Coherence-Prepared Scale Quantization Predicts Off-Diagonal Vacuum-Mode Covariance

Authors: Sterling Geisel, QBist Lab

Abstract

Ma and Ma formulate Planck-scale spacetime microstructure by quantizing local scale factors rather than the metric tensor. Their framework introduces first-order scale coordinates and second-order fluctuation amplitudes, constructs a scale-geometry action, and diagonalizes quadratic scale modes into harmonic oscillators with zero-point energy. This working paper applies one Pudding Theory Postulate: Vacuum Receptivity. The fit is narrow. The source paper already treats spacetime vacuum structure as a set of quantized scale modes. Pudding Theory adds the claim that such modes are not wholly passive. Under matched energy, curvature, and modal spectra, coherent boundary preparation should produce a residual off-diagonal covariance among near-degenerate scale modes. The proposed test is executable as a numerical protocol inside the Ma-Ma quadratic scale action, and later as an analogue experiment if scale-mode operators can be reconstructed. The distinguishing observable is a normalized covariance asymmetry, not a shift in mean vacuum energy.

Source Synopsis

Ma and Ma propose a scale-based route to spacetime quantization. Their starting point is standard. General relativity represents spacetime by a smooth differentiable manifold. Quantum mechanics implies that, near the Planck length, microscopic intervals cannot be treated as fixed classical quantities. Instead of directly quantizing the metric tensor, they promote local scale factors to fundamental dynamical variables.

The construction has two levels. The first-order scale manifold carries scale coordinates and a corresponding geometry. The second-order amplitude manifold carries fluctuation-amplitude coordinates. This hierarchy lets the authors represent local stretching, compression, and inhomogeneity of microscopic spacetime intervals while preserving covariance.

The source paper develops fluctuation-dependent differential operators and deformed commutators. From these, it derives a generalized uncertainty relation with scale-dependent coefficients. It then embeds scaled Klein-Gordon and Dirac equations into the same framework. The formal aim is not only to perturb known equations. It is to make microscopic scale variation part of the geometric substrate.

The quantization step is central. The first-order fluctuating scale factors are promoted to operators. A quadratic action is diagonalized by spectral decomposition. A stabilizing potential supplies finite modal frequencies and avoids the pathologies of a purely kinetic Hamiltonian. The resulting degrees of freedom are quantized as harmonic oscillators. Their ground-state terms define a scale-mode zero-point contribution.

This gives the paper its connection to vacuum energy. The modal zero-point spectrum is controlled by geometric renormalization-group flow. Ultraviolet fixed points suppress individual contributions, while infrared discrete fixed points supply finite structure. The authors suggest that this mechanism may help regulate ultraviolet divergences and may bear on the cosmological constant problem.

A second major result is a microscopic area operator built from the scale fluctuation modes. Its state counting can reproduce the leading Bekenstein-Hawking area law after matching microscopic parameters. The source paper therefore joins scale quantization, vacuum energy, ultraviolet regulation, and black-hole entropy in one formal program.

Postulate Lens

The applicable Postulate is Vacuum Receptivity. The source paper gives the technical object to which the Postulate can attach: a quantized vacuum sector made of discrete scale modes. Pudding Theory states that the vacuum receives, weighted by local coherence. Here that sentence has an operational meaning. It means that the covariance statistics of the scale-mode vacuum should depend weakly on coherent preparation, after ordinary spectral effects have been controlled.

This is not a claim about changing Ma and Ma’s field equations. It is not a claim about replacing the scale manifold. It is a claim about a residual response term in the second-order statistics of the quantized scale amplitudes.

The null model is the passive oscillator vacuum. Once the quadratic action, stabilizing potential, boundary spectrum, energy density, and curvature invariants are fixed, the covariance matrix of the modes is fixed. Pudding Theory predicts an additional coherence-weighted term. The mean zero-point energy may be unchanged. The covariance structure need not be.

The relevant mathematical object is

\[ C_{mn}=\langle 0|\Delta\hat a_m \Delta\hat a_n|0\rangle , \]

where \(\hat a_m\) and \(\hat a_n\) are quantized scale-mode amplitudes after spectral decomposition. In a passive vacuum, the controlled coherent and incoherent ensembles yield the same normalized off-diagonal covariance. In the Pudding Theory reading, they do not.

Pudding Theory Prediction

The prediction is a controlled covariance asymmetry among near-degenerate quantized scale modes. Consider a compact computational domain \(D\) in the first-order scale manifold. Use the Ma-Ma quadratic scale-sector action with the same kinetic matrix \(M\), restoring kernel \(K\), stabilizing potential, and reference scale in every trial. Diagonalize the quadratic operator to obtain modes \(u_n\), frequencies \(\omega_n\), and operators \(\hat a_n\).

Construct two boundary ensembles.

The coherent ensemble \(\mathcal{E}_c\) fixes a phase-aligned boundary profile over a selected mode band \(B\). Its coherence parameter is

\[ \mathcal{K}=\left|\frac{1}{N_B}\sum_{n\in B}e^{i\phi_n}\right|^2 , \]

with \(\mathcal{K}\geq 0.9\). The incoherent ensemble \(\mathcal{E}_i\) uses randomized phases in the same band, with \(\mathcal{K}\leq 0.1\). Both ensembles are constrained to have identical total boundary energy, identical mode-power spectrum \(|c_n|^2\), identical average curvature invariants on \(D\), and the same modal density. This prevents the test from confusing coherence with ordinary boundary-condition selection.

