I reviewed the uploaded PDF/TeX draft; line numbers below refer to 1.tex.

1

A. Must-fix items before circulation/submission

1. Active internal comments are still in the manuscript.
   Locations: lines 184–186; line 445; line 1412.
   Context:
   \wqy{I think we can name the paper as ...}
\dgw{TBD}
   Fig. 1 caption contains DGW/WQY discussion comments.
   We would like to thank ... for helpful discussions.
   Fix: remove all author comments and replace the acknowledgement placeholder.

2. Massless mode function has the wrong power of $H$.
   Location: Eq. (1.7), line 320.
   Current:

   $$\frac{H^2}{\sqrt{2k^3}}(1+ik\eta)e^{-ik\eta}.$$

   Issue: for a canonically normalized massless scalar in dS, this should be $H/\sqrt{2k^3}$, not $H^2/\sqrt{2k^3}$. The current expression makes later powers of $H$ inconsistent.
   Fix:

   $$u_k(\eta)=\frac{H}{\sqrt{2k^3}}(1+ik\eta)e^{-ik\eta}.$$

3. Schwinger–Keldysh bulk-to-bulk Green-function identity is missing the branch sign.
   Location: Eq. (1.8), line 331.
   Current:

   $$(\Box_\eta-m^2)D^{bc}(k,\eta,\eta')=i(-H\eta)^4\delta(\eta-\eta')\delta^{bc}.$$

   Issue: later in the paper, for example around Eq. (3.6), you use the branch-dependent sign $ib$.
   Fix:

   $$(\Box_\eta-m^2)D^{bc}(k,\eta,\eta')= i\,b\,(-H\eta)^4\delta(\eta-\eta')\delta^{bc},$$

   assuming $b=\pm$ is identified with $\pm1$.

4. The general action in Eq. (1.2) does not match the stated class of theories.
   Location: lines 270–272.
   Current text: “general interactions involving only first time derivatives of the fields.”
   Issue: Eq. (1.2) contains only a potential $V[\Phi_i]$, while later sections use derivative interactions, higher-derivative interactions, and boost-breaking operators.
   Fix: either restrict the sentence or write the action as something like

   $$S=\int d^4x\sqrt{-g}\left[\sum_i\left(-\frac12(\partial\Phi_i)^2-\frac12m_i^2\Phi_i^2\right)+\mathcal L_{\rm int}(\Phi_i,\nabla_\mu\Phi_i,\ldots)\right].$$

   Then state separately which derivative order is assumed in Sec. 2.

5. Principal/complementary mass thresholds are written dimensionally incorrectly.
   Locations: lines 279, 284, 247, 255.
   Current: $m>3H/2$ is rendered in several places as m > 3H 2; similarly for $m<3H/2$.
   Fix: write explicitly

   $$m_\sigma=m>\frac{3H}{2},\qquad m_\sigma=m<\frac{3H}{2}.$$

   In the TeX, use m > 3H/2 or m>\frac{3H}{2}.

6. Sec. 3 says the direct boundary differential-equation validation is in Appendix A, but Appendix A is about path-integral issues.
   Location: line 564.
   Current: “a procedure we rigorously perform in Appendix A for validation.”
   Fix: this should probably be Appendix B, or Appendix B/C depending on the intended reference.

7. Fig. 1 caption contains a substantive definition error for $E_{\rm int}$.
   Location: line 444.
   Current:
   $E_{\rm int}=-\lambda g(\phi^3\phi''+\frac12\phi^2(\phi')^2)$ “is the interaction part of the equation of motion.”
   Issue: the factor $\lambda$ comes from $f=\lambda\phi^2$, so the displayed object is closer to $E_{\rm int}f$, not $E_{\rm int}$.
   Fix: either write

   $$E_{\rm int}= -g\left(\phi\phi''+\frac12(\phi')^2\right)$$

   for the example, and then separately $E_{\rm int}f=-\lambda g(\cdots)$, or relabel the caption as “$E_{\rm int}f=\cdots$.”

8. Fig. 1 caption also has several obvious text bugs.
   Location: line 444.
   Current: “Blue sots”; “$E_0=(\partial^2-m^2)\phi$”; author comments remain.
   Fix: “Blue dots”; use $E_0=(\Box-m^2)\phi$; delete all comments.

