Theory Hardening Analysis: MCE/EME Theory — Inconsistencies, Contradictions, and Resolutions (v1.0)

Revision note (v13.0). Parts I to IV and the summary table are kept as the historical record of the v12.1 review. Several of the "Resolutions" recorded here were later found to be wrong, and they are not edited. Each issue whose resolution is withdrawn or superseded now carries a v13.0 status line, and the summary table has a v13.0 status column. A new Part V lists the errors found by the v13.0 audit, with the corrected values. The current architecture is in section 1 of the main document. Where Part V and an earlier part disagree, Part V is current. "EME" is the historical name of MCE.

Internal review document. The v12.1 review recorded below stated that all issues had been addressed. That statement is superseded by the v13.0 audit in Part V. The document is retained for transparency and as an audit trail.

Overview

This document records a systematic, adversarial review of the MCE/EME theory corpus (v12.0). Issues are rated by severity:

🔴 Critical — Would be fatal to the theory if unresolved; sufficient for rejection at peer review
🟠 Major — Significantly weakens the theory; must be addressed before submission
🟡 Minor — Weakens rigour; should be addressed but not immediately fatal
🟢 Presentational — Formatting, clarity, or terminology issues; easy to fix

Part I: Mathematical and Logical Errors

Issue 1: Circular Reasoning in the Derivation of κ 🔴

Location: Quantum-Mechanical Foundation and First-Principles Derivations, Section 1.2

Problem Identified: The document claims that κ is "derived from fundamental constants, not fitted." The given formula is:

$\kappa = \frac{1}{\sqrt{4\pi\epsilon_0}} \sqrt{\frac{G}{c^2}}$

This formula contains $G$ — Newton's gravitational constant. But the MCE theory's entire purpose is to provide a mechanistic replacement for gravity. If κ is defined in terms of G, then the theory has not derived gravity from electromagnetic first principles; it has merely renamed G and called κ a different symbol. This is circular: the theory assumes gravity (via G) to derive the parameter that is supposed to produce gravity.

Severity Assessment: 🔴 Critical — Any GR purist will identify this immediately and use it to dismiss the entire framework.

Resolution Applied: The correct framing, now explicitly stated in the main document, is as follows:

G is not derived by MCE; it is absorbed. Newton's G is an empirically measured proportionality constant that relates mass to force. The MCE theory explains the mechanism by which this force arises (QVP asymmetry → scalar field φ → attraction), but the magnitude of that force is set by κ, which must be fixed to observational data in exactly the same way that GR requires experimental measurement of G. The claim "κ is derived, not fitted" should be restated as: "κ is derived from G and fundamental constants under the requirement that MCE reproduces Newtonian gravity in the macroscopic limit." The derivation is a matching condition, not a first-principles prediction of G.

This is not a weakness — it is an honest acknowledgement that MCE is an EFT. GR itself doesn't predict G; it takes G from experiment. MCE does the same. The novel content is in the mechanism and the WEP-violating predictions at microscales, not in predicting a new value of G.

Action: Updated phrasing in main document Section 1.2 and QM Foundation document Section 1.2. κ is described as "derived via a matching condition from G and fundamental constants, not as a free fit parameter."

v13.0 status: Withdrawn. Matching κ to G only relabels G, so the circularity was not removed by this resolution. The value κ = 1.623×10⁻¹⁰ C/kg is also wrong (Part V, item V.1). In v13.0, G enters through the Einstein–Hilbert action as an experimental input (Level 0) and no parameter is fixed by matching G. κ may survive as notation for β₀/M_Pl, which has units of inverse mass.

Issue 2: Mathematical Error — Non-Local Propagator Ghost Poles 🔴

Location: Causality Proof for the EME Non-Local Operator, Sections 2.1, 3.1, 3.2

Problem Identified: The original causality proof used a polynomial regulator: $K(\square) = \frac{1}{\square + m^2} \left[1 + \frac{\square}{\Lambda^2}\right]^{-2}$

The proof then claimed: "The non-local term $[1 - p^2/\Lambda^2]^2$ is a polynomial in $p^2$ , which has no poles... the non-local part of the propagator is a non-singular function."

This is algebraically incorrect. The propagator is $D(p) = 1/G(p)$ where $G(p) = (-p^2+m^2)[1-p^2/\Lambda^2]^2$ . The zeros of G(p) at $p^2 = \Lambda^2$ are poles of D(p), not non-singularities. The polynomial $[1-p^2/\Lambda^2]^2$ being pole-free does not mean its reciprocal $[1-p^2/\Lambda^2]^{-2}$ is pole-free — the opposite is true. Any reviewer with a QFT background would immediately identify this error.

Furthermore, the "spacelike poles" argument used to dismiss these poles is non-trivial in Lorentzian signature and requires the machinery of distributional Green's functions and the Källén-Lehmann representation — none of which are provided.

Severity Assessment: 🔴 Critical — A demonstrably incorrect mathematical claim in the causality proof would invalidate the theory's claims of ghost-freedom.

Resolution Applied: The polynomial regulator has been replaced throughout with the exponential entire-function regulator: $K(\square) = \frac{e^{-\square/\Lambda^2}}{\square + m^2}$

The exponential function $e^{-\square/\Lambda^2}$ is entire — it has no poles anywhere in the finite complex plane. This eliminates all non-local poles from the propagator. The causality proof now reduces to the standard retarded Green's function argument for the single physical pole at $p^2 = m^2$ , which is rigorous and well-established. The ghost-freedom proof is now a single-line residue calculation (Appendix D, Section 4).

This approach is consistent with the broader non-local gravity literature (Biswas et al., Modesto) and is more parsimonious than the polynomial approach.

v13.0 status: Open. The single-pole property of the exponential form factor stands. The causality statement ("rigorous and well-established") does not: the factor e^{p²/Λ²} grows at timelike momenta, so the retarded-Green's-function argument is conditional pending expert review, and the "Lee–Wick" label does not apply to entire-function form factors (Part V, item V.5).

Issue 3: Coherence Length Scale Bridging Formula 🟠

Location: Quantum-Mechanical Foundation and First-Principles Derivations, Section 2

Problem Identified: The fundamental coherence length is computed as $\lambda_c^{\text{fund}} \approx 3.8 \times 10^{-13}$ m. The document then claims the macroscopic value $\lambda_c \approx 10^{-6}$ m is recovered via thermal decoherence using: $\lambda_c^{\text{eff}} \approx \lambda_c^{\text{fund}} \cdot \frac{E_{\text{ZPF}}}{k_B T}$

Two problems:

Numerical check: $3.8 \times 10^{-13} \times (0.511 \text{ MeV} / 0.026 \text{ eV}) = 3.8 \times 10^{-13} \times 1.96 \times 10^7 \approx 7.4 \times 10^{-6}$ m — this gives approximately 7 μm, not 1 μm. The discrepancy of a factor of ~7 is not acknowledged.
The relationship between the Lindblad master equation and this bridging formula is stated but not derived. The claim $m_{\text{eff}} \propto \Gamma$ and $\lambda_c^{\text{eff}} = \hbar/(m_{\text{eff}} c)$ requires a specific proportionality constant to yield the stated formula, which is not given.

Severity Assessment: 🟠 Major — The numerical discrepancy is visible to any reader who checks the arithmetic, and will be used to question the coherence scale estimate.

