Positive modes. A finite difference. An attractive force.

01 / 04

Start with a finite system

The finite-wave lab supplies a harmonic lattice and ordinary oscillator vacuum energies ℏω/2. Every raw mode energy is positive. A piston separates chambers of lengths a=1 and b=3, with units ℏ=c=1. Moving the boundary changes the finite total energy; a centered finite difference estimates the force at fixed grid spacing.

The largest fixed/fixed case contains 131,070 modes; the fixed/free case contains 131,073. Exact closed forms of finite trigonometric sums evaluate the largest grids. Smaller direct sums and matrix diagonalizations independently check those formulas. This is a specified one-dimensional scalar model, not three-dimensional electromagnetic matter.

02 / 04

Boundary conditions change the sign

The finest fixed/fixed force is −0.116355283601548, compared with the standard continuum piston benchmark −0.116355283466289. The relative difference is about 1.16×10⁻⁹. The fixed/free case gives +0.058177641806342, within about 1.26×10⁻⁹ of its continuum benchmark. A negative force points toward the shorter chamber; changing the piston boundary reverses the sign.

No zeta function or inserted −1/12 constant is used to compute those lattice forces. The model tests an energy difference under supplied dynamics and boundary assumptions. It does not derive the vacuum-energy postulate itself.

03 / 04

Where minus one twelfth actually appears

For a positive, exponentially damped integer sum, subtracting the specified leading reference 1/ε² leaves a finite remainder approaching −1/12. At ε=0.001, the finite sum is approximately 999999.9166666708, its reference is 1000000, and the remainder is −0.0833333291667. The ordinary partial sums 1+2+…+N=N(N+1)/2 still grow without bound.

A regularized or reference-subtracted remainder is a different mathematical operation from ordinary addition. Its physical use needs a justified comparison and response model. It cannot by itself imply negative total energy, a traversable wormhole, a time machine or a physical dimension count.

04 / 04

What was rerun for this edition

The archived lab was rerun locally for this update. All 4,113 finite model cases and six independent numerical checks passed, including matrix/spectral and direct-force/closed-sum comparisons. The source reference and numerical scope are recorded below. This repeats the same supplied model; it is not fresh experimental validation.

The grid is spatially finite while oscillator amplitudes and time remain continuous. The illustrative β-dependent lattice laws are inputs, not a derived D0RE law. A genuine transfer to material-specific electromagnetism would need the field content, dispersion, dissipation, geometry, boundaries and measurement response to be justified and then tested independently.

Evidence & sources

Source-bound editorial synthesis of the cited record. The claim classes above separate imported physics, conditional mathematics, finite simulations, historical measurements and prospective tests.

1. Finite-mode mathematical handoff — Finite_Wave_Mode_Handoff.md
   Reviewed source record · full text outside this preview
2. This edition’s finite-mode replay — Finite_Wave_Mode_Replay.json
   Reviewed source record · full text outside this preview