9. Eq. (3.21) uses a different normalization for the cubic derivative interaction than Sec. 3.1.
   Locations: line 764 versus lines 576, 588, 607, etc.
   Issue: Sec. 3.1 repeatedly uses $\frac12\lambda_3(\nabla\phi)^2\sigma$, but Eq. (3.21) writes $\lambda_3(\nabla\phi)^2\sigma$. This can shift factors of 2 in the field-redefinition discussion.
   Fix: use one convention throughout, or explicitly state that the $\lambda_3$ in Sec. 3.2 differs by a factor of 2.

10. The field-redefinition expansion in Eq. (3.23) should be checked for signs and factors after fixing the normalization above.
    Location: line 773.
    Issue: because the original interaction normalization is inconsistent, the induced terms

    $$-\frac{\lambda_3}{2}\nabla\sigma\nabla(\tilde\phi^2),\qquad \lambda_2\lambda_3(\dot{\tilde\phi}\sigma^2+\tilde\phi\sigma\dot\sigma)$$

    may not match the intended $\lambda_3$.
    Fix: redo this short expansion after choosing the convention for $\lambda_3$.

11. Eq. (3.33) appears to miss a curvature-commutator term.
    Location: line 853.
    Current identity:

    $$\int\sqrt{-g}\,(\nabla_\mu\nabla_\nu\phi)^2\sigma =-\int\sqrt{-g}\,(\nabla_\mu\phi)(\nabla^\mu\Box\phi)\sigma +\frac12\int\sqrt{-g}\,(\nabla_\mu\phi)^2\Box\sigma .$$

    Issue: in curved space, commuting covariant derivatives acting on $\nabla_\mu\phi$ produces a Ricci term. One expects a contribution proportional to

    $$-\int\sqrt{-g}\,\sigma\,R_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi,$$

    which in dS$_4$ is $-3H^2\int\sqrt{-g}\,\sigma(\nabla\phi)^2$.
    Fix: either add the curvature term and propagate it into the recursion relations, or state clearly that $\nabla^n\phi$ in Sec. 3.3 is a schematic/ordinary-derivative operator for which this commutator is not included.

12. The recursion relations in Sec. 3.3 may be affected by the missing curvature term.
    Locations: Eqs. (3.34)–(3.39), lines 862–899.
    Issue: if the covariant-derivative identity is corrected, extra lower-derivative exchange/contact terms proportional to $H^2$ should appear.
    Fix: rederive the recursion after deciding whether $\nabla^n\phi$ denotes genuinely covariant derivatives or a shorthand basis modulo curvature terms.

13. Eq. (4.1) has index typos in the most general boost-breaking operator.
    Location: lines 974–980.
    Issue: the third factor for $\sigma$ contains repeated/mismatched indices:

    $$\partial_{k_1}\cdots\partial_{k_{\gamma_3}}\, \partial^{l_1}\cdots\partial^{j_{\gamma_1}}\sigma .$$

    The $k$-indices should pair with $\gamma_2$, and the final $j_{\gamma_1}$ should be $l_{\gamma_1}$.
    Fix: use something like

    $$(\partial_\eta^{\alpha_3}(\partial_i^2)^{\beta_3} \partial_{k_1}\cdots\partial_{k_{\gamma_2}} \partial_{l_1}\cdots\partial_{l_{\gamma_1}}\sigma).$$

14. Massive-field EOM sign in Eq. (4.5) is likely wrong.
    Location: line 999.
    Current:

    $$c_\sigma^2k^2\sigma=-\partial_\eta^2\sigma+\frac2\eta\partial_\eta\sigma+\frac{m^2}{H^2\eta^2}\sigma.$$

    Issue: for

    $$\sigma''-\frac2\eta\sigma'+\left(c_\sigma^2k^2+\frac{m^2}{H^2\eta^2}\right)\sigma=0,$$

    solving for $c_\sigma^2k^2\sigma$ gives a minus mass term.
    Fix:

    $$c_\sigma^2k^2\sigma=-\sigma''+\frac2\eta\sigma'-\frac{m^2}{H^2\eta^2}\sigma .$$

15. Eq. (4.6) has multiple notation/sign problems.
    Location: lines 1001–1008.
    Issues:
    a'_{123} appears where $\alpha'_{123}$ is intended; powers switch between $(-H\eta)$ and $(H\eta)$; the sign of the derivative of the measure factor should be checked.
    Fix: rewrite the formula cleanly. For example, after IBP one expects a term proportional to

    $$(4-\alpha'_{123})H(-H\eta)^{\alpha'_{123}-5}$$

    up to the chosen sign convention, plus the two terms where one derivative is moved from $\sigma$ to the two $\phi$’s.

