Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Source and scope

Adapted from IDWT_04_Gravity.md, §3.10.2–3.10.5c, especially the explicit functional, near line 315. Status: checked algebra within a flat, scalar, fixed-norm mean-field model; not a derivation of the V2 master dynamics.

Functional and stationary equation

For real coordinates x,y in R^d, coupling g>0, kinetic coefficient set to one, and Q = integral |Psi|^2 fixed, retain the source functional

E[Ψ]=∫∣∇Ψ(x)∣2 dx+g2∬(x⋅y)2∣Ψ(x)∣2∣Ψ(y)∣2 dx dy. E[\Psi]=\int |\nabla\Psi(x)|^2\,dx +\frac g2\iint(x\cdot y)^2|\Psi(x)|^2|\Psi(y)|^2\,dx\,dy.

Variation of E-mu Q gives

[−Δ+g∫(x⋅y)2∣Ψ(y)∣2 dy]Ψ(x)=μΨ(x). \left[-\Delta+g\int(x\cdot y)^2|\Psi(y)|^2\,dy\right]\Psi(x)=\mu\Psi(x).

For an isotropic density, its second moment is integral y_i y_j |Psi(y)|^2 dy = Q delta_ij <r^2>/d. Thus V(x)=lambda |x|^2, with lambda=g Q <r^2>/d. The normalized oscillator ground state has <r^2>=d/(2 sqrt(lambda)), so

λ=(gQ/2)2/3,Ψ(x)=Q λd/8π−d/4exp⁡(−λ ∣x∣2/2). \lambda=(gQ/2)^{2/3},\qquad \Psi(x)=\sqrt Q\,\lambda^{d/8}\pi^{-d/4} \exp(-\sqrt\lambda\,|x|^2/2).

The historical formula lambda=(g/2)^(2/3) is the Q=1 specialization. Choosing Q=1 is not a derivation of quantum charge discreteness.

Excited-shell self-consistency

The same closure can be evaluated on an excited oscillator shell. Let N≥0N\geq0 be its total excitation degree, keep the conserved norm QQ fixed, and require the second moment to be isotropic. A single-shell oscillator state satisfies

μN=(2N+d)λ,⟨r2⟩N=2N+d2λ. \mu_N=(2N+d)\sqrt\lambda,\qquad \langle r^2\rangle_N=\frac{2N+d}{2\sqrt\lambda}.

The second identity follows from the oscillator virial identity, or by integrating the normalized Laguerre modes. Isotropy of the second moment is the extra condition that makes the self-sourced potential scalar. It is satisfied, for example, by the radial states with even NN. Self-consistency therefore gives

λN3/2=gQ(2N+d)2d,λN=[gQ(2N+d)2d]2/3. \boxed{\lambda_N^{3/2}=\frac{gQ(2N+d)}{2d},\qquad \lambda_N=\left[\frac{gQ(2N+d)}{2d}\right]^{2/3}.}

This includes the ground-state equation at N=0N=0 and has a positive solution for every positive gQgQ and every positive spatial dimension. It is a property of the specified flat scalar functional; a topological Euler characteristic does not enter this calculation.

At fixed g,Qg,Q, different occupied shells source different well widths. A common well for all particles at a depth consequently needs a sustaining field whose relevant moment remains common, or another derived feedback law. The single-mode closure alone does not implement that premise.

For comparison, hold one well fixed and use two isotropic unit-normalized shell profiles. The dot-square pair integral is

∬(x⋅y)2ρN(x)ρM(y) dx dy=(2N+d)(2M+d)4dλ. \iint(x\cdot y)^2\rho_N(x)\rho_M(y)\,dx\,dy =\frac{(2N+d)(2M+d)}{4d\lambda}.

With a fixed probe MM, the depth-six shell ratio for the assigned N=34N=34 and N=12N=12 is 74/30=37/1574/30=37/15. Their cumulative state-count ratio is instead (406)/(186)=206+13/17\binom{40}{6}/\binom{18}{6}=206+13/17. These describe different operators. A common bare coefficient does not by itself remove the spatial matrix elements from a physical coupling calculation.

geometric_channel_checks.py verifies the shell moments by independent radial Laguerre quadrature and substitutes them into the full mean-field coefficient at fixed norm. Neither the norm nor the field amplitude is identified with particle mass.

Transfer conditions

  • This produces an isotropic stationary solution, not a proof that it is the unique physical background or that its full fluctuation spectrum is stable.
  • The eigenvalue mu includes the full mean-field potential; the interaction appears with one-half its double integral in E. Do not equate mu Q with E.
  • When expanding this functional in oscillator modes, retain the kinetic matrix and quartic term, or explicitly subtract the background potential already included in oscillator eigenvalues. Otherwise it is double-counted.
  • A resonant-only quartic mode sum requires a justified rotating-wave or time-averaging reduction; index-resonance restrictions do not follow merely from expanding the static functional in a basis.
  • The formula as written uses a fixed origin. The centroid-relative completion defines and varies a centred version with translation covariance and zero net self-force. That completion remains a flat scalar auxiliary model.
  • This closure does not identify oscillator eigenvalues with cumulative state counts, select occupied particle indices, or supply a physical lock frequency.