{"id":"c778bc54-0bf5-4757-8cfd-4a056d7c55fc","arxiv_id":"2505.13030","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For the P(Phi)2 model on a circle, the Hamiltonian renormalisation flow with Dirichlet kernels has the known continuum theory as its fixed point, and a naively local discretised coupling flows to the correct quasi-local one in finitely many steps.","lead":"This paper treats a self-interacting scalar field on a circle with a Hamiltonian renormalisation scheme based on smooth Dirichlet kernels. It finds that the scheme's fixed point matches the known rigorous two-dimensional P(Phi)2 quantum field theory, including a finitely convergent flow for the interaction coupling.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The discretised flow supports the fixed-point claim; the projected-field flow is a consistency check, and kernel-dependence limits scope rather than correctness.","rationale":"The reader's weakest assumption correctly identifies that the projected-field flow is a consistency check: the Dirichlet-kernel intertwining identity makes the initial family the blocked-continuum family, so the fixed-point property there is a consequence of the setup. However, the discretised flow of §4.3 provides genuine nontrivial support: the naive local coupling is not a fixed point, and the flow corrects it to the quasi-local fixed point in finitely many steps. My check of the algebra confirms the flow equation (4.21) and the fixed-point verification (4.22) are correct; the convergence bound is also correct. I find that the coarse-graining ratio M′ = 3M is not load-bearing: the same argument works for any odd q > 1, yielding the same fixed point g_M, so this part of the reader's caveat is weaker than stated. The kernel-dependence remains the central scope limitation, but since the paper's claim is about its own flow, this does not falsify the claim. The one manuscript issue worth noting is the dimensionally inconsistent identity in §4.3 ('ω^{-1}_{3M} I_{M,3M} = I_{M,3M} ω^{-1}_{3M}' should read 'ω^{-1}_{3M} I_{M,3M} = I_{M,3M} ω^{-1}_M'); the subsequent flow equation shows the intended correct identity was used. Overall, no critical error was found, so the CONDITIONAL verdict stands, primarily for a clarification of the constructed versus emergent nature of the fixed point.","tokens_in":22601,"tokens_out":48095,"duration_ms":453766,"concrete_test":"Recompute the discretised flow of §4.3 with coarse-graining ratio q = 5 instead of q = 3, starting from the naive coupling (4.18). If the flow converges to the same fixed point g_M (4.15), with the convergence bound adjusted to r_k = 1 + ⌊ln(k/2)/ln 5⌋, the ratio choice is confirmed non-load-bearing. Separately, to test kernel-dependence, repeat the free-field blocking analysis of [5,6,7,8] with the Schwarz kernel and check whether the natural initial family (3.3) equals the blocked-continuum family; failure would confirm the scheme is kernel-specific.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the extent to which the fixed point is determined by the choice of coarse-graining kernel rather than discovered by the flow. In §4.2 the projected-field flow is fixed by construction: the intertwining identity ω·P_M = P_M·ω_M (4.13), valid because the Dirichlet kernel preserves L_M, forces the natural initial Fock family (3.3) to be both the blocked-continuum family and the fixed point. The nontrivial support is §4.3, where the naive local coupling (4.18) flows to the quasi-local g_M (4.15) in at most r_k = 1 + ⌊ln(k/2)/ln 3⌋ steps. This convergence is robust under the coarse-graining ratio: for any odd q > 1, the same g_M is reached because the modulo condition δ_{Σn,0 mod q^r M} becomes vacuous once q^r > k(M−1)/(2M). Thus M′ = 3M is not restrictive. The remaining kernel-dependence is real but is a scope limitation: the paper claims the result for its own scheme, and the authors flag that other kernels (e.g., Schwarz) fail even for free theories. No internal inconsistency or mathematical error was found in the flow computation, the finite-step bound, or the fixed-point verification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper applies the Hamiltonian renormalisation scheme developed in earlier works, with Dirichlet-kernel coarse graining, to the P(Φ)_2 model on a circle in finite volume. The