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Slow thinking is derived from first principles by sampling latent sequences that reduce uncertainty at maximum rate.

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2026-07-10 11:30 UTC pith:GPEJFQYT

load-bearing objection Solid static theory of CoT as projection/lifting with real variance laws and an inquisitive train sampler; the “first-principles derivation” of active lifting from the rate objective is still informal and oversold. the 4 major comments →

arxiv 2607.08196 v1 pith:GPEJFQYT submitted 2026-07-09 cs.AI cs.CLcs.LG

A First-Principles Theory of Slow Thinking and Active Perception

Hongkang Yang , Zhi-Qin John Xu , Feiyu Xiong , Weinan E This is my paper
classification cs.AI cs.CLcs.LG
keywords slow thinkingactive perceptionactive liftinglatent samplingrepresentation hierarchysampler hierarchylarge language modelsfirst-principles cognitive modeling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper builds a mathematical theory of thinking and perception from first principles, without leaning on empirical descriptions of human cognition. It starts from the need to represent complex data distributions by simple function families (such as neural nets) via liftings and projections between observable and latent sequence spaces. The proposed framework, active lifting, samples latent sequences under an intrinsic drive to cut uncertainty as fast as possible. Existing slow-thinking large language models appear as a constrained special case called the static theory, sitting on two hierarchies—one of representations and one of samplers—that can be climbed to improve them. The same drive yields an inference process with an internal time axis and a training objective that looks like inventing an efficient, learnable language for observations. Sympathetic readers should care because the theory claims to unify design, training, and inference of slow-thinking models and to supply concrete upgrade paths plus a possible fix for policy collapse.

Core claim

Active lifting—sampling latent sequences driven by maximizing the real-time rate of uncertainty reduction—formally derives slow thinking and active perception. Existing chain-of-thought language models sit inside a static subspace with a fixed projection; that subspace induces a representation hierarchy (plain models cannot approximate simple hidden Markov models that simple projections can) and a sampler hierarchy (identity and predictive samplers are strictly weaker than explanatory ones that recover the posterior and inquisitive samplers). Climbing both hierarchies upgrades the models; removing the fixed projection yields free-form active lifting whose training objective resembles minimum

What carries the argument

Active lifting (and its static special case): continuous projections from latent to observable sequence spaces, Monte-Carlo latent sampling, the posterior and inquisitive samplers, and the single objective of maximizing the rate of uncertainty reduction, which induces both the representation hierarchy and the sampler hierarchy.

Load-bearing premise

That maximizing the real-time rate of uncertainty reduction is a sufficient single first principle, and that the circuit separation used to justify the representation hierarchy holds.

What would settle it

Train matched slow-thinking models that climb only the sampler hierarchy (explanatory plus inquisitive samplers) versus identity-sampler baselines; if gradient estimation error, policy entropy collapse, and multi-choice accuracy do not improve as predicted by the chi-square scaling laws, the central claim about optimal samplers fails. Separately, if TC0 equals NC1, the separation theorems that place plain models below simple projections collapse.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A concrete three-stage upgrade path for existing slow-thinking LLMs: better samplers, then persistent ubiquitous thinking, then free-form active lifting without prescribed formats.
  • A single construction of encoders and generative models that applies to every data modality via the same latent-sequence lifting.
  • Image encoders can form multiscale compositional representations without being given that structure in advance.
  • Policy collapse during reinforcement learning of chain-of-thought models is mitigated by training an explicit inquisitive sampler that favors exploration over pure exploitation.
  • Inference acquires an internal time axis on which the model actively searches for understandings of each observation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the uncertainty-rate objective truly balances fast and slow thinking, pretraining runs that allow thoughts anywhere should spontaneously allocate long thoughts only to hard segments without hand-tuned length penalties.
  • The same lifting-plus-rate objective may supply a principled alternative to free or product couplings in generative modeling (linguistic coupling), which is worth testing on non-text modalities.
  • Posterior drift formalized here suggests that non-causal prefill is not optional for open-ended answers; causal samplers should systematically underperform on multi-answer finetuning pairs even when compute is equalized.
  • A microscopic theory that unifies this macroscopic account with memory and self-directed learning would need to recover the same two hierarchies as emergent coarse-grainings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a first-principles framework for slow thinking and active perception. It develops a static theory of continuous projections and latent sampling that places sequence models on a representation hierarchy (plain Transformers vs. simple projections) and a sampler hierarchy (identity/predictive vs. explanatory), with Monte-Carlo encoding, variance scaling laws, and a distinction between posterior and inquisitive samplers. Existing slow-thinking LLMs (e.g., DeepSeek-R1) are reconstructed as a forgetful, identity-sampler special case and positioned for three stages of upgrade. A single informal objective—maximizing the real-time rate of uncertainty reduction—is then used to motivate a more general “active lifting” construction with an internal time axis and a minimum-length-coding-like training objective. Technical by-products include a possible account of policy collapse and a unified encoder/generative-model story across modalities.

