{"id":"719761dd-d2b3-47f1-bd30-12dc10442ae3","arxiv_id":"2506.08831","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Massive Dirac operators in R^4 have L^1 to L^infty decay t^{-2} at regular thresholds, and t^{-1} plus a (log t)^{-1} finite-rank correction when threshold resonances or eigenvalues are present.","lead":"Four-dimensional Dirac equations with potentials are shown to have solutions decaying like t^{-2} in time at regular thresholds, with a precise correction term when resonances or eigenvalues sit exactly at threshold. The paper matters because it supplies the missing four-dimensional case of a standard dispersive toolkit used for nonlinear Dirac stability.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The non-regular branch of Theorem 1.1 rests on the unverified rank-two/orthogonality classification of threshold obstructions; if Corollaries 5.3–5.4 fail, Proposition 4.4 and the (log t)^{-1} bound collapse.","rationale":"The reader's weakest-assumption analysis identifies the same load-bearing point: the non-regular dispersive theorem depends on the precise finite-rank structure of the threshold obstructions. I read the full manuscript and found no contradiction in the regular-case argument or in the high-energy dyadic bounds, but the non-regular branch is genuinely delicate. The proof of Corollary 5.4 is compressed, Lemma 5.6 is explicitly omitted, and the negative-threshold adaptation is deferred; these are exactly the places where a hidden assumption could enter. Because these assertions are structural rather than mere technical refinements, the central theorem should be accepted only conditionally on completing and checking them. Since the reader already reached a CONDITIONAL verdict, my stress-test does not move the verdict; it reinforces it.","tokens_in":54412,"tokens_out":14785,"duration_ms":176165,"concrete_test":"Independently re-derive Corollaries 5.3 and 5.4 from the free resolvent kernels (14)–(17), then test Lemma 4.2 directly: for an orthonormal basis φ1, φ2 of Q = S1−S2, compute the 2×2 matrix A0 = (⟨G1v*φ_i, G1v*φ_j⟩) using G1 = (m/(2π^2)) M_uc and verify that it is nonsingular. Also verify that M_uc v*φ = 0 if and only if φ ∈ S2, and that the quotient S1/S2 has dimension equal to the rank of the map φ ↦ (∫[v*φ]_1, ∫[v*φ]_2). If these identities confirm rank(S1−S2) ≤ 2 and S2vG1 = 0, Proposition 4.4 stands; if a rank-three resonance quotient or a nonzero S2vG1 example is constructed, Theorem 1.1's non-regular claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1's non-regular case is carried by the inversion of B±(z) in Section 4 via the Jensen-Nenciu/Feshbach formula (51). The invertibility and the exact singular expansions in Propositions 4.3 and 4.4 require two structural facts from Section 5: S2vG1 = 0 (Corollary 5.3) and rank(S1−S2) ≤ 2 (Corollary 5.4). These give the block decomposition S1 = Q ⊕ S2 with Q of dimension at most two and vanishing off-diagonal G1 terms. In particular, Lemma 4.2 represents z^{-2}QA±(z)Q as g0(z)A0 + A1 with A0 a 2×2 matrix; its invertibility depends on G1v*φ1 and G1v*φ2 being linearly independent. If the resonance quotient had dimension at least three, no such 2×2 matrix exists and the 1/g± expansion for (QA±Q)^{-1}, and hence the definition of F_t in (60), fails. If S2vG1 ≠ 0, the off-diagonal terms in the Feshbach inversion (56) do not vanish and the claimed leading singularities of (B±(z))^{-1} change. Corollary 5.4's proof only shows that the non-L2 part of ψ lies in a two-dimensional space; it does not fully spell out why the map φ → M_uc v*φ has kernel exactly S2 and quotient dimension exactly two. The section also omits the proof of Lemma 5.6 and defers the negative threshold to 'straightforward modifications.' Since the non-regular result is the paper's main novelty, this is the load-bearing soft spot.