16. Undefined/misleading $k_s$ notation in Sec. 4.1.
    Locations: lines 1013 and 1059.
    Current: “internal powers $k_s^{2m}$” / “powers $k_s^2$.”
    Issue: $k_s$ is not defined; the internal momentum magnitude is denoted $s$.
    Fix: replace by $s^{2m}$, $s^2$, or “powers of the internal momentum magnitude $s$.”

17. The Schwinger–Keldysh sewing condition under field redefinitions is oversimplified.
    Location: Appendix A, Eq. (A.1), lines 1421–1430.
    Issue: after $\Phi=\tilde\Phi+f[\tilde\Phi]$, the final-time condition should transform as

    $$\Phi^+(\eta_0)+f[\Phi^+](\eta_0) = \Phi^-(\eta_0)+f[\Phi^-](\eta_0),$$

    not simply $\Phi^+(\eta_0)=\Phi^-(\eta_0)$, unless the redefinition is derivative-free and invertible pointwise on the boundary. This is especially important because Sec. 4.2 later discusses velocity-dependent redefinitions.
    Fix: restrict Appendix A explicitly to derivative-independent local field redefinitions, or formulate the derivation in phase space/canonical variables.

18. The Jacobian formula in Eq. (A.2) is mathematically misleading.
    Location: lines 1438–1440.
    Current:

    $$\exp\left[\int d^4x\log\left[1+f'[\Phi(x)]\delta^{(4)}(x-x)\right]\right].$$

    Issue: the $\delta(0)$ comes from the trace, not inside the logarithm in this way.
    Fix: for derivative-free $f$, write

    $$\ln J=\mathrm{Tr}\ln\!\left[(1+f'(\Phi(x)))\delta^{(4)}(x-y)\right] =\delta^{(4)}(0)\int d^4x\,\ln(1+f'(\Phi(x))).$$

    Also fix the extra parenthesis in the displayed expression.

19. The dimensional-regularization statement about $\delta^{(d)}(0)$ is inaccurate.
    Location: line 1447.
    Current: “In dimensional regularization, terms proportional to $\delta^{(d)}(0)$ and its derivatives are divergent as $1/\epsilon$.”
    Issue: in dim reg, scaleless contact quantities such as $\delta^{(d)}(0)=\int d^dp/(2\pi)^d$ are often set to zero; more generally they represent local UV counterterms.
    Fix: say “Such terms are local UV contact terms; in dimensional regularization scaleless ones vanish, and in other schemes they can be absorbed into local counterterms.”

20. Broken-phase field redefinition omits the identity term.
    Location: Eq. (A.9), line 1498.
    Current:

    $$\Delta\Phi\to f[\Delta\tilde\Phi+\bar\Phi]-f[\bar\Phi].$$

    Issue: this maps the fluctuation only to the nonlinear part, not to itself plus the nonlinear shift.
    Fix:

    $$\Delta\Phi = \Delta\tilde\Phi + f[\bar\Phi+\Delta\tilde\Phi]-f[\bar\Phi].$$

21. Contradiction in the second-order Schwinger–Dyson discussion.
    Locations: lines 1573 and 1598.
    Current: line 1573 says

    $$\Delta_r^2S=-2\Delta_r\Delta_aS,$$

    but line 1598 says they are equal.
    Fix: the cancellation should be described as

    $$\frac12\Delta_r^2S+\Delta_r\Delta_aS=0,$$

    with the factors appearing in Eq. (A.20).

22. Appendix B says “for $\Delta_\sigma=3$,” which conflicts with the paper’s definition.
    Location: line 1660.
    Issue: earlier $\Delta_\sigma=3/2-\nu$. A massive $\sigma$ does not generally have $\Delta_\sigma=3$.
    Fix: clarify what weight is meant here. If this is the conformal weight of the massless pair/source rather than $\sigma$, rename it.

