativity between successive transformations. Hence the Householder construction does not require three-dimensional rotations, or their higher-dimensional counterparts, to commute. Renaming ^= e= z and substituting Eqs. (19) and (20) into Hamiltonian (4) derives Hamiltonian (5)—the proof acts on the geometric arguments of the interactions, not on the interactions themselves—giving the Hamiltonian identity, Eq. (6). Thus a zero-field range-two O(n)O(n) chain of L spins is mapped exactly onto a simpler range-one O(n)O(n) chain of L−1L-1 spins with an axial single-spin potential. The same change of variables also preserves the open-chain measure. Let dμnd _n denote the normalized O(n)O(n)-invariant measure on Sn−1S^n-1. For fixed preceding variables, i+1=ii S_i+1= G_i σ_i is only an orthogonal transformation, and therefore dμn(i+1)=dμn(i).d _n( S_i+1)=d _n( σ_i). (21) The partition function Zrange 2OBC(L) Z_ range\,2 OBC(L) = = ∫∏i=1Ldμn(i)e−βℋrange 2OBC(L) _i=1^Ld _n( S_i)e^- _ range\,2 OBC(L) (22) = = ∫dμn(1)[∫∏i=1L−1dμn(i)e−βℋrange 1OBC(L−1)] d _n( S_1) [ _i=1^L-1d _n( σ_i)e^- _ range\,1 OBC(L-1) ] = = Zrange 2OBC(1)⋅Zrange 1OBC(L−1), Z_range\,2 OBC(1)· Z_ range\,1 OBC(L-1), where β=1/(kBT)β=1/(k_BT) with kBk_B being the Boltzmann constant and T the absolute temperature. Since ℋrange 1OBC(L−1)H_ range\,1 OBC(L-1) is independent of 1 S_1, the integral with respect to μn(1) _n( S_1) yields a factor that is nothing but Zrange 2OBC(1)Z_range\,2 OBC(1), which is unity if normalized dμn(1)d _n( S_1) is used. If instead the unnormalized surface measure dΩd is used, Zrange 2OBC(1)Z_range\,2 OBC(1) is the global-orientation factor Ωn−1=2πn/2Γ(n/2) _n-1= 2π^n/2 (n/2). Note Ωn−1=2,2π,4π _n-1=2,2π,4π for n=1,2,3n=1,2,3, respectively. We then arrive at Eq. (7). ■ The endpoints and historical special cases clarify what is new. For n=1n=1, S0=±1S^0=\± 1\ and the Householder matrices reduce to the scalars (σ)=σ H(σ)=σ. Equation (16) then gives σi=SiSi+1 _i=S_iS_i+1, and Eq. (20) becomes Dobson’s Ising bond identity [26]. The apparently special algebraic property Si2=1S_i^2=1 is therefore the O(1)O(1) endpoint of the general geometric construction. For n=2n=2, Harada introduced relative planar angles and showed that the competing-interaction chain is transformed into a nearest-neighbor planar chain in a magnetic field [27]. For n=3n=3, Harada and Mikeska established the same duality at the transfer-integral level, but explicitly argued that, because rotations in three dimensions do not commute, the duality could not be derived at the Hamiltonian level [28]. Equations (16)–(20) remove that obstruction: the construction uses reflections in the full orthogonal group O(n)O(n) and relies on their involutive and symmetric properties, not on commuting proper rotations. This distinction also sharpens the novelty of the present result. Householder reflections themselves are classical [32, 33], and reflection operations have long been used in classical-spin algorithms [35, 36]. Likewise, the term “moving frame” belongs to a much older geometric tradition [34]. What appears to be new is their synthesis in the recursion (16): an ordered sequence of local Householder reflections transports the spin frame along the chain and converts N geometry into N geometry. We have not found a prior use of this recursive Householder moving-frame transformation to derive (i) an exact Hamiltonian-level frustration–field/range-reduction mapping for arbitrary n≥1n≥ 1, (i) the arbitrary-function extension in Eqs. (4) and (5), or (i) the finite-size open-chain identity (7). I.3 Linear corollary: J1J_1-J2↔J_2 J-h for n-vector spins Choosing the uniform linear interaction functions Ui(x)=−J1x,Vi(x)=−J2xU_i(x)=-J_1x, V_i(x)=-J_2x (23) gives the Hamiltonian-level mapping between the J1J_1-J2J_2 and J−hJ-h chains with J=J2J=J_2 and h=J1h=J_1. The configuration-level identities (19) and (20) immediately give the observable correspondences. For the linear case, ⟨i⋅i+1⟩J1−J2 S_i· S_i+1 _J_1-J_2 = = ⟨^⋅i⟩J−h, z· σ_i _J-h, ⟨i⋅i+2⟩J1−J2 S_i· S_i+2 _J_1-J_2 = = ⟨i⋅i+1⟩J−h. σ_i· σ_i+1 _J-h. Thus a zero-field N bond correlation becomes a field-induced magnetization, whereas the N correlation becomes an N exchange correlation. I.4 Physical nonlinear example: single-ion anisotropy becomes biquadratic exchange The arbitrary-U,VU,V theorem is more than a formal generalization. The human collaborator recognized the following application. Consider the canonical vector model in a longitudinal field with uniaxial single-ion anisotropy, ℋfield=−J∑i=1L−2i⋅i+1−h∑i=1L−1σiz+D∑i=1L−1(σiz)2.H_ field=-J _i=1^L-2 σ_i· σ_i+1-h _i=1^L-1 _i^z+D _i=1^L-1( _i^z)^2. (24) Set Ui(x)=−hx+Dx2,Vi(x)=−Jx.U_i(x)=-hx+Dx^2, V_i(x)=-Jx. (25) The theorem gives the exactly equivalent zero-field frustrated model ℋspontaneous= _ spontaneous= − - J∑i=1L−2i⋅i+2−h∑i=1L−1i⋅i+1 J _i=1^L-2 S_i\!·\! S_i+2-h _i=1^L-1 S_i\!·\! S_i+1 (26) + + D∑i=1L−1(i⋅i+1)2. D _i=1^L-1( S_i\!