Constructive Count Clocks with Spatial Support and Reciprocal Response

Current interpretation. This article preserves an 18 October inverse construction. Its supplied count-valued spectral clocks and separately excited response modes are mathematical comparison models. They are not a realization of the current proposal of C portions sharing one fundamental frequency, whose additive average registration rate is C times that fundamental. Persistence and response conclusions below apply only to the action and domains actually written here; they do not transfer to a different action. The current particle-geometry account develops that distinction and the newer constructions.

Several useful requirements can coexist in explicit equations: a real spatial count spectrum, nonaffine current response, anisotropic supporting geometry, positive complete energy, and frequencies that survive a responding part of the same field. A separate local construction also accommodates a small fitted clock correction while retaining its complete reference inertia and bulk cone bounds. The three actions below establish these combinations under stated premises. Their properties must not be combined into an unwritten master law.

Physical space is ordinary higher-dimensional space, taken here as $\mathbb R^{10}$. No dimension is compactified. Field components and spectral coefficients are not additional spatial dimensions or particle-position probabilities. The electron's six-dimensional structure and its literal external trajectory around a nucleus remain distinct from the internal classical fields calculated here. A finite normal envelope supplies transverse localization for these examples; it does not derive exact confinement to a lower-dimensional layer.

These are constructive inverse results. Their frequency targets and supporting structures are explicit inputs whose consequences are calculated. They improve the set of consistent candidates without selecting the observed particles or completing the physical mass operator.

Real Spatial Counts and the Actual Assignments

Let $d\in\{2,3,4,5,6,10\}$, and use normalized real product Hermite functions on ordinary space, with a common length $\ell$ and Gaussian ground envelopes in the $10-d$ normal directions. In dimensionless coordinates define

$$a_i=\frac{y_i+\partial_i}{\sqrt2},\qquad N_P=\sum_{i=1}^d a_i^*a_i, \qquad \|B_df\|^2=\sum_{|\alpha|=d}\frac{d!}{\alpha!}\|a^\alpha f\|^2.$$

On finite Hermite sums the full symmetric derivative satisfies

$$M_d=\frac{B_d^*B_d}{d!}=\binom{N_P}{d}. \tag{1}$$

For an active multiindex $\beta$ of total degree $K$, this follows from $\sum_{|\alpha|=d}\prod_i\binom{\beta_i}{\alpha_i}=\binom Kd$, the coefficient of $t^d$ in $\prod_i(1+t)^{\beta_i}$. The componentwise annihilators also make the off-diagonal matrix elements vanish. This proof works at odd and even depths under the same rule.

Two distinct constructions produce the same integer:

$$\dim\{N_P\le N\}=\binom{N+d}{d},\quad N=n-1; \qquad M_d|_{W_K}=\binom Kd,\quad K=n+d-1. \tag{2}$$

The cumulative prefix and the derivative eigenspace $W_K$ are different spaces. Equality of their counts does not equate their occupation, point contact or kinetic laws. In particular, a second-order wave needs squared stiffness $\mu_d^2M_d^2$ to have frequency $\mu_dM_d$; stiffness $\mu_d^2M_d$ would instead give a square root.

These are supplied maintained assignments, not modes selected by equation (1). The real spatial contact construction gives another precise count realization with its own measure and kinetic law.

A Spatial Action with Current and Anisotropic Support

Choose a finite sum of positive eigenspaces $W_K$, $K\ge d$. For each $a=(i,j)\in\{0,1\}^2$, let $f_a$ be an independently variable real unit spatial profile in that sum. Its field-component vector $e_a$ is orthogonal to the other component vectors. Define

$$u(k)=\left(\sqrt{\frac{1+k}{2}},\sqrt{\frac{1-k}{2}}\right), \quad v(x)=\frac{(1,x)}{\sqrt{1+x^2}},\quad p_{ij}=u_i^2v_j^2, \quad \phi=r(\cos\theta,\sin\theta)\otimes\sum_a\sqrt{p_a}f_ae_a. \tag{3}$$

Use the positive component chart $-1<k<1,x>0,r>0$. At $x=0$, independent profile directions would acquire zero metric weight, so that boundary needs a different chart. The product weights and common temporal phase are retained ansatz constraints, not a parametrization of every possible field.