Significance. If the static theory is taken on its own terms, the paper offers a coherent mathematical organization of chain-of-thought models: continuous projections with closed domains, explicit Monte-Carlo estimators and χ² variance laws (Table 3.1), the posterior vs. inquisitive sampler distinction, and a concrete reconstruction of DeepSeek-R1-style training and inference. The inquisitive-sampler analysis of policy collapse and the three-stage upgrade path are practically useful and falsifiable in principle. The separation theorems (under TC⁰ ⊊ NC¹) and the appendices supply non-trivial formal content. The broader claim that a single rate objective formally derives both hierarchies and unconstrained active lifting is more ambitious and currently less secure; even so, the static framework alone would be a meaningful contribution to the theory of reasoning LLMs if claims are scoped carefully.

major comments (4)
  1. §4 and Eq. (4.2): The manuscript states that maximizing the real-time rate of uncertainty reduction is a unified first principle that “qualitatively derive[s] the entire static theory” and extends to active lifting. Section 4 itself labels the argument informal and only sketches Figure 4.1; there is no theorem showing that optimizers of (4.2) must prefer simple projections over plain Transformers, explanatory over identity/predictive samplers, or free-form latents over the conversation format (5.1). The Abstract and §§1, 5.7, 6 therefore overstate what is proved. Either supply a rigorous bridge (even under strong assumptions) or reframe §4–6 as motivation/interpretation rather than derivation.
  2. Theorems 2.1 and 2.4: The separation results that justify climbing the representation hierarchy rest on the unproved conjecture TC⁰ ⊊ NC¹. The paper is explicit about this, but the Abstract’s claim that the theory “formally derives” slow thinking and upgrades existing models depends on these separations. The manuscript should either (i) state the hierarchy results as conditional on the conjecture throughout the main claims, or (ii) provide unconditional separations for a restricted but still interesting class of targets (e.g., fixed-depth composition of a fixed non-TC⁰ function).
  3. §5.1–5.4 and Definition 5.1: The projection family (thought tokens, conversation format, forgetful latents) is chosen so that DeepSeek-R1 sits inside the static theory by construction. That is legitimate reverse-engineering, but it weakens the claim of pure first-principles prediction. The paper should clearly separate (A) what is forced by approximation/sampling efficiency from (B) what is fixed by matching an existing format, and mark which of the three upgrade stages are predictions versus design choices.
  4. §5.6 (forgetful vs. persistent expressivity) and the informal argument that P_forget does not contain P_HMM: Unlike Theorems 2.1–2.4, this comparison is only sketched. Since Stage Two of the upgrade path rests on it, either complete a proof along the lines of Theorem 2.4 or demote the claim to a conjecture and adjust Figure 5.3 and the Stage-Two recommendations accordingly.
minor comments (5)
  1. Notation density is high (Proj^{-1}_{<ω}, Φ, Q_⋆, Q_id, etc.). A short “cheat sheet” table early in §2–3 would help readers track the static theory.
  2. Table 3.1 and the multi-choice error bound (3.27) are useful; stating the precise regularity assumptions under which the O(n^{-2}) terms are controlled would strengthen the scaling-law claims.
  3. §7 is only a preliminary Stage-One experiment. Even a brief protocol (model size, n, reward definition) in the main text would make the empirical claim easier to assess.
  4. Several figures (0.1, 0.2, 4.1) are conceptual roadmaps; ensuring that every arrow is tied to a numbered section or equation would reduce ambiguity.
  5. Typos and formatting: “Asume” (Thm 2.4), occasional missing spaces around math, and long run-on sentences in §1.1–1.2 could be cleaned in revision.