Resolution Applied: The discrepancy is acknowledged explicitly in the updated document. The correct range from the bridging formula is $\lambda_c^{\text{eff}} \in [1, 10]$ μm depending on the precise value of $T_{\text{eff}}$ and the specific ZPF modes contributing to decoherence. The theory uses $\lambda_c = 1$ μm as a conservative lower bound (which maximises the suppression at macroscopic scales and is therefore the most conservative choice for WEP compatibility). The full derivation of the Lindblad proportionality constant is deferred to the UV completion paper, with an explicit note that the factor-of-7 ambiguity translates to only a factor-of-7 uncertainty in the predicted WEP signal magnitude at the microscale — which does not affect the falsifiability conclusion.

v13.0 status: Withdrawn. The "bridging formula" multiplies ħ/(m_e c) by m_e c²/(k_B T), so the electron mass cancels and the result is ħc/(k_B T) = 7.63 μm at 300 K. That is the "factor of 7". It contains nothing about gravity, and the choice of 1 μm as a "conservative lower bound" had no basis. The fixed coherence length λ_c is replaced by the environment-dependent range λ(ρ) of the screened scalar. The thermal wavelength may be mentioned only as a possible origin of a micrometre scale (Part V, item V.18).

Issue 4: Modified Friedmann Equation Contains G — Inconsistency with "Replace Gravity" Claim 🟠

Location: Cosmological Extension of the Electrostatic Mass Emergence (EME) Theory, Section 3

Problem Identified: The modified Friedmann equation is: $H^2 = \frac{8\pi G}{3}\left(\rho_b + \rho_r + \rho_\Lambda + \rho_{\text{EME}}\right)$

This equation still contains G. If the EME theory replaces gravity, why is G still present? A critic will immediately ask: "Is this the old gravity plus the EME fluid, or is EME the replacement for gravity?"

Severity Assessment: 🟠 Major — Creates conceptual confusion about what the theory claims to replace.

Resolution Applied: The cosmological extension explicitly clarifies that in the cosmological coarse-graining, $G$ appears because the Einstein-Hilbert term $R/(16\pi G)$ is retained in the action for metric consistency (the MCE theory is not a theory of quantum gravity and does not modify spacetime geometry at the perturbative level). The $G$ in the Friedmann equation is therefore a spacetime geometry parameter that fixes the relationship between matter energy and spacetime curvature, while the force of gravity — the acceleration experienced by test masses — is generated by the EME scalar field $\phi$ , not directly by spacetime curvature.

The more precise statement is: MCE is a theory about the source of the gravitational force, not a modification of the metric structure of spacetime. The metric responds to the total energy-momentum tensor (including the MCE field), but the MCE field is what generates the attractive force between masses. At the cosmological level, this means G remains as a conversion constant between energy density and spacetime curvature, while $\rho_{\text{EME}}$ is the novel component that modifies the expansion history relative to ΛCDM.

v13.0 status: Superseded (the conclusion stands, the wording does not). The equation keeps G, and G is the metric-sector constant (Level 0). The statement that the force of gravity "is generated by the EME scalar field, not directly by spacetime curvature" is withdrawn: gravity is the metric sector, and the scalar is an additional, screened, composition-dependent force. The claim that MCE "replaces gravity" is withdrawn.

Issue 5: Short-Range Yukawa Coupling α ≈ 10³⁶ — Extreme Fine-Tuning Unexplained 🟠

Location: Refinement of WEP Suppression and Short-Range Force Compatibility, Section 3.1

Problem Identified: The EME short-range force law parameters are stated as $\alpha \approx 10^{36}$ and $\lambda^\pm \approx 10^{-12}$ m. A Yukawa coupling strength of $10^{36}$ times gravity at the sub-nuclear scale, with a range of $10^{-12}$ m, implies that the EME force is stronger than the strong nuclear force at those scales (the strong force has $\alpha_{\text{strong}} \sim 1$ and $\lambda_{\text{strong}} \sim 10^{-15}$ m). This seems physically implausible and would have observable consequences in nuclear physics that are not discussed.

Severity Assessment: 🟠 Major — This number is quoted without justification and would draw immediate critical attention.

Resolution Applied: The Yukawa parameterisation in the EME context describes the residual non-screened bipolar structure of the EME force at sub-nuclear scales, not a new nuclear-force-scale interaction. The parameter $\alpha \approx 10^{36}$ is the ratio of the Yukawa contribution to the gravitational contribution at the range $\lambda^\pm \approx 10^{-12}$ m. At this scale, all interactions (electromagnetic, strong, weak) are enormously stronger than gravity — the electromagnetic coupling is $\sim 10^{36}$ times gravity at that range, which is precisely the well-known hierarchy of forces. The EME bipolar structure mimics the natural force hierarchy: at sub-nuclear scales, the EME field transitions from its macroscopic "gravity-like" behaviour to coupling that is commensurate with the nuclear scale electromagnetic vacuum.

This parameter is therefore not fine-tuning — it reflects the known hierarchy of fundamental forces. The exponential suppression $e^{-r/\lambda^\pm}$ at $r \gg \lambda^\pm \approx 10^{-12}$ m (i.e., at all hadronic and above scales) ensures that this sub-nuclear EME contribution is invisible at any currently probed length scale.

v13.0 status: Withdrawn. The number 10³⁶ is the ratio of the electromagnetic to the gravitational force (1.24×10³⁶ for two protons). It is electromagnetism, not an MCE force, and the resolution above identified it as such while still presenting it as an MCE Yukawa coupling. The Yukawa parameterisation with α ≈ 10³⁶ at 10⁻¹² m is removed (Part V, item V.19).

Issue 6: Duplicate Content in Suppression Function Document 🟡

Location: First-Principles Derivation of the Suppression Function S(rho), Lines 86–130

Problem Identified: Sections 3.1 through 3.3 of this document appear twice: once as the primary derivation (lines 43–80) and again as a nearly verbatim repetition (lines 86–130). This is a copy-paste error that would be immediately visible to any reader and would undermine the document's professionalism.

Severity Assessment: 🟡 Minor (but damaging to credibility)

Resolution Applied: The duplicate content has been removed. A brief normalisation note clarifying the factor-of-2 bookkeeping (the 1/2 absorbed into δ(Z,A)) has been added in its place.

Issue 7: Typographical Error in QM Foundation Document 🟢

Location: Quantum-Mechanical Foundation and First-Principles Derivations, end of Section 2

Problem Identified: The text reads "...the necessary QFT justification for the effective parameter $\lambda_c$ .nclusion" — a missing newline and capital letter resulting in "Conclusion" being rendered as ".nclusion".

Severity Assessment: 🟢 Presentational

Resolution Applied: Fixed to "...the necessary QFT justification for the effective parameter $\lambda_c$ .\n\n## 3. Conclusion"

Issue 8: Appendix J Referenced But Non-Existent 🔴

Location: Main document, Section 6 and Appendices list

Problem Identified: The main document references "Appendix J: Geometric Framework Neutrality and Dual Applications" as the foundation for the theory's toroidal field stance and geometric neutrality claims. This appendix did not exist in any content file, making these claims entirely unsupported by the document set.

Severity Assessment: 🔴 Critical — A theory document that references its own non-existent appendix is internally incoherent. Any reader following the reference would find nothing.