Actual spatial integration gives

$$\|d\phi\|^2=dr^2+r^2\left[d\theta^2+ \frac{dk^2}{4(1-k^2)}+\frac{dx^2}{(1+x^2)^2} +\sum_a p_a\|df_a\|^2\right]. \tag{4}$$

The first-order cross terms vanish because $\langle f_a,df_a\rangle=0$, but the derivatives of $p_a$ still enter the equations of motion. The current variable $k$ is an integrated component current. It is not the pointwise ratio of a local current and density.

A Nonaffine Response and a Count-Derived Restoring Coefficient

For a common $b>0$, take

$$w(k,x)=\frac{1+k^2}{2}x^2-bx+\frac{b^2}{2}\ge0, \quad x_*(k)=\frac b{1+k^2}, \quad f(k)=\min_x w=\frac{b^2k^2}{2(1+k^2)}.$$

The complete reduced curvature is

$$f''(k)=w_{kk}-\frac{w_{kx}^2}{w_{xx}} =\frac{b^2(1-3k^2)}{(1+k^2)^3}. \tag{5}$$

Thus positive response stiffness $w_{xx}=1+k^2$ does not force an affine current law. At $(0,b)$, $w=dw=0$ and its Hessian is $\operatorname{diag}(b^2,1)>0$. The outer-current branch can also be repaired: for $b=1/2$, replace $w$ by

$$w_0=w-f(k_0)-f'(k_0)(k-k_0)+\frac14(k-k_0)^2.$$

Since $f''\ge-1/16$ on $|k|<1$, the reduced curvature is then at least $7/16$. Its zero minimum is at $(k_0,b/(1+k_0^2))$; the checked control uses $k_0=3/4$. This is a declared change of one common constitutive law.

For the full four-component field put

$$m=\sum_a p_a\langle f_a,M_df_a\rangle,\quad z=\sum_a p_a\langle f_a,M_d^2f_a\rangle,\quad \mathcal V=z-m^2\ge0,$$ $$\mathcal B=m+\eta_d(m)\frac{\mathcal V}{m},\qquad \eta_d(m)=1+\frac{(d!m)^{1/d}}d,\qquad \Lambda=\mu_d\mathcal B(1+gw),\quad g>0. \tag{6}$$

This is the variance of the entire normalized field, including the spread between component mean counts. Replacing it by the average of the four within-component variances changes the action. Here $m\ge1$, so the law is smooth on its massive domain.

For a pure shell $c_K=\binom Kd$, a normalized perturbation toward a distinct shell $c_j$ has quadratic coefficient

$$\Delta_j\left(1+\eta_d(c_K)\frac{\Delta_j}{c_K}\right), \qquad \Delta_j=c_j-c_K. \tag{7}$$

Every such coefficient is positive. For a lower shell the worst threshold is the nearest lower count, and Pascal's identity gives $c_K/(c_K-c_{K-1})=K/d$. On the other hand,

$$d!c_K=\prod_{s=0}^{d-1}(K-s)\ge(K-d+1)^d, \qquad \eta_d(c_K)\ge\frac{K+1}{d}>\frac Kd. \tag{8}$$

This one law works at any chosen finite cutoff. A previous constant $\eta=128$ works on the specified hundred-shell comparison domain because its sufficient threshold is $101/2$. Equation (6) removes that numerical coefficient and its cutoff dependence. It is not a uniqueness theorem or an infinite-dimensional evolution result.

Complete Supporting Geometry and Variation

Let

$$q_{a,AB}=\ell^2\int\partial_Af_a\partial_Bf_a\,d^{10}y, \quad q=\sum_a p_aq_a,\quad h=F^TF,\quad F\in GL^+(10),$$ $$S_q=I+\frac{\chi}{a_s^2}q,\quad t_q=(\det S_q)^{-1/10}, \quad T=a_s^2t_qS_q,\quad V_s=\frac\kappa8\operatorname{tr}[(T^{-1}h-I)^2]. \tag{9}$$

All constants are positive and common within the declared law. $\det T=a_s^{20}$. The product in the trace is similar to a symmetric matrix, so $V_s\ge0$ even when $T$ and $h$ do not commute. A covariant law can therefore support anisotropic states without introducing a preferred fixed external axis.

