Gravitationally-Induced Quantum Phase and M_∞ Curvature

The COW experiment split a slow neutron beam into two paths inside a silicon crystal interferometer, one path running at a greater height than the other in the Earth's gravitational field. A neutron on the upper path acquires more gravitational potential energy than one on the lower path; this difference in potential energy shows up as a phase difference when the two paths are recombined. The resulting interference pattern depends on the area \(A\) enclosed between the two paths, the neutron mass \(m\), the gravitational acceleration \(g\), the neutron wavelength \(\lambda = h/mv\), and Planck's constant \(h\).

The phase accumulated along a path through a gravitational potential \(\Phi = gz\) (taking \(z\) as height and using the non-relativistic limit) is

\[ \delta\phi = \frac{m^2 g A \lambda}{2\pi\hbar^2} = \frac{m g A}{\hbar v} \]

The observed fringe shift matched the prediction to better than 1%. Since then, atom interferometers using rubidium, cesium, and other species have improved the measurement by orders of magnitude. Modern atom interferometers sensitive to gravitational acceleration at parts per billion measure this phase for systems with masses \(10^{10}\) times larger than the neutron, obtaining the same formula with the same \(g\), validating that the phase scales exactly linearly with mass across many decades.

The formula contains an apparent puzzle: it involves \(m^2/\hbar^2\) or equivalently \(m/\hbar v\) — both \(m\) as a gravitational coupling (gravity pulls the particle down) and \(m\) as an inertial quantity (it sets the de Broglie wavelength). The equivalence of gravitational and inertial mass is assumed, as in classical physics. The COW experiment is a direct quantum test of whether that equivalence holds at the level of quantum phases, not just classical trajectories.

In IDWT, the mass of a particle mode \((n,d)\) is

\[ m(n,d) = m_{{\rm scale},d} \times S(n,d) \]

where \(S(n,d) = \binom{n+d-1}{d}\) is the hockey-stick number — the count of sector microstates at mode level \(n\) in sector \(d\) — and \(m_{{\rm scale},d}\) is a sector-specific energy scale derived from the kernel coupling \(g_{dd}\). The mass is a microstate count multiplied by the sector's energy unit; it is not a free parameter.

This same quantity enters gravity in two separate ways.

As gravitational source. Gravity in IDWT is curvature of \(M_\infty\) sourced by mass. The effective Poisson equation for a mode \((n,d)\) is

\[ \nabla^2 \Phi = 4\pi G_N \cdot S(n,d) \cdot m_{{\rm scale},d} \cdot |\psi_{3D}(\mathbf{r})|^2 \]

The source is the mode mass — the dimensional complexity of the mode, its microstate count times its sector energy scale. A mode with a large \(S(n,d)\) distorts \(M_\infty\) more strongly.

As inertial mass. The equivalence principle in IDWT follows from the sector structure. The inertial mass enters through the sector eigenvalue \(m_{\rm inertial} = S(n,d) \times m_{{\rm scale},d}\); the gravitational mass enters through \(T_{\mu\nu}^{\rm eff}\), which is proportional to the same sector normalization factor \(\|{\chi}\|^2_\Xi = 1\) for L²-normalized modes. Both are the same quantity (Part 4 §3.6, ✅). The equivalence principle holds exactly and universally — not as a postulate, but because inertial and gravitational mass are both \(S(n,d) \times m_{{\rm scale},d}\).

The COW phase is therefore

\[ \delta\phi = \frac{S(n,d) \times m_{{\rm scale},d} \times g \times A}{\hbar v} \]

with the mass determined by the sector combinatorics. For the neutron, the relevant mode structure is the proton-neutron baryon at the confined scale; the hadronic chain gives the composite nucleon mass \(m_N = N_c \Lambda_{\rm QCD}(1+1/n_u^2) = 940.4\) MeV (idwt.py STEP 7), 0.09% above the PDG neutron mass. For a fixed interferometer geometry (\(\lambda = 1.445\) Å, \(A = 10\) cm² as a representative case) the predicted phase is 57.0 rad against 56.9 rad with the PDG mass — a +0.17% shift that is the composite-mass residual, not a separate effect, and well inside the precision of the COW-class tests (idwt.py STEP 57). The structure of the phase — its proportionality to the gravitational mass, which is the same as the inertial mass and the de Broglie mass, all equal to \(S(n,d) \times m_{{\rm scale},d}\) — is determined by the sector geometry, not assumed.

In standard quantum mechanics and general relativity, the equivalence of gravitational and inertial mass is an axiom. It holds experimentally to better than one part in \(10^{13}\) from Eötvös-type experiments, and it holds at the quantum phase level in COW-type experiments. But it is not derived — it is imposed as a fundamental assumption that connects the metric coupling in GR to the inertial mass in the Schrödinger equation.

In IDWT, the same formula \(m = S(n,d) \times m_{{\rm scale},d}\) appears on both sides because the mass formula is the sector eigenvalue. The mass is not a coupling constant that could in principle take different values in the gravitational field equation and in the kinetic energy term; it is a single counting integer \(S(n,d)\) multiplied by a single sector energy unit \(m_{{\rm scale},d}\). The equivalence principle is a theorem of the sector structure (Part 4 §3.6), not an input. There is no room in the IDWT mass formula for the gravitational and inertial masses to differ — they are the same sector eigenvalue.

