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

Mass §10, beginning near line 633, states that the squared lightest occupied mass scales as g_dd/g_66 relative to the electron. Preserve this as a historical prescription, not an established consequence of varying the displayed action.

The useful distinction

Write D_d*=S(n_min(d),d) for a fixed occupied reference count and m(n,d)=mu_d S(n,d). The historical reference condition is

(μdDd∗)2=gddg66me2,μd=meDd∗gddg66. (\mu_d D_d^*)^2=\frac{g_{dd}}{g_{66}}m_e^2, \qquad \mu_d=\frac{m_e}{D_d^*}\sqrt{\frac{g_{dd}}{g_{66}}}.

For the d=3,4,6 comparison the reference counts are 1,15,18564. Once fixed, D_d* is a sector constant, not the variable S(n,d) of whichever excited mode is being evaluated. Consequently mu_d=C v_d/D_d*, with v_d=sqrt(g_dd) and C=m_e/sqrt(g_66), is compatible with band blindness.

This supports the corrected common-anchor interpretation in the existing V2 notes: a fixed-reference relation is not ruled out merely because it contains a count. It does not prove the relation or select its reference bands.

Remaining derivation

Derive the stationary cross-sector equations and kinetic normalization that select the occupied-reference masses rather than the pre-count coefficients. Explain why these particular occupied modes define the reference. Do not extend this prescription indiscriminately: V1 also uses different neutrino, boson, and tau scale constructions despite some universal wording.

The scalar oscillator closure lambda proportional to (gQ)^(2/3) is a different equation. Converting it into this mass-reference condition requires an explicit operator and units dictionary, not a relabeling of lambda.