A two-scale coherence scalar for residual dark energy and the sound horizon

⚑Derivation Flags (34)

• high
  Section 1 and Section 3: definition and retuning of rho_0 — rho_0 is first declared equal to the observed residual density, then treated as a quantity to be retuned in the late-only scan. No equation reconciles these prescriptions for a finite, evolving scalar.

  If wrong: The stated late-only parameter scan is not a scan of the model as initially defined, and the reported CPL values cannot be assigned an unambiguous normalization prescription.

• high
  Section 1 normalization and Section 3 late-only scan — The same parameter ρ0 is first fixed to the observed residual dark-energy density and later retuned to impose the present scalar budget. No equation relates the two prescriptions for an evolving solution.

  If wrong: The quoted CPL scan may correspond to a different model normalization than the one defined in the setup; hence its reported (w0,wa) values and comparison to ΛCDM are not unambiguously consequences of the stated model.

• high
  Section 1, eq. 5 (rho_0 = rho_Lambda) vs Section 3 (retune rho_0) — The normalization parameter rho_0 is fixed to rho_Lambda in Section 1 but retuned in Section 3 to enforce Omega_n(0)=0.685. No explicit equivalence or normalization map is supplied.

  If wrong: The reported late-only scan (w0,wa) values depend on which normalization was used. If the fixed and retuned prescriptions differ, the central numerical conclusion about the model's closeness to DESI+Pantheon+ is unreliable.

• high
  Section 3, “Representative points” and “The most evolved healthy point in the scan” — The conversion from scalar-potential parameters and initial data to the displayed CPL parameters is presented as a numerical result without sufficient evolution equations, initial conditions, or fit definition to reproduce it.

  If wrong: The claimed proximity to the Pantheon+ CPL center, the “most evolved healthy point” designation, and the claimed failure to reach the other centers are unsupported.

• high
  Section 3, late-only CPL scan — The reported best-fit CPL point is a central numerical result, but the initial epoch, initial velocity, evolution equations for H, and regression weighting are omitted.

  If wrong: The claimed improvement in crude distance relative to ΛCDM and the characterization as a Pantheon+-leaning thaw are quantitatively unsupported.

• high
  Section 3, late-only CPL table — The reported (w0,wa) values are the output of an unshown numerical integration. No initial epoch, initial dot n, integrator, Friedmann components, or CPL fitting procedure (fit to w(z), rho_n(z), or distances; weighted or unweighted) are specified.

  If wrong: The central late-time result of the paper is not reproducible from the text. If the integration or fitting procedure differs from what is implied, the reported (w0,wa) values and the distance metric d would change.

• high
  Section 3, late-only DESI check — The reported CPL outputs are presented as results of a scan, but the numerical initial-value problem and fitting prescription are not specified sufficiently for reproduction.

  If wrong: The headline late-only result, including the claimed relative proximity to the Pantheon+ CPL center, is unsupported as a reproducible consequence of the model.

• high
  Section 3, representative points table (alpha, n_ini) -> (w0, wa) — The mapping from potential parameters to CPL coefficients is the output of an unshown background integration. No initial \dot n, no integrator, and no specification of the fitting target (w(z) directly, rho_n(z), or distance moduli) or weighting over 0 <= z <= 2.5.

  If wrong: The abstract's central late-time result (w0,wa) ~ (-0.83,-0.27) and the entire d-metric comparison table in Sec. 3 would be unsupported; the claim that the model is 'closer to the DESI DR2 + Pantheon+ central value than LCDM' would have no numerical basis, though the structural w >= -1 no-go would survive.

• high
  Section 4, “Early hilltop and r_s” table — The reported sound horizons, peak fractions/redshifts, and expansion-rate shifts are numerical integration outputs without a stated sound-horizon definition, hilltop normalization equation, or complete background evolution prescription.

  If wrong: The principal early-time conclusion—that a mild configuration reduces r_s to about 140 Mpc while the target configuration is excluded—does not follow from the submitted derivation.

• high
  Section 4, early hilltop and rs — The conversion from the specified frozen fraction fE to ρE and the numerical sound-horizon, diffusion-scale, and Hubble-rate entries are asserted without the defining equations or numerical procedure.

  If wrong: The claimed reduction from 147.14 Mpc to about 140 Mpc and the conclusion that the mild configuration remains a viable background-level option do not follow from the supplied derivation.

• high
  Section 4, early hilltop table (f_E, sigma_E, delta, z_peak, f_peak, r_s,d, Delta r_s, Delta H/H) — The mapping from f_E at z_c ~ 3500 to rho_E is not specified. The sound-horizon integral, drag-epoch criterion, and the numerical evolution producing z_peak and f_peak are not shown.

  If wrong: The mild/target classification and the paper's only surviving sound-horizon claim rest on this unverifiable mapping. If the mapping differs, the exclusion of 'target' and the viability of 'mild' would change.

• high
  Section 4, early-hilltop sound-horizon table — The relation between fE and ρE and the integration producing zpeak, fpeak, rs,d, and ΔH/H are not provided.

  If wrong: The main claim that a mild hilltop reduces the drag sound horizon to about 140 Mpc would be unsupported.

• high
  Section 4, early-hilltop table — The conversion from f_E to rho_E and the numerical evolution producing z_peak, f_peak, r_s,d, and Delta H/H are not defined.

  If wrong: The paper's main early-time conclusion that a mild configuration reduces the sound horizon to about 140 Mpc fails.

• high
  Section 4, hilltop normalization: f_E at z_c ~ 3500 -> rho_E; and f_peak values — The prescription 'set the height so that a field frozen at z_c ~ 3500 would carry a fraction f_E of the total density' does not uniquely fix rho_E without stating the background density at z_c and the frozen field value. The reported amplification from f_E = 0.10 to f_peak = 0.20 (mild) and 0.34 (target) has no derivation or explanation.

  If wrong: The r_s,d values (140.28 and 135.39 Mpc) and Delta H/H (+4.9%, +20.6%) would be unreliable, invalidating both the abstract's '~140 Mpc' claim and the mild-viable / target-excluded dichotomy that is the paper's only forward-going recommendation for Paper II.

• high
  Sections 1 and 3, late normalization — The plateau normalization is fixed exactly in Section 1 but described as retuned in Section 3; no relation between these two prescriptions is supplied.

  If wrong: The normalization and resulting trajectories behind the central late-only CPL table are ambiguous.

• high
  Sections 1 and 3, plateau normalization — The model first fixes ρ0 to ρΛ but later retunes ρ0 to impose the late scalar budget; no equivalence between these prescriptions is derived.

  If wrong: The field evolution, present budget, and reported late-only CPL points may correspond to a different model from the one defined in Section 1.