23. Second derivative with respect to $x_0$ is written incorrectly.
    Locations: Eq. (B.15), lines 1691, 1693, and line 1694 text; also Eq. (B.37), line 1833.
    Current: $x_0^2\partial_{x_0^2}$.
    Issue: this denotes derivative with respect to $x_0^2$, not the second derivative with respect to $x_0$.
    Fix: use

    $$x_0^2\partial_{x_0}^2-2x_0\partial_{x_0}.$$

24. Eq. (B.32) has an incorrect SK summation index.
    Location: line 1803.
    Current: \sum_{b=\pm}(ic) with $c$ not defined.
    Fix:

    $$\mathcal K_b(k_3,k_4,\eta_1,\eta_0) = g_3\sum_{c=\pm}(ic)\int_{\eta_0}^{-\infty}\cdots D^\sigma_{bc}(s,\eta_1,\eta_2)K_c^\phi(k_3,\eta_2)K_c^\phi(k_4,\eta_2).$$

25. Eq. (B.33) double-counts the SK branch sum unless the seed is redefined.
    Locations: lines 1810 and 1815.
    Issue: line 1810 contains $\sum_b(ib)\hat I$, while line 1815 defines $\hat I$ itself with $\sum_b(ib)$.
    Fix: either remove the sum from line 1810 or define $\hat I$ without the branch sum.

26. Eq. (B.36) evaluates a boundary derivative at the wrong argument.
    Location: line 1825.
    Current: $\partial_1\hat{\mathcal K}_b(x,x_0,v)$ inside a boundary term.
    Fix:

    $$\partial_1\hat{\mathcal K}_b(x_0,x_0,v).$$

27. Typo in the composite-propagator identity.
    Location: line 1840.
    Current:

    $$\partial_1(\hat{\mathcal K}_+(x_0,x_0,v)-\hat{\mathcal K}_+(x_0,x_0,v))=0.$$

    Fix:

    $$\partial_1(\hat{\mathcal K}_+(x_0,x_0,v)-\hat{\mathcal K}_-(x_0,x_0,v))=0.$$

28. Coefficient $d_{pq}$ in Eq. (B.41) is dimensionally inconsistent.
    Location: line 1859.
    Current denominator: $q^2-3q+m^2$.
    Issue: $m^2$ is dimensionful unless $H=1$ has been imposed. Elsewhere the dimensionless combination is $m^2/H^2$.
    Fix: replace by $m^2/H^2$, or define $m$ in this equation as dimensionless. Also check the numerator $m^2$.

29. Set-exclusion notation is wrong in Eq. (B.41).
    Location: line 1860.
    Current: $q\neq\{0,1\}$.
    Fix: $q\notin\{0,1\}$.

30. Undefined $\alpha_\pm$ appears after defining $\tilde\alpha_\pm$.
    Location: line 1900.
    Current: $H^2\alpha_+\alpha_-(\beta_0+b)/(\pi m^2\cos\pi\nu)$.
    Issue: Eq. (B.46) defines $\tilde\alpha_\pm$, not $\alpha_\pm$.
    Fix: use one notation consistently.

31. Higher-derivative contact relation is missing a power of $\Box$.
    Location: line 1927.
    Current:

    $$\phi^2(\nabla^l\phi)^2\simeq 2^{-l}\phi^2\Box(\phi^2).$$

    Issue: for general $l$, this should presumably involve $\Box^l$, consistent with the commented-out line 946.
    Fix:

    $$\phi^2(\nabla^l\phi)^2\simeq 2^{-l}\phi^2\Box^l(\phi^2),$$

    if this is the intended identity.

32. Appendix C has an $f_{12}/f_{22}$ typo.
    Location: line 2012.
    Current: “while $f_{22}$ again has a complicated form…”
    Issue: the surrounding equations are about $f_{12}$.
    Fix: replace $f_{22}$ by $f_{12}$.

33. Eq. (C.13) likely has the wrong function on the right-hand side.
    Location: line 2024.
    Current:

    $$f_{22}(u)=f_{21}(u)-3f_2(u).$$

    Issue: from the preceding relation $f_{21}=f_{11}-3f_1$, one would expect at the next order something like

    $$f_{22}=f_{12}-3f_2,$$

    not $f_{21}-3f_2$.
    Fix: verify the expansion of the hypergeometric solutions and correct this relation.

