{"id":"6c2083df-ec31-4ebb-bee6-29c224e98545","arxiv_id":"2411.15489","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The edge zeta function of the standard non-uniform PGL3 quotient is computed exactly as a rational function, yielding an exact count of closed geodesic cycles.","lead":"Mathematicians computed a precise formula for the number of closed geodesic loops in a geometric object attached to the group PGL3 over a function field. The formula, a rational expression in powers of q, is the first exact result of its kind for a non-compact higher-rank building quotient.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5.2 asserts det(I−uT_1)=lim_k det A'_k(u) without a determinant-convergence argument; the rational zeta formula and Corollary 5.2 stand or fall on this unproved truncation limit.","rationale":"The paper gives a plausible and largely internally consistent computation: the finite block determinants are manipulated carefully, the final rational function has the expected pole/zero structure, and the trace interpretation of N_m matches the logarithmic derivative of the zeta function. The reader's weakest assumption identifies the same central gap: the equality det(I−uT_1)=lim_k det A'_k(u) in Section 5.2 is asserted rather than proved. I agree that this is the load-bearing issue. It is not a contradiction with existing consensus; it is an omitted analytical justification. The likely fix is straightforward—showing that cycles of length m are already contained in the truncation X_k for k≥m—but until that argument is supplied, the rational formula and the corollary rest on an unproven limit interchange. The apparent typo in Corollary 5.2 is secondary; the main determinant limit is the structural gap. For these reasons I do not adjust the reader's CONDITIONAL verdict.","tokens_in":18358,"tokens_out":13875,"duration_ms":122039,"concrete_test":"Prove the stabilization statement: for every m≥1 and every k≥m, every type-1 closed admissible cycle of length m lies in the truncated complex X_k, and hence Tr(T_{1,k}^m)=Tr(T_1^m). This reduces the determinant limit to coefficient-wise convergence of exp(−∑ Tr(T_{1,k}^m)u^m/m) to exp(−∑ Tr(T_1^m)u^m/m), which would justify (5.3). As a numerical spot-check, compute Tr(T_{1,3}^3) directly from the explicit T_{1,3} formulas in §5.1 for q=2 and compare with N_3=3q^6−6q^4+3q^3=120; a mismatch would immediately invalidate the truncation argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The zeta formula is obtained by two nested limiting procedures whose interchange is never justified. Section 5.1 defines det A'_k(u) as lim_{N→∞} det M_{k,N}(u), a determinant of the finite block truncation of I−uT_{1,k}. Section 5.2 then writes det(I−uT_1)=lim_{k→∞} det A'_k(u) and evaluates the k-limit by passing inside infinite sums, for instance α_{k,p(1,t)}→α_t and the sums over j leading to (5.3). Lemma 4.4 proves only weak convergence of ∑ u^n T^n/n and traceability of T^n; it does not show that the truncated operators T_{1,k} converge to T_1 in a topology that preserves Fredholm determinants, nor that the two limits commute. The missing step is load-bearing because Theorem 5.1 and the counting formula of Corollary 5.2 are exactly the result of this limit interchange. A natural repair is to prove that every type-1 closed admissible cycle of length m is contained in X_k for k≥m, so that Tr(T_{1,k}^m)=Tr(T_1^m) for all k≥m, and then to pass coefficient-wise through det(I−uT)=exp(−∑ Tr(T^m)u^m/m). Such a proof is absent, and the same gap affects the type-2 determinant in §5.3, which is reduced to type 1 by an asserted symmetry.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper defines a geometric edge zeta function for the non-uniform quotient X = PGL(3,F_q[t])\\PGL(3,F_q((t^{-1})))/PGL(3,F_q[[t^{-1}]]) and computes it explicitly. The main theorem (Theorem 1.1) states that for Γ = PGL(3,F_q[t]), the zeta function Z_Γ(q^{-s}) converges for Re s > 2 and equals the rational function (1 - q^{4-3s})^2(1 - q^{4-6s})^2 / ((1 - q^{3-3s})(1 - q^{6-3s})(1 - q^{3-6s})(1 - q^{6-6s})). The proof proceeds by introducing truncated subcomplexes X_k, computing det(I - uT_{1,k}) by finite block-matrix manipulations, passing to k → ∞, and then using a symmetry to handle the type-2 operator. A corollary gives the weighted count of closed type-1 cycles: N_m = 3q^{6r} - 6q^{4r} + 3q^{3r} if m = 3r and 0 otherwise.","tokens_in":18644,"tokens_out":7994,"duration_ms":63593,"significance":"If the main formula is correct, this is a significant new result: it provides the first explicit rational zeta function for a non-compact higher-rank building quotient and a concrete cycle-counting statement, extending the finite-complex results of Kang–Li and the tree-lattice zeta functions of Deitmar–Kang. The computation is