• medium
  Section 1 and Section 4: total-potential “hilltop” at n_E — n_E is the maximum of the Gaussian term alone, but not generically a stationary point of the complete two-scale potential. The manuscript starts the field at n_E+delta and refers to an early hilltop without quantifying when the plateau slope may be neglected.

  If wrong: The claimed initial displacement from a hilltop, and therefore the interpretation of delta, z_peak, and the intended early-time release dynamics, may be shifted relative to the actual total-potential stationary point.

• medium
  Section 1, full potential and hilltop location — The parameter n_E centers the Gaussian term but is not generally the stationary hilltop of the complete two-term potential.

  If wrong: The interpretation of delta as displacement from the actual hilltop and the timing of early rolling require correction or a quantified negligible-slope approximation.

• medium
  Section 2, Lambda limit (n_ini >= 3/alpha) — The condition n_ini >= 3/alpha is treated as producing exact LambdaCDM behavior without defining a tolerance.

  If wrong: The sanity check and the table entry (alpha,n_ini)=(3.0,3.0) with w0=-1.000, wa=0.000 are approximate, not exact. The paper's claim of exact LambdaCDM behavior at the boundary is overstated.

• medium
  Section 2, Λ limit — The finite saturation condition is asserted to produce an indistinguishable ΛCDM background without a bound on the residual slope or subsequent field motion.

  If wrong: The claimed ΛCDM sanity check may fail at the accuracy intended, although the exact asymptotic limit αn→∞ remains valid.

• medium
  Section 3, naive two-parameter distance — The numerical distances are compatible with a plausible symmetrization of asymmetric errors, but the uncertainty convention and covariance treatment are not defined; the statistic is consequently not a calibrated sigma significance.

  If wrong: The ordering under the author's crude diagonal metric may survive, but the abstract's approximately 1.7 sigma interpretation would not be justified.

• medium
  Section 3, table of representative points and the identification of (alpha,n_ini)=(1.0,0.8) as 'the most evolved healthy point in the scan' — The paper presents a table of CPL fit values for four representative (alpha, n_ini) pairs and then identifies one point as 'the most evolved healthy point' without showing the full scan, the optimization criterion, or the numerical integration details. The word 'best' in the abstract is not backed by an explicit likelihood or chi-squared.

  If wrong: If the scan is not exhaustive or the optimization criterion is different, the claim that this point is the best late-only CPL fit would be unsupported, and the comparison with DESI central values would be unreliable.

• medium
  Section 4, diffusion-scale ratios — The changes in rd/rs after retuning θ* are stated without the diffusion integral, retuning rule, or resulting parameter values.

  If wrong: The limited conclusion that the damping ratio alone does not exclude either configuration would not be established, although the paper separately excludes the target using its asserted H(z*) change.

• medium
  Section 4, early hilltop table and the statement 'Target is excluded anyway: last scattering sits in the bump.' — The exclusion of the target hilltop is asserted based on the statement that last scattering sits in the bump, but the paper does not show the H(z) profile or the criterion for exclusion. The table gives Delta H/H=+20.6% at z=1100 for the target case, but the paper does not state what threshold of Delta H/H is excluded by observations.

  If wrong: If the exclusion criterion is not well-defined, the claim that larger hilltops are excluded by H(z*) is not rigorously established, and the promotion of only the mild hilltop to Paper II would be less justified.

• medium
  Section 4, hilltop definition from the full potential — The paper labels nE as the early Gaussian hilltop, but it is not generally a stationary point of the full two-scale potential.

  If wrong: The assumed frozen initial placement, rolling timing, and fE-to-zpeak mapping may not correspond to the described hilltop evolution.

• medium
  Section 4, r_s,d and r_d values; Delta(r_d/r_s) = +1.3% (mild), -0.2% (target) — Sound-horizon and diffusion-scale integrals are not written down. The reported Delta(r_d/r_s) is stated 'after retuning to fix theta_*' but the retuning (which h values) is not reported, and the text's claim that r_d 'changes in step with r_s' is not reconciled with a +1.3% ratio shift for the mild case.

  If wrong: The claim that 'Silk damping from that ratio alone is not an exclusion' would be unsupported, potentially removing the mild hilltop as a viable option and eliminating the paper's only surviving early-time configuration. Peripheral to the late-only DESI analysis.

• low
  Appendix, kernel-to-potential coarse-graining — The claim that a retarded, exponentially damped phase-locking kernel generates rho_0(1 - e^{-alpha n}) after coarse-graining is asserted with no derivation. The author explicitly flags this as motivation only and states uniqueness is not claimed, so it is peripheral.

  If wrong: Only the physical motivation for the potential's functional form is lost; all background results, which take V(n) as a postulate, are unaffected.

• low
  Section 2, finite plateau criterion — A finite saturation condition is used to state Lambda-like behavior without a quantitative tolerance.

  If wrong: The exact wording w_n=-1 and 'indistinguishable' at the stated boundary is too strong, although the asymptotic Lambda limit as alpha n tends to infinity remains valid.

• low
  Section 3, distance metric d — The distance d uses asymmetric reported CPL uncertainties without specifying whether upper, lower, or symmetrized uncertainties are used.

  If wrong: The comparison metric d, used to support the claim of relative closeness to DESI+Pantheon+, depends on the uncertainty convention. This is a secondary detail but should be clarified.

• low
  Section 4, 'Naive H0 rescaling at fixed theta_*' — The paper states that a naive H0 rescaling maps mild onto ~71 km/s/Mpc and target onto ~73 km/s/Mpc, but does not show the rescaling formula or the values of theta_* used. The paper labels these as 'scale checks, not fits,' which mitigates the severity.

  If wrong: If the rescaling is incorrect, the scale-check numbers would be misleading, but since they are explicitly not fits, the impact on the main conclusions is limited.

• low
  Section 4, Delta(r_d/r_s) = +1.3% (mild) and -0.2% (target) — The paper states 'the photon diffusion scale r_d changes in step with r_s' but then reports a +1.3% ratio shift for the mild case, which is not 'in step' in the sense of equal fractional changes.

  If wrong: This is a minor wording inconsistency that does not affect the central conclusions, but it should be reconciled.

• low
  Section 4, H0 rescaling: mild -> ~71, target -> ~73 km/s/Mpc — The naive fixed-theta_* mapping to H0 is not shown. The paper explicitly labels these 'scale checks, not fits', which mitigates the gap.

  If wrong: Nothing central; the paper explicitly disclaims solving the Hubble tension and labels these as order-of-magnitude checks.

• low
  Section 4, naive H_0 rescaling — The mapping from the sound-horizon reductions to H_0 approximately 71 and 73 is presented without the scaling equation or angular-diameter-distance calculation.

  If wrong: Only the explicitly labeled scale-check estimates would change; the scalar equations would remain unaffected.

• low
  Section 4, naive H0 rescaling — The mapping from sound-horizon changes to the stated H0 scale checks is not given.