authors define a natural discretised family in the Fock representation selected by the free Hamiltonian, with one-particle kernel ω_M^(0) = (p^2 − Δ_M)^(1/2). They show that this family coincides with the family obtained by blocking the continuum theory, and that the projected-field renormalisation flow is a fixed point because the Dirichlet projection P_M intertwines ω and ω_M (Eqs. (4.11)–(4.13)). To obtain a non-trivial check, they introduce a naive, perfectly local interaction coupling g_M^(0) in the χ-basis and derive the discrete blocking flow for the coupling constants (Eq. (4.21)). They prove by induction that after r_k = 1 + [ln(k/2)/ln 3] steps the coupling reaches the quasi-local fixed point g_M (Eq. (4.15)). The paper also gives an elementary proof in Appendix A that the normal-ordered interaction is a densely defined symmetric operator in the free Fock representation. The stated conclusion is that the scheme reproduces the known continuum P(Φ)_2 Hamiltonian at finite volume.","tokens_in":22842,"tokens_out":19810,"duration_ms":200753,"significance":"If the result stands, this is the first application of this particular Hamiltonian renormalisation scheme to an interacting quantum field theory and a non-trivial demonstration that the scheme can reproduce a known constructive-QFT fixed point. The paper's strengths include an explicit finite-step convergence bound independent of the resolution scale M; a closed-form expression for the quasi-local fixed-point coupling with a detailed analysis of its locality properties; a clear identification of the intertwining condition (4.11)/(4.13) that makes the natural initial family a fixed point; and a self-contained proof of the dense definability of the interaction in the free Fock representation. The authors are also candid about the kernel dependence of the construction, noting that the intertwining property is special to the Dirichlet kernel and fails for kernels such as the Schwarz kernel even in the free case. No internal inconsistency or mathematical error was found in the central flow computation, the finite-step bound, or the fixed-point verification.","major_comments":[{"comment":"The abstract's unconditional claim that the Hamiltonian renormalisation flow 'finds this theory indeed as a fixed point' is broader than what is demonstrated. In §4.2 the projected-field flow is a fixed point by construction: the initial Fock kernel (3.8) is the compression of the continuum kernel, and the intertwining identity (4.13) forces the natural family to be both the blocked-continuum family and the fixed point. The genuinely dynamical result is the convergence of the interaction coupling in §4.3, starting from the naive local coupling (4.18), and even there the Fock state and the free part are held fixed at the natural choice. The authors should state this scope explicitly in the abstract and conclusions, distinguishing the consistency check of §4.2 from the non-trivial flow of §4.3. This is a presentation issue, but it affects the central advertised claim and should be corrected before publication.","section":"Abstract; §4.2; §4.3"}],"minor_comments":[{"comment":"The displayed intertwining identity in the text has a type mismatch: it should read ω^{-1}_{3M} I_{M3M} = I_{M3M} ω^{-1}_M, not I_{M3M} ω^{-1}_{3M}. The subsequent equations use the correct version, so this is a typo, but it should be fixed for clarity.","section":"§4.3, text before Eq. (4.20)"},{"comment":"The stated number r_k = 1 + [ln(k/2)/ln 3] is an upper bound; when k/2 is an exact power of 3 the expression overestimates the minimal sufficient number of steps by one. The exact sufficient value is ceil(log_3(k/2)), with at least 1. Since the text says 'at most r_k steps', the claim is not wrong, but the formula could be made tighter.","section":"§4.3, bound after Eq. (4.26)"},{"comment":"The sentence 'the former is in the spirit of renormalisation schemes outside a lattice context while the former emphasises the traditional real space block spin interpretation' should read 'while the latter emphasises' in the second clause.","section":"§4, introductory paragraph"},{"comment":"The notation P(Φ)2 is easily misread; P(Φ)_2 would be clearer and consistent with the body of the paper.","section":"Title and Abstract"},{"comment":"The figures illustrating