Resolution Applied: Appendix J has been created as a comprehensive standalone document (see Appendix J), covering: formal proof of geometric neutrality; the Toroidal Field (TF) framework with specific boundary conditions and predictions; the Standard Heliocentric (SH) framework with full GR test compatibility table; the geomagnetic-QVP coupling mechanism with quantitative estimates; a discrimination table for TF vs SH within MCE; and a philosophical position statement on empirical priority over geometric dogma.

v13.0 status: Open. The appendix exists, but several of its numbers were wrong and are corrected in v13.0: the radius ratio R_T/r_T = 6.4, the quadrupole moment, the pole-asymmetry null, the geomagnetic estimate, the claim that odd harmonics are forbidden, and the statement that the heliocentric model is a consequence of MCE (Part V, items V.8 to V.10 and V.21).

Issue 9: Terminology Inconsistency — EME vs MCE 🟡

Location: All documents

Problem Identified: The theory is referred to interchangeably as "EME Theory" (Electrostatic Mass Emergence) and "MCE Theory" (Mass-Charge Emergence) throughout the document set. The main document title refers to "MCE Theory" and the executive summary uses both. Other appendices use "EME" exclusively. This creates confusion about whether these are the same theory or distinct variants.

Severity Assessment: 🟡 Minor

Resolution Applied: The canonical name is MCE Theory (Mass-Charge Emergence), with "EME" (Electrostatic Mass Emergence) retained as the historical/colloquial shorthand for the same theory. The main document now clarifies in its opening paragraph: "The terms EME and MCE refer to the same theory. EME reflects the historical naming from the theory's electrostatic origins; MCE is the updated name reflecting the full scalar-vector-tensor structure."

Part II: Theoretical Gaps and Missing Content

Issue 10: No Explicit Treatment of Antimatter 🟠

Location: Main document, Section 1.5

Problem Identified: The compatibility table in Section 1.5 states that "MCE predicts that antimatter will fall towards matter with the same acceleration as matter." The justification given is that "antimatter has positive mass-energy." This is correct as a statement, but is not a derivation from the MCE mechanism. The ALPHA experiment at CERN has now directly measured that antihydrogen falls downward at $g$ within experimental uncertainty. If MCE produces gravity from QVP asymmetries, the QVP contribution of an antiproton needs to be separately calculated — it is not obvious that $\rho_{\text{eff}}(\bar{p}) = \rho_{\text{eff}}(p)$ without a calculation.

Severity Assessment: 🟠 Major — With ALPHA and AEgIS constraining antimatter gravity, a theory with no antimatter QVP calculation is exposed.

Resolution Applied: The antimatter QVP calculation is now included. Key argument: CPT symmetry requires that the vacuum polarisation tensor $\Pi^{\mu\nu}(q^2)$ is identical for a particle and its antiparticle (since CPT maps one to the other and the vacuum is CPT-invariant). Since $\rho_{\text{eff}}$ is derived from the trace of $\Pi^{\mu\nu}$ , and since CPT invariance is exact in any local QFT, the effective charge $\rho_{\text{eff}}(\bar{p}) = \rho_{\text{eff}}(p)$ . Antimatter falls with the same acceleration as matter. This is now a derivation from CPT invariance, not an ad hoc assertion.

v13.0 status: Estimated. The argument stands as an argument from CPT invariance of the trace coupling. It is not a calculation of the antiproton's coupling, and "derivation" overstated it. The ALPHA free-fall result (2023) predates these documents, so agreement with it is a consistent retrodiction, not a prediction.

Issue 11: No Treatment of Gravitational Time Dilation in Detail 🟡

Location: Experimental Design..., Table in Section 7.2

Problem Identified: The table states that gravitational time dilation is "predicted as a consequence of the scalar field potential $\phi$ acting on the clock's energy levels" to $10^{-5}$ precision. But no calculation is shown. GPS clocks require corrections to 1 part in $10^{10}$ per day — much more precise than $10^{-5}$ . A reviewer will ask: is MCE actually consistent with GPS?

Severity Assessment: 🟡 Minor — The claim may be correct but the stated precision is misleading.

Resolution Applied: In the MCE framework, time dilation has two contributions: (1) the standard GR contribution from the background metric (which MCE inherits through the Einstein-Hilbert term), and (2) a novel MCE contribution from the scalar field potential $\phi$ modifying local clock frequencies. The dominant contribution is (1), which gives the standard Schwarzschild time dilation $\Delta t / t = GM/(rc^2)$ , reproducing GPS corrections exactly. Contribution (2) is suppressed by $S(r, \rho)$ and is negligible at GPS orbital altitudes. MCE is fully compatible with GPS because its novel predictions are suppressed to below $10^{-20}$ at macroscopic scales.

v13.0 status: Inherited (the result), Withdrawn (the reasoning). GPS clock corrections are a General Relativity result of the metric sector (Level 0). They are not derived from the scalar. The statement that the scalar contribution is "suppressed by S(r,ρ) to below 10⁻²⁰" is withdrawn together with S(r,ρ). A scalar contribution to clock rates is small only if thin-shell screening holds. A first-pass estimate for the Earth passes the MICROSCOPE requirement with a thin margin (Part V, item V.27). The scalar contribution to clock rates has not been calculated on its own, and the full screening calculation is open (see the screened scalar document, sections 4 and 8).

Issue 12: Bullet Cluster Treatment Incomplete 🟡

Location: Experimental Design..., Section 3.1

Problem Identified: The Bullet Cluster (1E 0657-56) is correctly identified as a key test case. The observed separation between the X-ray (baryonic) gas and the gravitational lensing mass is the most cited evidence for particle dark matter. The document states MCE must reproduce this "solely using the EME field structure generated by visible matter." However, no mechanism or even qualitative explanation is provided for how a purely baryonic-sourced EME field could produce a lensing mass distribution that is spatially offset from the visible baryons by several hundred kiloparsecs.

Severity Assessment: 🟡 Minor (but a very important gap for dark matter arguments)

Resolution Applied: The key MCE mechanism for the Bullet Cluster is the non-linear, density-dependent screening function $S_\rho(\rho)$ . In the collision region, the X-ray gas has density $\rho \gg \rho_c$ , so $S_\rho \approx 0$ and the EME material-dependent contribution is fully screened. The two galaxy subclusters (low density stellar matter), which have passed through each other, have $\rho \ll \rho_c$ , so $S_\rho \approx 1$ and the full EME field contribution is unsuppressed. The lensing mass (which is what weak gravitational lensing measures) follows the low-density stellar mass, not the high-density gas — which is exactly the observed offset. This is a qualitative prediction from MCE that is consistent with the Bullet Cluster observation, without invoking any dark matter particle. A full numerical simulation (as described in the experimental document) is needed to confirm the quantitative lensing profile.

v13.0 status: Withdrawn. A conformally coupled scalar does not bend light beyond General Relativity, so it cannot produce a lensing mass offset from the baryons. Screening the scalar force by density does not change this: it alters how matter moves, not how light is bent. Level 1 does not explain the Bullet Cluster (Part V, item V.20).

Part III: Theoretical Hardening Recommendations Beyond the Reviewer's Suggestions

The following items go beyond the reviewer's suggestions and represent independent improvements identified in this analysis.

Recommendation A: Vacuum Energy Cancellation — Toy Model Calculation 🟠

Background: Section 1.3 of the main document states: "The MCE theory addresses the cosmological constant problem by proposing a symmetry in the UV completion that cancels the bulk vacuum energy, leaving only the mass-induced QVP asymmetry as the source of the MCE field." This is stated but not demonstrated.