·\! S_i+1)^2. Thus, within this exact open-chain mapping, single-ion anisotropy ⟷ biquadratic exchange. Biquadratic exchange is a standard ingredient of effective multiorbital spin models, including models developed for iron-based superconductors [37]. This specific nonlinear correspondence also makes the discovery provenance tangible: neither the single-ion-anisotropy/biquadratic-exchange relation nor the arbitrary-U,VU,V hypothesis was in the human researcher’s AHS when the AI proposed the extension. It is therefore a concrete many-body consequence that lay beyond the human’s original line of sight. The theorem converts the field-controlled anisotropic problem into a spontaneous zero-field problem containing N, N, and biquadratic exchange without requiring either model to be solved in closed form. The configuration-level identities (19) and (20) give the additional exact dictionary ⟨(σiz)2⟩field=⟨(i⋅i+1)2⟩spontaneous. ( _i^z)^2 _ field= ( S_i· S_i+1)^2 _ spontaneous. For n=1n=1 (the Ising model) the squared bond is identically unity, so the nonlinear content begins at n>1n>1. I The q-state Potts chain The target open-chain Potts Hamiltonians generalized with arbitrary interaction functions Ui,ViU_i,V_i are ℋrange 2Potts _ range\,2 Potts = = ∑i=1L−1Ui(δσi,σi+1)+∑i=1L−2Vi(δσi,σi+2), _i=1^L-1U_i( _ _i, _i+1)+ _i=1^L-2V_i( _ _i, _i+2), (27) ℋrange 1Potts _ range\,1 Potts = = ∑i=1L−1Ui(δτi,P)+∑i=1L−2Vi(δτi,τi+1), _i=1^L-1U_i( _ _i,P)+ _i=1^L-2V_i( _ _i, _i+1), (28) where σi _i and τi _i denote one of q Potts states at site i, and P a specific Potts state. δσi,σi+1 _ _i, _i+1 is the Kronecker delta (1 if σi=σi+1 _i= _i+1 and 0 otherwise). The linear specialization, Eq. (23), gives the standard q-state Potts model. Because d∈0,1d∈\0,1\, every function satisfies Ui(d)=Ui(0)+[Ui(1)−Ui(0)]dU_i(d)=U_i(0)+[U_i(1)-U_i(0)]d, i.e., the arbitrary-function Potts extension is algebraically just an inhomogeneous standard Potts coupling plus a constant (see Supplemental Material Sec. VI.E [31]). Theorem 2. For every integer q≥2q≥ 2, every chain size L, and arbitrary real functions Ui(d),Vi(d)U_i(d),V_i(d) on d∈0,1d∈\0,1\, Eqs. (27) and (28) can be mapped exactly onto each other under the change of variables. Equivalently, the transformation is a bijection after retaining the decoupled variable σ1σ_1. The Hamiltonian mapping is temperature independent, and at every finite temperature the partition functions obey the exact identity Zrange 2Potts(L)=Zrange 2Potts(1)⋅Zrange 1Potts(L−1),Z_ range\,2 Potts(L)=Z_ range\,2 Potts(1)· Z_ range\,1 Potts(L-1), (29) where Zrange 2Potts(1)=qZ_ range\,2 Potts(1)=q and Zrange 1Potts(0)=1Z_range\,1 Potts(0)=1. The standard Potts interaction depends solely on equality of two states and therefore remains fully permutation invariant. However, the key to the mapping is temporarily putting the full qS_q symmetry aside and labeling the states by σi∈0,1,…,q−1 _i∈\0,1,…,q-1\ and P=0P=0 according to ℤqZ_q with all additions and subtractions understood modulo q. Introduce the N bond variables bi=σi−σi+1(modq),i=1,…,L−1,b_i= _i- _i+1 q, i=1,…,L-1, (30) and then perform the staggered relabeling τi=(−1)ibi(modq). _i=(-1)^ib_i q. (31) The N Potts invariant becomes δσi,σi+1=δbi,0=δτi,0. _ _i, _i+1= _b_i,0= _ _i,0. (32) For the N invariant, δσi,σi+2=δσi−σi+2,0=δτi−τi+1,0=δτi,τi+1. _ _i, _i+2= _ _i- _i+2,0= _ _i- _i+1,0= _ _i, _i+1. (33) Equations (32) and (33) are purely kinematic identities and give the exact microscopic mapping between Hamiltonians (27) and (28). Then, the partition function Zrange 2Potts(L) Z_ range\,2 Potts(L) = = ∑σ1,…,σL=0q−1e−βℋrange 2Potts(L) _ _1,…, _L=0^q-1e^- _ range\,2 Potts(L) (34) = = ∑σ1=0q−1[∑τ1,τ2,…,τL−1=0q−1e−βℋrange 1Potts(L−1)] _ _1=0^q-1 [ _ _1, _2,…, _L-1=0^q-1e^- _ range\,1 Potts(L-1) ] = = qZrange 1Potts(L−1). q\;Z_ range\,1 Potts(L-1). Since Hrange 1Potts(L−1)H_ range\,1 Potts(L-1) is independent of σ1 _1, the summation with respect to σ1 _1 gives the factor q=Zrange 2Potts(1)q=Z_ range\,2 Potts(1). ■ The simpler J-h Potts open chain has been solved in closed form [25, 30]. Therefore, Theorem 2 yields a closed-form exact solution to the 1D J1J_1-J2J_2 Potts open chain for every q≥2q≥ 2 and every L≥1L≥ 1. The exact finite-size partition function is written in a compact form ZJ1−J2Potts(L)=q⟨b|2×2L−2|b⟩,L≥2,Z_J_1-J_2 Potts(L)=q b|T_2× 2^L-2|b , L≥ 2, (35) where ⟨b|=|b⟩=(y,q−1) b|=|b T= (y, q-1 ), and 2×2=(xy2q−1yq−1yx+q−2).T_2× 2= pmatrixxy^2& q-1\,y\\[2.84526pt] q-1\,y&x+q-2 pmatrix. (36) where x=eβJ2x=e^β J_2 and y=eβJ1/2y=e^β J_1/2. 2×2T_2× 2 is the reduced transfer matrix in the 2×22× 2 MSS [1, 25], whose diagonalization is elementary. Other implications are remarked in the next section IV.3. IV Discussion Table 2: Four discovery types used in this work. Reflection on Types I-I motivated this work. Type IV was added retrospectively during the human–AI analysis. The classification is contextual rather than absolute: the same result can change type as the state of scientific awareness changes. Type Operational meaning I Solve a long-standing known unknown (KU). I Reveal a remote unknown unknown (U). I Systematically develop a landscape opened by I, I, or IV, leading to new I/I/IV breakthroughs. IV Recognize an unknown known (UK). IV.1 Organizing human-AI research We first describe the conceptual framework that motivated