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies L1-to-Linfinity dispersive estimates for massive Dirac operators D_m+V in four spatial dimensions with decaying self-adjoint matrix-valued potentials. Theorem 1.1 proves a t^{-2} bound at regular thresholds and, when a threshold is not regular, a bound with a finite-rank correction F_t satisfying ||F_t|| ≲ 1/log t and residual t^{-1}, with explicit decay assumptions (delta > 5, delta > 4, delta > 8 depending on the obstruction type). The proof combines expansions of the free Dirac resolvent near the thresholds, the symmetric resolvent identity, a Jensen-Nenciu/Feshbach inversion scheme for M_±(z), and a classification of threshold obstructions in terms of the subspaces S_1 and S_2 in Section 5. Theorems 1.2 and 1.3 add high-energy frequency-localized estimates and a near-optimal boundedness result.","tokens_in":54774,"tokens_out":14160,"duration_ms":163435,"significance":"If correct, the paper gives a complete low-energy dispersive description for four-dimensional massive Dirac operators with threshold obstructions, matching the Schrödinger analogue in [19] and extending earlier work in dimensions 1, 2, and 3. The paper is notable for its explicit decay thresholds, the explicit construction of the finite-rank operator F_t, and the absence of fitted parameters. The main risk is concentrated in the nonregular branch of Theorem 1.1, which depends on structural properties of the threshold subspaces proved in Section 5 and on the consistency of the + and - resolvent expansions. Those points need repair, but they appear to be fixable within the scope of the manuscript. The stress-test concern about the rank-two/orthogonality classification is real, although part of the missing argument can be reconstructed from Lemmas 5.1 and 5.2.","major_comments":[{"comment":"The definitions of g_+^1 and g_-^1 are inconsistent with their later use. Eq. (11) states g_+^1(z) = g_-^1(z) = z^2(a_1 log z + b_1) and Eq. (12) states g_+^2(z) = g_-^2(z) = z^4(a_2 log z + b_2). However, the paper repeatedly uses nonzero differences g_+^1 - g_-^1 and g_+^2 - g_-^2: after Eq. (19) one reads 2 Im(g_+^1(z)) = c z^2; Lemma 4.2 ends with the claim (g_+^1 - g_-^1) = c z^2; and the proof of the S_1 = S_2 case in Section 4 asserts (g_+^2(z) - g_-^2(z))/z^4 = 2 Im(b_2) ≠ 0. These statements are impossible under Eqs. (11)-(12) as written. Since the nonregular branch of Theorem 1.1, including the definition of F_t in Eq. (60), depends on the difference between the + and - resolvent expansions, please correct the definition (presumably b_1^+ ≠ b_1^- and b_2^+ ≠ b_2^-) and recheck all signs and leading terms in Propositions 4.1, 4.3, 4.4, and Corollary 4.5.","section":"Section 2, Eqs. (11)-(12)"},{"comment":"The proof of the rank bound dim(S_1 - S_2) ≤ 2 is too terse for a load-bearing statement. The calculation shows that for φ in S_1, the non-L2 part of ψ = -G_0 v* φ has the form <x>^{-2}(a_1, a_2, 0, 0)^T + O_{L2}(1). To conclude that the quotient S_1/S_2 has dimension at most two, one must also show that the map φ → M_uc v* φ has kernel exactly S_2 and that the quotient dimension equals the rank of this map. This quotient/rank step is not explicitly written. The missing step is load-bearing because Lemma 4.2 uses two-dimensionality to represent z^{-2} Q A_±(z) Q as a 2x2 matrix and to prove invertibility from the linear independence of G_1 v* φ_1 and G_1 v* φ_2. If the resonance quotient had dimension three or more, the block decomposition in Proposition 4.1, the expansions in Proposition 4.4, and the definition of F_t in Eq. (60) would fail. Please expand the proof or provide a precise reference that supplies the quotient argument in the four-dimensional setting.","section":"Section 5, Corollary 5.4"},{"comment":"The proof contains a false identity. Corollary 4.5 states that 'Definition 2.3 tells us that S_2 D_0 = S_1 D_0 = 0', but Definition 2.3(2) and its remarks state S_1 D_0 = D_0 S_1 = S_1. Consequently, the claim that the first term in the difference (M_+^{-1} - M_-^{-1}) is O_1(z^{2-}) is not justified; with S_1 D_0 = S_1, the leading contribution appears to be O(1), not O(z^2). This case drives the eigenvalue-only branch of Theorem 1.1, where F_t is claimed to be zero, so the argument needs to be redone. The final statement of Corollary 4.5 may still be true, but the proof as written does not establish it.","section":"Section 4, Corollary 4.5, S_1 = S_2 case"}],"minor_comments":[{"comment":"The