Proposed Addition: Consider a toy model with a real scalar field $\varphi$ (the "Higgs-sector-like" field) with a $\mathbb{Z}_2$ symmetry $\varphi \to -\varphi$ . In the symmetric phase, the vacuum energy is: $\rho_{\text{vac}}^{\text{sym}} = \frac{1}{2} \sum_k \omega_k + \frac{1}{2} \sum_k (-\omega_k) = 0$ where the second sum is over virtual antiparticles with opposite sign (the symmetry pairs virtual particle and antiparticle contributions exactly). The $\mathbb{Z}_2$ symmetry forces exact cancellation.

In the presence of mass $m_{\text{particle}}$ , the $\mathbb{Z}_2$ symmetry is explicitly broken by the coupling to the Higgs VEV. The residual vacuum energy is: $\rho_{\text{vac}}^{\text{residual}} = \frac{m_{\text{particle}}^2 c^2}{16\pi^2 \hbar^3} \Lambda^2$ This is proportional to $m^2 \Lambda^2$ . For $m = m_e$ (electron mass) and $\Lambda = \Lambda_{\text{EFT}} = 10^{10}$ eV: $\rho_{\text{vac}}^{\text{residual}} \approx \frac{(0.511 \times 10^6)^2 \times (10^{10})^2}{16\pi^2 \hbar^3 c^3} \approx 10^{-3} \text{ eV}^4$ This corresponds to $\Lambda_{\text{eff}}^4 \sim 10^{-3}$ eV $^4$ , which is 52 orders of magnitude smaller than the naive QFT vacuum energy $\Lambda_{\text{UV}}^4 \sim (10^{18} \text{ GeV})^4$ and close to the observed $\Lambda_{\text{obs}}^4 \sim (10^{-3} \text{ eV})^4$ . While not a perfect match, the symmetry argument dramatically reduces the cosmological constant problem from 120 to $\sim 4$ orders of magnitude — a significant improvement that merits further development in the UV completion paper.

v13.0 status: Withdrawn. The arithmetic is wrong and the conclusion does not follow. The stated formula gives 1.65×10²⁹ eV⁴ for Λ = 10¹⁰ eV, not about 10⁻³ eV⁴. The observed scale is (2.3 meV)⁴ = 2.8×10⁻¹¹ eV⁴, so the residual is about 40 orders of magnitude too large. The naive Planck-scale estimate is about 123 orders too large. The toy model therefore reduces 123 to about 40, not 120 to 4. It is speculative and not a solution (recomputed in Part V, item V.4).

Recommendation B: N-Body Simulation Integration Path 🟡

The theory would benefit from a concrete open-source code implementation plan. Recommended approach:

Integrate the MCE scalar field equation as a modified Poisson solver in the publicly available GADGET-4 N-body code.
The modification is: $\nabla^2 \phi = -4\pi G \rho_{\text{eff}} \cdot S_\rho(\rho)$ , replacing the standard $\nabla^2 \Phi_N = -4\pi G \rho$ .
Run the simulation on the Aquarius halo from the Millennium Simulation initial conditions.
Compare the resulting density profile with the standard NFW profile.

Predicted distinguishing result: The MCE potential has a slightly softer core (lower central density) than the NFW profile because $S_\rho(\rho) < 1$ at high densities, reducing the effective gravitational pull in overdense regions. This would produce galaxy rotation curves that are slightly shallower in the inner region — consistent with the observed "cusp-to-core" discrepancy in ΛCDM without requiring baryonic feedback.

v13.0 status: Withdrawn. The proposed modified Poisson equation used the withdrawn S_ρ, and the "predicted distinguishing result" (softer cores, shallower inner rotation curves) was asserted without a calculation. No N-body run exists. The integration path would now be a solver for the nonlinear screened scalar equation, with the linear-theory coupling G_eff(k,a) = G[1 + 2β²k²/(k² + a²m²(a))] as a check. Not yet computed.

Recommendation C: Phase-Diagram of MCE Observable Signatures 🟡

A combined "phase diagram" in the $(r, \rho)$ plane would be a powerful communication tool for the theory's testable predictions. This diagram would show:

The region where MCE = Newtonian gravity (large $r$ and large $\rho$ ): the suppression function $S(r,\rho) \approx 0$ .
The region of measurable WEP violation ( $r \lesssim \lambda_c$ , $\rho \lesssim \rho_c$ ): $S \approx 1$ .
The transition region ( $r \sim \lambda_c$ or $\rho \sim \rho_c$ ): the testable intermediate regime.
Overlay of existing experimental constraints and future experimental reach.

This diagram communicates at a glance why all macroscopic tests are consistent with MCE (they lie in the $S \approx 0$ region) and why micro-scale tests are needed (they target the $S \approx 1$ region).

v13.0 status: Retired. The diagram was drawn in the plane of a fixed λ_c and the tanh density function S_ρ, both of which are retired. A phase diagram would now be drawn in the plane of the environment-dependent range λ(ρ) and the thin-shell condition, and it has not been recomputed.

Recommendation D: Response to MICROSCOPE v2 Potential 🟡

The MICROSCOPE satellite completed its mission in 2018 with a result of $\eta < 10^{-15}$ . A MICROSCOPE successor mission (conceptual name MICROSCOPE-2 or STEP) could reach $\eta \sim 10^{-18}$ . MCE must demonstrate that its suppression mechanism is robust against even this improved sensitivity.

At the MICROSCOPE orbital altitude ( $h \approx 710$ km), the effective density of the test mass environment is the density of the test mass itself ( $\rho \approx 8.9 \times 10^3$ kg/m³ for Platinum). The density suppression is: $S_\rho = 1 - \tanh(8.9 \times 10^3 / 1.1 \times 10^3) \approx 1 - \tanh(8.1) \approx 2 \times 10^{-7}$ The spatial suppression at the scale of the test mass separation ( $r \approx 10^{-2}$ m): $S_r = e^{-10^{-2}/10^{-6}} = e^{-10^4} \approx 10^{-4343}$ The combined suppression is $S \approx 10^{-4343}$ , which is below $10^{-18}$ by an astronomical margin. MICROSCOPE-2 at $\eta \sim 10^{-18}$ would not detect the MCE signal at satellite altitude. The decisive experiment remains the microscale composition test at $r \approx 1$ μm.

v13.0 status: Withdrawn. Three errors. First, the density used for platinum, 8.9×10³ kg/m³, is not the density of platinum (about 21,450 kg/m³, a handbook value; the alloy flown on MICROSCOPE is to be checked). Second, S_ρ and S_r, and the product 10⁻⁴³⁴³, belong to the retired fixed-length model. Third, MICROSCOPE did not end in 2018 with "η < 10⁻¹⁵": its final result (Touboul et al., Physical Review Letters 129, 121102, 2022) is η(Ti,Pt) = [−1.5 ± 2.3 (stat) ± 1.5 (syst)]×10⁻¹⁵. The correct statement is that MICROSCOPE fixes the Earth thin-shell requirement, 3ΔR⊕/R⊕ ≲ 1.9×10⁻⁷ for the legacy benchmark. A first-pass estimate meets that requirement by a factor of about 1.5 to 4; the full calculation is open (Part V, items V.7 and V.27). A successor mission reaching η ~ 10⁻¹⁸ would tighten the requirement by a factor of about 2,700 (from 2.7×10⁻¹⁵), if the requirement scales linearly with the bound.

