{"id":"ccff6bfe-c86e-4c3d-89ab-331a48e7f6a8","arxiv_id":"2610.02138","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":4,"one_line_summary":"Exact analytic power spectrum and squeezed bispectrum for a strongly mixed two-field inflationary system with unequal sound speeds, expressed through Heun functions and verified numerically.","lead":"A calculation derives exact normalized modes for two mixed inflation fields that travel at different sound speeds, and extracts the power spectrum and primordial non-Gaussianity they produce. It shows that a much slower entropy field can strongly enhance curvature power in three distinct regimes, while a faster one decouples.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Integral identity (A.3) is actually derived in Appendix A.1, not merely numerically verified; the reader's keystone concern is largely defused.","rationale":"The reader's ACCEPT verdict with a modest gap flagged on identity (A.3) is correct in outcome but overstates the gap: the appendix actually proves the identity analytically, and the most delicate step (the hypergeometric bracket annihilating the kernel) checks out against the standard ODE of K. Load-bearing it is—everything in (1.8) rests on (5.13)—but the independent numerical confirmation at 2×10⁻¹¹ across orderings and real/imaginary ν, plus the internal consistency of the derivation, leaves no substantive risk. The only asserted-not-expanded step is the '(Du−Dv)K = −(u²+v²)[...]' factorization labeled 'by the chain rule'; this is mechanical and testable, not a conceptual gap. A further potential concern, the rc→1 limit (Eqs. 5.20–5.25) relying on asserted hypergeometric connection coefficients, is also validated by the numerical agreement at rc=1 against the gamma-function reference, so it does not rise to a load-bearing objection. I accordingly see no reason to adjust the ACCEPT verdict; the reader's weakest_assumption is partially agreed with (the identity is the keystone) but its characterization as 'verified only numerically' is too strong, since a valid closed-form derivation is present. The concrete test isolates the one remaining unexpanded step, settling the question definitively if run.","tokens_in":53048,"tokens_out":6920,"duration_ms":45026,"concrete_test":"Symbolically expand (Du−Dv)K(s(u,v)) with Du, Dv as in (A.6) and s(u,v) as in (A.2), verifying the claimed factorization into −(u²+v²)[4s(1−s)K″+6(1−s)K′+(ν²−1/4)K]. If the factorization holds, Eqs. (A.3) and (5.13) follow rigorously and the numerics in Fig. 6 are corroboration rather than evidence. As a secondary check, evaluate the right-hand side of (A.3) at a grid of u values for one heavy (ν=i, λ=1, rc=2) and one light (ν=0.8, λ=0.7, rc=0.5) case against direct numerical integration of (4.3) with the same normalization; agreement to machine precision settles any residual branch/contour subtlety.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader flags Eq. (A.3) as the keystone of the exact spectrum (5.15)/(1.8), asserting it is 'verified only numerically, with no independent closed-form proof.' That characterization is too strong. Appendix A.1 supplies an analytic derivation: the integral is shown to satisfy the spectral equation (via the operator identity Du−Dv acting on the kernel), to be analytic at u=rc with unit value, and—since the local analytic solution of the second-order ODE at the regular point rc is unique up to scale—to coincide with Ŷ1. The most delicate step is the claimed factorization (Du−Dv)K = −(u²+v²)[4s(1−s)K″+6(1−s)K′+(ν²−1/4)K] (Eq. A.7, 'by the chain rule'). I checked the bracket against the actual ODE of the kernel: with K(s)=s^{−1/2}2F1(−1/4+ν/2,−1/4−ν/2;1/2;s), the standard hypergeometric gauge transformation (leading exponent −1/2 ⇒ γ′=3/2) gives exactly s(1−s)K″+(3/2)(1−s)K′−(1/16−ν²/4)K=0, which multiplied by 4 reproduces the bracket in (A.7). So the identity's internal structure is consistent at its critical point; what remains asserted rather than displayed is only the factorization itself (the u²+v² prefactor), a tedious but mechanical computation. The identity is additionally supported by independent time-domain integration of the coupled field equations to better than 2×10⁻¹¹ in relative power across both sound-speed orderings and real and imaginary ν (Fig. 6). Consequently I find no hard objection to the central claim; the residual risk is concentrated in the single unexpanded chain-rule step, which is testable directly.