proof of Lemma 2.4 says 'we leave the details to the reader'. Since D_0 is used in the expansions in Lemma 2.5 and in boundary terms in Lemma 3.8, please include a fuller proof or cite the specific lemma in [20], [23], or [26] that covers this exact operator with the four-dimensional kernels.","section":"Section 2, Lemma 2.4"},{"comment":"In the sentence 'This follows from the relationship R_0^+ - R_0^+ = (D_m + sqrt(z^2 + m^2))[R_0^+ - R_0^-]', the first difference should presumably be R_0^+ - R_0^-.","section":"Section 2, paragraph after Eq. (19)"},{"comment":"The operator D_2 appears in the expansions (B_±(z))^{-1} = -D_2/z^2 + ... and M_±^{-1}(z) = -D_2/z^2 + ... but is never defined in Section 4. It appears again in Lemma 5.6. Please define D_2 explicitly, presumably as the inverse of S_2 v G_2 v* S_2 on S_2 L^2.","section":"Section 4, Propositions 4.3-4.4"},{"comment":"The phrase 'resonance at zero' and 'eigenvalue at zero' is used even though the thresholds under consideration are λ = m and λ = -m. Please say 'at a threshold' or 'at ±m' to avoid confusion with the spectral parameter z = 0.","section":"Theorem 1.1 and Theorem 1.2, items (i)-(iii)"},{"comment":"Lemma 5.6 is stated without proof. Even if the lemma is not directly invoked in Section 4, it is part of the advertised classification of threshold obstructions. Please provide the proof or a complete statement of the three-dimensional result being adapted, including how Eq. (67) and the logarithmic terms in four dimensions are used.","section":"Section 5, Lemma 5.6"},{"comment":"The negative threshold -m is dismissed with 'straightforward modifications' in Remark 2.6 and at the end of Section 5. Theorem 1.1 quantifies obstructions at both thresholds, and the total rank bound for F_t is twice the per-threshold rank. Please state the exact changes needed for the negative branch, including the analogues of Corollaries 5.3-5.4 and Proposition 4.4.","section":"Remark 2.6 and end of Section 5"}],"recommendation":"major_revision","confidential_remarks":"The paper's reliance on prior works is methodological rather than circular, and there is no issue with fitted parameters or invented entities. The main concerns are internal consistency of the + and - resolvent expansions, a false identity in Corollary 4.5, and the terseness of the Section 5 classification. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this paper closes the last open dimension for massive Dirac dispersive estimates, and the main theorems look right. Regular thresholds give t^-2 decay; with a resonance or eigenvalue you get a finite-rank F_t of size (log t)^-1 and a t^-1 error. The genuinely new technical content is the log-weighted resolvent expansion in 4D and the two-dimensional resonance classification at each threshold. The framework leans on [19] and the authors' own 2D/3D Dirac papers, but that is legitimate inheritance, not padding.\n\nWhat is good: the low-energy analysis is careful and detailed. The Born series truncation, the oscillatory integral lemmas, and the high-energy dyadic argument are all spelled out to a level a specialist can follow. The structural core, Section 5, is correct. I traced Corollary 5.4: Lemma 5.2 shows the map from S1 to the two coefficients in M_uc v^* phi has kernel exactly S2, so the quotient dimension is at most two. The orthogonality S2 v G1 = 0 follows from the same computation. The stress-test concern about a possible rank-three resonance space does not land on reading the paper—the rank argument is valid, just too short. Likewise the linear independence in Lemma 4.2 works, but the proof is compressed.\n\nWhere the paper is actually soft: exposition and omitted details, in exactly the places that carry the non-regular case. Corollary 5.4's proof is a paragraph that hides the kernel argument; a referee will want that expanded. Lemma 5.6 is stated with \"proof is standard\" and omitted; it is not load-bearing later, so minor. Lemma 2.4 leaves details to the reader—fine but annoying. The negative threshold is \"straightforward modifications\"—acceptable. Theorem 1.3's proof is a sketch, but the result is secondary.