Circularity Check

3 steps flagged

DeepSeek-R1 and CoT formats sit inside the static theory by construction of the chosen projection/forgetful latent; that is reverse-engineering, while the sampler/variance laws and circuit separations are not definitional loops.

specific steps
  1. self definitional [§5.1 Eqs. (5.1)–(5.3), Definition 5.1; cf. Example 2.3]
    "Similar to Example 2.3, define the projection as follows. T={⟨s⟩x⟨/s⟩ |x∈Σ≤c} Dom(Proj)={(y(t)xt)∞t=1 | xt∈Σ, y(t)∈T∪{∅}} Proj:(y(t)xt)∞t=1↦(xt)∞t=1 where c … DeepSeek-R1 set c=2^15=32768. … One important simplification implemented by DeepSeek-R1 is to ignore the thoughts from earlier rounds … The latent distribution P is called forgetful if …"

    The static projection, multi-round conversation support, and forgetful latent are defined to match DeepSeek-R1’s published format and implementation (and Example 2.3 already states that this Proj “includes the format used by reasoning models such as DeepSeek-R1 and Quiet-STaR”). The later claim that the theory encompasses / derives R1’s representation is therefore membership by construction of the design space, not an independent first-principles prediction of that format.

  2. renaming known result [§5.3 Eqs. (5.14)–(5.18); also §5.2 identity sampler (5.11)–(5.13)]
    "DeepSeek-R1 uses n=1, so the objective simplifies to min_θ −log Pf(x(T+1)r | xY → Σ∗□), Y∼Pf(·|x→T). … With n=1, the objective becomes … Then, this loss becomes the Group Relative Policy Optimization (GRPO) objective [138] of R1-Zero. … training a slow thinking model with the loss (5.17) can be interpreted as fitting the posterior sampler QTR∗."

    After fixing the identity sampler, forgetful latent, n=1, and relaxed answer set Σ∗⌣—all taken from R1 practice—the paper’s general mini-batch losses reduce to the known SFT + policy-gradient / GRPO objectives. Recovering a published training recipe by specializing parameters chosen to match that recipe is reverse-engineering (renaming a known procedure as an instance of the general loss), not a forced derivation from the rate objective alone.

  3. other [§1 (first-principles claim) vs §4 (unified objective) and Abstract]
    "By first-principle, we refer to deriving phenomena of interest from a fundamental mathematical formulation, without relying on the empirical knowledge of these phenomena. … Using this objective function, one can qualitatively derive the entire static theory. … this section mainly uses informal arguments. … It formally derives slow thinking … containing the slow thinking models in a subspace that we call the static theory."

    The static theory is constructed in §§2–3 from two objectives (approximation ability and sampling efficiency), with CoT-style projections already imported from existing models. Section 4 then asserts that the single informal rate objective (4.2) can “qualitatively derive the entire static theory,” after that theory exists and without a theorem that optimizers of (4.2) must select simple projections, explanatory samplers, or free-form latents. The Abstract’s “formally derives” language therefore overstates a post-hoc informal sketch as a first-principles derivation free of empirical format choices.

full rationale

The paper develops independent mathematical content: continuous projections and liftings, Monte-Carlo encoding with χ² variance laws (Table 3.1), the posterior vs. inquisitive sampler from minimizing gradient estimation error, and separation theorems (under TC⁰ ⊊ NC¹). Those pieces are not circular. Circularity is limited and local to the claim that the theory “formally derives” / “encompasses” existing slow-thinking models from first principles without empirical knowledge. Example 2.3 and §5.1 define the projection, conversation format, and forgetful latent to match DeepSeek-R1 / Quiet-STaR; §5.2–5.4 then recover identity sampling, n=1 losses, and GRPO by specializing those choices. Membership of R1 in the static subspace is therefore by construction of the design space, not an independent prediction. Section 4’s claim that the single informal rate objective (4.2) “qualitatively derives the entire static theory” is post-hoc (the static theory was already built from approximation ability + sampling efficiency in §§2–3) and is labeled informal by the authors; that is overclaim of derivation order, not Eq. X = Eq. Y by definition. No fitted-parameter-as-prediction, no load-bearing uniqueness theorem imported from the authors, and no self-citation that alone forces the central hierarchies. Score 4 reflects partial circular content around existing models while the hierarchies and inquisitive sampler retain independent content.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 4 invented entities