the quasi-local coupling would be easier to read with explicit axis labels and a statement of the colour scale; as presented, quantitative values are hard to extract from the interpolated surfaces.","section":"Figures 1–4"}],"recommendation":"major_revision","confidential_remarks":"The explicit computations in Section 4 are sound, and the convergence result for the interaction coupling is a genuine positive result. The main issue is framing: the headline claim that the flow 'finds' the fixed point is overstated because the fixed-point property of the state and free part is built into the choice of Dirichlet kernel and initial kernel. This is fixable by a careful revision of the abstract and conclusions, together with an explicit statement in §4.3 that the state flow is not tested from a deviating initial state. I would be happy to see the revised version; no concerns about novelty or citation practice."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is exactly what it looks like: the first time this Hamiltonian renormalisation scheme has been run on an interacting field theory, and on P(Phi)2, the standard test bed. The genuinely new technical result is in §4.3. There the authors take a naive, perfectly local initial coupling (4.18), run the discretised flow, and show it reaches the quasi-local fixed point (4.15) in at most r_k = 1 + floor(log(k/2)/log 3) steps for a polynomial of degree k. The calculation is explicit, the fixed point identity (4.22) checks out, and the finite-step bound is correct. The closed-form expression for the quasi-local coupling and the graphical analysis of its locality are real contributions.\n\nThe projected-field flow in §4.2 is a different matter. There the fixed point is built in: the initial Fock kernel is defined as the compression of the continuum operator, and the Dirichlet kernel satisfies the intertwining identity (4.13) because the derivative preserves L_M. So that part is a consistency check, not a discovery. The authors themselves say in §4.3 that one should use a deviating initial family to test the flow nontrivially, which is exactly what they then do. The abstract's phrasing 'the flow finds this theory as a fixed point' is defensible only because of §4.3; without that section it would be circular. They could make the abstract more precise.\n\nI also want to flag one thing the stress-test note got right: the choice M' = 3M is not a real weakness. The convergence argument only needs the coarse-graining ratio to be odd and greater than 1; the modulo condition goes away once 3^r M exceeds k(M-1)/2. So that detail is robust.\n\nThe genuine soft spots are scope limitations. The paper works in 1+1 dimensions on a compact circle, with the Dirichlet kernel, and it does not take the continuum limit; it shows the finite-resolution flow has the known blocked continuum theory as its fixed point. That is a consistency check on a known theory rather than a construction of a new one. The authors are clear about this, and the higher-dimensional discussion is explicitly labelled as expectation. Appendix A's proof that the interaction is densely defined in the Fock representation is a useful, elementary addition.\n\nWho is this for? People working on Hamiltonian renormalisation, constructive QFT, and canonical quantum gravity programmes that borrow these techniques. It deserves a serious referee: the algebra is sound, the main convergence claim is checked explicitly, and the interpretation issue is fixable with clearer framing. I would send it to review.\n\nMy recommendation: accept the paper for peer review, with the request that the authors distinguish the constructed fixed point of §4.2 from the emergent convergence of §4.3 in the abstract and introduction. The mathematics holds up; the framing just needs to be honest.","headline":"The real news is the finite-step convergence of a naive local coupling to the quasi-local fixed point in §4.3; the projected-flow part is a consistency check, and the paper is mostly honest about that, even if the abstract overstates it.","tokens_in":23370,"tokens_out":2186,"would_cite":true,"duration_ms":27183,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81T08","81T16","81T17"],"pacs":["11.10.Gh","11.10.Kk"],"model":"deepseek-v4-flash","headline":"The Hamiltonian renormalisation flow reproduces