this work and was subsequently refined by its outcomes. The recognition problem led to a context-dependent classification that now comprises four broad types: Types I–I motivated the present study, while Type IV emerged in hindsight and then gave rise to the fourth axis of the AHS, as the black axis illustrated in Fig. 1. Type I solves a longstanding problem—a known unknown (KU). Type I reveals an unknown unknown (U) that is far beyond the existing AHS. Type I systematically develops the landscape opened by Type-I, Type-I, or Type-IV discoveries through exploration, development, generalization, connection, and application, thereby creating conditions for subsequent Type-I, Type-I, or Type-IV breakthroughs. Type IV recognizes an unknown known (UK): a latent connection among already available ingredients. A repeated pattern of I → I/I/IV → new I → new I/I/IV →⋯→·s forms a helix of discovery. Table 2 summarizes this terminology. The classification focuses on well-documented timestamped recognition problems rather than less objective measures of difficulty or significance. Because a Type-I KU is longstanding, it can contain something beyond the AHS; its resolution can reveal Type-I breakthroughs. Conversely, recognizing a Type-I U can drive a Type-I breakthrough and sometimes even make a Type-I puzzle appear easy in the expanded AHS. In comparison, Type-I-stage KUs can be easier to identify and quicker to solve, resulting in two important ingredients: data and the implications of the data. Deep insight into those results provides an operational pathway to new AHS and next Type-I/I/IV breakthroughs. Figure 2: Schematics of (a) a 9d one-shot autonomous Type-I route vs (b) the human-AI co-development approach, where the research state, active context, and the agents’ roles are progressively updated as the research evolves from an initial Type-I AHS towards Type-I/I/IV breakthroughs. Both approaches start with intelligence equipped with known knowns (KKs). We then translated the second, methodological question as follows: can human–AI research be organized systematically so that AI participation in Type-I development increases the chance of reaching next Type-I/I/IV breakthroughs—by producing results whose deeper implications can subsequently be recognized by the human or the AI? To address this question about the AI’s role, we proposed a nine-dan AI-contribution framework [8] to replace a vague statement such as “AI assisted this research.” The framework is divided into two groups: (1) 1d through 5d, in which the human can complete the study without the AI and the AI is employed to dramatically accelerate the project, and (2) 6d through 9d, in which the human cannot complete the study without the AI and the AI acts as a scientific discoverer and an inspirational research partner. In the organization used here, levels one dan to five dan (1d–5d) form a Type-I track and levels 6d–9d a Type-I/I track, each with increasing autonomy and scientific impact. The newly added Type IV is primarily a recognition-oriented discovery and, when contributed by AI, can be assigned 2d or 6d depending on the AI’s role. In this work, the Potts Type-IV recognition is therefore classified as 6d, whereas the Type-IV extension to arbitrary inhomogeneous interactions arose from the human’s recognition of the implications of the AI-generated microscopic proof. If Type I is the cradle of Types I/I/IV, sustained AI participation in Type I—with progressive 1d–5d contributions—may likewise be a cradle for 6d–9d scientific performance, including autonomous hypothesis generation outside the AHS. Now for practical purposes we find it is necessary to distinguish an active problem space (APS) from an AHS. Problem statements define an APS, not necessarily an AHS. An AHS begins when concrete candidate explanations, representations, mechanisms, or solution routes are being actively entertained. This clarifies why Type I matters in this human-AI co-development approach. Type I does not merely keep the AI busy. It creates an actual AHS by progressively developing concrete structure. Operationally, we do not reveal the whole human APS to the AI, and only select those problems that the human can use to design the initial AHS. If the AI solves or makes a significant step in solving unreleased problems in the human APS—for which the human could not form an AHS—during the AHS course, that is an operationally identifiable event of AI going outside the AHS. Further, if the problems are outside the human APS, that could be a strong indicator of AI-driven or AI-led breakthroughs. Hence, the longitudinal human–AI co-development should be distinguished from one-shot autonomous problem solving: here the research state, active context, the agents’ roles are progressively updated, as illustrated in Fig. 2. At the center of this Y-shaped workflow is a human-designed initial Type-I AHS for AI to develop and maintain task-relevant scientific context across a progression of real research problems. In the 1d–5d Type-I track used here, the selected KUs are problems that the human could in principle complete without the AI; therefore, the