  If wrong: Only the illustrative H0 numbers are affected; the author already states that they are not fits.

⚑Derivation Flags (34)

• high
  Section 1 and Section 3: definition and retuning of rho_0 — rho_0 is first declared equal to the observed residual density, then treated as a quantity to be retuned in the late-only scan. No equation reconciles these prescriptions for a finite, evolving scalar.

  If wrong: The stated late-only parameter scan is not a scan of the model as initially defined, and the reported CPL values cannot be assigned an unambiguous normalization prescription.

• high
  Section 1 normalization and Section 3 late-only scan — The same parameter ρ0 is first fixed to the observed residual dark-energy density and later retuned to impose the present scalar budget. No equation relates the two prescriptions for an evolving solution.

  If wrong: The quoted CPL scan may correspond to a different model normalization than the one defined in the setup; hence its reported (w0,wa) values and comparison to ΛCDM are not unambiguously consequences of the stated model.

• high
  Section 1, eq. 5 (rho_0 = rho_Lambda) vs Section 3 (retune rho_0) — The normalization parameter rho_0 is fixed to rho_Lambda in Section 1 but retuned in Section 3 to enforce Omega_n(0)=0.685. No explicit equivalence or normalization map is supplied.

  If wrong: The reported late-only scan (w0,wa) values depend on which normalization was used. If the fixed and retuned prescriptions differ, the central numerical conclusion about the model's closeness to DESI+Pantheon+ is unreliable.

• high
  Section 3, “Representative points” and “The most evolved healthy point in the scan” — The conversion from scalar-potential parameters and initial data to the displayed CPL parameters is presented as a numerical result without sufficient evolution equations, initial conditions, or fit definition to reproduce it.

  If wrong: The claimed proximity to the Pantheon+ CPL center, the “most evolved healthy point” designation, and the claimed failure to reach the other centers are unsupported.

• high
  Section 3, late-only CPL scan — The reported best-fit CPL point is a central numerical result, but the initial epoch, initial velocity, evolution equations for H, and regression weighting are omitted.

  If wrong: The claimed improvement in crude distance relative to ΛCDM and the characterization as a Pantheon+-leaning thaw are quantitatively unsupported.

• high
  Section 3, late-only CPL table — The reported (w0,wa) values are the output of an unshown numerical integration. No initial epoch, initial dot n, integrator, Friedmann components, or CPL fitting procedure (fit to w(z), rho_n(z), or distances; weighted or unweighted) are specified.

  If wrong: The central late-time result of the paper is not reproducible from the text. If the integration or fitting procedure differs from what is implied, the reported (w0,wa) values and the distance metric d would change.

• high
  Section 3, late-only DESI check — The reported CPL outputs are presented as results of a scan, but the numerical initial-value problem and fitting prescription are not specified sufficiently for reproduction.

  If wrong: The headline late-only result, including the claimed relative proximity to the Pantheon+ CPL center, is unsupported as a reproducible consequence of the model.

• high
  Section 3, representative points table (alpha, n_ini) -> (w0, wa) — The mapping from potential parameters to CPL coefficients is the output of an unshown background integration. No initial \dot n, no integrator, and no specification of the fitting target (w(z) directly, rho_n(z), or distance moduli) or weighting over 0 <= z <= 2.5.

  If wrong: The abstract's central late-time result (w0,wa) ~ (-0.83,-0.27) and the entire d-metric comparison table in Sec. 3 would be unsupported; the claim that the model is 'closer to the DESI DR2 + Pantheon+ central value than LCDM' would have no numerical basis, though the structural w >= -1 no-go would survive.

• high
  Section 4, “Early hilltop and r_s” table — The reported sound horizons, peak fractions/redshifts, and expansion-rate shifts are numerical integration outputs without a stated sound-horizon definition, hilltop normalization equation, or complete background evolution prescription.

  If wrong: The principal early-time conclusion—that a mild configuration reduces r_s to about 140 Mpc while the target configuration is excluded—does not follow from the submitted derivation.

• high
  Section 4, early hilltop and rs — The conversion from the specified frozen fraction fE to ρE and the numerical sound-horizon, diffusion-scale, and Hubble-rate entries are asserted without the defining equations or numerical procedure.

  If wrong: The claimed reduction from 147.14 Mpc to about 140 Mpc and the conclusion that the mild configuration remains a viable background-level option do not follow from the supplied derivation.

• high
  Section 4, early hilltop table (f_E, sigma_E, delta, z_peak, f_peak, r_s,d, Delta r_s, Delta H/H) — The mapping from f_E at z_c ~ 3500 to rho_E is not specified. The sound-horizon integral, drag-epoch criterion, and the numerical evolution producing z_peak and f_peak are not shown.

  If wrong: The mild/target classification and the paper's only surviving sound-horizon claim rest on this unverifiable mapping. If the mapping differs, the exclusion of 'target' and the viability of 'mild' would change.

• high
  Section 4, early-hilltop sound-horizon table — The relation between fE and ρE and the integration producing zpeak, fpeak, rs,d, and ΔH/H are not provided.

  If wrong: The main claim that a mild hilltop reduces the drag sound horizon to about 140 Mpc would be unsupported.

• high
  Section 4, early-hilltop table — The conversion from f_E to rho_E and the numerical evolution producing z_peak, f_peak, r_s,d, and Delta H/H are not defined.

  If wrong: The paper's main early-time conclusion that a mild configuration reduces the sound horizon to about 140 Mpc fails.

• high
  Section 4, hilltop normalization: f_E at z_c ~ 3500 -> rho_E; and f_peak values — The prescription 'set the height so that a field frozen at z_c ~ 3500 would carry a fraction f_E of the total density' does not uniquely fix rho_E without stating the background density at z_c and the frozen field value. The reported amplification from f_E = 0.10 to f_peak = 0.20 (mild) and 0.34 (target) has no derivation or explanation.

  If wrong: The r_s,d values (140.28 and 135.39 Mpc) and Delta H/H (+4.9%, +20.6%) would be unreliable, invalidating both the abstract's '~140 Mpc' claim and the mild-viable / target-excluded dichotomy that is the paper's only forward-going recommendation for Paper II.

• high
  Sections 1 and 3, late normalization — The plateau normalization is fixed exactly in Section 1 but described as retuned in Section 3; no relation between these two prescriptions is supplied.

  If wrong: The normalization and resulting trajectories behind the central late-only CPL table are ambiguous.

• high
  Sections 1 and 3, plateau normalization — The model first fixes ρ0 to ρΛ but later retunes ρ0 to impose the late scalar budget; no equivalence between these prescriptions is derived.

  If wrong: The field evolution, present budget, and reported late-only CPL points may correspond to a different model from the one defined in Section 1.

• medium
  Section 1 and Section 4: total-potential “hilltop” at n_E — n_E is the maximum of the Gaussian term alone, but not generically a stationary point of the complete two-scale potential. The manuscript starts the field at n_E+delta and refers to an early hilltop without quantifying when the plateau slope may be neglected.