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper derives the canonically normalized Bunch–Davies mode functions of a two-field inflationary system with arbitrary constant quadratic mixing λ and unequal sound speeds cφ≠cσ (ratio rc=cσ/cφ), using a Laplace representation in which the coupled dynamics reduces to a single second-order Heun equation. The central result is Eq. (1.8)/(5.15): the exact late-time curvature power spectrum, valid for µ²>0 without expanding in λ or rc, expressed as a sum of two squared Heun functions with parameters (1.9). The spectrum recovers the unmixed result at λ=0 (Eq. 5.19) and the equal-speed gamma-function result continuously as rc→1 (Eq. 5.25). At rc≪1 the authors establish power-law enhancement ~λ²|C|² rc^{-2χ} with χ=√(ν²+λ²), logarithmic growth ln²(1/rc) at χ=0, and bounded oscillations in ln rc for imaginary χ; at rc≫1 the correction to the unmixed spectrum vanishes as λ² ln(rc)/rc². The same modes feed a tree-level isosceles bispectrum for three cubic interactions (α,β,γ), giving squeezed branches q^{3/2∓ν} with coefficients b_{I,±} and, for heavy fields, an oscillation of amplitude |b_{I,+}| and phase arg b_{I,+} at fixed frequency ρ=√(µ²/H²−9/4). The manuscript reports extensive internal verification: analytic λ→0 and rc→1 limits; independent numerical integration of the coupled time-domain equations agreeing below 2×10⁻¹¹ in relative power (Fig. 6); heavy-field bispectrum matches (Table 1) including reproduction of the equal-speed result of Ref.","tokens_in":53474,"tokens_out":19973,"duration_ms":113304,"significance":"If correct, this is a substantial advance for the cosmological-collider program: it is the first exact analytic treatment of the strongly mixed unequal-sound-speed system, free of expansions in λ and rc, and it produces falsifiable hierarchy laws (power-law, logarithmic, and oscillatory regimes at rc≪1; decoupling at rc≫1) with a clear physical interpretation in terms of the ordering of sound-horizon crossings and the intermediate mass combination m0²=µ²−λ²H². The bispectrum result shows non-monotonic rc-dependence of the collider amplitude and phase at fixed squeezed frequency (Fig. 1). I gave particular scrutiny to the keystone integral identity (A.3) in Appendix A.1, which the reader's report flagged as only numerically verified. That characterization is too strong: the identity is derived analytically by showing the integral satisfies the spectral equation, is analytic at u=rc with unit value, and therefore coincides with Ŷ1 by the uniqueness of the local analytic solution; I checked directly that the bracket in Eq.","major_comments":[],"minor_comments":[{"comment":"The factorization (D_u−D_v)K = −(u²+v²)[4s(1−s)K″+6(1−s)K′+(ν²−1/4)K] is stated \"by the chain rule\" without the computation. I verified that the bracketed expression is exactly the hypergeometric ODE for K(s)=s^{−1/2}2F1(−1/4+ν/2,−1/4−ν/2;1/2;s), so the bracket vanishes and the identity (D_u−D_v)K=0 holds regardless of the prefactor. Since this is the keystone of Eq. (5.13) and hence of (1.8), a two-line demonstration that the bracket is the hypergeometric equation would make the step transparent.","section":"A.1, Eq. (A.7)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The connection coefficients C_w in Eq. (5.22) are said to \"follow from the hypergeometric connection formulas for the merging singularities\" with no derivation. The result is consistent with the headline check (5.25) and with Fig. 6, but since the rc→1 limit is one of the two advertised consistency checks, a short derivation in Appendix A.2 would strengthen the manuscript.","section":"5.2, Eqs. (5.20)–(5.22)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The statement that the squeezed coefficients are \"specified explicitly in Eq. (7.22)\" is stronger than what is delivered: the b_{I,±} depend on the integrals M_{I,η} defined in Eqs. (7.18)–(7.20), which are evaluated numerically (Tables 1 and 2) rather than in closed form. Suggest softening to \"specified by Eqs. (7.22), (A.34), and (7.18)–(7.20)\".","section":"7.2, Eq. (7.22)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The caption does not list the parameter values (λ, ν, rc) of the six numerical agreement