AI can be well mentored. In this setting, the AI is expected not only to accelerate the project but also to generate hypotheses outside the AHS because of its broad knowledge base and its ability to connect representations and synthesize solution methods. The human remains engaged throughout to recognize the implications of the AI results and update the AHS accordingly. Like in a Socratic dialogue, the solution, or even the problem, emerges from the human-AI conversation. This makes a difference in practice: For a Type-I APS-centered approach, one may prefer the highest-performance reasoning option. By contrast, in a Type-I AHS-centered approach, a nonmaximal reasoning mode with faster response, better suited to sustaining rapid human–AI iteration, may be preferable. For example, among the five reasoning options available in our ChatGPT 5.6 interface at the time of this study—Instant, Medium, High, Extra High, and Pro—we used High for discovery and Extra High for critical manuscript review. IV.2 Human-AI research trajectory The physics progression and the AI-contribution progression are related but not identical. The discovery type describes what occurred relative to the scientific landscape; the dan level in Ref. [8] describes what role AI played in that particular research step. Fig. 1 illustrates the physics progression as an expansion from a single point [1] to a four-axis AHS: model choice from Potts to O(n)O(n), system size L, homogeneous arbitrary interactions, and inhomogeneity. Fig. 3 records the human–AI co-development trajectory and the main AI-contribution episodes. The two figures share the same AHS color scheme; thus, the physics progression and the AI-contribution progression can be compared. Figure 3: Schematics of human–AI co-development trajectory through four AHS stages represented by green, orange, red, and black rounded rectangles (RRs). The color scheme is the same as Fig. 1: Green stands for the human-designed initial Type-I AHS with the starting point being the q2→q^2→ q transfer-matrix reduction for the Potts chain, orange/red the AI-driven discovery space beyond the initial AHS (in which the human updated the AHS following the AI results), and black the final AHS for inhomogeneity. Blue R denotes the human’s initial APS, from which only one problem was exposed to the initial AHS, but the AI autonomously identified and solved two other problems (first two red RRs on the AI side); by contrast, orange RRs are outside the human’s initial APS. Question marks denote hypotheses under active consideration. Green arrows indicate the workflow and red arrows the human recognition of the implications of the AI results. L→∞L→∞ and ‘any L’ mean thermodynamic and Hamiltonian-level mapping problems, respectively. The discovery types and the dan ranks of the AI contribution are listed on the right side. The curved arrow highlights Type IV as a discovery mode that may accelerate breakthroughs. IV.2.1 Human APS, initial AHS, and a Type-I breakthrough In the present case, the Hamiltonian-level mapping questions for the Potts and O(3)O(3) chains belonged to the human APS but not to the human AHS, because the human had no active hypothesis or constructive route toward solving them. These questions were deliberately not supplied to the AI and therefore were absent from the AI’s initial problem context. Instead, the human selected the problem of the thermodynamic J1J_1-J2↔J_2 J-h mapping for n-vector spins to design the initial AHS as the landscape expanded along the model-choice axis: standard Potts → clock → XY → Heisenberg →O(n)→ O(n) (green axis in Fig. 1; green rounded rectangles in Fig. 3). Here the AI remained inside the research loop. The accumulated context eventually enabled the homogeneous arbitrary-U,VU,V hypothesis, which clearly lay outside the human’s initial AHS (orange axis in Fig. 1; orange rounded rectangles in Fig. 3), a Type-I discovery. The AI also autonomously tested its nonlinear structure numerically for discrete q-state clock spins by constructing the full q2×q2q^2× q^2 overlapping-pair matrix and the mapped q×q× q range-one matrix for U(x) U(x) = = 0.37x−0.81x2+0.23x3+0.11cos(1.7x), 0.37x-0.81x^2+0.23x^3+0.11 (1.7x), V(x) V(x) = = −0.44x+0.31x2+0.17cos(2.3x)+0.08x4. -0.44x+0.31x^2+0.17 (2.3x)+0.08x^4. For q=3,4,5,7,9q=3,4,5,7,9 and β=0.37,1,2.3β=0.37,1,2.3, the two transfer matrices and thus their largest eigenvalues agreed to relative errors below 4×10−154× 10^-15. This stress test is not part of the proof, and was not prompted by the human, but the AI’s own extra effort to distinguish precisely the nonlinear extension from the linear mapping. Under the task-level definition in Ref. [8], the arbitrary-U,VU,V hypothesis-generation and autonomous proof episode qualifies as 9d AI-led: the AI conceived the extension and completed its mathematical derivation outside the human’s initial AHS but also APS, while the human role was to challenge, verify, and recognize its geometrical connection to the zigzag construction [31] and one example of its nonlinear applications (Sec. I.4). IV.2.2 AI-reshaped AHS and a Type-I/I breakthrough During