self-contained in the sense that the determinant formula is derived rather than assumed, and no parameters are fitted to the answer. The final expression is internally consistent with the log-derivative computation in the proof of Corollary 5.2. The principal weaknesses are the unproved interchange of truncation limits and the unproved type-2 symmetry; both are local and appear repairable within the scope of the paper.","major_comments":[{"comment":"The equality det(I − uT_1) = lim_{k→∞} det A'_k(u), where det A'_k(u) = lim_{N→∞} det M_{k,N}(u), is asserted without a theorem justifying either the N-limit or the k-limit for determinants of infinite operators. Lemma 4.4 establishes only that the series ∑ u^n T^n/n converges weakly and that T^n is traceable; it does not show that the truncated operators T_{1,k} converge to T_1 in a topology that preserves Fredholm determinants, nor that the two limits commute. Since formula (5.3), Theorem 5.1, and Corollary 5.2 are exact consequences of this limit interchange, the gap is load-bearing. A natural repair is to prove that every type-1 closed admissible cycle of length m is contained in X_k for all k ≥ m, so that Tr(T_{1,k}^m) = Tr(T_1^m) for k ≥ m, and then to pass coefficient-wise through det(I − uT) = exp(−∑_m Tr(T^m)u^m/m).","section":"Section 5.2, Eq. (5.3)"},{"comment":"The sentence 'Under the definition of T2,k on Yk, the matrix representation of I − uT2,k coincides with I − uT1,k' is stated without proof. The type-2 factor of the zeta function, and therefore the full product formula in Theorem 1.1, depend on this identification. The paper should provide an explicit bijection between the edge sets E_1(X_k) and E_2(Y_k) (or a direct comparison of the matrices) to justify this claim.","section":"Section 5.3"},{"comment":"The proof of Lemma 4.4 asserts that there is α > 0 such that the series −∑_{n≥1} q^{-sn} T_k^n/n converges weakly and is traceable for |q^{-s}| < α, but no estimate on the growth of Tr(T_k^n) is given. Lemma 3.2 shows finiteness for each fixed n, not an exponential bound, and the positivity argument permits reordering sums but does not by itself imply convergence of the series. Since Proposition 4.5 and the subsequent determinant computations rely on this convergence, the lemma needs a proof or a reference for the required bound.","section":"Section 4.3, Lemma 4.4"}],"minor_comments":[{"comment":"There are several typos that should be corrected: 'trunction', 'on e of', 'disceterete', 'weigthed', and 'postive'.","section":"Abstract and Introduction"},{"comment":"The displayed formula reads N_m = 3q^{2r} − 6q^{4/3 r} + 3q^r if m = 3r, which is inconsistent with the proof just below it and with the abstract, both of which give N_m = 3q^{6r} − 6q^{4r} + 3q^{3r}.","section":"Corollary 5.2"},{"comment":"The phrase 'the operator T2,k on the set of oriented edges of color 1 in Yk' should presumably read 'type 2' rather than 'color 1'.","section":"Section 5.3"},{"comment":"Reference [CM94] contains the garbled string 'W. M/suppress Lotkowski'; the author name should be corrected to W. Mlotkowski.","section":"References"},{"comment":"The label 'en,m−n,3n' appears to be a typo for 'en,m−n,3'.","section":"Figure 10"},{"comment":"The definition of T2 is written as 'T2e = ∑_{e2→e1} w(e,e1)e1', which is confusing; the summation should be over same-type edges e' with s(e') = t(e), as in the definition of T1.","section":"Section 4.2"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the paper is likely correct in substance, but the load-bearing truncation-limit step and the type-2 symmetry assertion need to be addressed before publication. If the authors can supply the missing convergence argument and the explicit symmetry verification, the result would be a solid contribution suitable for the journal. I do not see grounds for rejection, but the current version is not yet complete."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hong and Kwon give the first explicit rational formula for an edge zeta function of a non-compact, higher-rank building quotient: the standard PGL(3,F_q[t]) quotient, plus a weighted counting formula for closed geodesic cycles. The zeta function for weighted complexes is a new object, and the truncation technique used to compute it is not a routine extension of the finite-complex work in KL14 or DKM20. The computation is long but careful, and the final formula is internally consistent with the log-derivative check.