Physical scale accounting for the normal carrier

The inverse carrier supplies a concrete geometric conversion between its surface radius and its simplex frequency unit. That conversion depends on the normal hopping strength. Equal frequency units therefore do not imply equal radii without an additional relation among the physical coefficients.

These are conditional scale constraints on the prescribed operator. The calculation does not determine a physical radius, introduce compactified dimensions, or fit a new particle assignment.

The frequency unit includes the spatial matrix factor

Let R_n be the ordinary radius of the surface feature, c the common wave speed, and gamma_n the dimensionless hopping coefficient of the gapped normal carrier. Its leading simplex frequency unit is

$$\omega_{0,n}=\frac{c}{R_n}K_n(\gamma_n),\qquad K_n(\gamma)=\frac{n!}{\gamma^{n-1}\prod_{l=0}^{n-2}h_l}. \tag{1}$$

The h_l are fixed spatial integrals. Every occupied degree at one depth uses the same R_n, gamma_n and h_l. Finite-gamma corrections still depend on the degree, and must be checked before using the leading simplex polynomial as an exact spectrum.

If G_n denotes the dimensional coefficient multiplying the bounded hopping term, then

$$G_n=\frac{c}{R_n}\gamma_n. \tag{2}$$

G_n is a coefficient of this inverse spatial operator. Its relation to the same-field interaction remains unproved. Holding gamma fixed gives omega_0 proportional to R^(-1). Holding dimensional G fixed instead gives, at leading order,

$$\omega_{0,n}= \frac{n!c^n}{G_n^{n-1}R_n^n\prod_l h_l}. \tag{3}$$

The source law must therefore say which coefficients remain fixed when the cavity scale varies. The conversion from a frequency unit to a radius cannot silently switch between these choices.

A shared leading frequency unit across depths

Suppose the depth-six and depth-ten constructions are required to share the same leading simplex frequency unit. Define

$$A=\frac{10!\prod h^{(6)}}{6!\prod h^{(10)}} \simeq19461.2049515. \tag{4}$$

Three additional premises give different consequences:

For gamma_6=10000, equal dimensionless hopping requires a radius ratio about 1.94612e-12. Equal radii instead require gamma_10 about 499.80044. Equal dimensional hopping gives gamma_10 about 674.39909 and a radius ratio about 0.0674399. These are alternative conditional relations, not measurements or derived physical source models.

A common radius across depths is an additional assumption. Combining it with equal dimensional hopping gives a stronger condition with an exact test.

An all-strength bound for equal radius and dimensional hopping

Under both equalities, gamma is common. The electron matrix has diagonal entries 13 through 18 and six normal modes. The tau matrix has diagonal entries 23 through 32 and ten normal modes. Their off-diagonal entries are gamma times their respective h_l.

For a lower-bidiagonal matrix with positive diagonal d_j and hopping gamma h_j, its inverse has entries

$$(T^{-1})_{ij}=(-1)^{i-j} \frac{\gamma^{i-j}\prod_{k=j}^{i-1}h_k} {\prod_{k=j}^{i}d_k},\qquad i\ge j. \tag{5}$$

Diagonal sign matrices remove the alternating signs without changing the operator norm. The smallest positive canonical frequency is one divided by that norm.

Compare the electron inverse with the leading six-by-six block of the tau inverse. The ratio of corresponding positive entries is independent of gamma:

$$R_{ij}= \prod_{k=j}^{i}\frac{23+k}{13+k} \prod_{k=j}^{i-1}\frac{h_k^{(6)}}{h_k^{(10)}}. \tag{6}$$

Let C=max R_ij. For nonnegative matrices, entrywise domination implies domination of the spectral norm. The norm of the tau block is at most that of its full inverse. Consequently,

$$\boxed{\frac{\omega_\tau}{\omega_e}\le C \simeq18.1922701} \tag{7}$$

for every nonnegative common gamma. The assigned simplex-count ratio 64512240/18564 is over three thousand. The equal-radius, equal-dimensional-hopping version of this projected carrier therefore cannot realize that target. This conclusion follows from the inverse entries, beyond the numerical parameter scan used as a check.

The result concerns this carrier and these additional scale equalities. It does not exclude common radius when dimensional coefficients differ by depth. Small controlled normal-complement corrections cannot bridge this discrepancy; abandoning their controlled regime would be a different operator test.

Finite hopping remains a distinct correction

For the equal-radius family in the table, gamma_6=10000 and gamma_10=499.80044 give a muon/electron ratio about 2.79e-5 below the simplex ratio, and a tau/electron ratio about 0.00786 below it. Equating the leading frequency units does not remove these corrections.

The following mathematical examples increase that family's strengths until both assigned ratios approach their simplex targets to the declared tolerance:

These tolerances are not observational uncertainty claims. Increasing the hierarchy is not a derivation of its physical value. The source must determine the coefficients and their depth dependence.

A frequency anchor fixes the remaining length unit

Let omega_e^phys be the selected physical electron frequency and ell_e=c/omega_e^phys its equivalent frequency length. Then

$$\frac{R_6}{\ell_e}=\omega_e^{\rm dimensionless}. \tag{8}$$

At gamma_6=10000 the dimensionless frequency is about 8.72770e-13. Choosing this hierarchy and anchoring the electron frequency therefore implies a radius that is that fraction of ell_e. Equation (8) supplies the physical conversion for the stated dimensionless operator.

The normal width a_6 additionally obeys a_6/R_6=1/Lambda_6. Its value is not determined by (8). A large Lambda used to certify a normal spectral approximation is an inverse hierarchy assumption, not a measurement of a physical profile width.

Verification

holomorphic-scale-accounting-20260914T070835_580367Z.json passes 77 checks of the factorial conversion, common-scale relations, inverse-entry bound and complete finite-chain frequencies. Seventy-digit singular calculations resolve the small relative corrections before conversion to floating-point output.

The earlier report and exact source remain preserved. Its last precision-target check was limited by subtracting nearly equal double-precision ratios. The refined calculation evaluates the factored corrections directly at high precision.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.