the Type-I development, the AI independently encountered the Hamiltonian-level mapping problem through the literature, brought it into its own APS, and produced a Hamiltonian-level proof for n=3n=3, thereby overcoming the 1990 stated challenge [28], a Type-I discovery. Thus, it was the AI who led the Hamiltonian-level mapping problem to the shared human–AI APS. This pivoted the research by introducing a new 1/L1/L axis to the AHS (red axis in Fig. 1; red rounded rectangle in Fig. 3), a Type-I. The AI contribution was temporally assigned 6d AI-reshaped and would be readjusted by its subsequent performance. The human recognized its potential and then conjectured an O(n)O(n) microscopic mapping (red plane in Fig. 1; red rounded rectangle on the human side in Fig. 3). The actual prompt was “Hi buddy, for the clock, XY, and Heisenberg models there is a remarkably clean microscopic geometric derivation, which exposes the mapping at the Hamiltonian level and a strong finite-size mapping for the partition functions for the open chains. This seems to hold for n-vector spins for all n and arbitrary interaction functions.” Note that the AI-generated arbitrary-U,VU,V result had by then already entered the shared AHS and had been adopted by this prompt. Then, the AI synthesized the recursive Householder moving-frame method to derive the proof, a Type-I discovery, classified as 9d under the definitions of Ref. [8]. One may argue that, since the human made a prompt, the AI contribution was not 9d AI-led but closer to 8d AI-driven. This provides a useful case study of the context-dependent nature of the nine-dan framework. 8d is the case in which AI autonomously generalizes human-supplied seed examples to the full solution. Here, the critical seed example—the Hamiltonian-level proof for n=3n=3—was supplied by the AI, not by the human. Hence, the AI solved a conjecture with its own seed and full solution; its contribution was clearly beyond 8d. Moreover, the Hamiltonian-level proof problem was introduced to the shared human-AI APS by the AI, while it had been hidden in the human’s private APS. Therefore the 9d AI-led assessment was not an exaggeration but a rational action. IV.2.3 Inhomogeneity, a Type-IV breakthrough Finally, the human recognized the implication of the geometry-based microscopic proof—which acts on the arguments of the interaction functions, not the interactions themselves—and introduced arbitrary inhomogeneous interaction functions. This expands the AHS to the fourth axis from the homogeneous specialization (black axis in Fig. 1; black rounded rectangle in Fig. 3) with no need to modify the proof—a Type-IV recognition breakthrough. This immediately brings a broader class of inhomogeneous geometries within the theorem, after a suitable one-dimensional ordering, including regular two-leg ladders, bond- and site-decorated chains, dimerized chains, and disordered chains, obtained by appropriate site dependence or vanishing of selected UiU_i and ViV_i. IV.3 Potts implication and Type-IV discovery Moreover, the intermediate 6d AI-reshaped event generated another autonomous microscopic proof for the q-state standard Potts chain (the first red rounded rectangle). This not only closes a knowledge gap but also provides an ideal illustration of the recognition problem: precisely because the solution is so simple, it exposes a broader lesson about representation. Historically, the q-state Potts model was introduced as a simpler generalization of the Ising model than the planar q-state model now known as the clock model [22]. It is invariant under any permutation of the q Potts states. Exploiting the qS_q symmetry or the q−1S_q-1 symmetry in a field that selects one state, say P, has taken the center stage. For example, the 2025 thermodynamic mapping was obtained by using qS_q symmetry to reduce the q2×q2q^2× q^2 transfer matrix of the J1J_1-J2J_2 Potts chain to the 2×22× 2 maximally symmetric subspace (MSS) containing the largest eigenvalue and matching it to the 2×22× 2 MSS of the q-state J-h Potts chain due to q−1S_q-1 symmetry [1]. The labeling of the Potts states including P has no physical meaning—they can be named arbitrarily. For example, the Potts states were conventionally labeled by σi∈0,1,…,q−1 _i∈\0,1,…,q-1\ and P=0P=0 [25, 38] or σi∈1,2…,q _i∈\1,2…,q\ and P=1P=1 [1, 30] with no rule imposed for their relative relationship. However, this is now recognized to be the blind spot in finding a microscopic mapping. In the present workflow, the recognition breakthrough itself followed the human-designed Type-I research. We deliberately stopped at an intermediate q-dimensional invariant subspace instead of the 2×22× 2 MSS (see Supplemental Material I.A). This weaker reduction is already sufficient to prove the thermodynamic mapping and is also the step that survives for the clock model and continuous vector spins—it was the actual starting point of the initial AHS (Fig. 3 the topmost green arrow marked by q2→q^2→ q). The reduced q×q× q transfer matrix and the mapping are certainly independent of how the Potts states are labeled, as verified by the human using symbolic