{"id":"afdf6a39-61b2-4079-bade-cfac7442ce5b","arxiv_id":"2601.19070","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A p-adic integral-equation formulation of deep networks is shown to have a unique hidden state under a contraction condition; the claimed thermodynamic limit and infinite-state bifurcation are not proven.","lead":"This paper defines a p-adic (tree-structured) integral-equation model of deep neural networks and proves a fixed-point theorem in a stable parameter region. It claims this reveals a thermodynamic limit and a critical bifurcation into infinitely many states, but that part is largely conjectural.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Thermodynamic-limit identification is asserted, not proved: Theorem 1 discretizes the continuous equation but never shows width→∞ DNNs (1.1) converge to (1.2).","rationale":"The reader's weakest assumption—that the continuum equation (1.2) is what finite DNNs converge to—is exactly the load-bearing point. The paper explicitly calls the passage from (1.1) to (1.2) an 'ansatz' and never supplies a convergence proof. The universal-architecture section gives only a finite-size recasting, and Theorem 1(ii) proves the reverse direction (discretizing the continuous equation), not the forward direction (finite DNNs approaching the continuous fixed point). If this limit fails, the fixed-point theorem, while mathematically clean, is a result about an isolated p-adic integral equation rather than about deep neural networks. The random-network prior section also relies on the same ansatz, so the Gaussian-expansion claims inherit the gap. I therefore agree with the reader's identification of the central weakness. The L_ϕ=∥ϕ∥∞ issue is real but secondary: it affects the precise boundary of the stable region for non-tanh sigmoids, but the contraction argument itself only needs the Lipschitz constant. The rejection stands: the advertised claims about DNNs are not established, even though the p-adic fixed-point theorem may be salvageable as a conditional contribution.","tokens_in":38131,"tokens_out":6399,"duration_ms":76336,"concrete_test":"Take the standard one-hidden-layer DNN h_i = ∑_{j=1}^n W_ij tanh(x_j)+ξ_i, with W_ij i.i.d. N(0, σ_w²/n) and ξ_i i.i.d. N(0, σ_ξ²). Apply the §6.3 embedding with p_n the smallest prime > n and p_n^{-2} normalization. For each n, define W_n(x,y), ξ_n(x) as the step functions on the p-adic tree obtained from the finite weight matrix and bias vector, and let h_n(x) be the embedded hidden state. Check (numerically and analytically) whether ∥h_n - h^*∥_{L²(Z_p)} → 0 in probability, where h^* solves (1.2) with kernels W_n, ξ_n. If it diverges or converges to a different object (e.g., a Gaussian process), the ansatz is false; if it converges, the paper still needs a proof, but at least this regime is consistent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim that (1.2) is the thermodynamic limit of finite DNNs (1.1) rests on the Section 1 ansatz, not on a theorem. Section 6.3 only recasts a finite DNN with max width N into a p-adic discrete DNN by choosing a prime p>N; the embedding depends on N and no limit N→∞ is taken or proven. Section 6.4's assertion that 'Z is dense in Z_p, and thus there are continuous functions' interpolating the matrices is not a convergence argument: denseness does not turn arbitrary finite-dimensional data into a limiting L² kernel W, and the p^{-L-∆} rescaling changes with p. Theorem 1(ii) goes in the opposite direction: it shows that Picard iterates of the continuous equation can be realized by p-adic discrete DNNs. That establishes consistency of the discretization, not that forward passes of standard DNNs (1.1) converge to h in any topology as widths grow. Without this convergence, the advertised 'critical organization'—also undercut by the paper's own admission that the precise study of (7.3) is open and that the infinite-state result is proven only for the toy model of Section 8—does not apply to DNNs. A secondary issue: Definition 1 sets L_ϕ=∥ϕ∥∞, which fails for general sigmoids (e.g., logistic has Lipschitz constant 1/4 but sup norm 1/2), so the stable-region condition in Theorem 1 is mis-stated unless ϕ=tanh.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a p-adic continuous DNN described by the integral equation h(x)=∫ W(x,y)ϕ(h(y))dy+∫ W_in(x,y)x(y)dy+ξ(x) and claims this is the thermodynamic limit of finite DNNs/RNNs with sigmoid activations. It proves uniqueness and continuous dependence of the L^2 solution in the region 0<L_ϕ‖W‖_2<1 by a contraction argument (Proposition 1, Theorem 1(i)), and shows that Picard iterates of the continuous equation can be realized by p-adic discrete DNNs (Theorem 1(ii)). It gives an algorithm to recast finite DNNs as p-adic discrete DNNs, and analyzes a toy-model edge detector for which the set of states is classified and exhibits a lattice structure. The second half develops a random version with generalized Gaussian parameters and derives formal finite-dimensional path-integral formulas and a power expansion for the output distribution.","tokens_in":38624,"tokens_out":11413,"duration_ms":116917,"significance":"The