A scale-response obstruction for the inverse current mass branch

The finite-chain inverse construction can make the occupied frequency proportional to the simplex product $P_d(n)=\prod_{j=0}^{d-1}(n+j)=d!S(n,d)$ at each spatial depth. Its zero-current reference coefficient is fixed by the finite component chain, rather than by a separately fitted mass for each particle:

$$\mu_d(h)=\frac{1}{(d-1)!\sqrt{Z_d(h)}},\qquad Z_d(h)=\sum_{j=0}^{d-1}\frac{h^{2j}}{(j!)^2}. \tag{1}$$

Here $h=g(R)R/c$ is the dimensionless reference edge at physical radius $R$. The proposed frequency unit on an occupied branch is

$$u_d(R)=\frac{c}{R}d!\mu_d(h(R)),\qquad \omega_{n,d}=u_d(R)S(n,d). \tag{2}$$

The inverse current function remains a conditional local design. Its master-field origin, spatial supporting energy and complete stability have not been derived. This note asks which one common positive function $g(R)$ could at least make the reference units stationary and agree between the six-dimensional electron–muon sector and ten-dimensional tau sector. The same function is also tested at the maintained depth-two boson indices. Equal physical gap coefficients across depths are an additional hypothesis of this test; the sector-nesting premise alone does not impose it.

Stationary radii and occupied-branch bounds

Define the logarithmic falloff of the common edge and the elasticity of the inverse reference unit by

$$\eta(R)=-\frac{d\log g}{d\log R},\qquad q_d(h)=-\frac{d\log\mu_d}{d\log h} =\sum_{j=0}^{d-1}j w_j, \quad w_j=\frac{h^{2j}/(j!)^2}{Z_d(h)}. \tag{3}$$

The weights are positive and sum to one. Consequently $0<q_d(h)<d-1$, and $dq_d/d\log h=2\operatorname{Var}_w(j)>0$. Direct differentiation of (2) gives the exact stationary-radius condition

$$\frac{d\log u_d}{d\log R} =-1+q_d(h)(\eta-1)=0, \qquad \boxed{\eta(R_d)=1+\frac{1}{q_d(h_d)}}. \tag{4}$$

The next derivative, evaluated at a stationary radius, is

$$\frac{d^2\log u_d}{d(\log R)^2} =q_d(h_d)\frac{d\eta}{d\log R} -2\operatorname{Var}_w(j)(\eta-1)^2. \tag{5}$$

A strict local minimum therefore requires positive $d\eta/d\log R$ at that radius. This is a necessary radial condition; it says nothing about angular stress or the complete field–metric Hessian.

The causal finite component chain has $|\partial_n\epsilon_n|\le1$. The inverse branch must therefore obey $\mu_d(h)P_d'(n)<1$ at every assigned index. The strongest assigned bounds in the relevant sectors are

$$h_2>\sqrt{191^2-1}=190.99738218\quad(n=95),\qquad h_6>53.57003192\quad(n=35),\qquad h_{10}>35.21386344\quad(n=23). \tag{6}$$

These are necessary slope bounds, not existence or stability proofs for the inverse branches. At the ten-dimensional threshold, $q_{10}=8.93218749>5$, and $q_{10}$ increases with $h$. Equation (4) thus gives $\eta(R_{10})<1.2$, whereas $\eta(R_6)>1.2$ because $q_6<5$. Similarly $\eta(R_2)>2$ and $\eta(R_6)<2$ on its occupied branch.

An exact bound for monotone falloff

Suppose $\eta(R)$ is nondecreasing with $R$ across the stationary radii. The endpoint inequalities just established force the ordering

$$R_{10}<R_6<R_2. \tag{7}$$

Since $\eta(R_{10})=1+1/q_{10}(h_{10})>10/9$, integration of $d\log h/d\log R=1-\eta$ between the first two radii gives, with $r=R_{10}/R_6<1$,

$$\frac{h_{10}}{h_6}\ge r^{-1/9}. \tag{8}$$

The same monotonicity gives $\eta>1$ between $R_6$ and $R_2$, so $h_6>h_2>190.99738218$. To turn (8) into a frequency bound, factor the highest power from (1):

$$\mu_d(h)=\frac{h^{-(d-1)}}{\sqrt{T_d(h)}},\qquad T_d(h)=\sum_{j=0}^{d-1} \left[\frac{(d-1)!}{j!}\right]^2h^{-2(d-1-j)}\ge1. \tag{9}$$