34. Wrong cross-reference in Appendix C.
    Location: line 2044.
    Current: “The $\cref{eq:master_eq_3pt}$ becomes…”
    Issue: the equation discussed is the four-point exchange equation, not the 3-point master equation.
    Fix: refer to the relevant four-point bootstrap equation, probably Eq. (B.37) or Eq. (B.39).

35. Eq. (3.20) correction term may have an inconsistent mixing operator.
    Location: line 740.
    Current: correction integral uses $I_{b,g_3\phi^2\sigma\times\lambda_2\phi\sigma}$, while the surrounding bispectrum uses $\lambda_2\phi\dot\sigma$.
    Fix: either restore the dot, or add a sentence explaining why the correction term is written with the undifferentiated mixing.

36. Eq. (3.16) defines a rescaled propagator using $k_3$ although the statement is general.
    Location: line 698.
    Current: $\hat K^\phi\equiv 2k_3^3K^\phi$.
    Fix: write

    $$\hat K^\phi(k,\eta)\equiv 2k^3K^\phi(k,\eta),$$

    or explicitly say this is for the $k_3$ leg.

37. The conversion from $k\partial_k$ to a $\dot\phi$ insertion misses normalization factors.
    Location: line 702.
    Issue: $\dot\phi=a^{-1}\phi'=-H\eta\,\phi'$, so $\eta\partial_\eta$ differs by $-H^{-1}\dot{}$ factors and possible signs.
    Fix: say “up to the corresponding factor of $-H$” or write the exact relation.

38. Boundary-pair notation in Sec. 4.2 looks like subtraction.
    Locations: lines 1203, 1204, 1256, 1278.
    Current:

    $$\phi'\phi\sigma-\phi^2\sigma',\quad \phi^2\sigma-\phi'\phi\sigma'$$

    etc.
    Issue: the dash denotes a pairing of two boundary vertices, not a subtraction.
    Fix: use $\times$, “paired with,” or comma notation:

    $$\phi'\phi\sigma \times \phi^2\sigma'.$$

39. Eq. (4.14) uses $a=\pm$ as a branch label, which conflicts visually with the scale factor $a(\eta)$.
    Location: line 1162.
    Fix: use $b=\pm$ or $s=\pm$ for the branch label.

40. Eq. (4.16) prefactor should be rechecked.
    Location: lines 1188–1194.
    Issue: the factor

    $$\frac{1}{H^4\eta_1\eta_2^3}$$

    is not obviously consistent with the stated boundary vertices $a^{-2}\phi'\phi\sigma'$ and $\phi^2\sigma$.
    Fix: rederive the powers of $a(\eta_0)$ in this example and state the vertex normalizations explicitly. This one is a “check” item rather than a definite error.

B. Text, grammar, and presentation fixes

41. Abstract: singular/plural mismatch.
    Location: line 167.
    Current: “a more systematic investigations”
    Fix: “a more systematic investigation” or “more systematic investigations.”

42. Abstract: awkward boost-breaking sentence.
    Location: lines 162–166.
    Current: “For theories with broken dS boost, through boundary terms, we reduce…”
    Fix:
    “For theories with broken dS boosts, we use IBP/EoM relations to reduce exchange correlators from bulk interactions to a basis of independent templates, and then classify the boundary-boundary and boundary-bulk contributions…”

43. Affiliation spacing.
    Locations: lines 135 and 139.
    Current: P.R.China
    Fix: P.R. China.

44. Intro: “integration-by-parts” as a noun.
    Location: line 196.
    Current: “an integration-by-parts”
    Fix: “integration by parts” or “an integration-by-parts relation.”

45. Intro: grammar around EoM terms.
    Location: line 196.
    Current: “the terms in the Lagrangian being proportional to the equation of motion”
    Fix: “terms in the Lagrangian proportional to the equations of motion.”

46. Intro: flat-space boundary condition phrasing.
    Location: line 198.
    Current: “the future and past infinity”
    Fix: “future and past infinity” or “past and future infinity.”

47. Intro: single-field phrasing.
    Location: line 207.
    Current: “For single field inflation scenario”
    Fix: “For single-field inflation scenarios” or “In the single-field inflation scenario.”

