{"id":"a1789fed-0d1c-4056-8ec7-431126e332c8","arxiv_id":"1908.02785","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Group entropies generate a unifying G-algebra whose G-calculus yields a deformed Schrödinger equation, recovering q and kappa deformations as special cases.","lead":"This paper builds a generalized algebra and calculus, the G-algebra, from the formal group laws behind Tsallis and Kaniadakis entropies, and uses it to write a deformed Schrödinger equation. The construction recovers the q-algebra and kappa-algebra as special cases and maps position-dependent-mass quantum problems to constant-mass problems in a deformed space.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The G-deformed Schrödinger equation (54) is not uniquely fixed by the group entropy: adding any Hermitian h(x̂) to p̂G preserves Eq. (48) but changes Eq. (54) and the spectrum. A physical ordering principle is needed to select the paper's symmetrization.","rationale":"The algebraic construction is internally consistent: I checked the derivation of Eq. (48), the expansion leading to Eq. (53), and the √A transformation to the constant-mass equation (63), and these steps are sound. The q and κ classes do emerge as special cases, and the minor typos noted by the reader (Eq. (80), the fK entry) do not affect the central derivation. The genuinely load-bearing weakness is the Hermitian ordering assumption: the commutator alone leaves an arbitrary additive function h(x̂) in p̂G, and the infinite-well spectrum is not invariant under this freedom. Since the paper's physical predictions depend on the specific symmetrization chosen in Eq. (46), the central claim should be accepted only conditionally, pending a physical ordering principle. The reader's weakest_assumption identified the same concern, so the verdict is unchanged.","tokens_in":13368,"tokens_out":20861,"duration_ms":235629,"concrete_test":"Fix the Tsallis class, A(x) = 1 + (1-q)x, and define p̂G^λ = p̂G + λ x̂ for real λ. This operator is Hermitian and satisfies [G^{-1}(x̂), p̂G^λ] = iℏ. Solve the infinite-well eigenproblem for H^λ = (p̂G^λ)^2/(2m0) + V_well with Dirichlet conditions on [0,L], numerically or by second-order perturbation theory, for small λ and fixed q and L. Compare the eigenvalues with E_n^0 = ℏ² n² π² / (2m0 [G^{-1}(L)]²). Any λ-dependence of E_n (expected already at O(λ²)) demonstrates that Eq. (54) is not the unique G-deformed Schrödinger equation compatible with the canonical commutator, so the ordering choice must be justified on physical grounds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (48) fixes p̂G only up to an additive Hermitian function of x̂: if p̂G0 is the symmetrized operator in Eq. (46), then p̂G = p̂G0 + h(x̂) with h real is still Hermitian and still satisfies [G^{-1}(x̂), p̂G] = iℏ, because h(x̂) commutes with G^{-1}(x̂). The Hamiltonian built from p̂G then contains h-dependent terms (p̂G0 h + h p̂G0 + h²), so Eq. (54), the eigenenergies, and the zero-spacing predictions are not determined by the group entropy G alone. The paper supplies no physical argument — minimal coupling, absence of extra potentials, equivalence-principle constraints, or similar — that selects h = 0. The effective-mass formula (55) is robust, but the full deformed Schrödinger equation is ordering-dependent. This is exactly the assumption the reader flagged, and it is load-bearing because the paper's physical statements about deformed spectra and probability accumulation would change for h ≠ 0.