\n\nThe main soft spot is the truncation limit in Section 5.2. The paper writes det(I - u T_1) = lim_k det A'_k(u) without a proof that finite-truncation determinants converge to the Fredholm determinant of the infinite operator. Lemma 4.4 only gives weak convergence of traces of powers, so the stress-test note is right to flag this. But the gap is not load-bearing. Lemma 3.2 implies that any closed admissible cycle of length m has bounded vertical coordinate; hence, for k large enough, all cycles of length m lie in the interior of X_k, and Tr(T_{1,k}^m) = Tr(T_1^m). Coefficient-wise passage through the log-determinant formula would close the gap. A referee should ask for that argument, but the ingredients are already in the paper.\n\nSection 5.3, on type-2 cycles, is more of a problem: it asserts a symmetry between the two edge types instead of proving it. Given how heavy the type-1 computation is, this needs at least a clear statement of the isomorphism or a separate truncation for type 2 before I'd call it fully rigorous.\n\nTwo smaller issues: Lemma 4.4 is terse about absolute convergence for complex s, and Corollary 5.2 contains an apparent typo—the correct counting formula appears in Corollary 1.2. Both are fixable without touching the main result.\n\nOverall, the core computation is plausible, the authors cite the prior literature carefully, and the result is a genuine extension of zeta-function theory to infinite higher-rank quotients. The missing determinant-limit proof is real but repairable. This deserves a serious referee; I'd expect the result to survive with minor revision.","headline":"A genuinely new computation of an edge zeta function for a non-compact PGL3 quotient, with a repairable gap in the determinant-limit step.","tokens_in":19189,"tokens_out":8201,"would_cite":true,"duration_ms":67521,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37E15","20E42","22E50","20G25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves an exact rational formula for the edge zeta function of the standard non-uniform quotient complex of the PGL(3) Bruhat–Tits building by the lattice PGL(3,F_q[t]), and derives from it a closed weighted count of closed…","keywords":["edge zeta function","weighted complex","Bruhat-Tits building","PGL(3)","function field","closed geodesic cycles","Ihara zeta function","truncation"],"falsifier":"For $q=2$, enumerate all type-1 admissible cycles of length 3 in the quotient using the weight diagram of Section 3 and compute $\\sum_c w(c)\\ell(c_0)$; the formula predicts 120, so any other value would disprove the rational zeta formula. The same check can be done by computing $\\operatorname{Tr}(T_1^3)$ directly from the operator.","tokens_in":18143,"feed_emoji":"♾️","tokens_out":12076,"duration_ms":94351,"temperature":0.7,"pith_summary":"This paper defines an edge zeta function for weighted complexes—infinite quotients where each directed edge carries a weight counting how many geodesic continuations it has—and proves an exact rational formula for the edge zeta function of the standard non-uniform quotient $\\Gamma\\setminus\\mathcal{B}$, where $\\Gamma=\\operatorname{PGL}(3,\\mathbb{F}_q[t])$ and $\\mathcal{B}$ is the Bruhat–Tits building of $\\operatorname{PGL}(3,\\mathbb{F}_q(\\!(t^{-1})\\!))$. The zeta function converges for $\\operatorname{Re}(s)>2$ and equals a rational function in $q^{-s}$ with four factors. From that formula the paper extracts a closed count of weighted closed geodesic cycles: $N_m=3q^{6r}-6q^{4r}+3q^{3r}$ when $m=3r$, and $N_m=0$ otherwise. The result matters because it carries the Ihara–Bass determinant method from finite graphs and infinite trees into a genuinely higher-rank, non-compact building quotient.","feed_headline":"Counting closed cycles in a PGL3 building: exact zeta formula","feed_subtitle":"Only geodesic cycles of length divisible by 3 contribute; their weighted count is computed exactly.","key_machinery":"The engine of the proof is the operator $T_k$ on the vector space formally spanned by the oriented type-$k$ edges of $\\Gamma\\setminus\\mathcal{B}$, with matrix entries $w(e,e')$ counting the number of lifts of $e'$ that continue a lift of $e$ along a geodesic without forming a chamber. A trace computation identifies $Z_{\\Gamma,k}(u)^{-1}$ with $\\det(I-u^kT_k)$. The determinant is evaluated by truncating the complex to the subcomplex $X_k$ of vertices with $n\\le k$, applying block Gaussian elimination to the finite matrix $I-uT_{1,k}$, and taking the limit in a specific order; the recurrence for the Schur-complement blocks $B_{k,\\ell}(u)$ produces $(1-q^3u^3)(1-q^6u^3)/(1-q^4u^3)^2$ for type 1 and the same expression in $u^2$ for type 2.","core_discovery":"The central claim is that for $\\Gamma=\\operatorname{PGL}(3,\\mathbb{F}_q[t])$, the edge zeta function is given by\n$$Z_\\Gamma($q^{{-s}}$)=\\frac{(1-$q^{{4-3s}}$)^2(1-$q^{{4-6s}}$)^2}{(1-$q^{{3-3s}}$)(1-$q^{{6-3s}}$)(1-$q^{{3-6s}}$)(1-$q^{{6-6s}}$)}$$\nand converges for $\\operatorname{Re}(s)>2$. The proof goes through the type-$k$ edge zeta functions $Z_{\\Gamma,k}(q^{-s})=1/\\det(I-q^{-sk}T_k)$, where $T_k$ is the weighted adjacency operator on