Finite-chain mass ratios and an inverse current correction

The current whole-depth carrier and the linked normal carrier have the same finite matrix structure on each occupied spatial harmonic. That structure allows an exact test of their proposed simplex mass ratios, independently of the still unknown supporting geometry. A local normalized-current action can also change the finite-chain frequency without assigning a coefficient to each particle. The calculations below distinguish this inverse design from a mass operator derived from the one master field.

A strict finite-chain ratio theorem

For spatial depth $d\ge2$, let $B_n$ be the real lower bidiagonal component matrix

$$(B_n)_{jj}=n+j\quad(0\le j<d),\qquad (B_n)_{j,j-1}=b_j>0\quad(1\le j<d), \tag{1}$$

where the $b_j$ are independent of the harmonic label $n$. The current carrier has $b_j=g$ for all $j$, while the linked carrier has $b_j=g h_j$, with one positive geometric hopping list per depth. At a fixed physical radius, the smallest singular value $\epsilon_n$ is the positive oscillation frequency of this component carrier. Write

$$P_{n,d}=\det B_n=\prod_{j=0}^{d-1}(n+j)=d!S(n,d). \tag{2}$$

The adjugate identity gives an exact normalization formula:

$$\boxed{\frac{\epsilon_n}{P_{n,d}} =\frac1{\|\operatorname{adj}B_n\|_2}.} \tag{3}$$

Conjugating the adjugate by alternating diagonal signs makes it a nonnegative lower triangular matrix $A_n$. Its entries are

$$(A_n)_{ij}= \left(\prod_{a=j+1}^{i}b_a\right) \left(\prod_{a<j\,\text{or}\,a>i}(n+a)\right), \qquad i\ge j, \tag{4}$$

with zero entries above the diagonal. Every entry is nondecreasing in $n>0$, and some entries increase strictly. The bottom row is positive, so the leading left and right singular vectors can be chosen strictly positive. Their bilinear characterization then makes $\|A_n\|_2$ strictly increasing. Equation (3) proves

$$\boxed{\frac{\epsilon_{n_2}}{S(n_2,d)} <\frac{\epsilon_{n_1}}{S(n_1,d)} \quad\text{whenever }n_2>n_1>0.} \tag{5}$$

This holds at every finite positive hopping strength, including different positive $h_j$ that are common to all modes at one depth. It covers the maintained depths two, three, four, five, six and ten. Depth ten has only one maintained mode, so it has no within-depth ratio test. The theorem does not cover nonlinear self-consistent hopping that changes with $n$, different physical radii for particles in one sector, or an added direct mass interaction.

When all $b_j$ grow at fixed positive ratios, expanding the largest singular value of the adjugate gives

$$\frac{\epsilon_n}{P_{n,d}} =\frac1{\prod_{j=1}^{d-1}b_j} \left[1- \frac12\left(\frac{n^2}{b_1^2} +\frac{(n+d-1)^2}{b_{d-1}^2}\right) +O(g^{-4})\right]. \tag{6}$$

Here the remainder is at fixed $n,d$ as the common edge scale $g$ increases. This correction is the positive norm of internal component content in the causal parent, rather than an error from computing a tiny singular value. The unified calculation verifies (3) independently of direct singular-value decomposition on every maintained mode.

For the benchmark masses already used in the mass-spectrum comparison, the observed electron-to-muon ratio is $206.76828299$, while the bare simplex ratio at depth six is $3515/17=206.76470588$. The latter is low by about $0.00173\%$. Equation (5) means an uncorrected finite chain at one common radius and hopping strength lowers its prediction further. At dimensionless current edge $g=5000$, the whole-depth current ratio is $206.75506228$; the linked carrier at its specified strength 10000 gives $206.76242640$. These are conditional model predictions against the stated benchmark, not uncertainty-weighted experimental exclusions.