points, nor which side of rc=1 and which real/imaginary-ν cases they cover; the 2×10⁻¹¹ claim would be reproducible from the caption alone if these were given. The text also refers to \"the accompanying numerical material\" without a pointer; a repository or DOI should be provided so the stated tolerances are checkable.","section":"Fig. 6 and footnote 2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence identifying the condition Reν<3/2 \"with the condition µ²>0\" may confuse readers unfamiliar with the heavy-field branch: for µ²>0 one has either real ν<3/2 or imaginary ν with Reν=0, so Reν<3/2 holds automatically. A one-line restatement of this would preempt misunderstanding.","section":"5.1, after Eq. (5.4)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The applicability condition q≪min(rc,1/rc) for the strict squeezed form is an important practical restriction, especially for strongly hierarchical sound speeds; a sentence in the Conclusions noting this would help readers who use only Eq. (1.13).","section":"7.3, Eq. (7.27)","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"accept","confidential_remarks":"The manuscript builds on the authors' own preprints Refs. [49] and [50] and on the Laplace-space formalism of Belrhali, Poisson and Renaux-Petel (Refs. [53], [68]); the unequal-speed content is genuinely new. The equal-speed spectrum and bispectrum are re-derived in-paper (Sections 2 and 7.4), so citation to the preprints is adequate rather than a gap. One remark for the editor only: the Acknowledgments include a statement about access to OpenAI models through a grant program; this is unusual but not a scientific issue, and does not affect my recommendation. The fit with JCAP's scope is appropriate, and the level of internal and numerical verification is well above the journal's typical standard."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"For your file on 2610.02138: this paper does what it says. It takes the equal-speed strong-mixing solutions of the authors' own Refs. [49, 50], relaxes the equal sound-speed assumption, and gives exact closed-form modes and a power spectrum with no expansion in either lambda or rc. The central object, Eq. (1.8), expresses the late-time spectrum as two squared Heun functions with the parameters in (1.9). It recovers the unmixed limit at lambda=0 and the equal-speed gamma-function result continuously as rc→1. The hierarchy laws in Sections 6.1–6.3 are the real content: power-law enhancement ~lambda^2 |C|^2 rc^{-2 sqrt(nu^2+lambda^2)} for rc<<1, ln^2 growth at chi=0, bounded oscillations in ln rc for imaginary chi, and the ln rc / rc^2 decoupling for rc>>1. These are new and testable. The squeezed bispectrum with exact mixing, including the phase-modifying hierarchy dependence, is a genuine extension of the collider program.\n\nThe numerical verification is strong and independent. The time-domain integration of the coupled field equations, which does not use the Heun construction, agrees below 2e-11 in relative power across both sound-speed orderings and real and imaginary nu, and the bispectrum coefficients agree to better than 1e-9 (Tables 1 and 2).\n\nOn the reader's flagged keystone: I think that concern is overstated. Identity (A.3) is not merely verified numerically — Appendix A.1 derives it. The contour representation, differentiation under the integral, the operator identity (A.7) transferring D_u−D_v onto the kernel, and the normalization at u=rc with unit value, together with uniqueness of the local analytic solution, constitute a real analytic argument. There is one asserted step: the chain-rule factorization in (A.7), where the (u^2+v^2) prefactor is stated but not expanded. I checked the bracket against the actual hypergeometric equation of K and it matches, so this looks like tedious-but-mechanical rather than wrong. A minor gap, not a load-bearing flaw.\n\nThe limits are not uniform in mu^2→0, and the squeezed formula carries a validity condition (7.27) with the rough estimate q << min(rc, 1/rc); both are stated by the authors, which is fair. Scope is acknowledged: isosceles triangles only, three representative cubic interactions. That is a sensible first pass.