48. Intro: “towards” should be “of/into.”
    Location: line 218.
    Current: “systematic investigation towards the boundary contributions”
    Fix: “systematic investigation of boundary contributions” or “into boundary contributions.”

49. Intro: hyphenation.
    Location: line 226.
    Current: “field-redefinitions”
    Fix: “field redefinitions.”

50. Intro: awkward verb phrase.
    Location: lines 226–227.
    Current: “and then have them compared and matched”
    Fix: “and then compare and match them order by order…”

51. Outline: singular/plural mismatch.
    Location: line 251.
    Current: “this type of boundary terms … yields”
    Fix: “this type of boundary term … yields” or “these boundary terms … yield.”

52. Notation: capitalization.
    Location: line 264.
    Current: “de Sitter Universe”
    Fix: “de Sitter universe.”

53. Notation: avoid $$...$$.
    Location: line 275.
    Fix: replace by \[…\] or a numbered equation environment.

54. Notation: line break command at paragraph end.
    Location: line 311.
    Issue: paragraph ends with \\.
    Fix: remove it.

55. Sec. 2 opening sentence is awkward.
    Location: line 337.
    Current: “which is concluded as several diagrammatic relations.”
    Fix: “which we summarize as several diagrammatic relations.”

56. Sec. 2 wording.
    Location: line 341.
    Current: “shares certain similarity with”
    Fix: “shares some similarities with.”

57. Sec. 2: formal tone.
    Location: line 413.
    Current: “Now let’s begin…”
    Fix: “We begin…”

58. Sec. 2: repeated “Then.”
    Location: lines 414–416.
    Fix: vary the sentence structure.

59. Sec. 2: possessive typo.
    Location: line 462.
    Current: “it’s spatial derivatives”
    Fix: “its spatial derivatives.”

60. Sec. 2.2: stray ~ line.
    Location: line 536.
    Fix: delete the standalone ~.

61. Sec. 3: informal opening.
    Location: line 547.
    Current: “Let’s start…”
    Fix: “We start…”

62. Sec. 3.1.1: quasi-single-field wording.
    Location: line 567.
    Current: “in quasi-single field scenario”
    Fix: “in the quasi-single-field scenario.”

63. Fig. 2 caption: punctuation.
    Location: line 571.
    Current: “interaction $\phi^2\sigma$ .”
    Fix: remove the space before the period.

64. Sec. 3.1.2: phrase with unnecessary comma.
    Location: line 630.
    Current: “It is important to notice that, the…”
    Fix: “It is important to note that the…”

65. Sec. 3.1.3: adjective order.
    Location: line 653.
    Current: “late-time derivative finite exchange”
    Fix: “late-time finite derivative exchange.”

66. Fig. 4 block: missing space after environment.
    Location: line 676.
    Current: \end{small}The
    Fix: add a blank line or space after \end{small}.

67. Sec. 3.1.4: sentence fragment.
    Location: line 690.
    Current: “And the relation is”
    Fix: “The relation is…”

68. Sec. 3.1.4: spacing before cross-reference.
    Location: line 704.
    Current: in ~\cref
    Fix: in~\cref.

69. Sec. 4.1: “generalization” spelling consistency.
    Location: line 1001.
    Current: “generalization” while paper mostly uses British “generalised” elsewhere.
    Fix: pick US or UK spelling consistently.

70. Appendix A: duplicated word.
    Location: line 1517.
    Current: “let’s first look at the the 1st variation”
    Fix: “we first look at the first variation.”

71. Appendix A: informal “let’s.”
    Locations: lines 1517 and elsewhere.
    Fix: replace by “we.”

72. Appendix A: typo in source labels.
    Locations: lines 1511, 1551, 1557, etc.
    Current: J_1^+, J_1^- appear in places where probably $J^+$, $J^-$ are intended.
    Fix: remove the spurious subscript 1 unless there is a source $J_1$ defined.

73. Appendix A: grammar around ghost fields.
    Location: line 1454.
    Current: “where $c$ is ghost field.”
    Fix: “where $c$ is a ghost field.”

74. Appendix A footnote 8: subject-verb agreement.
    Location: line 1459.
    Current: “the Jacobian only contribute”
    Fix: “the Jacobian only contributes.”

75. Appendix B: grammar.
    Location: line 1707.
    Current: “which could be understood as straight consequence”
    Fix: “which can be understood as a direct consequence.”