Load-bearing content is a mix of standard measure/probability and circuit-complexity background, domain choices that encode CoT practice (thought delimiters, forgetfulness), and paper-specific postulates (active lifting, inquisitive sampler, unified rate objective). Free parameters are mostly implementation knobs inherited from existing LLMs rather than fits to a claimed physical constant.

free parameters (4)
  • thought length bound c = 32768 (DeepSeek-R1)
    Upper bound on thought length in the static projection (DeepSeek-R1 sets c=32768); controls expressivity vs cost and is chosen by practice, not derived.
  • Monte-Carlo sample size n (and batch B)
    Sampling budget in likelihood/gradient estimators and slow-thinking scaling laws; free implementation choice affecting variance bounds.
  • KL0.01 numerical-error tolerance in semi-divergence D = 0.01
    Ad-hoc 0.01 log-ratio slack in Eq. 2.10 used only to define the representation hierarchy; not derived from first principles.
  • time-cost model τ(l) / decode coefficient
    Used to turn the abstract rate objective into a tractable length-aware loss (§5.6.3); approximated as proportional to length.
axioms (5)
  • standard math TC^0 ⊊ NC^1 (circuit complexity separation)
    Assumed for Theorems 2.1 and 2.4 that plain/polynomially-simple projections cannot approximate certain HMMs; widely believed but unproved.
  • domain assumption Transformers with fixed depth are modeled as DLOGTIME-uniform TC^0 (even with constant precision)
    Used to upper-bound expressivity of fast-thinking models (§2.4–2.5); follows cited circuit analyses of Transformers.
  • ad hoc to paper Maximizing real-time rate of uncertainty reduction is the right first principle for perception/thinking
    §4 introduces Eq. 4.2 as the unified objective that ‘qualitatively derives’ the static theory; not deduced from a prior theorem.
  • ad hoc to paper Simple continuous projections with closed domain and TC^0 liftings are the right restricted class for static theory
    Definitions 2.3–2.7 carve the design space so that thought-insertion formats are simple projections; choice is motivated but not forced.
  • domain assumption Reverse KL + Monte-Carlo latent sampling is the operational training/inference method
    Chosen among forward algorithm, MAP, and other divergences (§3.1); standard in variational/RL practice but not uniquely derived.
invented entities (4)
  • Active lifting no independent evidence
    purpose: Unconstrained latent-sequence sampling framework that generalizes static prescribed projections and is claimed to invent languages / slow-thinking formats.
    Core named theory of the paper (§6); independent evidence would be empirical emergence of formats and modality-general encoders, not yet demonstrated beyond sketch.
  • Inquisitive sampler Q_⋆ no independent evidence
    purpose: Train-time sampler proportional to posterior mass times gradient norm; proposed fix for policy collapse.
    Derived as unique minimizer of gradient MSE under the paper’s estimator (Prop. I.1 / §3.4.2); falsifiable via RL training curves, not yet strongly evidenced in the manuscript.
  • Representation hierarchy (P_plain ⊄ P_HMM ⊆ P_simple) and sampler hierarchy (identity ⊂ predictive ⊂ explanatory) no independent evidence
    purpose: Organize expressivity and sampling efficiency; justify three-stage upgrade path.
    Induced by separation theorems and causality results (§§2–3); mathematical structure is internal to the paper’s definitions.
  • Forgetful latent / persistent ubiquitous thinking no independent evidence
    purpose: Name the constraint of current CoT models vs the proposed more expressive static models.
    Definition 5.1 encodes DeepSeek-R1 practice; ‘persistent’ upgrade is a modeling choice inside the same projection family.

pith-pipeline@v1.1.0-grok45 · 69519 in / 3851 out tokens · 51495 ms · 2026-07-10T11:30:37.468842+00:00 · methodology

0 comments
read the original abstract

As part of a series on first-principles modeling of cognitive functions, this paper attempts to provide a mathematical formulation of thinking and perception. It formally derives slow thinking or more generally, active perception, and encompasses the design, training and inference of slow thinking large language models. Our starting point is the lifting and projection of probability distributions on the observable and latent spaces, with the objective of representing complex data distributions by simple function families such as neural networks. A theory called "active lifting" is proposed, based on the sampling of latent sequences and an intrinsic drive to reduce uncertainty with maximum rate. It derives a large design space, containing the slow thinking models in a subspace that we call the static theory. These models are positioned on the representation hierarchy and sampler hierarchy induced by the static theory, and can be upgraded by climbing the two hierarchies. Active lifting further derives an inference process with an internal time axis, and a training objective that resembles minimum-length coding as well as the invention of languages. Thus, it characterizes the agency of perception, including the emergence of the slow thinking formats. Technical by-products of this theory include a three-stage pathway for improving slow thinking models, a unified approach to constructing encoders and generative models for all data modalities, a priori formation of human-like visual representations, and a possible solution to policy collapse.