Equations (2), (8) and (9) then imply

$$\begin{aligned} \frac{u_{10}}{u_6} &=\frac{5040}{r}\frac{h_6^5}{h_{10}^9} \sqrt{\frac{T_6(h_6)}{T_{10}(h_{10})}}\\ &\le\frac{5040\sqrt{T_6(h_6)}}{h_6^4} <\boxed{3.788522\times10^{-6}}. \end{aligned} \tag{10}$$

The final function decreases with $h_6$, so the last inequality uses only the depth-two bound (6). Even without depth two, the depth-six bound alone gives $u_{10}/u_6<6.146628\times10^{-4}$. The benchmark electron and tau masses and maintained counts give $(m_\tau/S(23,10))/(m_e/S(13,6))\simeq1.000644$. Thus a common gap law with nondecreasing logarithmic falloff cannot supply this cross-depth scale relation within the exact inverse-current reference normalization. The result covers more than one chosen power-law response, but it does not exclude a different common operator, a depth-dependent geometric conversion, or a nonmonotone supporting law.

What a different common scale response would require

For any common law with $R_{10}<R_6$, define its mean falloff on that interval by

$$\bar\eta=\frac{1}{\log(R_6/R_{10})} \int_{R_{10}}^{R_6}\eta(R)\,d\log R. \tag{11}$$

The exact ratio in (10), with $T_{10}\ge1$, shows that matching a target ratio $U=u_{10}/u_6$ requires

$$\boxed{\bar\eta\le\frac{10}{9} +\frac{\log\!\left[ 5040\sqrt{T_6(h_6)}/(U h_6^4)\right]} {9\log(R_6/R_{10})}.} \tag{12}$$

At the illustrative radius ratio $R_{10}/R_6=0.2$, the occupied depth-six bound limits $\bar\eta$ to about $0.6006$, and including depth two limits it to about $0.2492$, for the stated benchmark $U$. These are necessary conditions, not a constructed supporting law. If instead $\eta\ge1$ throughout the interval, matching that benchmark requires $R_{10}/R_6<3.7861\times10^{-6}$ when depth two is included.

The stationary endpoints have $\eta(R_{10})>10/9$ and $\eta(R_6)>1.2$, while (12) demands a mean below $10/9$ at every finite radius ratio. A smooth successful common law with these ordered radii must dip below $10/9$ between them. If both stationary radii are strict local minima, (5) makes $\eta$ rise locally at both endpoints. The intervening dip then requires at least one local maximum and one local minimum of $\eta$. Deriving such scale dependence from one stable same-field action is a specific remaining task.

Positive power sums as candidate scale laws

Two common ways to build a positive reference edge from scale-homogeneous terms have opposite, exact effects on the radial curvature. First take a direct positive sum,

$$g(R)=g_0\sum_i a_i(R/R_0)^{-p_i},\qquad a_i>0.$$

The weights of its terms are positive and normalized. Differentiating their weighted mean gives $\eta=\sum_iw_ip_i$ and $d\eta/d\log R=-\operatorname{Var}_w(p_i)\le0$. Equation (5) is then strictly negative at every finite-gap stationary unit, because $\operatorname{Var}_w(j)>0$ and $\eta-1>0$ there. This entire additive positive-power class has only radial maxima, regardless of how many positive terms are added.

Now take the reciprocal of a positive stiffness sum,

$$g(R)=\frac{g_0}{\sum_i a_i(R/R_0)^{p_i}},\qquad a_i>0.$$

Its weighted mean has $d\eta/d\log R=+\operatorname{Var}_w(p_i)\ge0$. It can create a radial local minimum if the positive first term in (5) exceeds the spectral variance term. It then falls under the cross-depth bound (10), independently of how many positive powers are included. A single stable response with a positive power-sum stiffness and a radius-independent linear source would have this reciprocal scale form after elimination, but that reduction is only a conditional model: the same-field origin and current dependence must still be derived. If its coupling to the current is affine, the eliminated response additionally has nonpositive current curvature, the opposite sign from the required convex inverse-current correction.

The positive-sum batch checks three representative shared laws at two strengths across all six maintained depths. Its selected report contains 60 stationary points: 12 additive-gap maxima and 24 local minima plus 24 outer maxima for two reciprocal-stiffness examples. At the occupied-branch strength $\gamma=5000$, the reciprocal examples give ten-to-six unit ratios about $3.03\times10^{-10}$ and $1.01\times10^{-10}$. The analytic variance identities, rather than this finite scan, establish the class restrictions.