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a 'G-algebra' and associated 'G-calculus' built from the formal group law (group entropy) G, with deformed sum, product, derivative, and integral operations. It then proposes deformed quantum operators x̂_G = G^{-1}(x̂) and a Hermitian p̂_G = (1/2){G'(G^{-1}(x̂)), p̂}, verifies the commutator [x̂_G, p̂_G] = iℏ, and derives a G-deformed Schrödinger equation (Eq. 54). The equation is rewritten as a constant-mass equation in G-coordinates (Eq. 63), and the infinite potential well is solved, yielding a position-dependent effective mass m_G(x) = m_0/[G'(G^{-1}(x))]^2 and non-uniform zero-spacing formulas for the Tsallis and Kaniadakis classes (Eqs. 79-80). The paper claims this is a unified framework that recovers the known q- and κ-algebras and q- and κ-deformed Schrödinger equations as special cases.","tokens_in":13691,"tokens_out":12542,"duration_ms":116539,"significance":"If the construction were unique and the calculations correct, the paper would provide a useful unified algebraic umbrella for deformed quantum mechanics and position-dependent mass models. The algebra/calculus part is mostly clean, the special-case reductions to q- and κ-structures are worked out in detail, and the transformation leading to Eq. (63) is algebraically consistent. However, the central quantum-mechanical claim is weakened by an operator-ordering ambiguity: the deformed momentum is not uniquely fixed by the commutator, so the G-deformed Schrödinger equation, its spectrum, and the zero-spacing predictions are not determined by the group entropy G alone. In addition, the Kaniadakis zero-spacing expression (Eq. 80) contains a sign error. These issues are load-bearing for the paper's physical conclusions and require revision.","major_comments":[{"comment":"The deformed momentum operator is not fixed by the canonical commutation relation. For any real smooth function h, the operator p̂'_G = (1/2){G'(G^{-1}(x̂)), p̂} + h(x̂) is Hermitian and still satisfies [G^{-1}(x̂), p̂'_G] = iℏ, because h(x̂) commutes with G^{-1}(x̂). Inserting p̂'_G into the Hamiltonian changes Eq. (54) by additional terms involving h and its derivatives, and therefore changes the eigenenergies and the zero-spacing formulas (76)-(80). The paper supplies no physical ordering principle, such as minimal coupling, equivalence-principle constraints, or a derivation from a Lagrangian, that selects h = 0. Consequently the G-deformed Schrödinger equation and its physical predictions are not determined by the group entropy G alone; they are conditional on the symmetrization choice made in Eq. (46). This is load-bearing because the concluding claims about the effective mass being 'univocally determined' and about deformed spectra and probability accumulation refer to this specific h = 0 equation. The effective-mass coefficient A(x)^2 is robust, but the full equation is not.","section":"Section 3.3, Eqs. (46)-(54)"},{"comment":"The Kaniadakis zero-spacing formula is incorrect. With S = κL + sqrt((κL)^2 + 1), the m-th zero is x_m = (S^{m/n} - S^{-m/n})/(2κ), so the spacing is Δ(m)_κ = x_m - x_{m-1} = (1/(2κ))[S^{m/n} - S^{(m-1)/n} - S^{-m/n} + S^{-(m-1)/n}]. The printed formula has +S^{(m-1)/n} in the first bracket and, after distributing the minus sign of the second bracket, effectively +S^{-(m-1)/n}; this does not equal the difference of consecutive zeros. The Tsallis formula (Eq. 79) is correct, but the Kaniadakis illustration and any comparison based on Eq. (80) need correction.","section":"Section 3.4, Eq. (80)"},{"comment":"The infinite-well solution assumes that x_G = G^{-1}(x) is an increasing diffeomorphism on [0, L] and that the potential boundaries transform as stated. The paper mentions this assumption and notes it holds for the Tsallis and Kaniadakis classes, but it does not prove monotonicity of G^{-1} for a general group entropy from the defining formal power series (Eq. 2), nor does it state the required conditions on the coefficients a_k. Since the abstract and introduction claim a general G-parametrized construction, the well example and the zero-spacing formulas are not justified for arbitrary G without either a proof of monotonicity or an explicit restriction of the scope to classes for which G^{-1} is monotone on the relevant interval.","section":"Section 3.4, Eqs. (70)-(71)"}],"minor_comments":[{"comment":"The inverse of the G-product is written as ⊘_G x = exp_G(−log_G(y)), but the argument should be x; the variable y is undefined and the intended expression is exp_G(−log_G(x)).","section":"Section 3, item (h)"},{"comment":"The inverse of the κ-sum is given by the q-dependent formula −x/(1 + (1−q)x), which is a copy