For each ensemble, compute the vacuum covariance matrix after projecting onto the same physical spectral sector:

\[ C^{(c)}_{mn}=\langle\Delta\hat a_m\Delta\hat a_n\rangle_{\mathcal{E}_c}, \qquad C^{(i)}_{mn}=\langle\Delta\hat a_m\Delta\hat a_n\rangle_{\mathcal{E}_i}. \]

Estimate the normalized asymmetry for near-degenerate mode pairs \(m,n\in B\), \(m\ne n\), and \(|\omega_m-\omega_n|/\bar\omega<\epsilon_\omega\):

\[ \mathcal{A}_{mn} = \frac{ C^{(c)}_{mn}-C^{(i)}_{mn} }{ \sqrt{C^{(0)}_{mm}C^{(0)}_{nn}} }. \]

The passive null model predicts \(\mathcal{A}_{mn}=0\) within numerical error once the matched controls are imposed. Vacuum Receptivity predicts a positive band-structured mean,

\[ \overline{\mathcal{A}}_{B}>0, \]

localized to phase-aligned near-degenerate modes. It should vanish outside the prepared band and should not increase total vacuum energy. Its signature is redistribution of covariance, not energy production.

The numerical threshold should be set by the solver, not by rhetoric. A valid study must first run null simulations with randomized labels and phase-scrambled controls to determine the numerical floor \(\sigma_{\mathrm{null}}\). A detection requires \(\overline{\mathcal{A}}_{B}>5\sigma_{\mathrm{null}}\). A null result with \(\sigma_{\mathrm{null}}\leq10^{-6}\) would be a strong exclusion for this implementation.

Falsifiable Observable

The observable is the band-averaged normalized off-diagonal covariance asymmetry \(\overline{\mathcal{A}}_{B}\) between coherent and incoherent boundary ensembles with matched energy, curvature invariants, modal density, and mode-power spectrum. Current passive quantization predicts zero residual asymmetry after these controls. Pudding Theory predicts a positive residual in the prepared near-degenerate band. If \(\overline{\mathcal{A}}_{B}\) were measured to be \(0\pm10^{-6}\) across all controlled near-degenerate scale-mode bands, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The source paper gives a formal quantization of scale factors. It does not show a responsive vacuum. Your proposal risks renaming ordinary boundary-condition physics.

Sterling: That is the central risk. The protocol is designed around it. The coherent and incoherent ensembles have the same energy, curvature invariants, modal density, and mode-power spectrum. Only phase coherence differs.

Tanaka: Boundary phases can still change correlations in ordinary quantum theory.

Sterling: Yes. Then the null model must include exactly that. The claim survives only if a residual off-diagonal covariance remains after passive boundary effects are computed and subtracted.

Tanaka: The Planck scale is not experimentally accessible.

Sterling: Direct access is not required for the first test. The first test is internal. Use the source paper’s quadratic scale action as the apparatus. If the estimator returns no coherence-weighted residual in simulation, the application fails.

Tanaka: And if it returns a residual?

Sterling: Then the burden shifts. One must show it is not numerical leakage, degeneracy misclassification, or an uncontrolled spectral difference. The Postulate earns no credit unless it predicts a term that survives those checks.

Tanaka: That is at least a real standard.

Sterling: It is the only useful standard.

Discussion

This working paper does not claim that Ma and Ma endorse Pudding Theory. Their paper is a scale-geometry quantization program. Its criteria are covariance, operator consistency, stable harmonic modes, ultraviolet behavior, and consistency with black-hole entropy counting.

The Pudding Theory addition is small. It concerns whether the quantized scale-mode vacuum is passive or receptive to coherence. The correct target is covariance because mean energy is too blunt. A receptive vacuum can preserve total zero-point energy while redistributing correlations among near-degenerate modes.

The main limitation is operational. The fundamental scale modes are not presently measurable in a laboratory. The proposed protocol is therefore first a falsification test inside the mathematical model. Analogue systems may later test the logic, but not the Planck-scale claim directly.

The second limitation is degeneracy. Coherent preparation can masquerade as ordinary spectral selection. This is why the estimator must match mode power, energy, curvature, and modal density before comparing covariance.

The conclusion would change if the Ma-Ma formalism cannot define \(\hat a_m\) covariance unambiguously. It would also change if every apparent \(\overline{\mathcal{A}}_{B}\) reduces to passive boundary physics. In that case, Vacuum Receptivity adds no independent content here.

References

1. Weihu Ma and Yu-Gang Ma, “Scaling-Based Quantization of Spacetime Microstructure,” Nuclear Physics B 1022, 117282 (2026); arXiv:2601.15649v1. DOI: doi:10.1016/j.nuclphysb.2025.117282.

2. S. Ochs, “Pudding Theory: A Topological Theory of Information Fields,” QBist Lab, 2026.

3. P. Hořava, “Gravity at a Lifshitz Point,” Physical Review D 79, 084008 (2009). DOI: doi:10.1103/PhysRevD.79.084008. Cited in Ma and Ma as Ref. [31].

4. Steven Weinberg, “The Cosmological Constant Problem,” Reviews of Modern Physics 61, 1 (1989). DOI: doi:10.1103/RevModPhys.61.1. Cited in Ma and Ma as Ref. [69].

5. J. D. Bekenstein, “Black Holes and Entropy,” Physical Review D 7, 2333 (1973). DOI: doi:10.1103/PhysRevD.7.2333. Cited in Ma and Ma as Ref. [75].