the continuum $\\mathrm{P}(\\Phi)_2$ model on the circle as its fixed point.","keywords":["Hamiltonian renormalisation","P(Phi)2 model","Dirichlet kernel","fixed point","constructive quantum field theory","Fock representation","coarse graining","quasi-local coupling"],"falsifier":"Compute the free-theory matrix elements that define blocking from the continuum using a non-Dirichlet kernel, such as the Schwarz kernel, with the same initial Fock family; the paper predicts a mismatch, so a match would refute the claimed necessity of the Dirichlet kernel's intertwining property. The same comparison at finite resolution $M$ for the interacting theory would show whether the fixed point is an artefact of this kernel choice.","tokens_in":22295,"feed_emoji":"🎯","tokens_out":13313,"duration_ms":120282,"temperature":0.7,"pith_summary":"This paper extends a Hamiltonian renormalisation scheme, previously tested on free fields, to an interacting quantum field theory: the self-interacting scalar field with polynomial potential in two spacetime dimensions, $\\mathrm{P}(\\Phi)_2$, on a finite circle. Its central claim is that the renormalisation flow — a blocking procedure that removes high-resolution degrees of freedom — has the known continuum $\\mathrm{P}(\\Phi)_2$ theory as a fixed point. Because this theory is one of the few interacting QFTs with a rigorous non-perturbative construction, it gives the scheme a benchmark it did not have for interacting fields. The paper shows the natural Fock family is already the one blocked from the continuum, and that a separate discrete blocking calculation drives a naive local starting coupling to the correct quasi-local fixed point in a finite number of steps.","feed_headline":"Hamiltonian flow finds interacting P(Phi)2 as a fixed point","feed_subtitle":"With the Dirichlet-kernel coarse graining, the flow converges in finitely many steps to the known continuum theory.","key_machinery":"The central object is the Dirichlet kernel $P_M(x,y)=\\sum_{n\\in Z_M}e_n(x-y)$, an orthogonal projection onto the $M$-mode subspace $L_M$ spanned by Fourier modes with $|n|\\le (M-1)/2$; it acts as a smoothed replacement for the delta distribution. The load-bearing identity is $\\omega\\,P_M = P_M\\,\\omega_M$ with $\\omega=(p^2-\\partial^2)^{1/2}$ and $\\omega_M=(p^2-\\partial_M^2)^{1/2}$, which holds because the spatial derivative $\\partial$ preserves $L_M$. In the discretised version, the same property makes the coarse-graining map $I_M$ and its threefold interpolation $I_M^{3M}$ compatible with the derivative, so the flow equation (4.21) contracts the momentum-conservation delta function $\\delta_{n_1+\\cdots+n_k,0\\,(\\mathrm{mod}\\,3^r M)}$ until the modulo constraint drops out and the quasi-local coupling (4.15) is reached.","core_discovery":"On the paper's own terms, the discovery is that the Hamiltonian renormalisation flow has the continuum $\\mathrm{P}(\\Phi)_2$ model as a fixed point, in both formulations used. In the projected-field picture the flow is fixed immediately: because the Dirichlet-kernel subspace $L_M$ is invariant under the spatial derivative, the covariance satisfies $\\omega\\,P_M = P_M\\,\\omega_M$, which makes the natural Fock family of Section 3 identical to the family obtained by blocking the continuum theory of Section 2. In the discretised picture the statement is stronger: starting from the ultra-local coupling $g^{(0)}_{M;m_1,\\dots,m_k} = M^{k-1}\\prod_{s=1}^{k-1}\\delta_{m_s,m_k}$, the blocking equation drives the coupling after at most $r_k = 1 + \\lfloor \\ln(k/2)/\\ln 3\\rfloor$ steps to the quasi-local fixed point $g_{M;m_1,\\dots,m_k}$ of eq. (4.15). The authors read this as the first demonstration that their Hamiltonian renormalisation scheme, previously applied only to free fields, works for an interacting QFT.","pith_inferences":["If the fixed-point claim is correct, the known $\\mathrm{P}(\\Phi)_2$ answer becomes a calibration target: for theories without a known continuum solution, convergence of the same flow to a quasi-local fixed point could serve as evidence that a continuum Hamiltonian exists, while non-convergence would flag a problem.","The finite-step convergence with ratio 3 suggests a testable scaling law: for a coarse-graining ratio $q$, the required number of steps may grow like $\\ln(k)/\\ln