76. Appendix B: grammar.
    Location: line 1710.
    Current: “each $\Delta$ appears in the bootstrap equation indicates…”
    Fix: “each $\Delta$ appearing in the bootstrap equation indicates…”

77. Appendix B footnote 11: grammar.
    Location: line 1717.
    Current: “can no longer justify if”
    Fix: “can no longer determine whether.”

78. Appendix B: capitalization after subsection opening.
    Location: line 1794.
    Current: “In this subsection, We want…”
    Fix: “In this subsection, we…”

79. Appendix B: informal phrase.
    Location: line 1794.
    Current: “We want to compare…”
    Fix: “We compare…”

80. Appendix B: “It’s easy” tone.
    Location: line 1926.
    Fix: “It is straightforward…”

81. Appendix B: subject-verb agreement.
    Location: line 1927.
    Current: “the contact diagram … are order by order correspond”
    Fix: “the contact diagram … corresponds order by order…”

82. Appendix C: grammar.
    Location: line 1971.
    Current: “with especially interest”
    Fix: “with particular interest.”

83. Appendix C: subject-verb agreement.
    Location: line 1983.
    Current: “others gives”
    Fix: “the others give.”

84. Appendix C: typo.
    Location: line 2053.
    Current: “remained part”
    Fix: “remaining part.”

85. Appendix C: article.
    Location: line 2057.
    Current: “we find a inhomogeneous solution”
    Fix: “we find an inhomogeneous solution.”

86. Appendix C: subject-verb agreement.
    Location: line 2068.
    Current: “the $O(\epsilon^{-1})$ terms becomes”
    Fix: “the $O(\epsilon^{-1})$ terms become.”

C. Labels, references, and LaTeX hygiene

87. Several \label{...} entries contain spaces.
    Examples:
    \label{field redef}, \label{general EFT}, \label{table 1}, \label{table 2}, \label{SD eqn}, \label{bootstrap equality}, \label{modified bootstrap equation}, \label{contact bootstrap}, \label{table 6}, \label{light contact}, \label{contact light}, \label{momentum loop integral}.
    Issue: LaTeX tolerates some of these, but they are fragile and bad style.
    Fix: use labels such as sec:field-redef, sec:general-eft, tab:trispectrum-count, eq:sd, etc.

88. Duplicate bibliography entries.
    Locations: references [49]/[74] and [53]/[75].
    Issue: [49] and [74] appear to be the same Pimentel–Wang paper; [53] and [75] appear to be the same Wang–Pimentel–Achúcarro paper.
    Fix: merge duplicate entries and update citations.

89. Reference [48] and several later references appear to be future-dated relative to the current draft’s implied literature timeline.
    Location: references [48], [57]–[68], [78]–[80].
    Issue: these may be intended if the paper is being drafted later, but some arXiv IDs such as 2512... and 2605... should be double-checked before submission.
    Fix: verify each future-dated arXiv reference and ensure the bibliography is not carrying placeholder/generated entries.

90. Figure filename typo.
    Location: line 811.
    Current: fig/corresponce.pdf
    Fix: rename to fig/correspondence.pdf and update the TeX.

91. Figure filename with space.
    Location: line 1457.
    Current: fig/ghost diagram.png
    Fix: rename to fig/ghost_diagram.png for portability.

92. The uploaded source does not compile in a clean environment without jcapmod.sty.
    Location: preamble line 1.
    Issue: \documentclass[a4paper]{jcapmod} requires the style/class file to be present.
    Fix: include jcapmod.cls/jcapmod.sty with the source bundle, or switch to the standard JCAP class if appropriate.

93. \caption package is loaded twice.
    Locations: lines 20 and 22.
    Fix: keep only one \usepackage{caption}.

94. Manual page dimensions and geometry are both used.
    Locations: lines 57–61 and line 23.
    Issue: this can lead to unexpected layout conflicts.
    Fix: use geometry options only, or remove geometry if the class handles the layout.

95. Color name cyan is redefined.
    Location: line 104.
    Current: \definecolor{cyan}{RGB}{...}
    Issue: cyan is a standard color name; redefining it may cause confusion.
    Fix: use a custom name, e.g. \definecolor{mycyan}{RGB}{...}.