\n\nThe decay thresholds (delta > 5, delta > 8, delta > 4) are stated explicitly, which is good practice. I see no fitted parameters or circular reasoning. I do see honest dependence on prior work.\n\nThis paper is for specialists in dispersive estimates for Dirac and Schrodinger operators. It deserves a serious referee: the main results are new, plausibly correct, and the gaps are fillable. I would send it to review with a request to expand Section 5 and either prove or fully cite Lemma 5.6 and the negative-threshold case.\n\nMy verdict: send to referee. I want the referee to check the rank-two argument line by line, but I do not expect it to fail.","headline":"Closes the 4D massive Dirac threshold-obstruction case; main theorems look right, but Section 5 is too terse exactly where it carries the non-regular argument.","tokens_in":55330,"tokens_out":4214,"would_cite":true,"duration_ms":43133,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q41","35B40","35P25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the massive Dirac equation in four dimensions with a decaying self-adjoint potential, the paper proves that the low-energy evolution decays like $t^{-2}$ when the threshold energies are regular, and like $t^{-1}$ up to a finite-rank…","keywords":["Dirac equation","dispersive estimates","threshold resonances","threshold eigenvalues","four dimensions","resolvent expansions","finite-rank corrections","massive Dirac operator"],"falsifier":"Find a self-adjoint potential with $|V(x)|\\lesssim \\langle x\\rangle^{-\\delta}$, $\\delta>5$, whose four-dimensional Dirac operator has a threshold resonance space $S_1-S_2$ of dimension three, or for which $S_2vG_1\\neq 0$; then Corollaries 5.3 and 5.4 would fail, the expansion of $M^{-1}_\\pm(z)$ in Proposition 4.4 would not have the asserted form, and the finite-rank operator $F_t$ would need rank larger than two. A more direct test is to compute numerically the low-energy kernel of $e^{itH}\\chi(H)P_{\\mathrm{ac}}(H)-F_t$ for a potential with a rank-two resonance and check whether the $L^1\\to L^\\infty$ norm decays like $t^{-1}$ with a $(\\log t)^{-1}$ correction term.","tokens_in":54203,"feed_emoji":"📉","tokens_out":6142,"duration_ms":68865,"temperature":0.7,"pith_summary":"The paper aims to settle the low-energy dispersive behavior of the massive Dirac equation in four spatial dimensions with a decaying potential. The central result is that when the thresholds $\\pm m$ are regular and $|V(x)|\\lesssim \\langle x\\rangle^{-\\delta}$ with $\\delta>5$, the absolutely continuous part of the evolution satisfies $\\|e^{itH}\\chi(H)P_{\\mathrm{ac}}(H)\\|_{L^1\\to L^\\infty}\\lesssim \\langle t\\rangle^{-2}$. When a threshold is obstructed, the evolution is instead $t^{-1}$ away from an explicit finite-rank operator $F_t$ with $\\|F_t\\|_{L^1\\to L^\\infty}\\lesssim (\\log t)^{-1}$, and $F_t=0$ if there is an eigenvalue but no resonance. A sympathetic reader would care because this gives a complete dispersive description in four dimensions, matching the known Schr\\\"odinger picture and identifying exactly how threshold obstructions degrade the decay.","feed_headline":"4D Dirac waves decay like $t^{-2}$ at regular thresholds","feed_subtitle":"Threshold resonances add only a $(\\log t)^{-1}$ finite-rank term, and eigenvalue-only obstructions leave none.","key_machinery":"The argument is carried by the resolvent identity $(D_m-\\lambda)(D_m+\\lambda)=-\\Delta+m^2-\\lambda^2$, which expresses the Dirac resolvent through the four-dimensional Schr\\\"odinger resolvent, combined with expansions of the Schr\\\"odinger resolvent near zero energy that contain logarithmic terms and the operators $G_0,G_1,G_2$. Threshold regularity is defined through invertibility of $T_0=U+vG_0v^*$, and obstructions are classified by the Riesz projections $S_1$ onto $\\ker T_0$ and $S_2$ onto $\\ker(S_1T_1S_1)$ inside $S_1L^2$. The load-bearing structural facts are that $S_1-S_2$ has rank at most two and that $S_2vG_1=0$; these make the inverse $M^{-1}_\\pm(z)$ have the specific singular expansions in Proposition 4.4, from which the finite-rank operator $F_t$ and the $(\\log t)^{-1}$ bound are obtained through the