paper contains several genuinely useful explicit pieces: the contraction fixed-point theorem is clean; Lemma 4 gives an explicit discretization of the p-adic integral equation; Section 6.3 provides a constructive embedding of finite layered networks into p-adic tree structures; Section 8's toy model yields a complete, explicit classification of stationary states and their partial order; and Sections 10-11 compute the marginal network prior under stated Gaussian assumptions. If the thermodynamic-limit identification (1.1)→(1.2) were actually proved, the paper would establish a new bridge between DNNs and p-adic statistical field theories. However, that identification is introduced as an ansatz and never proved, the general infinite-state bifurcation is not proved, and the 'infinite-width' random prior is finite-dimensional. The significance is therefore prospective; the paper does not currently deliver the advertised theory.","major_comments":[{"comment":"The central premise of the paper, stated in the abstract and Introduction, is that Eq. (1.2) is the thermodynamic limit of the finite DNN (1.1) as layer widths tend to infinity. This is introduced only as 'We propose the ansatz' (Section 1). Theorem 1(ii) goes in the opposite direction: it shows that Picard iterates of the continuous equation (1.2) can be realized by p-adic discrete DNNs with finite L and Δ. It does not show that forward passes of (1.1) converge to any solution of (1.2) in any topology as widths grow. Section 6.3 fixes a prime p>N and embeds the finite network into G_{L+Δ}; p depends on N and no N→∞ limit is taken. Section 6.4 asserts that because Z is dense in Z_p 'there are continuous functions' interpolating the matrices; denseness alone is insufficient for continuous extension of arbitrary discrete data and, even with an extension, does not imply convergence of the d","section":"§1, Eq. (1.2); §6.3–6.4; Theorem 1"},{"comment":"The abstract and §7 claim that outside X_stable the unique state 'breaks into an infinite number of states'. For the general p-adic continuous DNN of Definition 2 no theorem in the paper establishes this. After Theorem 1 the paper defines constant-state sets X_α, conjectures chaotic behavior, and states that 'a precise study of the dynamics of the mentioned map is an open problem' (end of §7). The only rigorous infinite-state result is the toy model (8.3) for a>1, whose classification is quoted as Theorem 2 from the author's [26]. The general bifurcation from a unique state to infinitely many states is therefore not proven; this is a central advertised claim.","section":"Abstract; §7 (after Theorem 1); Theorem 2"},{"comment":"Definition 1 sets L_ω=‖ω‖_∞ whenever ω(0)=0. This equality is false for standard sigmoidal activations: for ω(s)=tanh(s/2), which is Lipschitz with constant 1/2 and satisfies ω(0)=0, one has ‖ω‖_∞=1. Consequently the stable region X_stable in Theorem 1(i), defined as {0<‖ϕ‖_∞‖W‖_2<1}, is not the contraction region for such activations; it is unnecessarily restrictive. The contraction argument in Proposition 1 is valid with the true Lipschitz constant L_ϕ, so the statement should use 0<L_ϕ‖W‖_2<1. The current mis-statement affects the parameter region that the paper calls critical.","section":"Definition 1; Theorem 1(i); Eq. (7.1)"},{"comment":"Section 9.2 is titled 'DNNs with infinite-width', but the integrals defining the prior in Theorem 4 are taken over D_{L+Δ}(Z_p), a space of dimension p^{L+Δ} (piecewise-constant functions on p^{-(L+Δ)}-balls). The fields h, e h, e y in the path integral are therefore finite-dimensional; no limit L+Δ→∞ is taken. Extending the parameter spaces W, W_in, ... to L^2(Z_p×Z_p) does not make the integration over h and y infinite-dimensional. Thus the 'power-type expansion' in §11 is an expansion of a finite-dimensional Gaussian-type integral, and the claim of an infinite-width network prior is not supported. This undermines the random-part portion of the abstract.","section":"§9.2–§11; Theorem 4; Remark 8"}],"minor_comments":[{"comment":"In the W_in term the notation 'Λ_{+∆,L}' appears; this should presumably be Λ_{L+∆,L}.","section":"Eq. (5.2)"},{"comment":"The heading 'The p-adic three-like structures are universal architectures' contains a typo: 'three' should be 'tree'.","section":"§6.3 heading"},{"comment":"The proof says 'there exists n0+1 such that ∥h−h_{n0+1}∥<ε'; the notation is confusing. It should be written as 'there exists n_0 such that ∥h−h_{n_0+1}∥<ε' with n_0 defined clearly.","section":"Theorem 1(ii) proof"},{"comment":"The ansatz in Remark 9 writes ϕ(h(x)) inside the integral over y; this should be ϕ(h(y)) for consistency with Eq. (1.2).","section":"Remark 9"}],"recommendation":"reject","confidential_remarks":"The paper is largely built on the author's own prior results ([14], [26]) and would need substantial new mathematical work to substantiate the central claims. I recommend rejection; a revised paper that explicitly frames the integral equation as a model rather than a proven thermodynamic limit, and that restricts the generality claims accordingly, could be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe honest one-liner: this paper contains a correct Banach fixed-point theorem for a p-adic integral equation, plus a clever embedding of finite DNNs into p-adic trees, but the headline claims—thermodynamic limit of DNNs and critical bifurcation into infinitely many states—are not established. The limit is explicitly an ansatz, and the bifurcation is proven only for the toy model of Section 8.