derivation within the Wolfram Mathematica 14.3 software package for q=2−10q=2-10 before releasing the initial AHS to the AI. However, unlike the easily comparable MSS (having only four elements), the q×q× q space demands an efficient, analytic way for bookkeeping and comparing the matrix elements for arbitrary q. This drove the AI to exploit those labels as elements of ℤqZ_q, with subtraction modulo q, as the relative variables needed for the mapping. Later on, the AI autonomously recognized its use in the microscopic mapping; the auxiliary ℤqZ_q labeling of the Potts states does not alter the physical qS_q invariance of the original zero-field Hamiltonian, while the mapped field model has the expected q−1S_q-1 symmetry fixing P. This cyclic labeling with a strict subtraction rule is not just another convenient coordinate choice, but the mathematical bridge to connect the two models microscopically at the Hamiltonian level. In a broader sense, the discovery tells us something about the organization of knowledge: apparently neighboring facts can remain disconnected because they are expressed in incompatible representations. The Potts model demonstrates a central lesson that unifies all the pieces of discovery in this work: the correct local variable can be hidden in the conventional representation. For Potts, the conventional permutation-symmetric qS_q description does not expose the relevant subtraction structure. For O(3)O(3), laboratory-frame spin coordinates hide the bond-relative geometry. For general O(n)O(n), a fixed global coordinate system hides the recursive local frame. In every case, once the appropriate relative-coordinate representation is recognized, the Hamiltonian reorganizes itself. That lesson gives the Householder result extra significance: it is not merely an O(n)O(n) generalization of an Ising or Potts trick, but a systematic answer to the question that the Potts episode dramatizes: How do we construct the relative variables that reveal hidden Hamiltonian equivalences? With these insights, the Potts lesson suggests a redefinition of “unknown knowns” (UK) as the fourth type of discovery to recognize that a scientific result may remain inaccessible not because the ingredients are unknown or the computation is difficult, but because the relevant known ingredients are encoded in representations that conceal their connection. Type-IV knowledge connection is distinct from Type-I knowledge synthesis: Type IV recognizes a latent connection among already available ingredients, whereas Type I introduces a genuinely new relation lying far beyond the relevant AHS. As demonstrated above, Type-IV recognition can seed new Type-I, Type-I, and Type-I developments while also providing a unifying interpretation of earlier results. Type IV ties directly to AI: a sufficiently broad reasoning system may be unusually well-suited for representation switching and cross-domain connection. Here, the ℤqZ_q cyclic labeling is well known in the context of the clock model, which possesses the ℤqZ_q symmetry rather than qS_q [22, 27]. It has even been used to map the 1D standard Potts model with multi-site interactions onto a standard 2D Potts model [39]. Yet, to recognize this as a Type IV for the Hamiltonian-level J1J_1-J2J_2 ↔ J-h mapping, the problem must first be cast into the APS and AHS. The AI did so exactly and autonomously. Explicitly recognizing Type IV as a discovery mode may accelerate the identification of UKs. The last step of this work could be an example: the human’s timely recognition of arbitrary inhomogeneous interactions added a new fourth axis to the AHS—black axis in Fig. 1—with no need to modify the proof. IV.4 External comparison The May 2026 autonomous resolution of the Erdős planar unit-distance problem provides a useful external comparison [2, 3, 4]. There the open problem was already known and selected, making the event a particularly clear Type-I autonomous breakthrough. The general-purpose reasoning model was not specially trained for that mathematical problem but evaluated across a collection of Erdős problems—effectively an APS, rather than being supplied with a problem-specific AHS—and it unexpectedly found a successful route using sophisticated algebraic-number-theoretic ideas [2]. The present case probes both a complementary route and a similar route: the initial AHS did not contain an arbitrary-interaction conjecture and a microscopic proof conjecture, but the AI worked on a collection of Type-I thermodynamic mapping problems. It enlarged the AHS while participating in the development of a much narrower theorem. In that sense, the homogeneous U,VU,V result is Type I, while the n=3n=3 microscopic proof qualifies as both Type I and Type I: Type I because it overcame the 1990 stated obstruction, and Type I because its constructive route lay far beyond the human AHS. The subsequent O(n)O(n) Householder construction generalized this result after the human formulated the O(n)O(n) hypothesis. The geometric proof may be viewed as a synthesis of two familiar ideas: the ordered propagation characteristic of one-dimensional transfer constructions and