How the reciprocal-density weight arises

Start with a smooth, strictly positive availability density in a periodic cube. Total inventory is its volume integral. Dividing availability by inventory gives a density whose volume integral is one. For the constant-phase state tested here, the wave amplitude is the positive square root of that normalized density, multiplied by a spatially constant phase factor. [1] [3]

With zero potential, the expectation comes from the kinetic operator. Integration by parts converts the Laplacian expectation to an integral of the squared gradient of the wave function. Opposite faces of the periodic cube cancel the boundary contributions. Differentiating the square root divides the gradient of availability by twice the square root of availability times the square root of inventory. Squaring produces the squared availability gradient divided by four times availability times inventory. [1]

The kinetic operator contributes its calibrated action scale squared divided by twice its kinetic mass. Together with the chain-rule factor, the multiplier of the reciprocal-weighted integral divided by inventory is the action scale squared divided by eight times the kinetic mass. With physical coordinates the normalized weighted integral has units of inverse length squared, giving energy after multiplication. The source keeps the operator action calibration distinct from MQ upper-count quantities. [1]

Table 1 compares expressions at fixed phase. A spatially varying phase contributes a further term through the squared gradient of the wave function. The calculation also uses strict positivity to justify the square-root derivative everywhere. Profiles with nodes would need separate control of the integrand and boundary conditions. [1]

The periodic cube and the two exact results

The source takes a density that varies only along one axis. Its mean is inventory divided by cube volume. The fractional cosine amplitude is a fixed positive profile parameter divided by the square root of inventory. The square of the parameter must be less than inventory, so that even the cosine trough has positive density. The wave number is two pi divided by the cube side. [1]

The unweighted integral divided by inventory is exactly twice pi squared times the profile parameter squared divided by the fifth power of the cube side. The squared derivative contributes the squared wave number, and the sine squared averages to one half over the period. The amplitude carries the square root of inventory before squaring, which cancels the final inventory division. [1]

For the Hamiltonian, the reciprocal density remains inside the integral. Integrating over the period gives a dimensionless kinetic expectation equal to one minus the square root of one minus the profile parameter squared divided by inventory. Here the reference energy is the operator action scale squared times pi squared divided by twice the kinetic mass and the cube side squared. One can obtain the dimensionless result by reducing the periodic average of the weighted sine squared using the integral of the reciprocal cosine profile. The result keeps the full contrast dependence, including terms beyond quadratic order in contrast. [1]

Two ways to test the conversion

Table 2 shows examples computed from Eq. (44ps). The kinetic column is in units of the fixed reference energy. The candidate column is in units of twice pi squared divided by the cube side to the fifth power. Each normalization is held fixed across the examples. The quotient of the displayed columns is consequently a diagnostic of proportionality, with fixed reference scales factored out. [1]

The first two rows vary inventory and hold the profile parameter fixed. The candidate stays at one while the source result falls. This already rules out a single constant conversion across those rows. An inventory-dependent conversion could match those two particular profiles. The first and third rows test that proposal by keeping inventory at four. The quotient changes with the profile, ruling out a conversion that depends only on inventory across this family. [1]

Inventory 4 · 0.133975

Inventory 16 · 0.031754

Figure 1 shows the first comparison on a shared scale from zero to 0.15. The bars visualize the normalized kinetic expectation alone. A useful algebraic check is to rationalize one minus the square root. After dividing by the parameter squared, the resulting quotient is the reciprocal of inventory times the sum of one and the square root. The explicit profile dependence survives at fixed inventory. [1]

Refining a trial mode and preserving a physical state

In the fixed-kernel construction, Gaussian refinement rescales total inventory by the square of the refinement factor and the complex trial mode by the factor itself. This preserves the trial mode normalized by the square root of inventory. The physical density divided by inventory behaves differently. Its departure from the homogeneous density falls in inverse proportion to the refinement factor. Thus the normalized mode and normalized physical density implement different equivalences. [1]

Uniform replication of all cell counts preserves the normalized source density. With the same phase, Hamiltonian and boundary conditions it preserves the source expectation as well. It need not preserve the Gaussian mode norm shell. The manuscript also considers increasing kernel contrast to compensate for Gaussian refinement. In the fixed positive-kernel family, the required contrast eventually reaches the positivity bound. [1]

A projected density and Hamiltonian can preserve the source expectation while leaving readout-invisible count directions unresolved. An energy depending only on those projected quantities gives no energetic preference among states that preserve them. Selecting a hidden configuration requires a physical constraint or additional terms with the relevant dependence. Section 3.88 treats the common preparation, admissible count update, realized density and phase, and physical energy as requirements of the same reference rule. [1]

Reporting the scope and date

The obstruction applies to the proposed inventory-only energy identification under the fixed-kernel assumptions. A profile-dependent coefficient chosen after inspecting the result would match a selected energy value without supplying a law for probabilities, phase or updates. A physical timing claim needs the action and energy of the same realization. This work supplies a mathematical comparison and a criterion for a future derivation. [1] [4]

October 23, 2026 records the manuscript version. November 3, 2026 is the preparation date of this Brief. The complete comparison is in Section 3.87 of the technical manuscript available through the Institute, with related public reading linked below. [1] [2]