Part IV: Responses to the External Reviewer's Specific Suggestions

Reviewer Point 1: Full Renormalisation Analysis ✅ Addressed

The reviewer requested explicit beta functions and renormalisation group flow analysis. This has been provided in full in Appendix L: Renormalisation Group Analysis and UV Stability, including one-loop beta functions for all three MCE parameters, a fixed-point analysis, and a demonstration of radiative stability.

v13.0 status: Withdrawn (in part). The running of κ is withdrawn, because κ is no longer fixed by G and the running of β₀ has not been computed. The sign statement ("asymptotically free") was wrong, and the arithmetic had errors. The "demonstration of radiative stability" is not made. Appendix L was revised; see Part V, items V.13 to V.15.

Reviewer Point 2: Non-Local Operator Ghost/Instability Proof ✅ Addressed (Superseded)

The reviewer suggested a "perturbative expansion proving no ghosts or instabilities." The approach taken here is more rigorous: the polynomial regulator has been replaced with an exponential entire-function regulator (Appendix D, v2), which eliminates the ghost problem at the level of the operator definition rather than through perturbative argument. This is a stronger result.

v13.0 status: Open. The entire-function form factor adds no new poles. Whether the theory is ghost-free and causal in the full Lorentzian sense is conditional pending expert review (Part V, item V.5). "A stronger result" is withdrawn.

Reviewer Point 3: Material Dependence from Lattice QCD ✅ Addressed

The reviewer suggested tying $C_{\text{QFT}} \approx 0.03$ to lattice QCD data. This connection is now established in Appendix L (Section 5.2), which shows that $C_{\text{QFT}}$ is protected by isospin symmetry to be proportional to $(m_d - m_u)/\Lambda_{\text{QCD}}$ , a quantity directly measured by lattice QCD to 5% precision. The predicted experimental value (accounting for 14% QCD running) is $\Delta a/a \approx 6.0 \times 10^{-9}$ .

v13.0 status: Open (the link), Withdrawn (the number). The proportionality of C to (m_d − m_u)/Λ_QCD was asserted, not derived, and the lattice values quoted are marked to be verified in Appendix L. The v13.0 definition of δ already contains ε = (m_n − m_p)/m_p, so a further isospin-breaking factor in C may count the same breaking twice. The value 6.0×10⁻⁹ includes a QCD running factor of 0.86 that is probably applied twice and has no error bar that can be defended; it is retained only as a legacy reference point of about 6 to 7×10⁻⁹ (Part V, items V.15 and V.17).

Reviewer Point 4: Microscale Experimental Roadmap ✅ Already Present + Enhanced

The phased experimental roadmap was already present in the experimental design document. The atom interferometry protocol with Casimir force discrimination is detailed (Section 2.1, 7.1.1, 7.1.2). The new addition: the Phase-Diagram recommendation (Recommendation C above) provides a visual framework for the experimental roadmap.

Reviewer Point 5: Cosmological Forecasts for Euclid/JWST ✅ Already Present + Enhanced

The $P(k)$ suppression prediction and CMB damping tail shift are present in the cosmological extension document. The addition: an explicit note that the RG-improved prediction for $\Delta a/a$ (6.0 × 10⁻⁹ vs 7 × 10⁻⁹) also modifies the cosmological $P(k)$ suppression amplitude by 14%, which should be included in any Euclid forecast.

v13.0 status: Withdrawn. The cosmological forecasts had inputs that were wrong by many orders of magnitude (Part V, item V.12). The "14% modification of P(k)" followed from the double-counted running factor and from no P(k) calculation. The cosmological extension now states that no CMB or P(k) calculation has been done.

Reviewer Point 6: Toroidal Field Appendix ✅ Addressed (Exceeded)

The reviewer suggested a "speculative appendix" treating the toroidal field as a minor anisotropy within heliocentrism. We go significantly further. Appendix J provides a rigorous treatment of the Toroidal Field framework as a complete, internally consistent application of MCE with its own dedicated observational predictions (pole asymmetry, toroidal harmonics, geomagnetic-gravity coupling), while simultaneously demonstrating full compatibility with the heliocentric framework. The TF framework is not treated as "fringe-adjacent" speculation but as a legitimate alternative global boundary condition for the MCE field equations, with testable signatures that distinguish it from spherical models using existing satellite gravimetry data. This elevates the toroidal discussion from a footnote to a scientific programme.

v13.0 status: Withdrawn (for the nominal parameters). Tested against the measured quadrupole moment, the nominal toroidal parameters give J₂ about 350 times the observed value, and the pole-asymmetry protocol used a null of zero where standard geodesy already gives about 45 m (Part V, items V.9 and V.10; Global Geometry Hypothesis Tests). The toroidal framework stays listed as a falsifiable branch until a parameter set that reproduces the measured J₂ is proposed. "Exceeded" and "rigorous" are withdrawn.

Summary of Changes Made

The "v12.1 status" column is the historical record and is not edited. The "v13.0 status" column uses the status vocabulary of the main document.

Issue	Severity	v12.1 status (historical)	v13.0 status
1. Circular κ derivation	🔴 Critical	✅ Resolved — reframed as matching condition	Withdrawn
2. Ghost poles in causality proof	🔴 Critical	✅ Resolved — exponential regulator adopted	Open (conditional on expert review)
3. λ_c bridging formula discrepancy	🟠 Major	✅ Resolved — factor-of-7 acknowledged, conservative bound justified	Withdrawn
4. G in cosmological equations	🟠 Major	✅ Resolved — G retained as geometric parameter, clarified	Inherited (G is the metric-sector constant); wording Withdrawn
5. α ≈ 10³⁶ without justification	🟠 Major	✅ Resolved — natural force hierarchy argument applied	Withdrawn (it is electromagnetism)
6. Duplicate content in S(ρ) document	🟡 Minor	✅ Resolved — duplicate removed, normalisation note added	n/a (editorial)
7. Typographical error	🟢 Presentational	✅ Fixed	n/a (editorial)
8. Appendix J non-existent	🔴 Critical	✅ Created in full (16-page document)	Open (numbers corrected in v13.0)
9. EME/MCE terminology inconsistency	🟡 Minor	✅ Resolved — canonical name established	n/a (editorial)
10. Antimatter not derived	🟠 Major	✅ Resolved — CPT argument provided	Estimated (an argument, not a calculation)
11. GPS compatibility unclear	🟡 Minor	✅ Resolved — dominant contribution from GR metric	Inherited (reasoning Withdrawn)
12. Bullet Cluster mechanism missing	🟡 Minor	✅ Resolved — density screening mechanism applied	Withdrawn
A. Vacuum energy toy model	🟠 Major	✅ Added — 4-order reduction of cosmological constant problem	Withdrawn (about 123 to about 40 orders)
B. N-body simulation path	🟡 Minor	✅ Added — GADGET-4 integration plan	Withdrawn
C. Phase diagram recommendation	🟡 Minor	✅ Added — observable signature map	Retired
D. MICROSCOPE-2 robustness	🟡 Minor	✅ Added — suppression confirmed to $10^{-4343}$	Withdrawn

Net result (v12.1, historical): The MCE theory v12.1 (post-hardening) was recorded as having addressed all identified critical and major issues, added two new appendices (J and L), corrected three existing documents (causality proof, suppression function, QM foundation), and added a comprehensive hardening analysis for transparency. The statement that the theory was "in a significantly stronger position for peer review" is superseded. The v13.0 audit in Part V found further errors, and several of the v12.1 resolutions were themselves wrong.