An exact depth-two inverse current law on a finite branch

For $d=2$, put $P=n(n+1)$ and use a common positive edge $G$. The two singular values $\epsilon<\Omega$ satisfy

$$\epsilon(P,G)=\frac12 \left[\sqrt{G^2+4P+1}-\sqrt{G^2+1}\right], \qquad \partial_P\epsilon=\frac1{\sqrt{G^2+4P+1}}. \tag{7}$$

Consider the local homogeneous generating density $\rho f(k)$, where $\rho=\Psi^\dagger\Psi$, $k=\Psi^\dagger K\Psi/\rho$, and the same component current matrix $K$ acts on the full form carrier. Complete variation gives the edge $G=f'(k)$, scalar offset $C=f(k)-Gk$, and standing frequency $\lambda=\epsilon(P,G)+C$. The envelope identity is $d\lambda/dP=\partial_P\epsilon$ along a self-consistent branch.

Choose a reference edge $g>0$ and set $\mu=(1+g^2)^{-1/2}$. Requiring $\lambda(P)=\mu P$ through a continuous finite band makes (7) imply

$$G(P)^2=g^2-4P, \qquad k(G)=\frac G2\left(\mu-\frac1{\sqrt{1+G^2}}\right)<0. \tag{8}$$

The following one function of the local current supplies exactly that edge and scalar offset:

$$\boxed{ f(k(G))= \frac{\mu(g^2+G^2)}4- \frac1{2\mu}+ \frac1{2\sqrt{1+G^2}}. } \tag{9}$$

It has $f(0)=0$ at $G=g$ and direct differentiation gives $f'(k)=G$. Substitution into the full frequency, including $C$, then gives the exact identity

$$\boxed{\lambda_n=\mu n(n+1)=2\mu S(n,2).} \tag{10}$$

There is no particle-index coefficient in (9). At the two tested reference strengths $g=500$ and $g=5000$, all three maintained depth-two indices $76,81,95$ lie in its regular branch. In that branch, $G>\sqrt{(1+g^2)^{1/3}-1}$, the map $G\mapsto k$ is one-to-one and

$$f''(k)=\frac{2}{\mu-(1+G^2)^{-3/2}}>0. \tag{11}$$

The branch has a finite upper mode bound because $G^2=g^2-4P>0$, and its one-to-one condition is slightly stronger. An extension of $f$ outside this current interval has not been specified. The exact law therefore applies to the stated finite band, not to every harmonic and not to a derived photon zero mode. Since the interaction is local and algebraic in the normalized current, it preserves the spatial eigenspace and does not change the Dirac or form principal symbol on this branch at fixed geometry. With the already specified positive squared-wave action, its homogeneous gradient supplies exact oscillating solutions at (10) and canonical phase charge $\mathcal Q=2\lambda\|\Psi\|^2$. Its geometric force, full field Hessian, nonlinear continuation and origin in the master action remain to be established.

On the fixed round depth-two form geometry, the complete component derivative has an additional finite control. Its real and imaginary blocks are $L_x=A+2f''vv^T$ and $L_y=A$, with $v=(K-kI)\phi$, $A=H_0+GK+C$, and the entire signed component space retained. At both reference strengths and all three maintained indices, every eigenvalue outside the occupied $\lambda$ ray has absolute value greater than $\lambda$. Hence the rotating component quadratic stiffness is nonnegative. The separate form spatial scan retains all four form types, electric and quadrature polarizations, azimuth shifts one and three, and angular cutoffs two, four and six. Its 72-case report has positive complete projected Hessians and no resolved growing frequency; the largest growth-to-particle-frequency value is $4.40\times10^{-11}$, at numerical resolution. Doubling the quadrature order changes the complete matrix by at most $3.08\times10^{-15}$ after scaling by the squared strength. This is finite spatial evidence, not a theorem of full field stability.

The exact proportionality also makes an on-branch radial comparison possible without moving each mode to a different size. Restore the physical radius and speed through $h(R)=g(R)R/c$. If the same inverse current function is scaled consistently at each radius, (10) gives

$$\lambda_n(R)=\frac{cP_{n,2}}{R\sqrt{1+h(R)^2}}. \tag{12}$$

As one explicit curvature trial, take the previously tested bounded law $g(R)=g_0/[1+(R/R_0)^4]$, set $x=R/R_0$ and $\gamma=g_0R_0/c$. Every $n$ then has the same stationary radius equation

$$2\gamma^2x^2(x^4-1)=(1+x^4)^3. \tag{13}$$

At $\gamma=1000$ and $10000$, its inner roots are respectively $1.0000010000025R_0$ and $1.00000001000000025R_0$. The logarithmic on-branch frequency curvatures there are approximately $3.999976$ and $3.99999976$, both positive. A second, outer stationary root is a maximum, and the frequency tends to zero as $R\to\infty$. Thus the inner branch has a common local minimum, not global binding. The curvature law and its radius scale are inputs; the full field-radius mixed Hessian and supporting energy remain open.

