Plancks: Dimensional Unfolding, Force-Originated Energy, and the Smallest Building Block of the Universe

We place on a rigorous mathematical foundation the proposal that every building block in the universe arises from the different dimensional organizations of a single fundamental unit — the Planck unit. The background space is always three-dimensional and invariant; the only thing that changes is the extent to which a Planck unit’s axes unfold within this fixed three-dimensional fabric. We quantify this unfolding by a continuous field n, with n ∈ (0,4). Dimension is quantized at integer values (d = ⌊n⌋, d = 0,1,2,3); non-integer intermediate values are not a dimension but an energy state accumulated within that dimension before the next jump. As a unit opens n dimensions, its energy grows according to the ledger Eₙ = xⁿ, where x is a dimensionless quantity associated with the base Planck unit (x > 1). The central result is that this energy ledger need not be postulated: it can be derived from the integral of the work done by the force acting along the dimensional coordinate — energy is the work of the force. Using the workenergy theorem we show that the work of the force F(n) = (ln x)xⁿ yields exactly Eₙ = xⁿ. It follows naturally why the Planck base (n → 0) is the lowest-energy, motionless, and timeless state: because no force is unfolded, no work can be done, and the energy remains at its minimum (E = 1). The closing of a dimension (the collapse of a 3D particle to a Planck) is a four-jump, finite, and singularity-free process; the total absorbed energy is x⁴ − 1. Singularity is rejected: the universe exists neither with zero nor with infinity, but with finite numbers. Promoting n to a dynamical scalar field, we give a field-theoretic formulation in which the Euler–Lagrange equation of a single Lagrangian reproduces the same force and ledger, yields the dispersion relation governing stability and structure formation, and — with the kinetic weight K(n) restored — renders the Planck base a dynamical attractor that makes the collapse singularityfree. We further address the physical readings of x (a single unit and the total number of Plancks in the universe, N ≈ 8.5 × 10¹⁸⁴), the dimensional region of validity of the speed-of-light limit, and the demonstration via a Bell test that long-range correlation between distant units is not entanglement. All results are supported by eight figures and numerical checks; what is derived, what is a consistency check, and what remains open are honestly separated.