\n\nBottom line: a careful, honest derivation with real internal and numerical checks. The citation pattern is unproblematic — it builds on the authors' own equal-speed results and the Laplace formalism of Ref. [53] (and citations to published sources like DLMF are fine). I would cite it and would bring it to the reading group. It deserves a serious referee; the one residual ask is that the chain-rule step in (A.7) be displayed in full rather than asserted.","headline":"Extends the equal-speed strong-mixing solutions to unequal sound speeds with an exact Heun-function spectrum and explicit hierarchy laws; the reader's keystone worry is mostly defused because identity (A.3) is actually derived, not just numerically checked.","tokens_in":54128,"tokens_out":2322,"would_cite":true,"duration_ms":17456,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"This paper derives the exact late-time curvature power spectrum of a two-field inflationary system at arbitrary quadratic mixing and unequal sound speeds, showing it reduces to a sum of two squared Heun functions whose parameters are…","keywords":["inflation","cosmological collider","primordial non-Gaussianity","two-field inflation","unequal sound speeds","Heun equation","primordial power spectrum","squeezed bispectrum"],"falsifier":"Independently integrate the original coupled field equations in the time domain at masses and mixings outside the six tested points, especially near the resonant cases $\\chi = 0$, $\\nu = 0$, $\\nu = 1/2$ and $\\nu = 1$, and compare the resulting curvature power against the Heun-formula spectrum (1.8); agreement would support the central identity (A.3), while a discrepancy at the level of its numerical uncertainty would reveal a branch or contour issue. For the bispectrum, testing squeezed configurations with $r_c$ far from unity, where $q \\ll \\min(r_c, 1/r_c)$ may fail, would settle whether the quoted hierarchy estimates for the amplitude and phase hold.","tokens_in":54425,"feed_emoji":"🌌","tokens_out":8398,"duration_ms":46340,"temperature":0.7,"pith_summary":"This paper claims that the two-field inflationary system with constant quadratic mixing $\\lambda$ and unequal sound speeds $c_\\varphi$, $c_\\sigma$ admits an exact, closed-form late-time curvature power spectrum for any positive entropy mass $\\mu^2 > 0$, with no expansion in the mixing or the sound-speed ratio. The spectrum reduces to a sum of two squared Heun functions whose parameters are fixed by the mass, the mixing, and the ratio $r_c = c_\\sigma/c_\\varphi$, recovering the unmixed result at $\\lambda=0$ and the equal-speed gamma-function result as $r_c\\to 1$. At fixed mixing and $r_c\\ll 1$ the power is enhanced as $r_c^{-2\\sqrt{\\nu^2+\\lambda^2}}$, grows as $\\ln^2(1/r_c)$ at vanishing exponent, or oscillates in $\\ln r_c$, while for $r_c\\gg 1$ the correction to the unmixed spectrum vanishes as $\\lambda^2\\ln r_c/r_c^2$. The same normalized modes give tree-level squeezed bispectra for three cubic operators, with the heavy-field oscillation frequency set strictly by the final entropy mass. This matters because strong mixing can bring inflationary collider signals within observational reach, and unequal sound speeds change both the primordial power and the collider amplitude and phase.","feed_headline":"Unequal sound speeds tamed into one exact power-spectrum formula","feed_subtitle":"Curvature and entropy couple at any mixing strength; closed form shows power-law growth or decay between sound horizons.","key_machinery":"The Laplace representation: the coupled field equations are recast through contour integrals over plane waves, reducing the dynamics to a single second-order differential equation in a spectral variable $u$, which for unequal sound speeds is a general Heun equation with four regular singular points at $u = \\pm1$ and $u = \\pm r_c$. Contour choices select the positive-frequency incoming curvature and entropy waves, with Bunch\\textendash Davies conditions and the conserved inner product fixing the two normalized mode functions $Y_1$ and $Y_2$. The exact spectrum follows once the frozen amplitudes are reduced to Heun values at the origin through the integral identity (A.3), which connects the even imaginary-axis spectral solution to the endpoint value at the entropy singularity; the identity is derived in Appendix A.1 and verified numerically to about $2\\times10^{-11}$ in relative power at six representative points.","core_discovery":"The central claim is Eq. (1.8): for $\\mu^2 > 0$, the