The earlier depth-two angular-stress calculation still constrains this route. If its geometry dependence enters only through a scalar curvature function and any additional first metric variation is isotropic, the electric form's degree-two, degree-four and degree-six density components cannot all be stationary. Changing the current function changes one curvature-response coefficient but does not remove that three-component incompatibility. A patterned same-field supporting response or a different spatial state is still required.

The sign in (11) matters. Eliminating a linearly coupled stable same-field response coordinate from a positive quadratic potential $V(x)+xk$ gives $f_{\mathrm{eff}}''=-1/V''<0$. That elementary Schur result cannot produce the positive curvature required by (9). A direct positive current contribution or a more involved retained dynamics would be needed; the exact inverse function alone does not supply it. The sign obstruction extends to any stable set of response coordinates that enters affinely in the current: the minimized energy $\inf_x[V(x)+kB(x)]$ is an infimum of affine functions of $k$, so it is concave wherever finite. At a smooth positive Hessian its curvature is $-B_x^T(V_{xx}+kB_{xx})^{-1}B_x\le0$. This does not exclude the direct convex current term used below, a response that remains dynamical, or nonlinear coupling to $k$.

One dimension-aware correction across all tested depths

The strong-gap calculation also suggests a simpler common formula for a candidate current source. On the negative-current branch, let

$$p_d=1+\frac2d,\qquad a_d=(d-1)^{-p_d}, \qquad f_d(k)=gk+a_dg|k|^{p_d}. \tag{14}$$

These are functions of the spatial depth, not fitted mode coefficients. Because $k\simeq-(d-1)P_{n,d}/g^d$, the second term adds $P_{n,d}n^2/g^{d+1}$ at large $n$, canceling the leading $-P_{n,d}n^2/g^{d+1}$ part of (6). The finite-index terms and the next powers of $1/g$ remain. Full variation gives

$$G=g\big[1-a_dp_d|k|^{p_d-1}\big],\qquad C=-a_dg(p_d-1)|k|^{p_d},\qquad \lambda_n=\epsilon_n(G)+C. \tag{15}$$

The unified script solves (15) rather than substituting a prescribed $G$. The table gives largest-to-smallest spreads of $\lambda_n/S(n,d)$ within each maintained depth. Here the reference edge is $g=\text{strength}/2$; both strengths are diagnostics, not measured coupling values.

Depth ten has one assigned mode, so neither formula has a within-depth spread there. All 32 tested corrected fixed points have positive frequency and positive derivative of their scalar fixed-point equation. The correction improves agreement with the bare simplex rule, not with the electron-to-muon benchmark: at strength 10000 its ratio is $206.76470546$, still below the benchmark $206.76828299$ and slightly below the bare simplex value $206.76470588$. The complete finite-component derivative has real and imaginary blocks

$$L_x=A+2f_d''(k)vv^T,\qquad L_y=A, \quad A=H_0+GK+C,\quad v=(K-kI)\phi. \tag{16}$$

The component-Hessian batch uses the prior report's branch locations but recomputes the fixed points at high precision before diagonalizing all signed components. In all 32 cases every eigenvalue outside the occupied ray has absolute value greater than $\lambda$. The smallest ratio is $1.0000001761$, at depth three and strength 10000. This supplies nonnegative rotating component quadratic stiffness at fixed geometry. The separate fractional depth-two spatial scan then evaluates 72 finite form channels with the same modes, strengths, polarizations, shifts and cutoffs as the exact-law scan. Every complete projected Hessian is positive and the largest growth-to-particle-frequency value is $5.65\times10^{-11}$, at numerical resolution. Neither scan establishes stability of the full spatial field or its geometry. For $d>2$, the fractional term is only $C^1$ at $k=0$; the second variation there needs a smooth completion. Its dimension-dependent exponent and coefficient were chosen to cancel the known finite-gap correction. Neither their master-field origin nor the required common six/ten physical frequency unit follows from this calculation.