  If wrong: The claimed initial displacement from a hilltop, and therefore the interpretation of delta, z_peak, and the intended early-time release dynamics, may be shifted relative to the actual total-potential stationary point.

• medium
  Section 1, full potential and hilltop location — The parameter n_E centers the Gaussian term but is not generally the stationary hilltop of the complete two-term potential.

  If wrong: The interpretation of delta as displacement from the actual hilltop and the timing of early rolling require correction or a quantified negligible-slope approximation.

• medium
  Section 2, Lambda limit (n_ini >= 3/alpha) — The condition n_ini >= 3/alpha is treated as producing exact LambdaCDM behavior without defining a tolerance.

  If wrong: The sanity check and the table entry (alpha,n_ini)=(3.0,3.0) with w0=-1.000, wa=0.000 are approximate, not exact. The paper's claim of exact LambdaCDM behavior at the boundary is overstated.

• medium
  Section 2, Λ limit — The finite saturation condition is asserted to produce an indistinguishable ΛCDM background without a bound on the residual slope or subsequent field motion.

  If wrong: The claimed ΛCDM sanity check may fail at the accuracy intended, although the exact asymptotic limit αn→∞ remains valid.

• medium
  Section 3, naive two-parameter distance — The numerical distances are compatible with a plausible symmetrization of asymmetric errors, but the uncertainty convention and covariance treatment are not defined; the statistic is consequently not a calibrated sigma significance.

  If wrong: The ordering under the author's crude diagonal metric may survive, but the abstract's approximately 1.7 sigma interpretation would not be justified.

• medium
  Section 3, table of representative points and the identification of (alpha,n_ini)=(1.0,0.8) as 'the most evolved healthy point in the scan' — The paper presents a table of CPL fit values for four representative (alpha, n_ini) pairs and then identifies one point as 'the most evolved healthy point' without showing the full scan, the optimization criterion, or the numerical integration details. The word 'best' in the abstract is not backed by an explicit likelihood or chi-squared.

  If wrong: If the scan is not exhaustive or the optimization criterion is different, the claim that this point is the best late-only CPL fit would be unsupported, and the comparison with DESI central values would be unreliable.

• medium
  Section 4, diffusion-scale ratios — The changes in rd/rs after retuning θ* are stated without the diffusion integral, retuning rule, or resulting parameter values.

  If wrong: The limited conclusion that the damping ratio alone does not exclude either configuration would not be established, although the paper separately excludes the target using its asserted H(z*) change.

• medium
  Section 4, early hilltop table and the statement 'Target is excluded anyway: last scattering sits in the bump.' — The exclusion of the target hilltop is asserted based on the statement that last scattering sits in the bump, but the paper does not show the H(z) profile or the criterion for exclusion. The table gives Delta H/H=+20.6% at z=1100 for the target case, but the paper does not state what threshold of Delta H/H is excluded by observations.

  If wrong: If the exclusion criterion is not well-defined, the claim that larger hilltops are excluded by H(z*) is not rigorously established, and the promotion of only the mild hilltop to Paper II would be less justified.

• medium
  Section 4, hilltop definition from the full potential — The paper labels nE as the early Gaussian hilltop, but it is not generally a stationary point of the full two-scale potential.

  If wrong: The assumed frozen initial placement, rolling timing, and fE-to-zpeak mapping may not correspond to the described hilltop evolution.

• medium
  Section 4, r_s,d and r_d values; Delta(r_d/r_s) = +1.3% (mild), -0.2% (target) — Sound-horizon and diffusion-scale integrals are not written down. The reported Delta(r_d/r_s) is stated 'after retuning to fix theta_*' but the retuning (which h values) is not reported, and the text's claim that r_d 'changes in step with r_s' is not reconciled with a +1.3% ratio shift for the mild case.

  If wrong: The claim that 'Silk damping from that ratio alone is not an exclusion' would be unsupported, potentially removing the mild hilltop as a viable option and eliminating the paper's only surviving early-time configuration. Peripheral to the late-only DESI analysis.

• low
  Appendix, kernel-to-potential coarse-graining — The claim that a retarded, exponentially damped phase-locking kernel generates rho_0(1 - e^{-alpha n}) after coarse-graining is asserted with no derivation. The author explicitly flags this as motivation only and states uniqueness is not claimed, so it is peripheral.

  If wrong: Only the physical motivation for the potential's functional form is lost; all background results, which take V(n) as a postulate, are unaffected.

• low
  Section 2, finite plateau criterion — A finite saturation condition is used to state Lambda-like behavior without a quantitative tolerance.

  If wrong: The exact wording w_n=-1 and 'indistinguishable' at the stated boundary is too strong, although the asymptotic Lambda limit as alpha n tends to infinity remains valid.

• low
  Section 3, distance metric d — The distance d uses asymmetric reported CPL uncertainties without specifying whether upper, lower, or symmetrized uncertainties are used.

  If wrong: The comparison metric d, used to support the claim of relative closeness to DESI+Pantheon+, depends on the uncertainty convention. This is a secondary detail but should be clarified.

• low
  Section 4, 'Naive H0 rescaling at fixed theta_*' — The paper states that a naive H0 rescaling maps mild onto ~71 km/s/Mpc and target onto ~73 km/s/Mpc, but does not show the rescaling formula or the values of theta_* used. The paper labels these as 'scale checks, not fits,' which mitigates the severity.

  If wrong: If the rescaling is incorrect, the scale-check numbers would be misleading, but since they are explicitly not fits, the impact on the main conclusions is limited.

• low
  Section 4, Delta(r_d/r_s) = +1.3% (mild) and -0.2% (target) — The paper states 'the photon diffusion scale r_d changes in step with r_s' but then reports a +1.3% ratio shift for the mild case, which is not 'in step' in the sense of equal fractional changes.

  If wrong: This is a minor wording inconsistency that does not affect the central conclusions, but it should be reconciled.

• low
  Section 4, H0 rescaling: mild -> ~71, target -> ~73 km/s/Mpc — The naive fixed-theta_* mapping to H0 is not shown. The paper explicitly labels these 'scale checks, not fits', which mitigates the gap.

  If wrong: Nothing central; the paper explicitly disclaims solving the Hubble tension and labels these as order-of-magnitude checks.

• low
  Section 4, naive H_0 rescaling — The mapping from the sound-horizon reductions to H_0 approximately 71 and 73 is presented without the scaling equation or angular-diameter-distance calculation.

  If wrong: Only the explicitly labeled scale-check estimates would change; the scalar equations would remain unaffected.

• low
  Section 4, naive H0 rescaling — The mapping from sound-horizon changes to the stated H0 scale checks is not given.

  If wrong: Only the illustrative H0 numbers are affected; the author already states that they are not fits.