More generally, suppose a retained source and a stable stiffness give $g(R)=J(R)/D(R)$, where both $J$ and $D$ are positive sums of radius powers. Their exponent distributions obey

$$\eta=\langle p\rangle_D-\langle s\rangle_J, \qquad \frac{d\eta}{d\log R} =\operatorname{Var}_D(p)-\operatorname{Var}_J(s),$$

where $p$ and $s$ are the denominator and numerator exponents. To produce the required dip, the numerator variance must exceed the denominator variance somewhere; at each selected strict radial minimum, equation (5) requires the denominator variance to dominate by the positive spectral threshold. This specifies a multi-scale source structure, but does not provide its field origin. With only affine current coupling to a stable eliminated amplitude, its Schur contribution remains concave in the current, so a separate direct convex current invariant is still required.

A nonmonotone control and its competing branch

The obstruction depends on monotonicity. To test its scope with one common law across all six maintained depths, consider the positive but nonmonotone reference edge

$$g(R)=g_0\frac{[1+(R/10R_0)^4]^{B/4}} {[1+(R/R_0)^4]^{1.15/4} [1+(R/CR_0)^4]^{(B+1.2)/4}}.$$

The three transitions, at $R_0$, $10R_0$ and $CR_0$, make $\eta(R)$ rise, dip and rise again. This is an inverse diagnostic, not a field-derived supporting law. For each stated trial choice of $C$ and depth-two reference gap, the common amplitude $\gamma=g_0R_0/c$ is set by that gap at its radial minimum, then one common shape parameter $B$ is fitted to the benchmark $u_{10}/u_6\simeq1.000644$. No parameter is assigned to an individual mode. The six-depth fit finds:

The selected report records the stationary radius and gap at every depth. All selected radii are strict local minima of the reference frequency unit, and every assigned mode passes the necessary singular-slope bound. Direct high-precision singular-matrix roots exist at all 16 assigned modes; sampled intermediate points retain decreasing edge and current, as required for a convex inverse density on those samples. This verifies discrete inverse-branch feasibility more strongly than the slope bound, but it does not prove regularity or stability on the whole interval. The first two trial shapes make $g$ rise with radius across part of the interval. The third keeps $g$ strictly decreasing everywhere: for $B<1.15$ and $C>10$, its first positive sigmoid in $\eta$ is at least the middle sigmoid, so $1.15\sigma(R/R_0)-B\sigma(R/10R_0)>0$, where $\sigma(y)=y^4/(1+y^4)$; the final term is also positive. Thus a negative $\eta$ is a feature of the first two fits, not a general requirement of the cross-depth unit match. The third uses a wider separation between its positive scale transitions. Its $\eta$ has a local maximum at $3.191R_0$ and a local minimum at $28.733R_0$, satisfying the two-turning-point requirement from the stationary-frequency calculation.

The fit also produces two local depth-ten minima. The small-radius minimum carries the fitted frequency unit, while the outer minimum has the far smaller unit shown in the table. The frequency curve alone has no rule selecting the inner branch. Its outer barrier is also not a complete particle energy barrier: the supporting field's energy and metric stress remain absent. The fitted equality is therefore an existence control within the inverse reference equations, not a prediction of the observed tau mass.

The first fit exposes a further spatial constraint. At $h_2=220$, the 36-case depth-two form scan finds no resolved growth for $n=76,81$, but all 12 finite electric and quadrature controls at $n=95$ have growth around $0.862$ times the particle frequency. The value is stable across angular cutoffs two, four and six and two azimuth shifts. Positive projected Hessian eigenvalues in those cases do not remove the gyroscopic growth. A one-channel gap scan places its disappearance between $h_2=220$ and $240$ for the electric, shift-one, cutoff-six control. That scan does not establish the exact boundary for all spatial perturbations.

For both fits at $h_2=500$, the existing 36-case exact depth-two scan already tests the same inverse-current field state at that gap and passes its finite form controls. The new radius law does not alter that fixed-metric matrix, but its own field–metric variations have not been included. The nonmonotone fit still needs a master-field source, a selection rule among its radii, compatible nested spatial states, positive complete joint stability and physical inertial normalization.