Jensen--Nenciu inversion formula.","core_discovery":"The central claim is that threshold obstructions for the four-dimensional massive Dirac operator are rare and structurally simple: at each threshold there is an at most two-dimensional resonance space and finitely many eigenfunctions. With regular thresholds, the low-energy part of the evolution has the natural $t^{-2}$ decay. When the threshold is not regular, the paper constructs a time-dependent finite-rank operator $F_t$, of rank at most two per threshold, such that $\\|e^{itH}\\chi(H)P_{\\mathrm{ac}}(H)-F_t\\|_{L^1\\to L^\\infty}\\lesssim t^{-1}$ for $t>2$, with $\\|F_t\\|\\lesssim (\\log t)^{-1}$. The eigenvalue-only case is singled out: there $F_t=0$, so no finite-rank correction is needed. The discovery is thus a complete dichotomy: either the threshold is regular and decay is $t^{-2}$, or it is obstructed and the obstruction contributes only a slowly decaying finite-rank term.","pith_inferences":["The rank-at-most-two threshold classification suggests that the four-dimensional massive and massless Dirac equations may share the same dispersive lifespan; a natural test is whether sending $m\\to 0$ reproduces the massless threshold structure and whether the $(\\log t)^{-1}$ correction survives.","The $(\\log t)^{-1}$ rate appears tied to the logarithmic terms in the resolvent expansion, so it may be optimal: a potential whose resonance space saturates the two-dimensional bound should be the extremal case where no further cancellation is available.","The proof requires $\\delta>8$ in the eigenvalue case, while the low-energy boundedness argument of Theorem 1.3 is run with only $\\delta>5/2$; a testable extension is whether a refined selective iteration could lower the eigenvalue-case decay assumption toward the $\\delta>4$ range.","The explicit form $F_t$ in (60) suggests concrete numerical checks: for a given resonance potential, one can compute the $L^1\\to L^\\infty$ norm of the corrected evolution and verify that the correction term indeed saturates the $(\\log t)^{-1}$ size."],"forward_implications":["If the thresholds are regular, the low-energy Dirac evolution in four dimensions has the natural $\\langle t\\rangle^{-2}$ decay, and pairing this with the high-energy dyadic bounds gives global dispersive bounds with $\\langle H\\rangle^{-11/2-}$ smoothness on the initial data.","If a threshold resonance or eigenvalue is present, the evolution is $t^{-1}$ up to a finite-rank term of size $(\\log t)^{-1}$, which is the strongest uniform decay one can expect in the obstructed case from this method.","When there is an eigenvalue but no resonance, $F_t=0$, so the sole effect of the eigenvalue is to reduce the rate from $t^{-2}$ to $t^{-1}$ without introducing a separate correction term.","The resolvent expansions imply a limiting absorption principle near the thresholds and, consequently, that only finitely many eigenvalues lie in the spectral gap $[-m,m]$.","The dispersive bounds yield Strichartz estimates through the standard $T^*T$ argument."],"supporting_citations":[{"why":"Supplies the four-dimensional Schr\\\"odinger resolvent expansions with logarithmic terms that are adapted here to the Dirac structure at threshold.","marker":"[19]"},{"why":"Supplies the classification of threshold resonances for three-dimensional massive Dirac operators that this paper adapts to four dimensions.","marker":"[26]"},{"why":"Supplies the streamlined inversion scheme for $B_\\pm(z)$ in the presence of resonances and eigenvalues, used at the non-regular threshold.","marker":"[21]"},{"why":"Provides the Jensen--Nenciu inversion formula that yields the singular expansion of $M^{-1}_\\pm(z)$ when $T_0$ is not invertible.","marker":"[39]"},{"why":"Establishes the Stone's formula treatment and prior two-dimensional Dirac dispersive framework that this paper extends to four dimensions.","marker":"[24]"},{"why":"Gives the limiting absorption principle for Dirac operators used in the high-energy dyadic estimates.","marker":"[20]"},{"why":"Provides the oscillatory integral lemma used to obtain $t^{-2}$ decay without spatial weights.","marker":"[23]"},{"why":"Supplies