{"id":"683dc900-7df0-4693-b38f-206e242fae23","arxiv_id":"2608.03458","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper introduces a fractional Fibonacci number system and argues that densities with finite base-phi expansions are always localized in the many-body Aubry-André model.","lead":"An extension of the Fibonacci number system, called the fractional Fibonacci system, is introduced and used to classify particle densities in the many-body Aubry-André model. This classification is claimed to predict which densities are always localized and which undergo a metal-insulator transition.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The criterion is inferred from one special density: the paper's only numerical example of the always-localized class is the single-digit density F_{n-1}/F_n, while the claimed generalization to finite sums of Fibonacci ratios is unverified and no derivation from the Hamiltonian is given.","rationale":"The reader's weakest assumption is that the three numerical examples generalize to all densities. This is precisely the load-bearing gap in the paper: the only always-localized example shown is the single-digit density F_{n-1}/F_n, and the extension to finite sums of Fibonacci ratios is asserted without proof. The missing connection to spectral gap labeling is likely the correct way to prove the claim, but the paper does not make it. The thermodynamic-limit non-uniqueness acknowledged in Sec. 5 reinforces the concern, since gamma_+ and gamma_- approach the same base-phi number as gamma but give a different phase. These issues do not make the claimed criterion obviously false; the number-theoretic structure is coherent, and the physics is plausible via known properties of the almost Mathieu operator. However, the current manuscript provides insufficient evidence for the full classification, so the conditional verdict is appropriate: the central claim should be accepted only after the missing numerical or analytic support is supplied.","tokens_in":11453,"tokens_out":23500,"duration_ms":224420,"concrete_test":"For L = F_15, F_16, F_17, compute the single-particle spectrum and the many-body M2 for densities with finite base-phi expansions containing more than one nonzero digit, e.g., N/L = (F_{n-1} + F_{n-3})/F_n and N/L = (F_{n-1} + F_{n-4})/F_n. Check whether the Fermi energy lies in a spectral gap at W = 0.5t and whether the M2 size-scaling exponent is zero; repeat at W = 3t. In parallel, scan all densities N/L at L = F_17, compare the digit-count criterion with the gap-labeling criterion, and record any mismatch. If any finite-expansion density is metallic for W < 2t, or if any infinite-expansion density lies in a gap, the criterion as stated fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central classification rests on the assertion, made after three numerical examples, that 'the results were found to be general.' More concretely, the only member of the always-localized class that is actually computed is the density F_{n-1}/F_n, whose fractional Fibonacci representation is the single digit 0.1_{F_n}. The paper then extends the claim to all densities whose base-phi expansion has finitely many nonzero digits (finite sums of Fibonacci ratios). No numerical example or analytic argument is provided for any two-digit or multi-digit finite expansion, such as 0.101_phi or 0.1001_phi. The gap-labeling theorem for the almost Mathieu operator would likely supply the missing link, since finite base-phi expansions are exactly the numbers {m/phi} = {m alpha} that label spectral gaps, but the paper does not invoke or prove this connection. Without it, the classification is an extrapolation from a single data point. The thermodynamic-limit discussion in Sec. 5 compounds the problem: the paper itself shows that gamma_+ and gamma_- approach 0.1_phi yet exhibit a metal-insulator transition at W=2t, so whether a density is 'always localized' depends on the sequence used to approach it. Thus the central claim, as stated for the thermodynamic limit, lacks a unique limiting prescription.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a 'fractional Fibonacci number system' that represents fractions of the form N/F_n by writing N in the Zeckendorf representation and shifting its digits relative to the denominator F_n. The author shows that as n→∞ the fractional Fibonacci representations converge to base-φ expansions and proves several supporting number-theoretic statements in the appendix. The physical claim is a localization criterion for the non-interacting many-body Aubry-André model: for system size L=F_n, a density whose fractional Fibonacci representation is invariant as n increases is localized at all finite potential strengths W, while all other densities show a metal-insulator transition at W=2t. In the thermodynamic limit the paper