⚑Derivation Flags (34)

• high
  Section 1 and Section 3: definition and retuning of rho_0 — rho_0 is first declared equal to the observed residual density, then treated as a quantity to be retuned in the late-only scan. No equation reconciles these prescriptions for a finite, evolving scalar.

  If wrong: The stated late-only parameter scan is not a scan of the model as initially defined, and the reported CPL values cannot be assigned an unambiguous normalization prescription.

• high
  Section 1 normalization and Section 3 late-only scan — The same parameter ρ0 is first fixed to the observed residual dark-energy density and later retuned to impose the present scalar budget. No equation relates the two prescriptions for an evolving solution.

  If wrong: The quoted CPL scan may correspond to a different model normalization than the one defined in the setup; hence its reported (w0,wa) values and comparison to ΛCDM are not unambiguously consequences of the stated model.

• high
  Section 1, eq. 5 (rho_0 = rho_Lambda) vs Section 3 (retune rho_0) — The normalization parameter rho_0 is fixed to rho_Lambda in Section 1 but retuned in Section 3 to enforce Omega_n(0)=0.685. No explicit equivalence or normalization map is supplied.

  If wrong: The reported late-only scan (w0,wa) values depend on which normalization was used. If the fixed and retuned prescriptions differ, the central numerical conclusion about the model's closeness to DESI+Pantheon+ is unreliable.

• high
  Section 3, “Representative points” and “The most evolved healthy point in the scan” — The conversion from scalar-potential parameters and initial data to the displayed CPL parameters is presented as a numerical result without sufficient evolution equations, initial conditions, or fit definition to reproduce it.

  If wrong: The claimed proximity to the Pantheon+ CPL center, the “most evolved healthy point” designation, and the claimed failure to reach the other centers are unsupported.

• high
  Section 3, late-only CPL scan — The reported best-fit CPL point is a central numerical result, but the initial epoch, initial velocity, evolution equations for H, and regression weighting are omitted.

  If wrong: The claimed improvement in crude distance relative to ΛCDM and the characterization as a Pantheon+-leaning thaw are quantitatively unsupported.

• high
  Section 3, late-only CPL table — The reported (w0,wa) values are the output of an unshown numerical integration. No initial epoch, initial dot n, integrator, Friedmann components, or CPL fitting procedure (fit to w(z), rho_n(z), or distances; weighted or unweighted) are specified.

  If wrong: The central late-time result of the paper is not reproducible from the text. If the integration or fitting procedure differs from what is implied, the reported (w0,wa) values and the distance metric d would change.

• high
  Section 3, late-only DESI check — The reported CPL outputs are presented as results of a scan, but the numerical initial-value problem and fitting prescription are not specified sufficiently for reproduction.

  If wrong: The headline late-only result, including the claimed relative proximity to the Pantheon+ CPL center, is unsupported as a reproducible consequence of the model.

• high
  Section 3, representative points table (alpha, n_ini) -> (w0, wa) — The mapping from potential parameters to CPL coefficients is the output of an unshown background integration. No initial \dot n, no integrator, and no specification of the fitting target (w(z) directly, rho_n(z), or distance moduli) or weighting over 0 <= z <= 2.5.

  If wrong: The abstract's central late-time result (w0,wa) ~ (-0.83,-0.27) and the entire d-metric comparison table in Sec. 3 would be unsupported; the claim that the model is 'closer to the DESI DR2 + Pantheon+ central value than LCDM' would have no numerical basis, though the structural w >= -1 no-go would survive.

• high
  Section 4, “Early hilltop and r_s” table — The reported sound horizons, peak fractions/redshifts, and expansion-rate shifts are numerical integration outputs without a stated sound-horizon definition, hilltop normalization equation, or complete background evolution prescription.

  If wrong: The principal early-time conclusion—that a mild configuration reduces r_s to about 140 Mpc while the target configuration is excluded—does not follow from the submitted derivation.

• high
  Section 4, early hilltop and rs — The conversion from the specified frozen fraction fE to ρE and the numerical sound-horizon, diffusion-scale, and Hubble-rate entries are asserted without the defining equations or numerical procedure.

  If wrong: The claimed reduction from 147.14 Mpc to about 140 Mpc and the conclusion that the mild configuration remains a viable background-level option do not follow from the supplied derivation.

• high
  Section 4, early hilltop table (f_E, sigma_E, delta, z_peak, f_peak, r_s,d, Delta r_s, Delta H/H) — The mapping from f_E at z_c ~ 3500 to rho_E is not specified. The sound-horizon integral, drag-epoch criterion, and the numerical evolution producing z_peak and f_peak are not shown.

  If wrong: The mild/target classification and the paper's only surviving sound-horizon claim rest on this unverifiable mapping. If the mapping differs, the exclusion of 'target' and the viability of 'mild' would change.

• high
  Section 4, early-hilltop sound-horizon table — The relation between fE and ρE and the integration producing zpeak, fpeak, rs,d, and ΔH/H are not provided.

  If wrong: The main claim that a mild hilltop reduces the drag sound horizon to about 140 Mpc would be unsupported.

• high
  Section 4, early-hilltop table — The conversion from f_E to rho_E and the numerical evolution producing z_peak, f_peak, r_s,d, and Delta H/H are not defined.

  If wrong: The paper's main early-time conclusion that a mild configuration reduces the sound horizon to about 140 Mpc fails.

• high
  Section 4, hilltop normalization: f_E at z_c ~ 3500 -> rho_E; and f_peak values — The prescription 'set the height so that a field frozen at z_c ~ 3500 would carry a fraction f_E of the total density' does not uniquely fix rho_E without stating the background density at z_c and the frozen field value. The reported amplification from f_E = 0.10 to f_peak = 0.20 (mild) and 0.34 (target) has no derivation or explanation.

  If wrong: The r_s,d values (140.28 and 135.39 Mpc) and Delta H/H (+4.9%, +20.6%) would be unreliable, invalidating both the abstract's '~140 Mpc' claim and the mild-viable / target-excluded dichotomy that is the paper's only forward-going recommendation for Paper II.

• high
  Sections 1 and 3, late normalization — The plateau normalization is fixed exactly in Section 1 but described as retuned in Section 3; no relation between these two prescriptions is supplied.

  If wrong: The normalization and resulting trajectories behind the central late-only CPL table are ambiguous.

• high
  Sections 1 and 3, plateau normalization — The model first fixes ρ0 to ρΛ but later retunes ρ0 to impose the late scalar budget; no equivalence between these prescriptions is derived.

  If wrong: The field evolution, present budget, and reported late-only CPL points may correspond to a different model from the one defined in Section 1.

• medium
  Section 1 and Section 4: total-potential “hilltop” at n_E — n_E is the maximum of the Gaussian term alone, but not generically a stationary point of the complete two-scale potential. The manuscript starts the field at n_E+delta and refers to an early hilltop without quantifying when the plateau slope may be neglected.