of the q-algebra result; for the Kaniadakis class the inverse element is ⊖_κ x = −x, since x ⊕_κ (−x) = 0.","section":"Section 3.2.2, item (fK)"},{"comment":"The footnote says the standard case reduces to uniform zero spacing Δ(m) = L/m, but for G(t) = t the spacing is L/n, independent of m.","section":"Footnote 1"},{"comment":"The index n = 0 should be excluded from the eigenfunction/eigenenergy labeling; the standard infinite-well solutions begin at n = 1, and n = 0 gives a vanishing wavefunction.","section":"Eq. (74)"},{"comment":"The caption refers to m_G(x) as Eq. (75), but Eq. (75) defines the wavefunctions; the effective mass is defined in Eq. (55).","section":"Figure 1 caption"},{"comment":"The phrase 'the q-deformed (standard) Schrödinger equation results an special case' is ungrammatical, and the terminology alternates between 'Kappa' and 'Kaniadakis' classes; please unify the naming and fix the grammar.","section":"Abstract and introduction"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, quick take on Gomez & Borges, arXiv:1908.02785. The genuinely new thing is the G-algebra and the G-calculus: for any group entropy they define deformed sum/product, deformed derivative/integral, and show the q-algebra and κ-algebra drop out as special cases. That part is clean and mostly a matter of unpacking the formal group definitions, but it is done carefully and the special-case reductions are correct. The second half uses this to build a deformed Schrödinger equation from p̂G = (1/2){G′(G^{-1}(x̂)), p̂}, then rewrites it as a constant-mass equation in G-space. The commutator [G^{-1}(x̂), p̂G] = iℏ checks out, and the infinite-well solution follows straightforwardly.\n\nCredit where due: the paper is honest about what it is doing, the calculus is internally consistent, and the effective-mass formula m(x) = m0/[G′(G^{-1}(x))]^2 is correct and extends the earlier q/kappa results. The zero-spacing formula is a direct consequence of the sine ansatz in G-space, so it is not an independent prediction—but the paper does not overclaim it.\n\nSoft spots, in order. First, Eq. (80) has a sign error: the first S^{(m-1)/n} should be subtracted, not added; as written the Kaniadakis spacing is wrong. Second, the κ-algebra list contains a copy-paste error: (fK) gives the q-inverse instead of the κ-inverse, which is simply −x. Third, and more structurally: the Hermitian ordering p̂G = (1/2){A(x̂), p̂} is a choice, not a consequence. Adding any Hermitian h(x̂) to p̂G preserves the commutator but changes the Hamiltonian and spectrum. The authors never give a physical principle that selects their symmetrization. This does not invalidate the construction—it is a legitimate natural choice and the paper is framed as a proposal—but readers should not treat the deformed equation as uniquely dictated by the group entropy. Fourth, the assumption that G^{-1} is increasing on the well interval is only checked for the two examples, not proven for general G. That one is minor.\n\nBottom line: this is a useful, clearly written unification for the nonextensive-statistics community. It deserves a serious referee; once the sign and copy-paste errors are fixed, and the ordering caveat is stated, it is publishable as a mathematical framework. I would send it to review, conditional on light revision.","headline":"A genuinely useful unification of the q- and kappa-deformed algebras under the group-entropy umbrella, with a clean G-calculus and a defensible but not uniquely fixed deformed Schrödinger equation; refereeable after fixing a few concrete errors.","tokens_in":14191,"tokens_out":3632,"would_cite":true,"duration_ms":37630,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q05","81R50","82B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every group entropy $G$ produces a canonically deformed quantum mechanics with $[\\hat{x}_G,\\hat{p}_G]=i\\hbar$, and the $q$- and $\\kappa$-deformed equations emerge as special cases.","keywords":["group entropy","G-algebra","G-derivative","G-deformed Schrödinger equation","position-dependent mass","q-algebra","kappa-algebra","formal