the Householder reflection of numerical linear algebra. The cumulative frame i=1(1)H(2)⋯(i−1) G_i= G_1 H( σ_1)H( σ_2)·s H( σ_i-1) transports local bond geometry along the chain, converting the N scalar product of the original spins into the N scalar product of the transformed spins. This cross-domain synthesis is reminiscent, at a methodological level, of those recent AI-generated mathematical discoveries in which established tools from a distant field were brought unexpectedly to bear on a longstanding problem [2, 3, 4]. These events illustrate why autonomous hypothesis generation deserves separate attention from autonomous problem solving and how they are connected in one project. Maximizing single-shot reasoning power and maximizing longitudinal co-development efficiency are not necessarily the same optimization problem. Together, they sharpen the practical question: rather than waiting for isolated spectacular successes, can a research process be designed to cultivate the conditions from which such high-autonomy contributions emerge? Last but not least, the developmental hypothesis provided by this paper is only a proof of concept, not a causal demonstration that continuous Type-I involvement will generally produce Type-I/I/IV contributions. A single longitudinal case cannot establish an incidence rate, a necessary sequence of dan levels, or a universal relationship between discovery type and contribution level. The four discovery types are contextual, and the dan labels are task-level attributions. A rigorous research program in understanding the effectiveness of this approach should therefore test the hypothesis prospectively across multiple domains, with complete human–AI interaction records, predefined criteria for provenance and novelty, adversarial validation, and comparison against workflows in which AI participation is intentionally interrupted. We anticipate more reports of human–AI research conducted in the proposed Type-I-based discovery framework. V Summary In collaboration with a general-purpose AI, we have proved a theorem that connects spontaneous and field-induced mechanisms of collective behavior in classical spin chains. A zero-field O(n)O(n)-vector open chain with arbitrary inhomogeneous nearest- and next-nearest-neighbor interaction functions Ui(i⋅i+1)U_i( S_i· S_i+1) and Vi(i⋅i+2)V_i( S_i· S_i+2) is microscopically, via a Hamiltonian-level mapping, equivalent to a simpler O(n)O(n) open chain with nearest-neighbor interaction Vi(i⋅i+1)V_i( σ_i· σ_i+1) and axial single-spin potential Ui(σiz)U_i( _i^z) for every integer n≥1n≥ 1 and every system size L≥1L≥ 1. An analogous theorem holds when the continuous O(n)O(n) spins are replaced by the q-state Potts spins with the standard Potts interaction, implying a closed-form exact solution of the J1J_1-J2J_2 Potts open chain for every q≥2q≥ 2 and every L≥1L≥ 1. In comparison, the homogeneous, linear, thermodynamic mapping program constituted the human-designed initial AHS exposed to the AI for its progressive development. Both the homogeneous arbitrary-interaction theorem and the Hamiltonian-level proofs for the n=3n=3 and Potts cases were generated autonomously by the AI far beyond the initial AHS. The human updated the AHS by conjecturing an O(n)O(n) microscopic mapping, which AI proved by synthesizing a Householder moving-frame method. The final extension to arbitrary inhomogeneous interactions was recognized and formulated by the human, highlighting the critical role of human recognition and understanding of the AI results. The work demonstrates that AI can generate a scientific hypothesis outside the human collaborator’s AHS and even APS within such a longitudinal human–AI co-development workflow, and provides a proof of concept—not causal evidence—that sustained AI participation in an actual Type-I AHS may increase the chance of such breakthroughs. Acknowledgements. The author is grateful to Kevin Wang for helpful discussions that were related to the Potts implication. Brookhaven National Laboratory was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences (BES), Division of Materials Sciences and Engineering under Contract No. DE-SC0012704. The AI collaborator was a general-purpose OpenAI GPT-5.6 Sol reasoning model: the High mode was activated in chatting for discovery (4d, 6d, 9d) and the Extra High mode in critical manuscript review (2d). The human author reviewed and verified all AI-generated scientific content and assumes full responsibility for the manuscript. Data availability The timestamped human–AI interaction records supporting the discovery-provenance analysis, together with other supporting data, will be openly available. References [1] W. Yin, Exact solution of the frustrated Potts model with next-nearest-neighbor interactions in one dimension via AI bootstrapping, Phys. Rev. B 112, 094424 (2025). [2] OpenAI, An OpenAI model has disproved a central conjecture in discrete geometry, OpenAI Research (2026), published 20 May 2026. [3] W. Sawin, An explicit lower bound for the unit distance problem, arXiv:2605.20579 [math.CO] . [4] N. Alon, T. F. Bloom, W. T. Gowers, D. Litt, W. Sawin, A. Shankar, J. Tsimerman, V. Wang, and