  If wrong: The claimed initial displacement from a hilltop, and therefore the interpretation of delta, z_peak, and the intended early-time release dynamics, may be shifted relative to the actual total-potential stationary point.

• medium
  Section 1, full potential and hilltop location — The parameter n_E centers the Gaussian term but is not generally the stationary hilltop of the complete two-term potential.

  If wrong: The interpretation of delta as displacement from the actual hilltop and the timing of early rolling require correction or a quantified negligible-slope approximation.

• medium
  Section 2, Lambda limit (n_ini >= 3/alpha) — The condition n_ini >= 3/alpha is treated as producing exact LambdaCDM behavior without defining a tolerance.

  If wrong: The sanity check and the table entry (alpha,n_ini)=(3.0,3.0) with w0=-1.000, wa=0.000 are approximate, not exact. The paper's claim of exact LambdaCDM behavior at the boundary is overstated.

• medium
  Section 2, Λ limit — The finite saturation condition is asserted to produce an indistinguishable ΛCDM background without a bound on the residual slope or subsequent field motion.

  If wrong: The claimed ΛCDM sanity check may fail at the accuracy intended, although the exact asymptotic limit αn→∞ remains valid.

• medium
  Section 3, naive two-parameter distance — The numerical distances are compatible with a plausible symmetrization of asymmetric errors, but the uncertainty convention and covariance treatment are not defined; the statistic is consequently not a calibrated sigma significance.

  If wrong: The ordering under the author's crude diagonal metric may survive, but the abstract's approximately 1.7 sigma interpretation would not be justified.

• medium
  Section 3, table of representative points and the identification of (alpha,n_ini)=(1.0,0.8) as 'the most evolved healthy point in the scan' — The paper presents a table of CPL fit values for four representative (alpha, n_ini) pairs and then identifies one point as 'the most evolved healthy point' without showing the full scan, the optimization criterion, or the numerical integration details. The word 'best' in the abstract is not backed by an explicit likelihood or chi-squared.

  If wrong: If the scan is not exhaustive or the optimization criterion is different, the claim that this point is the best late-only CPL fit would be unsupported, and the comparison with DESI central values would be unreliable.

• medium
  Section 4, diffusion-scale ratios — The changes in rd/rs after retuning θ* are stated without the diffusion integral, retuning rule, or resulting parameter values.

  If wrong: The limited conclusion that the damping ratio alone does not exclude either configuration would not be established, although the paper separately excludes the target using its asserted H(z*) change.

• medium
  Section 4, early hilltop table and the statement 'Target is excluded anyway: last scattering sits in the bump.' — The exclusion of the target hilltop is asserted based on the statement that last scattering sits in the bump, but the paper does not show the H(z) profile or the criterion for exclusion. The table gives Delta H/H=+20.6% at z=1100 for the target case, but the paper does not state what threshold of Delta H/H is excluded by observations.

  If wrong: If the exclusion criterion is not well-defined, the claim that larger hilltops are excluded by H(z*) is not rigorously established, and the promotion of only the mild hilltop to Paper II would be less justified.

• medium
  Section 4, hilltop definition from the full potential — The paper labels nE as the early Gaussian hilltop, but it is not generally a stationary point of the full two-scale potential.

  If wrong: The assumed frozen initial placement, rolling timing, and fE-to-zpeak mapping may not correspond to the described hilltop evolution.

• medium
  Section 4, r_s,d and r_d values; Delta(r_d/r_s) = +1.3% (mild), -0.2% (target) — Sound-horizon and diffusion-scale integrals are not written down. The reported Delta(r_d/r_s) is stated 'after retuning to fix theta_*' but the retuning (which h values) is not reported, and the text's claim that r_d 'changes in step with r_s' is not reconciled with a +1.3% ratio shift for the mild case.

  If wrong: The claim that 'Silk damping from that ratio alone is not an exclusion' would be unsupported, potentially removing the mild hilltop as a viable option and eliminating the paper's only surviving early-time configuration. Peripheral to the late-only DESI analysis.

• low
  Appendix, kernel-to-potential coarse-graining — The claim that a retarded, exponentially damped phase-locking kernel generates rho_0(1 - e^{-alpha n}) after coarse-graining is asserted with no derivation. The author explicitly flags this as motivation only and states uniqueness is not claimed, so it is peripheral.

  If wrong: Only the physical motivation for the potential's functional form is lost; all background results, which take V(n) as a postulate, are unaffected.

• low
  Section 2, finite plateau criterion — A finite saturation condition is used to state Lambda-like behavior without a quantitative tolerance.

  If wrong: The exact wording w_n=-1 and 'indistinguishable' at the stated boundary is too strong, although the asymptotic Lambda limit as alpha n tends to infinity remains valid.

• low
  Section 3, distance metric d — The distance d uses asymmetric reported CPL uncertainties without specifying whether upper, lower, or symmetrized uncertainties are used.

  If wrong: The comparison metric d, used to support the claim of relative closeness to DESI+Pantheon+, depends on the uncertainty convention. This is a secondary detail but should be clarified.

• low
  Section 4, 'Naive H0 rescaling at fixed theta_*' — The paper states that a naive H0 rescaling maps mild onto ~71 km/s/Mpc and target onto ~73 km/s/Mpc, but does not show the rescaling formula or the values of theta_* used. The paper labels these as 'scale checks, not fits,' which mitigates the severity.

  If wrong: If the rescaling is incorrect, the scale-check numbers would be misleading, but since they are explicitly not fits, the impact on the main conclusions is limited.

• low
  Section 4, Delta(r_d/r_s) = +1.3% (mild) and -0.2% (target) — The paper states 'the photon diffusion scale r_d changes in step with r_s' but then reports a +1.3% ratio shift for the mild case, which is not 'in step' in the sense of equal fractional changes.

  If wrong: This is a minor wording inconsistency that does not affect the central conclusions, but it should be reconciled.

• low
  Section 4, H0 rescaling: mild -> ~71, target -> ~73 km/s/Mpc — The naive fixed-theta_* mapping to H0 is not shown. The paper explicitly labels these 'scale checks, not fits', which mitigates the gap.

  If wrong: Nothing central; the paper explicitly disclaims solving the Hubble tension and labels these as order-of-magnitude checks.

• low
  Section 4, naive H_0 rescaling — The mapping from the sound-horizon reductions to H_0 approximately 71 and 73 is presented without the scaling equation or angular-diameter-distance calculation.

  If wrong: Only the explicitly labeled scale-check estimates would change; the scalar equations would remain unaffected.

• low
  Section 4, naive H0 rescaling — The mapping from sound-horizon changes to the stated H0 scale checks is not given.

  If wrong: Only the illustrative H0 numbers are affected; the author already states that they are not fits.