fractional-integral and weighted-$L^2$ bounds used to show the resolvent error terms are absolutely bounded.","marker":"[36]"}],"fun_headline_variants":["4D Dirac dispersion: t^{-2} regular, log^{-1} finite-rank corrections","Threshold resonances in 4D Dirac: only slow log decay corrections","Dirac waves in 4D: eigenvalue-only thresholds require no finite-rank term","Four-dimensional Dirac: obstructions are at most 2D resonances","4D Dirac decay: regular thresholds t^{-2}, obstructed log^{-1}"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands or falls on the threshold classification: at each threshold the resonance subspace is at most two-dimensional and the eigenfunction subspace satisfies the orthogonality condition $S_2vG_1=0$, and if a potential produced a larger resonance space or broke that orthogonality, the constructed finite-rank correction $F_t$ and the $(\\log t)^{-1}$ rate would not follow from the stated expansions.","fun_headline_variants_meta":{"raw":{"variants":["4D Dirac dispersion: t^{-2} regular, log^{-1} finite-rank corrections","Threshold resonances in 4D Dirac: only slow log decay corrections","Dirac waves in 4D: eigenvalue-only thresholds require no finite-rank term","Four-dimensional Dirac: obstructions are at most 2D resonances","4D Dirac decay: regular thresholds t^{-2}, obstructed log^{-1}"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000482,"raw_usage":{"total_tokens":2398,"prompt_tokens":976,"completion_tokens":1422,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":1318}},"tokens_in":592,"tokens_out":1422,"duration_ms":15581,"temperature":1.0,"reasoning_tokens":1318,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:01:13.201526+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a self-adjoint potential with $|V(x)|\\lesssim \\langle x\\rangle^{-\\delta}$, $\\delta>5$, whose four-dimensional Dirac operator has a threshold resonance space $S_1-S_2$ of dimension three, or for which $S_2vG_1\\neq 0$; then Corollaries 5.3 and 5.4 would fail, the expansion of $M^{-1}_\\pm(z)$ in Proposition 4.4 would not have the asserted form, and the finite-rank operator $F_t$ would need rank larger than two. A more direct test is to compute numerically the low-energy kernel of $e^{itH}\\chi(H)P_{\\mathrm{ac}}(H)-F_t$ for a potential with a rank-two resonance and check whether the $L^1\\to L^\\infty$ norm decays like $t^{-1}$ with a $(\\log t)^{-1}$ correction term.","supporting_citations":[{"cited_title":"B., Goldberg, M, and Green, W","cited_arxiv_id":null,"evidence_quote":"Supplies the four-dimensional Schr\\\"odinger resolvent expansions with logarithmic terms that are adapted here to the Dirac structure at threshold."},{"cited_title":"B., Green, W","cited_arxiv_id":null,"evidence_quote":"Supplies the classification of threshold resonances for three-dimensional massive Dirac operators that this paper adapts to four dimensions."},{"cited_title":"B., Goldberg, M, and Green, W","cited_arxiv_id":null,"evidence_quote":"Supplies the streamlined inversion scheme for $B_\\pm(z)$ in the presence of resonances and eigenvalues, used at the non-regular threshold."},{"cited_title":"Nenciu.A unified approach to resolvent expansions at thresholds","cited_arxiv_id":null,"evidence_quote":"Provides the Jensen--Nenciu inversion formula that yields the singular expansion of $M^{-1}_\\pm(z)$ when $T_0$ is not invertible."},{"cited_title":"B., and Green, W","cited_arxiv_id":null,"evidence_quote":"Establishes the Stone's formula treatment and prior two-dimensional Dirac dispersive framework that this paper extends to four dimensions."},{"cited_title":"B., Goldberg, M, and Green, W","cited_arxiv_id":null,"evidence_quote":"Gives the limiting absorption principle for Dirac operators used in the high-energy dyadic estimates."},{"cited_title":"B., and Green, W","cited_arxiv_id":null,"evidence_quote":"Provides the oscillatory integral lemma used to obtain $t^{-2}$ decay without spatial weights."},{"cited_title":"Duke Math","cited_arxiv_id":null,"evidence_quote":"Supplies fractional-integral and weighted-$L^2$ bounds used to show the resolvent error terms are absolutely bounded."}],"review_version":1}