\n\nWhat is genuinely new: the p-adic continuous DNN (Definition 2) and the universal-architecture construction (Section 6.3) that maps any finite feedforward network into a p-adic discrete DNN without increasing parameter count. Theorem 1(i) is a solid contraction argument in L^2(Z_p) giving a unique state that depends continuously on parameters. That part is self-contained and correct, modulo the caveat below.\n\nThe soft spots are real. First, the paper never shows that finite DNNs (1.1) converge to the continuous equation (1.2) as width → ∞. The intro says 'we propose the ansatz,' and Section 6.4's denseness-of-Z-in-Z_p remark is not a convergence argument. So the thermodynamic-limit claim is a heuristic, not a theorem. Second, the infinite-state bifurcation is proven only for the scalar/multiplicative toy model via prior work [26]; the general model's dynamics outside the stable region is openly left as a conjecture. Third, Definition 1 sets L_ϕ = ||ϕ||_∞, which is not generally the Lipschitz constant for sigmoids like the logistic function; the stable-region condition in Theorem 1 is therefore misstated (conservatively so). Fourth, the 'infinite-width' prior in Sections 9–11 is computed on finite-dimensional D_{L+Δ}(Z_p), so the name is misleading.\n\nWho should read this: anyone working on p-adic hierarchical network models or formal analogies between DNNs and statistical field theory. The fixed-point theorem and the embedding algorithm are worth knowing. But the paper overclaims in the abstract and would need substantial revision to separate the proven mathematical core from the conjectural thermodynamic-limit story.\n\nRecommendation: send it to peer review—the core theorem and the embedding are referee-worthy—but the referee should insist that the abstract and title be rewritten to accurately represent what is proven. As it stands, the central advertised result is an ansatz.\n\nBest.","headline":"A clean fixed-point theorem for a p-adic integral equation is buried under an unproven 'thermodynamic limit' framing; the advertised critical bifurcation is only proven in a toy model.","tokens_in":38983,"tokens_out":3248,"would_cite":false,"duration_ms":35986,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","11S80","46S10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that the true infinite-width limit of deep sigmoid networks is a single integral equation over p-adic integers, and that this equation displays a bifurcation where a unique hidden state becomes infinitely many states.","keywords":["deep neural networks","thermodynamic limit","p-adic integers","hierarchical tree structures","critical organization","bifurcation","random neural networks","network prior"],"falsifier":"Check a concrete infinite sequence of finite sigmoid networks with fixed p-adic tree wiring and widths going to infinity, and see whether the hidden states converge in L^2 to the unique fixed point of the integral equation in the stable region; if some sequence fails to converge or converges elsewhere, the thermodynamic-limit ansatz is wrong. For the multiplicity claim, exhibit two different L^2 solutions of the fixed-point equation for a single parameter set with ∥φ∥∞∥W∥_2 > 1.","tokens_in":38048,"feed_emoji":"🌳","tokens_out":7314,"duration_ms":80704,"temperature":0.7,"pith_summary":"The paper tries to establish that the thermodynamic limit of deep sigmoid networks is not a Euclidean continuum but an integral equation over p-adic integers, numbers organized as an infinite rooted tree. In the region where the activation growth constant times the L2 norm of the weight kernel is below 1, the paper proves a unique hidden state exists and depends continuously on the parameters. Outside that region, the paper argues the unique state gives way to infinitely many states, a bifurcation it identifies with the critical organization seen in neural systems near phase transitions. It also derives a network prior for random infinite-width networks: a power-series expansion whose leading term is Gaussian. If right, this would give a rigorous framework for phase transitions and criticality in deep learning.","feed_headline":"One state becomes many at a deep network's critical threshold","feed_subtitle":"At infinite width, sigmoid networks become a p-adic tree; a threshold turns one state into many.","key_machinery":"The load-bearing object is the p-adic continuous DNN defined in Definition 2: one integral equation on Z_p in which the kernel W plays the role of all weight matrices and the activation φ is applied inside the integral. The finite analogues live on quotient trees G_l = Z_p/p^lZ_p, and the discretization