⚑Derivation Flags (34)

• high
  Section 1 and Section 3: definition and retuning of rho_0 — rho_0 is first declared equal to the observed residual density, then treated as a quantity to be retuned in the late-only scan. No equation reconciles these prescriptions for a finite, evolving scalar.

  If wrong: The stated late-only parameter scan is not a scan of the model as initially defined, and the reported CPL values cannot be assigned an unambiguous normalization prescription.

• high
  Section 1 normalization and Section 3 late-only scan — The same parameter ρ0 is first fixed to the observed residual dark-energy density and later retuned to impose the present scalar budget. No equation relates the two prescriptions for an evolving solution.

  If wrong: The quoted CPL scan may correspond to a different model normalization than the one defined in the setup; hence its reported (w0,wa) values and comparison to ΛCDM are not unambiguously consequences of the stated model.

• high
  Section 1, eq. 5 (rho_0 = rho_Lambda) vs Section 3 (retune rho_0) — The normalization parameter rho_0 is fixed to rho_Lambda in Section 1 but retuned in Section 3 to enforce Omega_n(0)=0.685. No explicit equivalence or normalization map is supplied.

  If wrong: The reported late-only scan (w0,wa) values depend on which normalization was used. If the fixed and retuned prescriptions differ, the central numerical conclusion about the model's closeness to DESI+Pantheon+ is unreliable.

• high
  Section 3, “Representative points” and “The most evolved healthy point in the scan” — The conversion from scalar-potential parameters and initial data to the displayed CPL parameters is presented as a numerical result without sufficient evolution equations, initial conditions, or fit definition to reproduce it.

  If wrong: The claimed proximity to the Pantheon+ CPL center, the “most evolved healthy point” designation, and the claimed failure to reach the other centers are unsupported.

• high
  Section 3, late-only CPL scan — The reported best-fit CPL point is a central numerical result, but the initial epoch, initial velocity, evolution equations for H, and regression weighting are omitted.

  If wrong: The claimed improvement in crude distance relative to ΛCDM and the characterization as a Pantheon+-leaning thaw are quantitatively unsupported.

• high
  Section 3, late-only CPL table — The reported (w0,wa) values are the output of an unshown numerical integration. No initial epoch, initial dot n, integrator, Friedmann components, or CPL fitting procedure (fit to w(z), rho_n(z), or distances; weighted or unweighted) are specified.

  If wrong: The central late-time result of the paper is not reproducible from the text. If the integration or fitting procedure differs from what is implied, the reported (w0,wa) values and the distance metric d would change.

• high
  Section 3, late-only DESI check — The reported CPL outputs are presented as results of a scan, but the numerical initial-value problem and fitting prescription are not specified sufficiently for reproduction.

  If wrong: The headline late-only result, including the claimed relative proximity to the Pantheon+ CPL center, is unsupported as a reproducible consequence of the model.

• high
  Section 3, representative points table (alpha, n_ini) -> (w0, wa) — The mapping from potential parameters to CPL coefficients is the output of an unshown background integration. No initial \dot n, no integrator, and no specification of the fitting target (w(z) directly, rho_n(z), or distance moduli) or weighting over 0 <= z <= 2.5.

  If wrong: The abstract's central late-time result (w0,wa) ~ (-0.83,-0.27) and the entire d-metric comparison table in Sec. 3 would be unsupported; the claim that the model is 'closer to the DESI DR2 + Pantheon+ central value than LCDM' would have no numerical basis, though the structural w >= -1 no-go would survive.

• high
  Section 4, “Early hilltop and r_s” table — The reported sound horizons, peak fractions/redshifts, and expansion-rate shifts are numerical integration outputs without a stated sound-horizon definition, hilltop normalization equation, or complete background evolution prescription.

  If wrong: The principal early-time conclusion—that a mild configuration reduces r_s to about 140 Mpc while the target configuration is excluded—does not follow from the submitted derivation.

• high
  Section 4, early hilltop and rs — The conversion from the specified frozen fraction fE to ρE and the numerical sound-horizon, diffusion-scale, and Hubble-rate entries are asserted without the defining equations or numerical procedure.

  If wrong: The claimed reduction from 147.14 Mpc to about 140 Mpc and the conclusion that the mild configuration remains a viable background-level option do not follow from the supplied derivation.

• high
  Section 4, early hilltop table (f_E, sigma_E, delta, z_peak, f_peak, r_s,d, Delta r_s, Delta H/H) — The mapping from f_E at z_c ~ 3500 to rho_E is not specified. The sound-horizon integral, drag-epoch criterion, and the numerical evolution producing z_peak and f_peak are not shown.

  If wrong: The mild/target classification and the paper's only surviving sound-horizon claim rest on this unverifiable mapping. If the mapping differs, the exclusion of 'target' and the viability of 'mild' would change.

• high
  Section 4, early-hilltop sound-horizon table — The relation between fE and ρE and the integration producing zpeak, fpeak, rs,d, and ΔH/H are not provided.

  If wrong: The main claim that a mild hilltop reduces the drag sound horizon to about 140 Mpc would be unsupported.

• high
  Section 4, early-hilltop table — The conversion from f_E to rho_E and the numerical evolution producing z_peak, f_peak, r_s,d, and Delta H/H are not defined.

  If wrong: The paper's main early-time conclusion that a mild configuration reduces the sound horizon to about 140 Mpc fails.

• high
  Section 4, hilltop normalization: f_E at z_c ~ 3500 -> rho_E; and f_peak values — The prescription 'set the height so that a field frozen at z_c ~ 3500 would carry a fraction f_E of the total density' does not uniquely fix rho_E without stating the background density at z_c and the frozen field value. The reported amplification from f_E = 0.10 to f_peak = 0.20 (mild) and 0.34 (target) has no derivation or explanation.

  If wrong: The r_s,d values (140.28 and 135.39 Mpc) and Delta H/H (+4.9%, +20.6%) would be unreliable, invalidating both the abstract's '~140 Mpc' claim and the mild-viable / target-excluded dichotomy that is the paper's only forward-going recommendation for Paper II.

• high
  Sections 1 and 3, late normalization — The plateau normalization is fixed exactly in Section 1 but described as retuned in Section 3; no relation between these two prescriptions is supplied.

  If wrong: The normalization and resulting trajectories behind the central late-only CPL table are ambiguous.

• high
  Sections 1 and 3, plateau normalization — The model first fixes ρ0 to ρΛ but later retunes ρ0 to impose the late scalar budget; no equivalence between these prescriptions is derived.

  If wrong: The field evolution, present budget, and reported late-only CPL points may correspond to a different model from the one defined in Section 1.

• medium
  Section 1 and Section 4: total-potential “hilltop” at n_E — n_E is the maximum of the Gaussian term alone, but not generically a stationary point of the complete two-scale potential. The manuscript starts the field at n_E+delta and refers to an early hilltop without quantifying when the plateau slope may be neglected.