Part V: v13.0 audit addendum

The v13.0 audit recomputed every number in the v12 document set that could be recomputed. This part lists the errors that the v12.1 review did not find. Each row gives where the error occurred, what was wrong, the corrected value, and the new status. Numbers were computed with Python's math module from CODATA constants. External numbers are those listed in section 7 of the repair specification and in the screened scalar document; anything else is marked "to be verified".

ID	Where	What was wrong	Corrected value	New status
V.1	Quantum-Mechanical Foundation §1.2; Field Roles; Refinement of WEP Suppression; main document v12; Appendix J §4.2; Appendix L §3.1; Toroidal Field Framework §6	The value $\kappa=1.623\times10^{-10}$ C/kg, "fixed by matching $G$ ". The written formula $\kappa=(4\pi\epsilon_0)^{-1/2}(G/c^2)^{1/2}$ gives $2.58\times10^{-9}$ C/kg. The Coulomb-type match $\kappa^2/(4\pi\epsilon_0)=G$ gives $8.62\times10^{-11}$ C/kg. No reading gives $1.623\times10^{-10}$ . Matching to $G$ only relabels $G$ . For a coupling $\kappa\phi T$ , $\kappa$ has mass dimension $-1$ , so units of C/kg are also wrong	$\kappa$ withdrawn. It may be used as notation for $\beta_0/M_{\rm Pl}$ (inverse mass). No parameter is fixed by matching $G$	Withdrawn
V.2	Quantum-Mechanical Foundation; Field Roles; Refinement of WEP Suppression; Suppression Function derivation; main document v12	The coefficient $2.36\times10^{-7}$ was quoted as $C\varepsilon$ . With $C=0.03$ and $\varepsilon=(m_n-m_p)/m_p=1.378\times10^{-3}$ , $C\varepsilon=4.13\times10^{-5}$ , which is 175 times larger. The old unsuppressed difference $1.9\times10^{-8}$ equals $2\beta_0^2\Delta\delta$ with an unstated factor $2\beta_0^2=5.7\times10^{-3}$	$C\varepsilon=4.13\times10^{-5}$ ; $\Delta(Z/A)_{\rm Al\text{-}Au}=0.0807$ ; $\Delta\delta_{\rm Al\text{-}Au}=3.34\times10^{-6}$ ; $\beta_0\approx0.053$ if the factor is $2\beta_0^2$ . That reading is an inference	Form of $\delta$ : Postulated ( $C$ is an input). Factor $2\beta_0^2$ : Estimated (inference)
V.3	Quantum-Mechanical Foundation §2; Appendix L §3; Appendix O (v12 items 1.5, 3.3)	A scalar with $m_\phi\approx10^{10}$ eV was used for a micrometre-scale range and for long-range gravity. Its Compton length is $\hbar c/(m_\phi c^2)=1.97\times10^{-17}$ m. A range of 1 μm needs $m=0.197$ eV, a factor $5.1\times10^{10}$ (10.7 orders) below $10^{10}$ eV. A range of 1 AU (about 93 million miles) needs $m\lesssim1.3\times10^{-18}$ eV, 27.9 orders below. A field cannot be both this heavy and long-range	The range is $\lambda(\rho)=\hbar/(m_{\rm eff}(\rho)c)$ and depends on the environment. $\Lambda_{\rm EFT}$ is a separate free scale	Withdrawn
V.4	Recommendation A above; main document v12 §1.3; Appendix O (v12 item 1.2)	The formula $m_e^2\Lambda^2/(16\pi^2)$ with $m_e=0.511$ MeV and $\Lambda=10^{10}$ eV was evaluated as " $\approx10^{-3}$ eV $^4$ ". It gives $1.65\times10^{29}$ eV $^4$ , 32 orders larger. The observed scale is $(2.3\ {\rm meV})^4=2.8\times10^{-11}$ eV $^4$ . The comparison "52 orders below $(10^{18}\ {\rm GeV})^4$ " does not follow from the stated numbers (they give 111 orders)	The toy-model residual is 39.8, about 40, orders too large. The naive Planck-scale estimate $M_{\rm P}^4$ is 122.9, about 123, orders too large (120.1 with the reduced Planck mass). The model reduces about 123 to about 40, not 120 to 4. The $\mathbb{Z}_2$ cancellation was assumed, not derived	Withdrawn as a claim of progress
V.5	Causality Proof, v12 text (non-locality length; the document has since been revised); main document v12; Appendix O	The length $\ell_{\rm NL}=\hbar c/\Lambda$ was given as $2\times10^{-26}$ m. That has the size of the time $\hbar/E=6.6\times10^{-26}$ s, not of a length. The regulator $e^{p^2/\Lambda^2}$ grows at timelike momenta, so the retarded-Green's-function argument is conditional. The "Lee–Wick" label does not apply to entire-function form factors	$\ell_{\rm NL}=1.97\times10^{-17}$ m for $\Lambda=10^{10}$ eV. The causality result is conditional pending expert review	Open
V.6	EFT Validity and Coarse-Graining Sketch, cutoff paragraph	A non-local length of $10^{-9}$ m was said to put the cutoff "in the GeV range"	$\hbar c/(1\ {\rm nm})=197$ eV, about 6.7 orders below 1 GeV. $\Lambda_{\rm EFT}$ is a separate free scale	Withdrawn
V.7	Recommendation D above; Appendix O (v12 item 1.6)	$S_\rho=1-\tanh(\rho/\rho_c)$ was evaluated with $\rho=8.9\times10^{3}$ kg/m³ for platinum (that is not platinum's density; platinum is about 21,450 kg/m³, a handbook value, and the alloy flown on MICROSCOPE is to be checked). With $\rho_c=1.1\times10^{3}$ kg/m³, $1-\tanh(8.1)=1.9\times10^{-7}$ for the density used and $2.3\times10^{-17}$ for 21,450 kg/m³. The quoted combined $10^{-4343}$ is $S_r$ alone; including $S_\rho$ gives $10^{-4350}$	Both $S_\rho$ and $S_r$ are retired. The Earth requirement is thin-shell screening, $3\,\Delta R_\oplus/R_\oplus\lesssim1.9\times10^{-7}$ for the legacy benchmark. The first-pass estimate is item V.27	$S_\rho$ , $S_r$ : Retired. The numbers: Withdrawn
V.8	Appendix J §4.2 and §4.3 table; Toroidal Field Framework §6; Appendix O	The geomagnetic estimate $\Delta g/g\sim GB_T^2/(\mu_0c^4\kappa^2)$ was quoted as $2\times10^{-14}$ . With $B_T=10^{-3}$ T and $\kappa=1.623\times10^{-10}$ , evaluated with $c^4$ , it is $2.5\times10^{-25}$ . The text's denominator used $8.99\times10^{16}$ , which is $c^2$ not $c^4$ (that gives $2.2\times10^{-8}$ ). Neither equals $2\times10^{-14}$ . With $\kappa$ in C/kg the expression is not dimensionless	$2.5\times10^{-25}$ as written. The estimate is withdrawn together with $\kappa$	Withdrawn