A positive rational scale control

The logarithmic falloff can have the required two turning points with ordinary positive power sums, without the fractional exponents in the preceding example. Take one shared reference law

$$\boxed{\frac{g(R)}{g_0} =\frac{1+(R/bR_0)^4} {[1+(R/R_0)^4][1+(R/CR_0)^4]}.}$$

Its logarithmic falloff is $\eta=4[\sigma(R/R_0)-\sigma(R/bR_0)+\sigma(R/CR_0)]$, where $\sigma(y)=y^4/(1+y^4)$. Since $1<b<C$, the first difference and final term are strictly positive: $g(R)$ decreases at every positive radius. The first bump can place the depth-ten stationary minimum before the middle dip, while depths two through six use the later rise. This is one function across all depths. When the outer transition is well separated, that first bump peaks at $4\tanh(\log b)$. Choosing $b=1.34$ puts its peak near $1.138$, between the occupied depth-ten stationary falloff near $1.111$ and the depth-six value above $1.2$. This explains the separate radius branches but is still an inverse choice.

In the rational-law batch, set $b=1.34$, fix the depth-two gap at $h_2=500$, and fit the common outer scale $C$ to the benchmark six-to-ten unit ratio. The resulting $C=18.00753138$ and $\gamma=179.0387654$ give $R_{10}/R_6=0.07325$, $h_{10}=116.6397$, and $u_{10}/u_6=1.00064435538$. All six selected radii are local minima; the exact inverse singular-matrix roots exist at every assigned mode and sampled intermediate points. The two falloff turning points occur at $1.1576R_0$ and $4.6846R_0$, with the latter at $\eta=0.03655>0$. The full values and source hashes are in the selected report. The four-point robustness batch varies $b$ from $1.332$ to $1.36$, refitting only $C$ each time. All four retain the local six-depth reference checks and two depth-ten minima; $C$ ranges from $17.740$ to $18.734$.

This algebraic law is simpler than the previous fractional-power trial and identifies a concrete source-to-stiffness shape: $J=1+(R/bR_0)^4$ and $D=[1+(R/R_0)^4][1+(R/CR_0)^4]$. No master-field reduction has produced these functions or the required direct convex current interaction. The inner depth-ten unit still competes with an outer local minimum whose unit relative to depth six is $4.84\times10^{-8}$; its selection and supporting energy remain unresolved. Matching the benchmark by fitting $C$ is a conditional existence test, not a mass prediction. For this rational law $g(R)\sim R^{-4}$ at large radius, so $h(R)\to0$ and $u_d(R)\sim cd/R\to0$. Its local minima are not global radial binding states without the complete supporting energy.

Constraint from projecting a scalar master contact

The proposed scalar master contact offers a direct action-level test. Suppose a fixed linear, canonically normalized map $P_{\rm sec}$ embeds a sector field $\psi$ into one master field, with no nonzero background and no eliminated dynamical modes. A pure scalar contact $U_4[\Psi]\propto\int(\bar\Psi\Psi)^2$ remains fourth-order after that projection:

$$U_4[P_{\rm sec}(t\psi)]=t^4U_4[P_{\rm sec}\psi].$$

The inverse-current generating density used here instead obeys

$$U_2[\psi]=\int\rho f(j/\rho),\quad \rho=\psi^\dagger\psi,\quad j=\psi^\dagger K\psi, \qquad U_2[t\psi]=t^2U_2[\psi].$$

These nonzero functionals cannot agree on an open amplitude cone. Linear projection of that pure quartic contact therefore does not produce this quadratic-homogeneous inverse mass current by itself. A fixed norm, a nonzero same-field background, a nonlinear reduction, or another master invariant can change the effective degree; each must enter the actual action and kinetic normalization rather than being identified with the required interaction after the fact.

For example, writing $\Psi=\Psi_0+P_{\rm sec}\psi$ inside the quartic contact produces a quadratic term proportional to $J_0J_\psi+2[\operatorname{Re}(\bar\Psi_0P_{\rm sec}\psi)]^2$, where $J_0=\bar\Psi_0\Psi_0$ and $J_\psi=\overline{P_{\rm sec}\psi}P_{\rm sec}\psi$. The background can supply a linearized operator, but its fixed quadratic term cannot by itself reproduce the nonlinear function $f(j/\rho)$ across an occupied band. A same-field reduction that does produce it must also account for the background phase response, the omitted higher terms and the positive current curvature. This is a restriction on the stated pure-contact linear-projection route, not on every possible master interaction.