M. M. Wood, Remarks on the disproof of the unit distance conjecture, arXiv:2605.20695 [math.CO] . [5] M. Krenn, R. Pollice, S. Y. Guo, M. Aldeghi, A. Cervera-Lierta, P. Friederich, G. d. P. Gomes, F. Hase, A. Jinich, A. Nigam, Z. Yao, and A. Aspuru-Guzik, On scientific understanding with artificial intelligence, Nat. Rev. Phys. 4, 761 (2022). [6] M. Ceriotti, M. Krenn, A. B. Lee, N. Marzari, B. Nachman, M. Pettee, and O. A. von Lilienfeld, Publishing physical sciences in the era of AI, PRX Intelligence 1, 010001 (2026). [7] W. Yin, Site-decorated model for unconventional frustrated magnets: Ultranarrow phase crossover and two-dimensional spin reversal transition, Phys. Rev. B 113, 104418 (2026a). [8] W. Yin, Half-ice, half-fire–driven ultranarrow phase crossover in one-dimensional decorated q-state Potts ferrimagnets: An AI-co-led exploration, APS Open Sci. 1, 000047 (2026b). [9] W. Yin and A. M. Tsvelik, Metamagnetic transition in low-dimensional site-decorated quantum Heisenberg ferrimagnets, arXiv:2511.06442 [cond-mat.str-el] . [10] H. E. Stanley, Exact solution for a linear chain of isotropically interacting classical spins of arbitrary dimensionality, Phys. Rev. 179, 570 (1969). [11] D. C. Mattis, The Theory of Magnetism I (Springer, Berlin, 1981). [12] D. C. Mattis, The Theory of Magnetism I (Springer, Berlin, 1985). [13] D. C. Mattis and R. Swendsen, Statistical Mechanics Made Simple, 2nd ed. (World Scientific, Singapore, 2008). [14] S. A. Kivelson, J. M. Jiang, and J. Chang, Statistical Mechanics of Phases and Phase Transitions (Princeton University Press, Princeton, NJ, 2024). [15] A. P. Ramirez, Strongly geometrically frustrated magnets, Annual Review of Materials Research 24, 453 (1994). [16] C. Lacroix, P. Mendels, and F. Mila, eds., Introduction to Frustrated Magnetism: Materials, Experiments, Theory (Springer, Berlin, Heidelberg, 2011). [17] H. A. Kramers and G. H. Wannier, Statistics of the two-dimensional ferromagnet. Part I, Phys. Rev. 60, 252 (1941). [18] Z. Glumac and K. Uzelac, Critical behaviour of the 1d q-state Potts model with long-range interactions, Journal of Physics A: Mathematical and General 26, 5267 (1993). [19] M. Blume, P. Heller, and N. A. Lurie, Classical one-dimensional Heisenberg magnet in an applied field, Phys. Rev. B 11, 4483 (1975). [20] I. Harada and H. J. Mikeska, Thermodynamic properties of the classical Heisenberg chain with nearest- and next-nearest-neighbor interactions, Zeitschrift für Physik B Condensed Matter 72, 391 (1988). [21] E. Ising, Beitrag zur Theorie des Ferromagnetismus (Contribution to theory of ferromagnetism), Zeitschrift für Physik 31, 253 (1925). [22] R. B. Potts, Some generalized order-disorder transformations, Mathematical Proceedings of the Cambridge Philosophical Society 48, 106 (1952). [23] G. S. Joyce, Exact results for the one-dimensional, anisotropic classical heisenberg model, Phys. Rev. Lett. 19, 581 (1967). [24] M. E. Fisher, Magnetism in one-dimensional systems—the Heisenberg model for infinite spin, American Journal of Physics 32, 343 (1964). [25] Z. Glumac and K. Uzelac, The partition function zeros in the one-dimensional q-state Potts model, Journal of Physics A: Mathematical and General 27, 7709 (1994). [26] J. F. Dobson, Many‐neighbored Ising chain, J. Math. Phys. 10, 40 (1969). [27] I. Harada, One-dimensional classical planar model with competing interactions, Journal of the Physical Society of Japan 53, 1643 (1984). [28] I. Harada and H. J. Mikeska, Thermodynamics and correlations of a helical spin chain with non-magnetic impurities, Journal of Physics: Condensed Matter 2, 953 (1990). [29] Here ‘spontaneous’ denotes interaction-generated zero-field collective behavior, not finite-temperature spontaneous symmetry breaking in the short-range one-dimensional chain. [30] S.-C. Chang and R. Shrock, Structure of the partition function and transfer matrices for the Potts model in a magnetic field on lattice strips, Journal of Statistical Physics 137, 667 (2009), arXiv:0907.0925 [cond-mat.stat-mech] . [31] See Supplemental Material for the MSS/transfer-integral-based proof of thermodynamic mappings and other technical details. [32] A. S. Householder, Unitary triangularization of a nonsymmetric matrix, J. ACM 5, 339 (1958). [33] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed. (Johns Hopkins University Press, Baltimore, 2013). [34] M. Fels and P. J. Olver, Moving coframes: I. regularization and theoretical foundations, Acta Appl. Math. 55, 127 (1999). [35] U. Wolff, Collective Monte Carlo updating for spin systems, Phys. Rev. Lett. 62, 361 (1989). [36] J. Kent-Dobias and J. P. Sethna, Cluster representations and the Wolff algorithm in arbitrary external fields, Phys. Rev. E 98, 063306 (2018). [37] A. L. Wysocki, K. D. Belashchenko, and V. P. Antropov, Consistent model of magnetism in ferropnictides, Nature Physics 7, 485 (2011). [38] F. Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982). [39] L. Turban, One-dimensional q-state Potts model with multi-site interactions, Journal of Physics A: Mathematical and Theoretical 50, 205001 (2017).