claims that densities with finite base-φ expansions are always localized and all other densities transition at W=2t, with the caveat that the limit can depend on the approximating sequence. The numerical support consists of three densities (ρ=1/2, ρ=F_{n-1}/F_n, ρ=1/√3) and two additional sequences γ_± approaching the golden-ratio density from above and below.","tokens_in":11733,"tokens_out":8111,"duration_ms":71786,"significance":"If the proposed classification were established, it would provide a compact number-theoretic characterization of the localization transition in a canonical quasiperiodic model, connecting Zeckendorf and base-φ representations to finite-size scaling and many-body localization. The fractional Fibonacci number system itself is a genuine construction, the limit Eq. (13) is proved correctly, and the paper is commendably explicit about the sequence-dependence of the thermodynamic limit through the γ_± example. The deficit is that the physical criterion is asserted from three numerical examples and a single member of the 'always localized' class; the claimed generality is not demonstrated either numerically or analytically. The paper therefore has real potential but as written does not yet support its central universal claim.","major_comments":[{"comment":"The central classification is asserted rather than established. The only density in the 'always localized' class that is actually computed is ρ = F_{n-1}/F_n, whose fractional Fibonacci representation is the single digit 0.1_{F_n}. No numerical example with a multi-digit finite base-φ expansion, such as 0.101_φ or 0.1001_φ, is presented, and no derivation from the Hamiltonian is given. The sentence 'These are example calculations, but the results were found to be general' is a bare assertion. Please provide either a systematic numerical scan over many densities and system sizes or an analytic argument (for instance, linking finite base-φ expansions to the gap-labeling of the almost Mathieu operator) before the universality claim can be regarded as supported.","section":"Model and three numerical example calculations; Table I"},{"comment":"As stated, the thermodynamic-limit criterion is not well-defined. The sequences γ_+ and γ_- both converge, as real numbers, to the same density 0.1_φ, yet both show a clear transition at W=2t in Fig. 2, in contrast to the sequence F_{n-1}/F_n. Hence localization behavior is not a function of the limiting base-φ expansion alone; it depends on the sequence of finite-system representations. The discussion acknowledges this dependence for γ_±, but the abstract and introduction claim that the criterion 'remains' in the thermodynamic limit, which is at least misleading. Please reformulate the thermodynamic-limit claim with an explicit quantifier over approximating sequences, or prove that the criterion applies to a distinguished class of sequences.","section":"The thermodynamic limit; Eq. (22); Fig. 2"},{"comment":"The proof that every real number has a base-φ expansion is incomplete for irrational numbers: the greedy algorithm described ('continue until we obtain zero') terminates only for numbers with finite expansions. For an irrational R one must instead define the infinite series, prove its convergence, and show that the no-consecutive-ones condition is preserved in the limit. This is standard and fixable, but as written the appendix does not prove Eq. (25) in the claimed generality.","section":"Appendix; proof of Eq. (25)"}],"minor_comments":[{"comment":"There are several typographical errors: 'Zeckedorf' should be 'Zeckendorf', 'Not that lim' should be 'Note that lim', 'Regardig' should be 'Regarding', 'sytem' should be 'system', and 'effect' should be 'effect'.","section":"Throughout"},{"comment":"The base-φ expansions in this section and in Table I omit the overline for repeating digits: ρ=1/2 is 0.\\overline{010}_φ, not 0.010_φ, and γ_- is 0.\\overline{01}_φ, not 0.01_φ, which as written equals φ^{-2}≈0.3819 and not 1/φ. The notation should be corrected because the argument that γ_- converges to the same number as γ rests on this identification.","section":"The thermodynamic limit"},{"comment":"The enumeration of densities for L=5 contains a duplicate: the list should be 0.0001, 0.001, 0.01, 0.101 for the four densities, and the subscripts should refer to F_5 rather than F_4 if the system size is 5.","section":"The thermodynamic limit"},{"comment":"Equation (11) is written as an infinite sum even though for finite M only finitely many coefficients c_j are nonzero. State explicitly that the sum is finite for fixed M, with the infinite form intended only for the n→∞ limit.","section":"Fibonacci number system, fractional Fibonacci number system, and base-φ numbers; Eq. (11)"},{"comment":"The symbols γ_+ and γ_- are defined as limits in Eq. (22) but are then used for