V.9	Appendix J §5, §5.1, §5.2, §8 (summary table); Toroidal Field Framework §7	$R_T/r_T=6.4$ was said to match Earth's radius ratio. The Earth's equatorial-to-polar radius ratio is 1.0034. A uniform torus with $R/r=6.4$ has $J_2=0.376$ (reference radius $R+r$ ) or $0.503$ (reference radius $R$ ) against the measured $1.083\times10^{-3}$ , 348 and 465 times too large. The pole-asymmetry null " $0\pm3$ mm" ignores that standard geodesy already gives about 45 m (EGM2008 values of +14.9 m and −30.1 m, quoted from the Global Geometry document). The $J_3$ term alone ( $J_3=-2.53\times10^{-6}$ ) gives 32 m	The toroidal prediction of 0.72 m would have to appear as a residual after the full standard model, not against zero. The nominal toroidal parameters are excluded	Withdrawn for the nominal parameters
V.10	Appendix J §5 table and §6 table	Odd-degree, odd-order harmonics were called forbidden in any spherical model and zero for the heliocentric case. The real Earth has measured nonzero odd harmonics. The predicted $\delta C_{3,1}\approx2\times10^{-10}$ is also far below the size of the known $C_{3,1}$ term (order $10^{-6}$ , to be verified against ICGEM)	A toroidal signal would have to be a residual after subtracting the full standard model	Withdrawn
V.11	Appendix P §1 table and §1.2 (MAGIS-100, Stanford, Eöt-Wash, MICROSCOPE rows); Appendix P milestone table; Experimental Design §7; Appendix J §5	MAGIS-100 "2025 bound $<3\times10^{-12}$ ": the instrument is under construction at Fermilab (installation due late 2027, commissioning 2028) and has no results. "Stanford AI 2022 $<7\times10^{-9}$ " and "Eöt-Wash 2023" could not be matched to published results. MICROSCOPE " $\eta\leq1.3\times10^{-15}$ " is not the result. "HUST-Grace2030" does not exist. "LRI 80 pm/√Hz": the published requirement is 80 nm/√Hz. HUST-Grace2026s is a real model (ESSD preprint essd-2026-53; DOI 10.5880/icgem.2026.001; degree and order 180), but the quoted noise floors are not from that source. The signal-to-noise tables were invented	MICROSCOPE: $\eta({\rm Ti,Pt})=[-1.5\pm2.3\,({\rm stat})\pm1.5\,({\rm syst})]\times10^{-15}$ , about $2.7\times10^{-15}$ combined. Other rows are replaced by the verified bounds in the screened scalar document or removed	Withdrawn
V.12	Appendix P Euclid forecast note; Appendix N (kernel with $k_c=1/\lambda_c$ ); Experimental Design (N-body modification)	$k_c=2\pi/\lambda_c$ for $\lambda_c=1\ \mu$ m is $6.3\times10^{6}$ m⁻¹. It was given as $\approx6\times10^{6}\ h$ /Mpc, a unit error. In Mpc⁻¹ the value is $1.9\times10^{29}$ , a factor $3.09\times10^{22}$ (about 22 to 23 orders). Also $\kappa^2C\sim10^{-21}$ cannot produce per cent effects	The Euclid, DESI and MACS J0025 forecast tables and the GRACE-FO "toroidal coupling" forecast are withdrawn. The method is the growth formula $G_{\rm eff}(k,a)=G[1+2\beta^2k^2/(k^2+a^2m^2(a))]$ . Not yet computed	Withdrawn
V.13	Appendix L §3.1, §4.1, §4.2, §6	$\beta_\kappa=+\kappa^3/(12\pi^2)$ is positive, so the coupling grows with energy and, for a dimensionless coupling, has a Landau pole at $\ln(\mu/\Lambda)=6\pi^2/\kappa^2$ . The text called this "asymptotically free" and stated "no Landau pole". In §4.2 the labels "marginally irrelevant" and "marginally relevant" were attached to the wrong directions of flow	A positive one-loop beta function means growth in the UV. Not asymptotically free. The running of $\kappa$ is withdrawn because $\kappa$ is no longer fixed by $G$ ; the corresponding statement for $\beta_0$ has not been computed	Withdrawn
V.14	Appendix L §3.1, §3.3, §3.2, §8.2 to §8.6	$\Delta\kappa/\kappa$ : $(8.4\times10^{-3})(1.623\times10^{-10})^2(23)=5.1\times10^{-21}$ , not $5\times10^{-24}$ (treating $\kappa$ as a pure number). $\Delta m^2=\kappa^2\Lambda^2\ln(\Lambda/\mu)/(16\pi^2)=0.146$ eV², not $1.5\times10^{-3}$ eV² (a factor 97), and $\Delta m^2/m^2=1.5\times10^{-21}$ , not $10^{-23}$ . Error budget, third term: $0.2\times\ln50/(2\pi/0.1179)=1.5\%$ , not 3.6%; total 11.0%, not 11.4%; variance shares 56%, 42%, 2%, not 52%, 39%, 9%. Lattice-improvement table: the formula gives 8.2% and 7.5%, not 7.2% and 5.5%; the $\Lambda_{\rm QCD}$ term alone is 7.1%. QCD running with fixed $\alpha_s=0.118$ and $\gamma_C=-2$ : $-14.7\%$ (linear), factor 0.863 (exponential)	The $\kappa$ and $m_\phi$ running is withdrawn. The corrected budget is in Appendix L	$\kappa$ , $m_\phi$ running: Withdrawn. Corrected budget: Estimated
V.15	Appendix L §7, §8.2, §8.4; Appendix N; main document v12	The QCD running factor 0.86 (Λ_EFT to $\mu_{\rm QCD}$ ) was applied to $\Delta a/a$ although $C$ is defined at $\mu_{\rm QCD}$ ( $C(\mu_{\rm QCD})=0.03$ in §8.2). If 0.03 is the value at $\mu_{\rm QCD}$ , no factor applies and the reference point is $7.0\times10^{-9}$ . If 0.03 is the value at $\Lambda_{\rm EFT}$ , running to the IR with the stated $\beta_C$ gives $1/0.863=1.16$ , not 0.86, which gives $8.1\times10^{-9}$ . The accompanying script anchors $C$ at $\mu_{\rm QCD}$ and applies no factor. With one-loop $\alpha_s(\mu)$ instead of fixed $\alpha_s(m_Z)$ the factor is 0.63 (0.2 to 10 GeV) or 0.84 (1 to 10 GeV)	Which reading is intended is not stated in the source. Unresolved	Open
V.16	Appendix O (v12 items 1.6, 4.3); main document v12; Appendix P	"MCE predicted MICROSCOPE / GW170817 / ALPHA" and "predictions precede tests". Those results (2022, 2017, 2023) predate the documents (2026)	They are consistent retrodictions. MICROSCOPE is a constraint on the model, not a prediction	Withdrawn
V.17	Appendix L §7 and §8.4; Appendix N; main document v12	The headline $(6.0\pm0.7)\times10^{-9}$ equals $1.9\times10^{-8}\,e^{-1}=6.99\times10^{-9}$ multiplied by 0.86, which is possibly double-counted. The $\pm0.7$ came from lattice inputs passed through a relation, $C\propto(m_d-m_u)/\Lambda_{\rm QCD}$ , that was not derived	A legacy reference point of about 6 to $7\times10^{-9}$ for $2\beta_0^2=5.7\times10^{-3}$ , $C=0.03$ , $f=e^{-1}$ , without error bars	Withdrawn as a prediction. Estimated as a reference point
V.18	Issue 3 above	The "bridging formula" equals $\hbar c/(k_BT)$ , because the electron mass cancels. It is 7.63 μm at 300 K. It was presented as a derivation of $\lambda_c$ and the factor 7 as an ambiguity	The thermal wavelength may be mentioned only as a possible origin of a micrometre scale	Withdrawn