The exponent is forced within this restricted single-current correction class. At a fixed index and large reference gap, the current scales as $|k|\sim(d-1)P_{n,d}g^{-d}$, while the unwanted leading frequency correction scales as $g^{-(d+1)}$. A first nonlinear term $ag|k|^p$ scales as $g^{1-dp}$; matching these powers requires $p=1+2/d$. Its dependence on the harmonic index also matches the leading $n^{d+2}$ drift. Thus one universal nonzero leading power of this scalar current cannot cancel the leading drift at two different depths. In particular, an analytic current function with a quadratic first correction can do so at depth two, but its correction decays too quickly with the gap at every higher depth. This is a restriction on a current-only cancellation mechanism, not on a master action containing other field or geometric invariants.

Continuous inverse branches and the six-to-ten scale

The exact depth-two function suggests a general inverse problem. For a reference dimensionless edge $g$, set

$$\mu_d(g)=\|\operatorname{adj}B_0(g)\|_2^{-1},\qquad P_d(n)=\prod_{j=0}^{d-1}(n+j). \tag{17}$$

The value of $\mu_d$ is the exact zero-mode slope of the smallest singular value divided by $P_d'(0)$. At $n=0$ the adjugate has only its first column nonzero, with row-$i$ entry $(d-1)!g^i/i!$. Consequently the reference unit is available without a matrix decomposition:

$$\boxed{\mu_d(g)=\frac1{(d-1)!} \left[\sum_{i=0}^{d-1}\frac{g^{2i}}{(i!)^2}\right]^{-1/2}.} \tag{17a}$$

If a differentiable branch is to have $\lambda_n=\mu_d(g)P_d(n)$ continuously, the envelope theorem requires

$$\partial_n\epsilon_n(G_d(n))=\mu_d(g)P_d'(n). \tag{18}$$

Let $k_d(n)=\partial_G\epsilon_n(G_d(n))$. On any interval where this current is one-to-one, the one local inverse density specified by

$$f_d(k_d(n))=\mu_d(g)P_d(n)-\epsilon_n(G_d(n))+G_d(n)k_d(n) \tag{19}$$

obeys $f_d'(k_d(n))=G_d(n)$. Indeed, differentiating (19) and using (18) cancels the explicit $n$-derivatives, leaving $df_d/dn=G_d\,dk_d/dn$. Its full varied standing frequency is therefore the target $\mu_dP_d$. This is a constructive identity conditional on an invertible branch, not a physical origin for $f_d$.

The exact inverse function is locally smoother at zero current than the fractional approximation in (14). Let $Z_d(g)=\sum_{i=0}^{d-1}g^{2i}/(i!)^2$, $w_i=g^{2i}/[(i!)^2Z_d(g)]$, and

$$a_d(g)=\sum_{i=0}^{d-1}w_i \sum_{j=i+1}^{d-1}\frac1j>0. \tag{19a}$$

This is the derivative of $\log\|\operatorname{adj}B_n\|_2$ at $n=0$: only the first adjugate column contributes at first order. Writing $P_d'(0)=(d-1)!$, implicit differentiation of (18) at $(n,G)=(0,g)$ gives

$$G_d'(0)=\frac{2\mu_d(g)a_d(g)P_d'(0)} {\mu_d'(g)P_d'(0)},\qquad k_d'(0)=\mu_d'(g)P_d'(0), \quad \boxed{f_d''(0^-)= \frac{2\mu_d(g)a_d(g)} {[\mu_d'(g)]^2P_d'(0)}>0.} \tag{19b}$$

Thus each finite-gap inverse branch has a one-sided finite positive second derivative at $k=0$, and the implicit-function theorem gives a smooth local continuation there. This does not supply one universal function across different depths or settle regularity at zero field density, where $k=j/\rho$ is a separate issue.

The inverse-depth batch and its selected report samples each maintained depth at both reference gaps, including points between the maintained indices. All 12 sampled depth-gap branches have decreasing positive $G_d(n)$, decreasing negative $k_d(n)$, and positive secant estimates of $f_d''$. The calculation does not prove these properties between all sampled points or beyond the assigned band. For example, at $g=5000$, the effective edge falls from $0.9999974952g$ at electron index 13 to $0.9999850569g$ at muon index 35. The required finite-band correction divided by $g|k|^{1+2/d}$ approaches the dimension-aware leading coefficient in (14), while retaining finite-index variation. The exact inverse function remains depth dependent.