finite-n values. Introduce explicit finite-n notation such as γ_+^{(n)} and γ_-^{(n)} to distinguish the sequence members from their limits.","section":"The thermodynamic limit; Eq. (22)"}],"recommendation":"major_revision","confidential_remarks":"The fractional Fibonacci number system and its limit to base-φ are sound and potentially useful, but the paper's main physical claim is supported by only one example in the 'always localized' class and a bare assertion of generality. The γ_± example also shows that the thermodynamic-limit criterion is sequence-dependent, which needs to be confronted explicitly in the abstract and introduction. The manuscript could be strengthened substantially by a systematic numerical scan and, ideally, a connection to known gap-labeling results; without that, the criterion remains a phenomenological observation rather than an established result. I recommend major revision rather than rejection because the gap is plausibly fixable within the paper's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's the short version: Hetényi has a genuinely new number-theoretic packaging for a known physical classification, but the localization criterion that is the paper's headline is asserted rather than established. The fractional Fibonacci number system is well-defined, the appendix proof that it limits to base-phi is correct, and the paper does a good job organizing previous numerical results (Refs [41,44]) into a cleaner rule. Credit where due: the self-similar structure of base-phi representations is neat, and the author is honest that the criterion loses strict meaning in the thermodynamic limit.\n\nThe soft spots are real. The always-localized class is numerically demonstrated for exactly one density, rho = F_{n-1}/F_n, whose fractional Fibonacci representation is the single digit 0.1_Fn. The leap from that to \"densities with finite base-phi expansions are always localized\" rests on the sentence \"These are example calculations, but the results were found to be general\" and no systematic scan across densities or system sizes. No analytic argument from the Hamiltonian is supplied, and the gap-labeling theorem for the almost Mathieu operator, which would be the natural bridge since finite base-phi expansions are precisely the rotations {m/phi}, is never invoked. That is a missed connection, not a fatal flaw.\n\nThe thermodynamic limit discussion compounds the issue. The paper itself shows that gamma_+ and gamma_- both converge to 0.1_phi but exhibit a metal-insulator transition at W=2t, so whether a density is \"always localized\" depends on the sequence used to approach it. The author acknowledges this, but it undercuts the clean thermodynamic-limit statement in the abstract.\n\nIs it worth refereeing? Yes. The number system is new, the mathematics is correct as far as it goes, and the result would be cited if the criterion could be supported. A serious referee should ask for either a derivation for a wider family of densities or a systematic numerical sweep across many densities and system sizes, plus an explicit discussion of the gap-labeling connection. The paper is not in shape to be published as is; the central claim is a conjecture presented as a result. But it is a serious conjecture from a working physicist, not hand-waving.\n\nWho is this for? People working on quasiperiodic localization, especially the many-body Aubry-André phase diagram, and anyone who likes number-theoretic reinterpretations of spectral problems. I would bring it to reading group but would not cite it in my own work until the criterion gets tested more broadly.\n\nRecommendation: send out for peer review, but flag clearly that the localization criterion needs either proof or much more numerical evidence.","headline":"A genuinely new number-theoretic framing for the many-body Aubry-André localization diagram, but the central criterion is asserted from a single density and needs much more support before it can be trusted.","tokens_in":12214,"tokens_out":2040,"would_cite":false,"duration_ms":17559,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that in the non-interacting many-body Aubry-André model, a filling is localized for every potential strength exactly when its density has a finite base-$\\phi$ expansion, and every other filling undergoes a metal-insulator…","keywords":["fractional Fibonacci number system","base-phi number system","Zeckendorf theorem","Aubry-André model","localization transition","many-body localization","golden ratio","thermodynamic limit"],"falsifier":"Take a filling with a finite but non-ratio base-$\\phi$ expansion, such as $\\rho=\\phi^{-2}+\\phi^{-5}$, approximate it by $N/L$ with $L=F_n$, and compute the size-scaling exponent of the polarization variance for $W/t<2$. The paper's criterion predicts a zero exponent (localized for all $W$); observing the metallic size scaling with exponent one and a transition at $W=2t$ would falsify the generalization beyond the three examples.","tokens_in":11251,"feed_emoji":"🔢","tokens_out":15412,"duration_ms":124383,"temperature":0.7,"pith_summary":"This paper aims to explain a puzzling feature of the non-interacting many-body Aubry-André model: certain particle densities are localized for any potential strength, while most others undergo a metal-insulator transition at $W=2t$. The explanation is a number system. The author introduces a fractional Fibonacci number system for densities $N/F_n$, built by taking the Zeckendorf decomposition of the numerator and dividing by $F_n$, and shows that it becomes the base-$\\phi$ representation in the thermodynamic limit. The proposed criterion is that a filling stays localized for all finite $W$ exactly when its digits in this representation do not grow as the system size increases; for all other fillings a genuine transition occurs at $W=2t$. If correct, this turns the phase diagram into an arithmetic statement: the always-localized fillings are the limits of Fibonacci ratios and finite sums of their base-$\\phi$ digits.","feed_headline":"Densities with finite Fibonacci digits never delocalize","feed_subtitle":"Finite base-φ fillings stay localized for every W; all other densities turn metallic at W=2t.","key_machinery":"The central object is the fractional Fibonacci number system. For a density $\\rho=N/L$ with $L=F_n$, decompose the numerator $N$ into a sum of distinct, non-consecutive Fibonacci numbers $F_2,\\dots,F_{n-1}$ (the Zeckendorf decomposition), then divide by $F_n$; the resulting expression $\\rho=\\sum_j a_j F_j/F_n$ gives the fractional-Fibonacci digits $a_j$. The system interpolates between the integer Fibonacci (Zeckendorf) representation and the base-$\\phi$ representation, because $F_{j+m}/F_{n+m}\\to\\phi^{j-n}$ as $m\\to\\infty$, so the digits shift past the radix point and the expansion becomes $\\rho=\\sum_{j=1}^\\infty b_j\\phi^{-j}$. The localization criterion is read directly from these digits: a stable finite digit sequence under increasing $n$ means the filling is always localized, while a digit sequence that keeps acquiring new entries signals a metal-insulator transition at $W=2t$. The digits obey the no-consecutive-ones constraint, which gives the expansion its self-similar structure.","core_discovery":"For a system of size $L=F_n$ and particle number $N$, write the density as $\\rho=N/L$ in the fractional Fibonacci system: decompose $N$ into a sum of non-consecutive Fibonacci numbers (Zeckendorf), then divide each term by $F_n$, producing digits $a_j$ in $\\rho=\\sum_{j=2}^{n-1}a_j F_j/F_n$. The paper's central claim is that if this digit sequence is invariant as $n$ grows, the ground state is localized for every finite potential strength, with a vanishing size-scaling exponent of the polarization variance; if new digits keep appearing, the system is metallic for $W/t<2$ and insulating for $W/t>2$, with a transition at $W=2t$. In the thermodynamic limit, because $F_{j+m}/F_{n+m}\\to \\phi^{j-n}$, the expansion becomes $\\rho=\\sum_{j=1}^\\infty b_j\\phi^{-j}$ in base-$\\phi$. There the claim is that densities with finite base-$\\phi$ expansions—those arising as limits of Fibonacci ratios and finite sums of such terms—are always localized, while all other densities, rational or irrational, transition at $W=2t$. Three numerical examples are presented: $\\rho=1/2$ and $\\rho=1/\\sqrt{3}$ show size dependence below $W=2t$, while $\\rho=F_{n-1}/F_n\\to 1/\\phi$ does not. The paper also shows that densities $\\gamma_\\pm=(F_{n-1}\\pm 1)/F_n$, which approach $1/\\phi$ from above and below, do show a $W=2t$ transition, so the thermodynamic-limit phase can depend on the direction from which the filling is approached.","pith_inferences":["If the criterion is general, it implies the filling axis is intermingled at every scale: any density can be approximated arbitrarily well by a finite base-$\\phi$ truncation, so an always-localized filling lies arbitrarily close to a metallic one. This fractal-in-filling structure is not drawn explicitly in the paper.","A direct numerical test would scan many densities with short base-$\\phi$ expansions (e.g. sums of two or three $\\phi^{-j}$ terms) and check that the variance size-scaling exponent is zero for all $W/t$; the paper only shows three examples.","The same interpolation between an integer Zeckendorf system and an irrational-base system may generalize to other quadratic-irrational modulations, such as silver-ratio potentials, where an analogous recurrence-based numeration would predict the anomalous fillings without fitting parameters."],"forward_implications":["For finite Fibonacci-sized systems, fillings whose fractional-Fibonacci digits are unchanged as $L=F_n$ grows are localized at every finite $W$; no size-scaling transition occurs.","In the thermodynamic limit, the always-localized fillings are exactly those with finite base-$\\phi$ expansions, i.e. limits of Fibonacci ratios and finite sums of their digits; all rational fillings and all other irrational fillings transition at $W=2t$.","Approaching the localized filling $1/\\phi$ with one extra or one missing particle ($\\gamma_\\pm$) restores a $W=2t$ transition, so the infinite-size phase diagram can depend on how the thermodynamic limit is taken.","The fractional Fibonacci representation supplies a parameter-free finite-size criterion: localization can be predicted from the Zeckendorf table of $N$ alone, without diagonalizing the Hamiltonian."],"supporting_citations":[{"why":"defines the Aubry-André-Harper Hamiltonian and the single-particle localization transition at $W=2t$ that the many-body phase diagram is compared against.","marker":"[4]"},{"why":"establishes the density-potential-strength phase diagram for the many-particle model, including rational densities transitioning and some irrational densities localizing.","marker":"[41]"},{"why":"supplies the bulk-polarization variance and its scaling that the numerical localization criterion uses.","marker":"[43]"},{"why":"preceding numerical study of the same localization transition, extended here to the fractional Fibonacci criterion and larger system sizes.","marker":"[44]"},{"why":"introduces the base-$\\phi$ number system that the thermodynamic-limit form of the criterion relies on.","marker":"[48]"},{"why":"supplies the Zeckendorf theorem, the unique Fibonacci decomposition on which the fractional Fibonacci representation is built.","marker":"[52]"}],"fun_headline_variants":["Finite Fibonacci digits freeze localization","Fibonacci density code predicts localization","Zeckendorf digits decide metallic vs insulating","Base-phi fillings: finite stays localized, others switch"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the three numerical examples shown are representative of all densities: the paper asserts the results are general, but it gives no derivation from the Hamiltonian and no systematic scan along the filling axis.","fun_headline_variants_meta":{"raw":{"variants":["Finite Fibonacci digits freeze localization","Fibonacci density code predicts localization","Zeckendorf digits decide metallic vs insulating","Base-phi fillings: finite stays localized, others switch"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000236,"raw_usage":{"total_tokens":1549,"prompt_tokens":1034,"completion_tokens":515,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":650,"completion_tokens_details":{"reasoning_tokens":461}},"tokens_in":650,"tokens_out":515,"duration_ms":5009,"temperature":1.0,"reasoning_tokens":461,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:49:01.464829+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a filling with a finite but non-ratio base-$\\phi$ expansion, such as $\\rho=\\phi^{-2}+\\phi^{-5}$, approximate it by $N/L$ with $L=F_n$, and compute the size-scaling exponent of the polarization variance for $W/t<2$. The paper's criterion predicts a zero exponent (localized for all $W$); observing the metallic size scaling with exponent one and a transition at $W=2t$ would falsify the generalization beyond the three examples.","supporting_citations":[{"cited_title":"Aubry and G","cited_arxiv_id":null,"evidence_quote":"defines the Aubry-André-Harper Hamiltonian and the single-particle localization transition at $W=2t$ that the many-body phase diagram is compared against."},{"cited_title":"Jeon and S","cited_arxiv_id":null,"evidence_quote":"establishes the density-potential-strength phase diagram for the many-particle model, including rational densities transitioning and some irrational densities localizing."},{"cited_title":"Kerala Varma and S","cited_arxiv_id":null,"evidence_quote":"supplies the bulk-polarization variance and its scaling that the numerical localization criterion uses."},{"cited_title":"Het´ enyi, ”Scaling of the bulk polarization in exten ded and localized phases of a quasiperiodic model” Phys","cited_arxiv_id":null,"evidence_quote":"preceding numerical study of the same localization transition, extended here to the fractional Fibonacci criterion and larger system sizes."},{"cited_title":"Huang, D","cited_arxiv_id":null,"evidence_quote":"introduces the base-$\\phi$ number system that the thermodynamic-limit form of the criterion relies on."},{"cited_title":"Vajda, ”Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications”, Ellis Hor- wood Limited, Chichester, U.K., 1989","cited_arxiv_id":null,"evidence_quote":"supplies the Zeckendorf theorem, the unique Fibonacci decomposition on which the fractional Fibonacci representation is built."}],"review_version":2}