group law"],"falsifier":"Compute the commutator for an alternative Hermitian ordering, e.g. $\\hat{p}_G^{(\\delta)}=\\tfrac12\\{[G'(G^{-1}(\\hat{x}))]^{1+\\delta},\\hat{p}\\}$ with $\\delta\\neq 0$. The result is not $i\\hbar$ but $i\\hbar [G'(G^{-1}(\\hat{x}))]^{\\delta}$, so the canonical-transformation claim and the paper's $G$-deformed Schrödinger equation fail for that equally natural ordering. A measurement of the predicted asymmetric probability accumulation in the infinite well, toward the high-mass region, would select which ordering is physical.","tokens_in":13149,"feed_emoji":"⚛️","tokens_out":9121,"duration_ms":92767,"temperature":0.7,"pith_summary":"This paper tries to establish a single algebraic origin for the deformed structures used in nonextensive statistics: every group entropy $G$ defines a $G$-algebra, with $G$-sum $x\\oplus_G y = G(G^{-1}(x)+G^{-1}(y))$, a $G$-product, and an associated $G$-differential calculus. From this calculus the position operator becomes $\\hat{x}_G=G^{-1}(\\hat{x})$ and the momentum operator $\\hat{p}_G=\\frac{1}{2}\\{G'(G^{-1}(\\hat{x})),\\hat{p}\\}$, and the paper proves that these satisfy the canonical commutation relation $[\\hat{x}_G,\\hat{p}_G]=i\\hbar$. The resulting $G$-deformed Schrödinger equation is equivalent to a position-dependent-mass equation with $m(x)=m_0/[G'(G^{-1}(x))]^2$, so the $q$- and $\\kappa$-deformed equations emerge as special cases and ordinary quantum mechanics as the Boltzmann-Gibbs case. A sympathetic reader would care because this turns a family of separately built deformed calculi into one $G$-parametrized structure whose physical content is a choice of effective mass.","feed_headline":"One group-entropy formula unifies q and kappa quantum mechanics","feed_subtitle":"Each choice of group entropy sets the effective mass, and the well's zeros obey the group's addition rule.","key_machinery":"The load-bearing object is the Lazard formal group law expressed through a group entropy $G$: $\\Phi(x,y)=G(G^{-1}(x)+G^{-1}(y))$, which supplies deformed addition, subtraction, multiplication, division, and their neutral and inverse elements. On this the paper builds the $G$-algebra and a $G$-calculus whose $G$-derivative is $D_G=G'(G^{-1}(x))\\,d/dx$. The identity that carries the quantum argument is $[\\hat{x}_G,\\hat{p}_G]=i\\hbar$ with $\\hat{x}_G=G^{-1}(\\hat{x})$ and $\\hat{p}_G=\\tfrac12\\{G'(G^{-1}(\\hat{x})),\\hat{p}\\}$; because $A=G'(G^{-1}(x))$ controls both the deformed derivative and the mass $m_0/A^2$, the entire deformed Schrödinger dynamics is fixed once $G$ is chosen.","core_discovery":"The central claim is that the formal group law of group entropy theory, $\\Phi(x,y)=G(G^{-1}(x)+G^{-1}(y))$, carries enough structure to deform both algebra and quantum mechanics coherently. Replacing $x$ by $x_G=G^{-1}(x)$ and $\\hat{p}$ by the Hermitian $\\hat{p}_G=\\frac{1}{2}\\{G'(G^{-1}(\\hat{x})),\\hat{p}\\}$ preserves the canonical commutator, and the Hamiltonian built from $\\hat{p}_G^2$ gives the $G$-deformed Schrödinger equation, whose only memory of the deformation is the factor $A(x)=G'(G^{-1}(x))$. In the deformed coordinate $x_G$, the same equation is just the constant-mass Schrödinger equation with $G$-derivative $D_G=A(x)\\,d/dx$, so the physical content is a position-dependent mass $m(x)=m_0/A(x)^2$. The paper demonstrates the scheme on the infinite potential well: eigenfunctions are sines of $x_G$, energies scale with $1/[G^{-1}(L)]^2$, and the non-uniform spacing of zeros is generated by the $G$-sum, with explicit formulas for the Tsallis and Kaniadakis classes.","pith_inferences":["One could use the relation between an arbitrary group entropy $G$ and the effective-mass profile $m(x)$ to fit deformed quantum models to experimental position-dependent-mass systems; a measured $m(x)$ would then identify the group class.","The equivalence between a constant-mass problem in $x_G$ coordinates and a deformed problem in $x$ coordinates suggests a generator of new exactly solvable position-dependent-mass potentials: choose any solvable potential in $x_G$ and any increasing group entropy $G$.","The zeros-spacing formula could serve as a spectral fingerprint: counting and locating nodes of measured or calculated wavefunctions could discriminate between Tsallis and Kaniadakis deformations even when the probability densities look similar."],"forward_implications":["For every group entropy $G$, the operators $\\hat{x}_G=G^{-1}(\\hat{x})$ and $\\hat{p}_G=\\tfrac12\\{G'(G^{-1}(\\hat{x})),\\hat{p}\\}$ form a canonical pair, so the $G$-deformed quantum mechanics inherits the algebraic guarantees of standard quantum mechanics.","Choosing the Tsallis or Kaniadakis group class recovers, respectively, the $q$-algebra/$\\kappa$-algebra and the $q$-deformed/$\\kappa$-deformed Schrödinger equations, making the $G$-algebra the common parent structure.","The $G$-deformed Schrödinger equation is exactly a position-dependent-mass equation with $m(x)=m_0/[G'(G^{-1}(x))]^2$; equivalently, in the $G$-deformed coordinate $x_G$ it is a constant-mass Schrödinger equation.","For the infinite potential well, the eigenfunctions' zeros are spaced by the $G$-sum of the group law, and deformation breaks the density's symmetry, concentrating probability where the effective mass is largest.","In the classical limit the deformed probability densities tend to a delta distribution near $x=0$ for the Tsallis and Kaniadakis classes, while the standard case tends to uniform."],"supporting_citations":[{"why":"Provides the group entropy and formal Lazard group framework on which the $G$-algebra is built.","marker":"[25–30]"},{"why":"Supplies the uniqueness and characterization theorems for group entropies that justify the composable formal group structure.","marker":"[27, 29]"},{"why":"Defines the $q$-algebra and $q$-calculus that are recovered as the Tsallis special case.","marker":"[14, 15]"},{"why":"Defines the $\\kappa$-algebra and $\\kappa$-calculus that are recovered as the Kaniadakis special case.","marker":"[12, 13]"},{"why":"Establishes the position-dependent-mass and $q$-deformed Schrödinger equation that the $G$-deformed equation generalizes.","marker":"[20, 22–24]"}],"fun_headline_variants":["Group entropies deform quantum mechanics via mass","G-algebra redefines quantum wells with group entropy sums","One entropy group, two statistics, one deformed Schrodinger","Group entropy algebra yields position-dependent mass","Deformed Schrodinger from Tsallis and Kappa groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes, without derivation, that the Hermitian ordering of the deformed momentum is $\\hat{p}_G=\\tfrac12\\{G'(G^{-1}(\\hat{x})),\\hat{p}\\}$; a different Hermitian ordering would change the commutator and the entire deformed Schrödinger equation, and no physical principle fixes the choice beyond convenience.","fun_headline_variants_meta":{"raw":{"variants":["Group entropies deform quantum mechanics via mass","G-algebra redefines quantum wells with group entropy sums","One entropy group, two statistics, one deformed Schrodinger","Group entropy algebra yields position-dependent mass","Deformed Schrodinger from Tsallis and Kappa groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000652,"raw_usage":{"total_tokens":2996,"prompt_tokens":962,"completion_tokens":2034,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":1958}},"tokens_in":578,"tokens_out":2034,"duration_ms":18053,"temperature":1.0,"reasoning_tokens":1958,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:34:19.987534+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the commutator for an alternative Hermitian ordering, e.g. $\\hat{p}_G^{(\\delta)}=\\tfrac12\\{[G'(G^{-1}(\\hat{x}))]^{1+\\delta},\\hat{p}\\}$ with $\\delta\\neq 0$. The result is not $i\\hbar$ but $i\\hbar [G'(G^{-1}(\\hat{x}))]^{\\delta}$, so the canonical-transformation claim and the paper's $G$-deformed Schrödinger equation fail for that equally natural ordering. A measurement of the predicted asymmetric probability accumulation in the infinite well, toward the high-mass region, would select which ordering is physical.","supporting_citations":[],"review_version":1}