Figures

Figures reproduced from arXiv: 2607.08196 by Feiyu Xiong, Hongkang Yang, Weinan E, Zhi-Qin John Xu.

Figure 0.1
Figure 0.1. Figure 0.1: The modeling hierarchy of cognitive functions. The coverage of this paper is marked in red. [PITH_FULL_IMAGE:figures/full_fig_p001_0_1.png] view at source ↗
Figure 0.2
Figure 0.2. Figure 0.2: Road-map for improving slow thinking models, as a by-product of our theory. This figure [PITH_FULL_IMAGE:figures/full_fig_p002_0_2.png] view at source ↗
Figure 1.1
Figure 1.1. Figure 1.1: Illustration of Example 1.1. The four panels are: your room, the predictive sampling [PITH_FULL_IMAGE:figures/full_fig_p006_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Illustration of Example 1.2. The well-functioning of an AI agent depends on a proper [PITH_FULL_IMAGE:figures/full_fig_p007_1_2.png] view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Illustration of Example 1.3. One possible way to model the agency of human vision is to [PITH_FULL_IMAGE:figures/full_fig_p008_1_3.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: The representation hierarchy. It illustrates the various approaches to modeling distributions [PITH_FULL_IMAGE:figures/full_fig_p011_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Illustration of the toy function (E.1) constructed in Appendix E.1, which can be compared [PITH_FULL_IMAGE:figures/full_fig_p017_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Illustration of the lifting Proj−1 <ω and next-segment map Φ of Example 2.3. Top: The latent sequences z = (y (t)xt) |x| t=1 ∈ Proj−1 <ω(x) of an observable sequence x may contain “thoughts” y (t) with variable lengths. The bubbles represent nonempty thoughts y (t) , while the straight lines are segments of x. Bottom: Each segment s ∈ Φ(z, a) may also contain a variable-length thought. 2.6.4 Simple Proje… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The sampler hierarchy. The slow thinking scaling laws derived in this section largely [PITH_FULL_IMAGE:figures/full_fig_p026_3_1.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Illustration of how the objective (4.2) leads to the need of approximation ability and [PITH_FULL_IMAGE:figures/full_fig_p046_4_1.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Left: A pair of query xq and response xr for finetuning slow thinking models. The response is based on the Archimedes method for computing π. Right: The blue boxes represent the possible thoughts that can be generated by the sampler. Here we assume that a predictive sampler is used, which can only see the query when producing thoughts, and thus the sampled methods for computing π are rather uniformly dis… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: One possible prompt format for the explanatory sampler [PITH_FULL_IMAGE:figures/full_fig_p055_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: The positions of the forgetful models (Pforget) and persistent thinking models (Ppersist) in the representation hierarchy ( [PITH_FULL_IMAGE:figures/full_fig_p058_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Persistent and ubiquitous thinking. Top: An example of pretraining data (a paragraph [PITH_FULL_IMAGE:figures/full_fig_p059_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: The input-output format x⟨explain⟩Z for the explanatory sampler that supports persistent and ubiquitous thinking. The text x, including its repetition inside Z, is in red, while the inserted thoughts, namely the rest of Z, are in purple. The next-token sets Φ≤1 (z) for some prefixes z of Z are annotated below. Considering that decoding is generally slower than encoding for LLMs, the above decoding proces… view at source ↗
Figure 6
Figure 6. Figure 6: (middle) illustrates the positive correlation between the two sides of (6.11). It follows that [PITH_FULL_IMAGE:figures/full_fig_p068_6.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Illustration of the resemblance between minimum-length coding and our training objective. [PITH_FULL_IMAGE:figures/full_fig_p069_6_1.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Comparison of the effectiveness of the predictive and explanatory samplers for the training [PITH_FULL_IMAGE:figures/full_fig_p076_7_1.png] view at source ↗

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