A minimal retained-amplitude calculation makes the needed direct term explicit. At fixed positive local field density $\rho$, let $\rho U(x,k;R)$ be a candidate generating energy density, where $x$ is a dimensionless real ratio of retained components of that same field. This homogeneous form is an additional premise. With $D(R)>0$, write the energy per unit density near the reference as

$$U(x,k;R)=\tfrac12D(R)x^2-J(R)x+b xk+F(k;R).$$

Eliminating only this amplitude gives

$$x_*(k,R)=\frac{J(R)-bk}{D(R)},\qquad U_{\rm eff}(k;R)=F(k;R)-\frac{[J(R)-bk]^2}{2D(R)}, \qquad \boxed{\partial_k^2U_{\rm eff}=F_{kk}-\frac{b^2}{D}.}$$

At zero current the background supplies a reference edge $F_k(0;R)+bJ(R)/D(R)$. Choosing a multi-scale source-to-stiffness ratio can make that edge's logarithmic falloff nonmonotone, but stability of the eliminated amplitude subtracts $b^2/D$ from the needed positive inverse-current curvature at fixed $\rho$. The master action must therefore supply a direct $F_{kk}>b^2/D$ on the occupied branch, as well as a geometric source and kinetic normalization for $x$. The exact inverse normalization also has $f(0;R)=0$. This reduction would require the separate value condition $F(0;R)=J(R)^2/[2D(R)]$, in addition to its slope and curvature conditions. Otherwise the retained background adds a scalar frequency offset and spoils the fixed simplex unit. That counterterm and the background's radial energy cannot be inferred from the fitted gap coefficient alone. This calculation does not identify an available $x,J,D,F$ in $\Psi_\infty$; it is an exact requirement on such a proposed reduction.

A separate lead for the nonlinear current power

The earlier approximate current correction has power $p_d=1+2/d$. A simple phase-space minimization reproduces that exponent without reproducing its normalization. For occupation $0\le v(p)\le1$ of momentum states in $d$ ordinary spatial directions, minimize the kinetic moment at fixed number density:

$$Q=(2\pi)^{-d}\int v(p)\,d^dp, \qquad T=(2\pi)^{-d}\int |p|^2v(p)\,d^dp. \tag{13}$$

Exchanging an occupied larger momentum with an empty smaller momentum lowers $T$, so the minimizer fills a ball $|p|\le p_F$. With $V_d=\pi^{d/2}/\Gamma(d/2+1)$, direct integration gives

$$Q=\frac{V_dp_F^d}{(2\pi)^d},\qquad \boxed{T=A_d Q^{1+2/d}},\qquad A_d=\frac{d}{d+2}(2\pi)^2V_d^{-2/d}. \tag{14}$$

For depths $2,3,4,5,6,10$, the coefficient $A_d$ in these canonical momentum units is respectively about $6.283,9.116,11.848,14.512,17.126,27.281$. The inverse-current leading coefficients $(d-1)^{-(1+2/d)}$ are respectively $1,0.31498,0.19245,0.14359,0.11696,0.07160$. This shows that the power alone is not a normalization derivation. The IDWT component current $k$ has not been identified with the occupation density $Q$, and a filled many-state momentum ball is an added physical premise. A field projection and kinetic normalization would have to supply that identification and its coefficient before this lead could be used as a mass interaction.

Reproducible comparison

The radius-law batch evaluates local stationary radii for one bounded $g(R)$ at every maintained depth. Its selected report contains 120 depth cases and 20 complete six-depth comparisons. The scale-bound and kinetic batch computes the exact thresholds and bound above, scans 14 additional shared powers at the depth-two necessary boundary, and compares all six kinetic coefficients. Its selected report records the inputs, checks and source hash. For bounded laws $g(R)=g_0/[1+(R/R_0)^q]$, the boundary scan runs from $q=2.0001$ to $1000$; its largest sampled ten-to-six unit ratio is $3.77982\times10^{-6}$. The formal step-response limit is $(10/6)\sqrt{Z_6(h_2)/Z_{10}(h_2)} =3.78431\times10^{-6}$ at the necessary depth-two gap boundary. The finite scan illustrates (10); the proof does not depend on its resolution. Neither calculation derives the scale law or proves a complete supported particle state.