q$; running the flow for $q\\neq 3$ would show whether the bound $1+\\lfloor\\ln(k/2)/\\ln q\\rfloor$ is generic or special to this choice.","The paper leaves open non-polynomial and unbounded potentials; an editorially suggested extension is to run the same discretised flow for a potential with infinitely many terms, where finite-step convergence in the degree is no longer automatic."],"forward_implications":["At every finite resolution $M$, the fixed-point family of the flow equals the family obtained by blocking the continuum $\\mathrm{P}(\\Phi)_2$ theory, so the scheme is consistent with the known rigorous construction.","A naive ultra-local starting coupling is not preserved by blocking: after one step it becomes quasi-local, and after finitely many steps it has reached the fixed-point coupling $g_{M;m_1,\\dots,m_k}$.","For a polynomial of degree $k$, the number of blocking steps needed is at most $1+\\lfloor\\ln(k/2)/\\ln 3\\rfloor$, so convergence is logarithmically fast in the polynomial degree and independent of the resolution $M$.","The Dirichlet kernel's smoothness and derivative-invariance are responsible; with position-more-local kernels such as the Schwarz kernel, even free scalar theories fail to have the natural family as the blocked family.","In higher dimensions with a Fock representation adapted to the free Hamiltonian, the same flow would find at best a quadratic form as its fixed point rather than an operator, so additional dressing transformations would be needed to promote it to an operator."],"supporting_citations":[{"why":"supplies the known rigorous continuum $\\mathrm{P}(\\Phi)_2$ theory that the flow is claimed to reproduce as its fixed point.","marker":"[12]"},{"why":"supplies the Dirichlet-kernel coarse-graining construction and its wavelet-theoretic motivation.","marker":"[14]"},{"why":"derives the Hamiltonian projection scheme from a Euclidean reconstruction theorem; this is the method being applied to an interacting theory.","marker":"[5]"},{"why":"sets up the general Hamiltonian renormalisation flow and the consistency conditions whose fixed points define continuum theories.","marker":"[4]"},{"why":"provides the earlier free scalar-field example in 1+1 dimensions whose fixed-point analysis this paper extends to the interacting case.","marker":"[6]"}],"fun_headline_variants":["Hamiltonian flow lands on continuum P(Phi)2 fixed point","Finite-step Hamiltonian renormalisation hits P(Phi)2 continuum","P(Phi)2 emerges as fixed point of Hamiltonian renormalisation flow","Dirichlet kernel steers Hamiltonian flow to known QFT fixed point","Interacting scalar field fixed point reached by Hamiltonian flow"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result depends on the coarse-graining kernel being the Dirichlet kernel, whose projection leaves the derivative-invariant mode subspace intact; if that property fails, the natural starting family is not the blocked continuum family even for free theories, and only the separate discrete-blocking calculation supports the fixed-point claim.","fun_headline_variants_meta":{"raw":{"variants":["Hamiltonian flow lands on continuum P(Phi)2 fixed point","Finite-step Hamiltonian renormalisation hits P(Phi)2 continuum","P(Phi)2 emerges as fixed point of Hamiltonian renormalisation flow","Dirichlet kernel steers Hamiltonian flow to known QFT fixed point","Interacting scalar field fixed point reached by Hamiltonian flow"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000182,"raw_usage":{"total_tokens":1279,"prompt_tokens":882,"completion_tokens":397,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":306}},"tokens_in":498,"tokens_out":397,"duration_ms":3948,"temperature":1.0,"reasoning_tokens":306,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:22:51.320539+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the free-theory matrix elements that define blocking from the continuum using a non-Dirichlet kernel, such as the Schwarz kernel, with the same initial Fock family; the paper predicts a mismatch, so a match would refute the claimed necessity of the Dirichlet kernel's intertwining property. The same comparison at finite resolution $M$ for the interacting theory would show whether the fixed point is an artefact of this kernel choice.","supporting_citations":[],"review_version":1}