oriented type-$k$ edges of the quotient. The determinant is evaluated by truncating the infinite complex in a horizontal direction, computing the finite determinants with a block-matrix recurrence, and taking the limit; type 2 is handled by the same computation with $u$ replaced by $u^2$. The same calculation yields the weighted closed-cycle count $N_m=3q^{6r}-6q^{4r}+3q^{3r}$ for $m=3r$ and $N_m=0$ otherwise.","pith_inferences":["If the truncation-limit computation is as robust as it appears, the same 'truncate in one direction, take Schur complements, then let the truncation grow' scheme should yield rational edge zeta functions for other non-cocompact lattices in $\\operatorname{PGL}_n$ that admit a finitely described fundamental domain.","The count $N_m=0$ for $m$ not divisible by 3 is strong enough to test by direct enumeration of admissible cycles in the finite truncated graphs for small $q$; a single mismatch would localize exactly where the infinite-determinant limit fails.","A natural next step, already anticipated in the paper's closing remarks, is to define a chamber zeta function for this infinite quotient and test whether the edge-chamber zeta identity proved for finite complexes survives in the infinite setting."],"forward_implications":["The exact closed-cycle count follows directly: a geodesic cycle of type 1 contributes only when its geometric length is a multiple of 3, with $N_{3r}=3q^{6r}-6q^{4r}+3q^{3r}$.","The Euler product over primitive geodesic cycles converges in the half-plane $\\operatorname{Re}(s)>2$, giving a well-defined zeta function for this non-compact quotient.","The full edge zeta function is the product of two identical rational factors, one for each edge type, confirming the type-1/type-2 symmetry of the building.","The determinant formula extends the Bass–Ihara relation to an infinite weighted complex in higher rank, where the operator $T_k$ is traceable although the complex is non-compact."],"supporting_citations":[{"why":"Supplies the finite-complex edge zeta definition, algebraic length, and the determinant formula that the paper extends to the infinite weighted quotient.","marker":"[KL14]"},{"why":"Supplies the weighted tree-lattice zeta function and its convergence for cuspidal lattices, the geometric model the paper adapts to higher rank.","marker":"[DK18]"},{"why":"Used for Lemma 2.2, the fact that a type-1 geodesic segment in the building has q^2 possible geodesic continuations, fixing the weight normalization.","marker":"[CM94]"},{"why":"Provides the reduction theory over a rational function field that gives the fundamental-domain double coset decomposition used throughout Section 3.","marker":"[Pr03]"},{"why":"Earlier computation of the weight adjacency operator on this non-uniform quotient, used for the fundamental domain and weight patterns.","marker":"[HK24]"},{"why":"Establishes the Bass–Ihara determinant formula for tree lattices in the compact case, the finite-graph analogue this paper's determinant formula generalizes.","marker":"[Ba92]"}],"fun_headline_variants":["Exact closed-cycle count from PGL3 building zeta function","Edge zeta function yields exact geodesic cycle counts","PGL3 building: rational zeta function and cycle counts","Zeta formula for non-uniform complex: exact cycle number","Closed cycles in PGL3 complex counted exactly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the determinant of the infinite operator $I-uT_1$ is the limit of the determinants of the finite truncated operators $I-uT_{1,k}$; the paper states this limit in Section 5.2 without proving the interchange, and the convergence lemma it proves covers only traces of powers of $T_k$, not determinants.","fun_headline_variants_meta":{"raw":{"variants":["Exact closed-cycle count from PGL3 building zeta function","Edge zeta function yields exact geodesic cycle counts","PGL3 building: rational zeta function and cycle counts","Zeta formula for non-uniform complex: exact cycle number","Closed cycles in PGL3 complex counted exactly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00075,"raw_usage":{"total_tokens":3326,"prompt_tokens":917,"completion_tokens":2409,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":2328}},"tokens_in":533,"tokens_out":2409,"duration_ms":15885,"temperature":1.0,"reasoning_tokens":2328,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:13:41.199722+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $q=2$, enumerate all type-1 admissible cycles of length 3 in the quotient using the weight diagram of Section 3 and compute $\\sum_c w(c)\\ell(c_0)$; the formula predicts 120, so any other value would disprove the rational zeta formula. The same check can be done by computing $\\operatorname{Tr}(T_1^3)$ directly from the operator.","supporting_citations":[],"review_version":1}