lemma shows the finite recursion is exactly the integral equation evaluated at tree nodes with a scale factor. The uniqueness proof is a Banach contraction argument on L^2: the fixed-point map for the hidden state contracts when the product of the activation's growth bound and the kernel norm is below 1. In the toy model, the state space for a>1 is parametrized by pairs of subsets (I+, I−) an","core_discovery":"The paper's central claim is that the thermodynamic limit of a deep or recurrent network with sigmoid activations is a single nonlinear integral equation over the ring of p-adic integers Z_p, h(x)=∫ W(x,y)φ(h(y))dy + ∫ W_in(x,y)x(y)dy + ξ(x). Z_p is described as the leaves of an infinite rooted tree, so the layers of a DNN become levels of the tree and the matrix products of the discrete recursion become kernel composition over the tree. The paper proves (Theorem 1) that when ∥φ∥∞∥W∥_2 < 1 the equation has a unique hidden state in L^2(Z_p), depending continuously on parameters and input, and that this state is approximated by discrete p-adic networks. It then argues that when the inequality","pith_inferences":["One testable prediction is that as finite-width sigmoid networks approach the threshold, collective fluctuations of the hidden state should slow down or diverge; this can be probed in ordinary training runs without p-adic machinery.","The p-adic tree structure suggests a natural renormalization-group operation: truncating the p-adic expansion coarse-grains the network, and critical exponents for the bifurcation could be extracted from how the kernel transforms under this operation.","The exclusion of ReLU matters: the contraction machinery depends on bounded activations with φ(0)=0; if a ReLU analogue has different limits, the p-adic thermodynamic limit may be specific to sigmoidal networks.","The lattice of states in the toy model implies a concrete signature for edge detectors: at criticality, the output depends on the choice of I±, making the network sensitive to initial conditions in a way that could be tested on grayscale images."],"forward_implications":["If the ansatz holds, infinite-width sigmoid networks have a well-defined continuum state whenever the weight kernel is small enough, and the state is stable under small parameter changes.","The threshold ∥φ∥∞∥W∥_2 = 1 becomes a predicted phase transition: below it the input determines the output uniquely, above it the network hosts many states.","Because any discrete DNN or RNN with sigmoids can be re-expressed as a p-adic tree network without adding parameters, hierarchical architectures are the natural setting for the thermodynamic limit, not a special case.","The random-network calculation yields an explicit prior for infinite width: the output distribution is a power series whose constant term is Gaussian, giving a concrete interface with statistical field theory.","The same bifurcation analysis extends formally to the partition function of the corresponding field theory, so the critical organization is a property of the whole network family."],"fun_headline_variants":["Sigmoid nets at infinite width: one state splits into many at a critical point","p-adic trees explain critical organization in deep networks","Infinite-width networks: a unique state bifurcates at a critical threshold","Deep nets at thermodynamic limit: criticality splits one state into many","When sigmoid nets become p-adic trees, criticality turns one state into many"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the paper's ansatz—stated but not proven—that when the number of neurons per layer tends to infinity, the discrete network dynamics converge to the continuum p-adic integral equation; if that convergence fails, the critical bifurcation picture does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Sigmoid nets at infinite width: one state splits into many at a critical point","p-adic trees explain critical organization in deep networks","Infinite-width networks: a unique state bifurcates at a critical threshold","Deep nets at thermodynamic limit: criticality splits one state into many","When sigmoid nets become p-adic trees, criticality turns one state into many"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00022,"raw_usage":{"total_tokens":1325,"prompt_tokens":825,"completion_tokens":500,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":402}},"tokens_in":569,"tokens_out":500,"duration_ms":5654,"temperature":1.0,"reasoning_tokens":402,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T07:46:17.123925+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check a concrete infinite sequence of finite sigmoid networks with fixed p-adic tree wiring and widths going to infinity, and see whether the hidden states converge in L^2 to the unique fixed point of the integral equation in the stable region; if some sequence fails to converge or converges elsewhere, the thermodynamic-limit ansatz is wrong. For the multiplicity claim, exhibit two different L^2 solutions of the fixed-point equation for a single parameter set with ∥φ∥∞∥W∥_2 > 1.","supporting_citations":[],"review_version":1}