96. Many unused packages/macros appear in the preamble.
    Examples: tikz-feynman, youngtab, tensor, appendix, custom title machinery, several author-comment macros.
    Fix: remove packages/macros not needed for the final manuscript to reduce warnings and dependency problems.

97. Hyphenation consistency: “late time” versus “late-time.”
    Locations: throughout; e.g. “late time divergence” appears in Appendix B/C, while the main text often uses “late-time divergence.”
    Fix: use “late-time” adjectivally: “late-time divergence,” “late-time boundary,” “late-time limit.”

98. Hyphenation consistency: “non derivative/non-derivative,” “boost breaking/boost-breaking,” “single field/single-field.”
    Fix: standardize to “non-derivative,” “boost-breaking,” “single-field” when used adjectivally.

99. Notation consistency: $\phi$ versus $\varphi$.
    Issue: the manuscript mostly uses \varphi, but some figure labels/captions and comments use \phi.
    Fix: use one notation consistently in the final figures and captions.

100. Notation consistency: $dS$, dS, and de Sitter.
     Issue: the text alternates among “dS boost,” “dS-invariant,” “de Sitter Universe,” etc.
     Fix: use “de Sitter” in prose and “dS” in technical compounds after defining it once.

101. “EoM” singular/plural.
     Issue: the paper uses “EoM terms,” “equation of motion,” and “equations of motion” somewhat interchangeably.
     Fix: for multiple fields/operators, use “equations of motion” or “EoM terms”; for one field, use “the equation of motion.”

102. Use of $\propto$ in Eqs. (3.9) and (3.10) despite giving explicit equalities.
     Locations: lines 642 and 645.
     Issue: the equations write \propto before explicit right-hand sides with coefficients.
     Fix: either use equality and define the exact normalization, or keep $\propto$ and remove exact numerical coefficients.

103. Ambiguous use of “boundary contributions caused by IBP and EoM.”
     Location: conclusion lines 1378–1380.
     Issue: EoM terms do not “cause boundary correlators” in the same way as IBP boundary terms; they collapse exchange diagrams into contact terms.
     Fix: say “we identify late-time divergent contributions generated by the IBP/EoM decomposition, including boundary terms and contact terms from EoM insertions.”

104. “In the end” is too informal for the final paragraph.
     Location: line 1404.
     Fix: “Finally,” or “Another interesting direction is…”

105. Comma splice in BEFT paragraph.
     Location: line 1397.
     Current: “… reheating surface, For instance…”
     Fix: replace comma with period and lowercase “for”: “… reheating surface. For instance…”

106. “This type of divergences” grammar.
     Location: line 1406.
     Fix: “This type of divergence” or “These divergences.”

107. “Boundary conformal anomalies” should be defined before being used as an interpretive phrase.
     Locations: Sec. 3 opening and Appendix B.
     Issue: the term is used as if already standard in the paper.
     Fix: add one explanatory sentence near the start of Sec. 3 or Appendix B: e.g. “By boundary conformal anomaly we mean the anomalous source term in the momentum-space conformal Ward identity induced by the late-time cutoff.”

108. Some displayed equations use inconsistent cutoff limits $\eta_0$ versus $\eta_0^-$.
     Examples: Eqs. (2.4), (2.8), (2.9), (2.10), Appendix A.
     Fix: choose one convention and state it once. Use $\eta_0^-$ only where avoiding boundary delta-functions is essential.

109. Several instances of “strictly vanish” / “vanishes strictly” should specify whether this is due to the SK branch sum, the late-time limit, or exact mode-function identities.
     Locations: Secs. 3.1.2 and 4.2.
     Fix: add a parenthetical reason in the first occurrence of each mechanism.

110. The paper uses both $\Box$ and $\partial^2$ in places where only $\Box$ is correct on dS.
     Locations: Fig. 1 caption and a few schematic passages.
     Fix: use $\Box$ for covariant d’Alembertian; reserve $\partial^2$ for flat or spatial derivatives with explicit definition.

D. Suggested triage order

I would fix in this order: first the active comments/placeholders; then the normalization and SK-sign issues in Eqs. (1.7), (1.8), (3.21); then the Sec. 3.3 curvature/recursion issue; then the boost-breaking operator/EOM issues in Sec. 4; then the Appendix A field-redefinition/Jacobian claims; finally the bibliography, labels, and copyediting.