V.19	Issue 5 above; Refinement of WEP Suppression §3.1	The Yukawa strength $\alpha\approx10^{36}$ at $10^{-12}$ m. $10^{36}$ is the electromagnetic to gravitational ratio (for two protons, $1.24\times10^{36}$ ). It is electromagnetism, not an MCE force	Removed	Withdrawn
V.20	Issue 12 above; Experimental Design §3.1; Appendix N; Appendix P	The Bullet Cluster was said to be explained by density screening	A conformally coupled scalar does not bend light beyond GR, so it cannot create a lensing mass offset from the baryons. Level 1 does not explain it	Withdrawn
V.21	Appendix J §3.2; Standard Heliocentric Framework §4	"The heliocentric model is a consequence of MCE", derived through $\kappa^2/(4\pi)=G/c^2$	The field equations are geometry-neutral. The Newtonian potential comes from the metric sector with $G$ from experiment	Withdrawn
V.22	Appendix J §3.1; Standard Heliocentric Framework §3	Mercury, light bending and Shapiro delay were attributed to the scalar. A pure scalar sourced by the trace $T$ does not couple to photons ( $T=0$ ) and gives the wrong perihelion advance and light deflection	These results are Level 0 and are inherited from General Relativity	Scalar derivation: Withdrawn. Results: Inherited
V.23	Cosmological Extension §1, §4 to §6	The scope paragraph called the theory a "local, terrestrial model" with a "strict no space/universe mechanisms" rule, which contradicts the rest of the document. The "unique signature" and "falsifiable CMB damping tail" were stated without any CMB or $P(k)$ calculation	Scope rewritten. No CMB or $P(k)$ calculation has been done. The dark-matter and dark-energy analogues are speculative	Withdrawn (claims); analogues Open
V.24	Appendix L §5.1; Appendix L §4.3; Appendix O items 1.1 and 4.4	A scalar mass term is diffeomorphism invariant, so "diffeomorphism invariance prohibits a mass term" is wrong. The AS check used the withdrawn $\kappa$ and $m_\phi$ . QCD was described as having a Landau pole in the UV, but QCD is asymptotically free. The exponential regulator was called "the unique mathematically minimal choice" and said to be used in string field theory, without support	Statements removed or corrected in the revised appendices	Withdrawn
V.25	Main document §3; Appendix O critique 3.2; screened scalar document §6	v12 compared a fractional signal $\Delta a/a\sim7\times10^{-9}$ with an atom-interferometer sensitivity quoted as a fraction of $g$ , and concluded that the micrometre near-source test is within reach of current technology	For an atom near a local source, $a$ is the Newtonian pull of that source. A 1 cm aerogel slab at 10 kg/m³ gives $2\pi G\sigma=4.2\times10^{-11}$ m/s². The legacy signal is $7.0\times10^{-9}\times4.2\times10^{-11}=2.9\times10^{-19}$ m/s² (about $3\times10^{-19}$ ). Asenbaum et al. (2020) resolve $1.4\times10^{-11}\,g=1.4\times10^{-10}$ m/s² per shot. A $5\sigma$ detection needs about $5\times10^{18}$ shots, about $3\times10^{12}$ years at 15 s per shot. Even for $2\beta_0^2=1$ the signal is $1.4\times10^{-16}$ m/s² and the time is about $10^{7}$ years. If $a$ is taken as local $g$ , the observable is the Earth-sourced test already bounded by MICROSCOPE and torsion balances	Withdrawn (the reach claim). Absolute signal: Estimated
V.26	Global Geometry Hypothesis Tests, flat-disc test; simulations page, flat-disc text	After the first v13.0 pass, the retired matching $\kappa^2/(4\pi)=G/c^2$ was still the stated source of the Newtonian limit in the Global Geometry document and in the simulation text	Removed in v13.0 audit pass 2. The Newtonian limit, including the flat-disc test, comes from the metric sector. No parameter is fixed by matching $G$	Withdrawn
V.27	Screened scalar document §4; Appendix O critiques 1.3 and 1.6; suppression-function note §6	Earth and Sun screening was described as not computed	First-pass estimate for uniform spheres and one ambient density, $n=1$ , $\Lambda=2.4$ meV, $\beta_0=0.053$ : Earth $3\Delta R/R=1.2\times10^{-7}$ (galactic ambient $1.7\times10^{-21}$ kg/m³) and $5.1\times10^{-8}$ (interplanetary $10^{-20}$ kg/m³), against $\lesssim1.9\times10^{-7}$ . Passes by a factor of about 1.5 to 4. Sun $3\Delta R/R\approx4\times10^{-11}$ . For $n=1$ the factor scales as $\Lambda^{5/2}\beta^{-3/2}\rho_{\rm ambient}^{-1/2}$ . It fails by about 20 times at $\Lambda=10$ meV and by about 8 times at $\beta_0=0.01$ . The full calculation (density profile, atmosphere, ambient field value, Moon, the Sun's field at the Earth, Cassini) is open	Estimated (first-pass). Full calculation: Open
V.28	Appendix O; screened scalar document §7	Uncorrected Jaffe et al. arXiv figures were still in use: anomalous acceleration $(11\pm24)$ nm/s², one-tailed bound below 50 nm/s², and $M<2.7\times10^{-3}M_{\rm Pl}$	Jaffe et al., Nature Physics 13, 938 (2017), author correction Nature Physics 19, 1946 (2023): 0.19 kg tungsten source; $a_{\rm anomaly}=(41\pm24)$ nm/s²; one-tailed $a_{\rm anomaly}<81$ nm/s² (95%); for $\Lambda=2.4$ meV and $n=1$ , excludes $M<1.7\times10^{-3}M_{\rm Pl}$ . The uncorrected arXiv numbers are superseded	Withdrawn (uncorrected figures)

Items that could not be resolved in the v13.0 revision. (a) Which of the two readings of the QCD running factor is intended (V.15). (b) An earlier audit note gave the scalar-mass gap as 17 orders. That figure was wrong. Item V.3 already records the registry values: 10.7 orders between $m=10^{10}$ eV and a 1 μm range, and 27.9 orders between that mass and a 1 AU range. The exponent 17 appears only in the Compton length $1.97\times10^{-17}$ m. (c) The sources of the lattice and PDG inputs quoted in Appendix L (marked to be verified there). (d) The EGM2008 pole values (+14.9 m, −30.1 m), which are quoted from the Global Geometry document and were not re-derived here. (e) The screened scalar document states the Earth requirement as $\lesssim2\times10^{-7}$ in the prose of Sections 4 and 8, and as $\lesssim1.9\times10^{-7}$ in the Section 4 table. This addendum and the main document use $1.9\times10^{-7}$ . The two figures are the same requirement at the stated precision; the author should make the prose match the table.