exact late-time curvature power spectrum of the two-field system at arbitrary quadratic mixing $\\lambda$ and sound-speed ratio $r_c$ is a sum of two squared Heun functions, $\\mathrm{HeunG}(1-r_c^{-2}, Q_1; \\tfrac14-\\tfrac{\\nu}{2}, \\tfrac14+\\tfrac{\\nu}{2}, 1, \\tfrac12; 1)$ and $\\mathrm{HeunG}(1-r_c^2, Q_2; \\tfrac54-\\tfrac{\\nu}{2}, \\tfrac54+\\tfrac{\\nu}{2}, 2, \\tfrac12; 1)$, times fixed gamma-function factors, with accessory parameters $Q_1$, $Q_2$ defined in Eq. (1.9). As the mixing vanishes the spectrum reduces continuously to the unmixed result, and as $r_c\\to 1$ it tends to the equal-speed gamma-function expression of the earlier exact solutions. The associated hierarchy laws state that at fixed nonzero mixing the small-$r_c$ regime is controlled by the combination $\\chi = \\sqrt{\\nu^2+\\lambda^2}$: the entropy contribution grows as $\\lambda^2|C(\\lambda,\\nu)|^2 r_c^{-2\\chi}$, grows as $\\ln^2(1/r_c)$ at $\\chi=0$, and oscillates as a bounded function of $\\ln r_c$ for imaginary $\\chi$, while for $r_c\\gg 1$ the leading correction to the unmixed spectrum is $\\lambda^2[2\\ln r_c + B(\\nu)]/r_c^2$ plus a subleading $r_c^{-3}$ term, a decoupling limit that holds even at large mixing. For the squeezed bispectrum of the three cubic operators the paper obtains coefficients $b_{I,\\pm}$, which for a heavy field with $\\nu = i\\rho$ combine into an oscillation of logarithmic frequency $\\rho$ fixed by the final entropy mass; unequal sound speeds modify the amplitude and phase, with the amplitude non-monotonic in $r_c$, but preserve the frequency and the $q^{3/2}$ squeezed scaling.","pith_inferences":["If the integral identity (A.3) admits a closed-form proof, it would constitute a new connection formula for the general Heun equation, a special-function result applicable well beyond cosmology.","Because the heavy-field oscillation frequency is robust to $r_c$ while the amplitude and phase vary, inferring the entropy mass from collider detections should marginalize over the sound-speed ratio, which otherwise enters only amplitude and phase.","The non-monotonic amplitude with $r_c$ implies that templates fixed at $r_c = 1$ could substantially misestimate collider amplitudes; full-shape surveys would need to scan $r_c$ alongside mass and mixing.","The two-sound-horizon picture suggests the $\\chi = \\sqrt{\\nu^2+\\lambda^2}$ enhancement is generic to any multi-field setup whose extra field crosses its sound horizon before curvature, so the hierarchy laws may transfer to models with derivative couplings beyond the constant-mixing case."],"forward_implications":["The two-field power spectrum at any mixing and sound-speed ratio is available in closed form, so template building for CMB and large-scale-structure searches need not expand in either parameter.","At slow entropy speed ($r_c \\ll 1$) the spectrum can be strongly enhanced as $r_c^{-2\\sqrt{\\nu^2+\\lambda^2}}$, grow as $\\ln^2(1/r_c)$, or oscillate in $\\ln r_c$, depending only on the combination $\\nu^2+\\lambda^2$.","At fast entropy speed ($r_c \\gg 1$) the system decouples: the correction to the unmixed spectrum vanishes as $\\lambda^2[2\\ln r_c + B(\\nu)]/r_c^2$ even for order-one mixing.","For heavy entropy fields the squeezed bispectrum is an oscillation in $\\ln(k_S/k_L)$ with frequency fixed by $\\mu/H$; the sound-speed hierarchy changes the amplitude and phase, and can delay the approach to the squeezed regime."],"supporting_citations":[{"why":"Supplies the equal-speed exact mode functions and power spectrum that this paper extends to unequal sound speeds; its normalization and phase convention are inherited.","marker":"[49]"},{"why":"Provides the equal-speed bispectrum framework and the three cubic interactions whose normalization the squeezed coefficients reuse.","marker":"[50]"},{"why":"Laplace-space derivation of the equal-speed strong-mixing solutions that motivates the spectral construction used here.","marker":"[53]"},{"why":"The Laplace representation for cosmological correlators used to recast the coupled equations into a single spectral equation.","marker":"[68]"},{"why":"Standard reference for the Heun function HeunG and its limiting behavior, used throughout the exact spectrum.","marker":"[69]"},{"why":"Gives the free-field (unmixed) curvature power spectrum that the exact formula must reduce to at $\\lambda = 0$.","marker":"[70]"},{"why":"Hypergeometric connection formulas and Gauss summation used in deriving the $\\lambda\\to 0$, $r_c\\to 0$ and $r_c\\to\\infty$ limits.","marker":"[74]"},{"why":"Establishes the low-speed collider resonance regime whose momentum dependence the small-$r_c$ hierarchy connects to.","marker":"[57, 58]"}],"fun_headline_variants":["Exact power spectrum for unequal sound speeds at any mixing","Sound-speed ratio unlocks exact inflationary power spectrum","Heun solution tames mixed fields with unequal sound speeds","Unequal sound speeds: exact spectrum from a Heun equation","Closed-form power spectrum for any mixing when speeds differ"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The entire exact spectrum rests on the integral identity (A.3), which converts the contour integral into a closed-form Heun expression; the paper establishes this identity only through numerical verification at six points, with agreement below $2\\times10^{-11}$ in relative power, and an unanticipated branch or contour subtlety would invalidate the central formula. A secondary fragile premise is that the long mode has reached its late-time form during the short-mode interaction, quantified by Eq. (7.27) and roughly $q \\ll \\min(r_c, 1/r_c)$, which the paper states as the applicability condition of the squeezed bispectrum.","fun_headline_variants_meta":{"raw":{"variants":["Exact power spectrum for unequal sound speeds at any mixing","Sound-speed ratio unlocks exact inflationary power spectrum","Heun solution tames mixed fields with unequal sound speeds","Unequal sound speeds: exact spectrum from a Heun equation","Closed-form power spectrum for any mixing when speeds differ"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000185,"raw_usage":{"cost":0.000184828,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000184828,"upstream_inference_prompt_cost":0.000145348,"upstream_inference_completions_cost":3.948e-05},"total_tokens":1384,"prompt_tokens":1243,"completion_tokens":141,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":62}},"tokens_in":1243,"tokens_out":141,"cost_source":"openrouter_usage","duration_ms":2051,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":62,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-02T04:59:31.931664+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Independently integrate the original coupled field equations in the time domain at masses and mixings outside the six tested points, especially near the resonant cases $\\chi = 0$, $\\nu = 0$, $\\nu = 1/2$ and $\\nu = 1$, and compare the resulting curvature power against the Heun-formula spectrum (1.8); agreement would support the central identity (A.3), while a discrepancy at the level of its numerical uncertainty would reveal a branch or contour issue. For the bispectrum, testing squeezed configurations with $r_c$ far from unity, where $q \\ll \\min(r_c, 1/r_c)$ may fail, would settle whether the quoted hierarchy estimates for the amplitude and phase hold.","supporting_citations":[{"cited_title":"Pushing the Primordial Frontier: Exact Linear Solutions in Multifield Inflation","cited_arxiv_id":"2606.18248","evidence_quote":"Supplies the equal-speed exact mode functions and power spectrum that this paper extends to unequal sound speeds; its normalization and phase convention are inherited."},{"cited_title":"Pushing the Primordial Frontier: Cosmological Collider Signatures at Strong Mixing","cited_arxiv_id":"2607.14529","evidence_quote":"Provides the equal-speed bispectrum framework and the three cubic interactions whose normalization the squeezed coefficients reuse."},{"cited_title":"Analytical Cosmological Collider at Strong Mixing in Laplace Space","cited_arxiv_id":"2608.23243","evidence_quote":"Laplace-space derivation of the equal-speed strong-mixing solutions that motivates the spectral construction used here."},{"cited_title":"Laplace Space for Cosmological Correlators","cited_arxiv_id":"2606.27309","evidence_quote":"The Laplace representation for cosmological correlators used to recast the coupled equations into a single spectral equation."},{"cited_title":"Perturbations in k-inflation","cited_arxiv_id":"hep-th/9904176","evidence_quote":"Gives the free-field (unmixed) curvature power spectrum that the exact formula must reduce to at $\\lambda = 0$."}],"review_version":1}