The COW experiment and modern atom interferometry are therefore evidence that the mass formula \(m = S(n,d) \times m_{{\rm scale},d}\) is simultaneously the right formula for gravity (gravitational coupling) and the right formula for quantum kinematics (de Broglie wavelength, quantum phase). This is the statement that the sector eigenvalue is the physical mass in every context, with no correction terms that distinguish the two roles. Experiments at higher and higher precision — current atom interferometers achieve gravitational sensitivity at parts per billion, and proposals exist for orders of magnitude further improvement — continue to test this identification.

The most direct IDWT-specific content of the COW test is its universality. The gravitational phase formula

\[ \delta\phi = \frac{m \, g \, A}{\hbar v} \]

must hold for every species with the same phase-to-mass ratio \(g A / \hbar v\). In IDWT, this is forced: every mode's gravitational and inertial mass are both \(S(n,d) \times m_{{\rm scale},d}\), so the ratio \(\delta\phi / m\) is species-independent. If any particle species were found to have a different phase-to-mass ratio — if a neutron and a proton in the same interferometer geometry and the same velocity showed different gravitational phase shifts per unit mass — that would contradict the IDWT equivalence principle proof.

Modern experiments test this. Atom interferometry with rubidium and potassium in the same apparatus (dual-species interferometers) measures the ratio of their gravitational accelerations to parts per million. The COW-type test can in principle be extended to antihydrogen — if gravity acts on antimatter with the same acceleration as on matter, the gravitational phase for an antihydrogen atom would be the same as for hydrogen at the same velocity. IDWT predicts no difference: antimatter in IDWT is the conjugate spinor \(\bar{\Psi}_\infty\), which has the same sector eigenvalues and therefore the same mass formula. The gravitational coupling is through \(T_{\mu\nu}^{\rm eff}\), which is positive-definite for both particle and antiparticle.

The ALPHA-g and AEGIS experiments at CERN are currently measuring the gravitational acceleration of antihydrogen. IDWT's prediction is unambiguous: the acceleration is \(g\) downward, the same as for hydrogen, and the COW phase for antihydrogen is the same as for hydrogen at the same velocity.

The COW experiment sits at an unusual intersection: it is a quantum experiment (interference, phase, de Broglie wavelength) that is sensitive to gravity. In standard physics this intersection is described by grafting Newtonian gravity onto the Schrödinger equation — the gravitational potential enters the Hamiltonian as \(mgz\), and the rest of quantum mechanics is unchanged. The quantum theory of gravity itself (how gravity behaves at the Planck scale, whether gravitons exist) plays no role in the COW calculation.

In IDWT, the situation is structurally different because gravity is not a quantum field. Gravity is curvature of \(M_\infty\) sourced by mass (Part 4 §3.1). There are no gravitons because there is no gravitational field to quantize — there is only the \(M_\infty\) geometry, which is not a dynamical quantum object. The sector-space geometry \(h_{ab}(\xi)\) is a fixed classical background; it is not quantized, has no fluctuations, and does not produce a KK graviton tower (Part 4 §3.4).

What does this mean for the COW experiment? The gravitational potential \(\Phi = gz\) that enters the neutron's Schrödinger equation is the low-energy, \(d=3\)-observer's read of \(M_\infty\) curvature sourced by the Earth's mass distribution. The COW phase is the quantum response of the neutron mode — a sector eigenstate of \(\Psi_\infty\) — to that curvature. The neutron is not a test particle in a classical gravitational field; it is a mode of the wave \(\Psi_\infty\) propagating through the curved \(M_\infty\) geometry that the Earth's sector microstates source.

No prediction differs from standard quantum mechanics for this experiment — the phase formula is the same. But the ontological picture is different: the COW fringe shift is not evidence that gravity and quantum mechanics are separate structures that happen to interact via \(mgz\). In IDWT it is a single phenomenon: the mode \(\Psi_\infty\) of the neutron propagating through \(M_\infty\) curvature that its own sector structure (and the Earth's) defines.

The equivalence principle proof in IDWT is established (✅, Part 4 §3.6): \(m_{\rm grav} = m_{\rm inertial} = S(n,d) \times m_{{\rm scale},d}\), following from the same sector normalization factor appearing in both the gravitational source and the kinetic energy. The prediction that the COW phase formula holds with the correct species mass is therefore a consequence of an established result, not a conjecture.

What is open (🔶): the mass formula for composite systems — baryons, atoms — requires identifying the relevant collective mode index \((n,d)\) for the composite, and the kernel-level confinement energy computation that produces the proton mass 938 MeV from the quark mode structure is an open problem (Part 8 §11). For atomic species used in interferometers (rubidium, cesium), the atomic mass enters as a measured input rather than a prediction from the sector structure. The COW test confirms the equivalence principle holds for these measured masses; it does not yet test whether the masses themselves are correctly predicted by IDWT.

The antimatter prediction — equal gravitational acceleration for matter and antimatter — is unambiguous (🔵, numerically consistent with no observed deviation, mechanism established). ALPHA-g's confirmation of downward gravitational acceleration for antihydrogen (2023) is consistent with this. Future experiments at higher precision will continue to test it.

The full account of gravity in IDWT is Part 4 of the notes: doi:10.5281/zenodo.19767493.