  If wrong: The claimed initial displacement from a hilltop, and therefore the interpretation of delta, z_peak, and the intended early-time release dynamics, may be shifted relative to the actual total-potential stationary point.

• medium
  Section 1, full potential and hilltop location — The parameter n_E centers the Gaussian term but is not generally the stationary hilltop of the complete two-term potential.

  If wrong: The interpretation of delta as displacement from the actual hilltop and the timing of early rolling require correction or a quantified negligible-slope approximation.

• medium
  Section 2, Lambda limit (n_ini >= 3/alpha) — The condition n_ini >= 3/alpha is treated as producing exact LambdaCDM behavior without defining a tolerance.

  If wrong: The sanity check and the table entry (alpha,n_ini)=(3.0,3.0) with w0=-1.000, wa=0.000 are approximate, not exact. The paper's claim of exact LambdaCDM behavior at the boundary is overstated.

• medium
  Section 2, Λ limit — The finite saturation condition is asserted to produce an indistinguishable ΛCDM background without a bound on the residual slope or subsequent field motion.

  If wrong: The claimed ΛCDM sanity check may fail at the accuracy intended, although the exact asymptotic limit αn→∞ remains valid.

• medium
  Section 3, naive two-parameter distance — The numerical distances are compatible with a plausible symmetrization of asymmetric errors, but the uncertainty convention and covariance treatment are not defined; the statistic is consequently not a calibrated sigma significance.

  If wrong: The ordering under the author's crude diagonal metric may survive, but the abstract's approximately 1.7 sigma interpretation would not be justified.

• medium
  Section 3, table of representative points and the identification of (alpha,n_ini)=(1.0,0.8) as 'the most evolved healthy point in the scan' — The paper presents a table of CPL fit values for four representative (alpha, n_ini) pairs and then identifies one point as 'the most evolved healthy point' without showing the full scan, the optimization criterion, or the numerical integration details. The word 'best' in the abstract is not backed by an explicit likelihood or chi-squared.

  If wrong: If the scan is not exhaustive or the optimization criterion is different, the claim that this point is the best late-only CPL fit would be unsupported, and the comparison with DESI central values would be unreliable.

• medium
  Section 4, diffusion-scale ratios — The changes in rd/rs after retuning θ* are stated without the diffusion integral, retuning rule, or resulting parameter values.

  If wrong: The limited conclusion that the damping ratio alone does not exclude either configuration would not be established, although the paper separately excludes the target using its asserted H(z*) change.

• medium
  Section 4, early hilltop table and the statement 'Target is excluded anyway: last scattering sits in the bump.' — The exclusion of the target hilltop is asserted based on the statement that last scattering sits in the bump, but the paper does not show the H(z) profile or the criterion for exclusion. The table gives Delta H/H=+20.6% at z=1100 for the target case, but the paper does not state what threshold of Delta H/H is excluded by observations.

  If wrong: If the exclusion criterion is not well-defined, the claim that larger hilltops are excluded by H(z*) is not rigorously established, and the promotion of only the mild hilltop to Paper II would be less justified.

• medium
  Section 4, hilltop definition from the full potential — The paper labels nE as the early Gaussian hilltop, but it is not generally a stationary point of the full two-scale potential.

  If wrong: The assumed frozen initial placement, rolling timing, and fE-to-zpeak mapping may not correspond to the described hilltop evolution.

• medium
  Section 4, r_s,d and r_d values; Delta(r_d/r_s) = +1.3% (mild), -0.2% (target) — Sound-horizon and diffusion-scale integrals are not written down. The reported Delta(r_d/r_s) is stated 'after retuning to fix theta_*' but the retuning (which h values) is not reported, and the text's claim that r_d 'changes in step with r_s' is not reconciled with a +1.3% ratio shift for the mild case.

  If wrong: The claim that 'Silk damping from that ratio alone is not an exclusion' would be unsupported, potentially removing the mild hilltop as a viable option and eliminating the paper's only surviving early-time configuration. Peripheral to the late-only DESI analysis.

• low
  Appendix, kernel-to-potential coarse-graining — The claim that a retarded, exponentially damped phase-locking kernel generates rho_0(1 - e^{-alpha n}) after coarse-graining is asserted with no derivation. The author explicitly flags this as motivation only and states uniqueness is not claimed, so it is peripheral.

  If wrong: Only the physical motivation for the potential's functional form is lost; all background results, which take V(n) as a postulate, are unaffected.

• low
  Section 2, finite plateau criterion — A finite saturation condition is used to state Lambda-like behavior without a quantitative tolerance.

  If wrong: The exact wording w_n=-1 and 'indistinguishable' at the stated boundary is too strong, although the asymptotic Lambda limit as alpha n tends to infinity remains valid.

• low
  Section 3, distance metric d — The distance d uses asymmetric reported CPL uncertainties without specifying whether upper, lower, or symmetrized uncertainties are used.

  If wrong: The comparison metric d, used to support the claim of relative closeness to DESI+Pantheon+, depends on the uncertainty convention. This is a secondary detail but should be clarified.

• low
  Section 4, 'Naive H0 rescaling at fixed theta_*' — The paper states that a naive H0 rescaling maps mild onto ~71 km/s/Mpc and target onto ~73 km/s/Mpc, but does not show the rescaling formula or the values of theta_* used. The paper labels these as 'scale checks, not fits,' which mitigates the severity.

  If wrong: If the rescaling is incorrect, the scale-check numbers would be misleading, but since they are explicitly not fits, the impact on the main conclusions is limited.

• low
  Section 4, Delta(r_d/r_s) = +1.3% (mild) and -0.2% (target) — The paper states 'the photon diffusion scale r_d changes in step with r_s' but then reports a +1.3% ratio shift for the mild case, which is not 'in step' in the sense of equal fractional changes.

  If wrong: This is a minor wording inconsistency that does not affect the central conclusions, but it should be reconciled.

• low
  Section 4, H0 rescaling: mild -> ~71, target -> ~73 km/s/Mpc — The naive fixed-theta_* mapping to H0 is not shown. The paper explicitly labels these 'scale checks, not fits', which mitigates the gap.

  If wrong: Nothing central; the paper explicitly disclaims solving the Hubble tension and labels these as order-of-magnitude checks.

• low
  Section 4, naive H_0 rescaling — The mapping from the sound-horizon reductions to H_0 approximately 71 and 73 is presented without the scaling equation or angular-diameter-distance calculation.

  If wrong: Only the explicitly labeled scale-check estimates would change; the scalar equations would remain unaffected.

• low
  Section 4, naive H0 rescaling — The mapping from sound-horizon changes to the stated H0 scale checks is not given.

  If wrong: Only the illustrative H0 numbers are affected; the author already states that they are not fits.