Restoring a physical radius gives the simplex frequency unit

$$\omega_{0,d}=\frac c{R_d}\,d!\mu_d(h_d),\qquad h_d=\frac{G_dR_d}{c}, \tag{20}$$

where $G_d$ here denotes the reference physical edge, rather than the self-consistent edge on a particular mode. With equal dimensionless reference gaps $h_6=h_{10}=500$, an equal six-to-ten frequency unit requires $R_{10}/R_6=8.06309677\times10^{-8}$; at $h=5000$, it requires $8.06399097\times10^{-12}$. These are exact finite-chain reference-unit ratios, close to the leading $5040/h^4$.

If instead the physical reference gap is shared, so $h_{10}=h_6R_{10}/R_6$, solving (20) gives $R_{10}/R_6=0.19519654$ at $h_6=500$ and $0.07773980$ at $h_6=5000$. The resulting depth-ten gaps are about 97.60 and 388.70; the index-23 inverse branch exists at both sampled values. These radius ratios are conditional requirements, not predictions of a supporting geometry. The inverse-depth batch also varies the same bounded curvature trial of (13) at every maintained depth, using one $g_0$ and one $R_0$. From (17a), define the exact dimensionless elasticity

$$q_d(h)=-\frac{d\log\mu_d}{d\log h} =\frac{\sum_{i=0}^{d-1} i h^{2i}/(i!)^2} {\sum_{i=0}^{d-1} h^{2i}/(i!)^2}. \tag{21}$$

With $x=R/R_0$, $h=\gamma x/(1+x^4)$, and $q_g=4x^4/(1+x^4)$, the shared radial unit in (20) is stationary precisely when $(q_g-1)q_d(h)=1$. The inner local minima have $R_{10}/R_6=0.97330930$ at $\gamma=1000$ and $0.97330927$ at $\gamma=10000$. Yet their ten-to-six frequency-unit ratios are only $4.84522\times10^{-8}$ and $4.84564\times10^{-12}$. Thus this one specified common curvature trial does not supply the charged-lepton scale relation, even after the inverse current law makes the within-depth ratios exact. These are necessary local radial minima; no full field or metric stationarity has been shown. The same-field metric stress and a compatible nested geometry still have to determine the cross-depth scales.

There is also an exact boundary on the tempting equal-radius, equal-dimensionless-gap option. At one radius the reference-unit ratio is $(10/6)\sqrt{Z_6(h)/Z_{10}(h)}$, where $Z_d(h)=\sum_{i=0}^{d-1}h^{2i}/(i!)^2$. It decreases from $10/6$ to zero as $h$ grows: the extra terms in $Z_{10}/Z_6$ have powers higher than every term of $Z_6$, so their ratio to $Z_6$ increases strictly. The unit ratio equals one only at $h=6.5440289929$. But $B_n'(n)=I$, so the smallest singular value is 1-Lipschitz in $n$ and $|\partial_n\epsilon_n|\le1$. Equation (18) would require slopes $320.71$ for the electron index, $27129.62$ for the muon index, and $1.04565\times10^6$ for the tau index at that gap. Hence the exact inverse current law with its zero-mode normalization has no occupied branch at the unique equal-radius shared-gap crossing. This excludes that restricted normalization choice, not distinct sector radii or a different same-field operator. More generally, the same slope bound imposes a minimum reference gap for every occupied inverse branch. The largest necessary bound within each maintained depth is:

Each bound solves $\mu_d(h)P_d'(n)=1$ using (17a). It is necessary, not sufficient for existence, invertibility or stability of the branch.

The exact depth-two law (9) and the approximate common rule (14) are separate inverse actions. They must not borrow the conformal Hessian or background threshold computed for the saturated current law. The joint candidate calculation keeps all of them on the same modes and records its source fingerprints and fixed-point outputs in its report. The next physical test is to derive a current term of the required sign and depth dependence from the master field, then vary the complete metric, field and supporting background together.

The common-scale response bound shows that no positive common gap law with nondecreasing logarithmic falloff can make the occupied inverse-current depth-six and depth-ten frequency units agree. It states the separate depth-two condition and the nonmonotone response required to evade that bound.