The deformation of a spatial field is $(U_{F,X}\phi)(X+Fy)=(\det F)^{-1/2}\phi(y)$. Its material derivative is $D_\tau\Psi=\partial_\tau\Psi-(\partial_\tau U)U^{-1}\Psi$. This specified connection gives $\|D_\tau\Psi\|=\|\phi'\|$; it is not the bare Cartesian derivative. Material motion is paid for by the $I_F$ term in the single action

$$L=\frac{r'^2}{2}+\frac{r^2}{2}\left[ \theta'^2+G_k k'^2+G_x x'^2+\sum_a p_a\|f_a'\|^2-\Lambda^2\right] +\frac{I_F}{2}\|F'\|_F^2-V_s, \tag{10}$$

where $G_k=1/[4(1-k^2)]$ and $G_x=1/(1+x^2)^2$. The supporting energy has a chosen zero minimum, not an uncounted positive offset.

For reproducible full variation, write $Z=T^{-1}$. Then

$$A_h=\frac\kappa4(ZhZ-Z),\quad A_T=\frac\kappa4(ZhZ-ZhZhZ),\quad C_q=\chi t_q\left[A_T-\frac{A_T:S_q}{10}S_q^{-1}\right],$$ $$\delta V_s=A_h:\delta h+C_q:\delta q, \qquad \partial_F V_s=2FA_h. \tag{11}$$

The new coefficient must also be differentiated:

$$\eta_d'(m)=\frac{(d!m)^{1/d}}{d^2m},\quad \mathcal B_m=1+\eta_d'\frac{\mathcal V}{m}-\eta_d-\eta_d\frac z{m^2}, \quad \mathcal B_z=\frac{\eta_d}{m}. \tag{12}$$

With $A=\mathcal B_mM_d+\mathcal B_zM_d^2$, the unit-profile gradient is $\nabla_a\mathcal B=2p_a[A f_a-\langle f_a,Af_a\rangle f_a]$. For $s=k,x$, $\mathcal B_s=\sum_a p_{a,s}(\mathcal B_m m_a+\mathcal B_z z_a)$ and $q_s=\sum_a p_{a,s}q_a$. The supporting profile gradient is the finite-band tangent projection of $-2p_a\ell^2\partial_A(C_{q,AB}\partial_B f_a)-2p_a(C_q:q_a)f_a$.

The current equations include the kinetic load $-r^2\sum_a p_{a,s}\|f_a'\|^2/2$, as well as $r^2\Lambda\Lambda_s+C_q:q_s$. The profile equations are

$$r^2p_a\nabla_\tau f_a'+(r^2p_a)'f_a' +r^2\Lambda\nabla_a\Lambda+\nabla_aV_s=0. \tag{13}$$

These terms prevent the simultaneous current, shape and support dynamics from being replaced by separate frozen-coordinate tests.

References, Energy and the Joint Hessian

Take any four unit profiles in the same $W_K$, the zero-minimum current, and $F^TF=T(q)$. Then $\mathcal B=c_K$, its constrained first variation vanishes, and

$$\omega=\mu_dc_K,\quad \theta'=s\omega,\quad r_0^2=A_0/\omega, \quad Q=r^2\theta'=sA_0,\quad E=\omega A_0,\quad s=\pm1. \tag{14}$$

All equations, including support and profile equations, are satisfied. The complete energy is the sum of the positive kinetic terms in (10), $r^2\Lambda^2/2$, and $V_s$. The common $A_0$ is an action normalization, not a separate fitted mass amplitude.

At fixed $Q$, put $V_Q=Q^2/(2r^2)+r^2\Lambda^2/2+V_s$, $\delta\zeta=(\delta k,\delta x)$, and let $\xi_{a,j}$ denote the profile tangent in shell $j$. Its complete reference Hessian is

$$\begin{aligned} \delta^2V_Q={}&4\omega^2(\delta r)^2 +r_0^2\omega^2g\,\delta\zeta^T\operatorname{Hess}(w)\delta\zeta\\ &+2r_0^2\omega\mu_d\sum_a p_a\sum_{j\ne K} \Delta_j\left(1+\eta_d(c_K)\frac{\Delta_j}{c_K}\right)\|\xi_{a,j}\|^2\\ &+\frac\kappa4\operatorname{tr} \left[T^{-1}(\delta h-\delta T)T^{-1}(\delta h-\delta T)\right]\ge0. \end{aligned} \tag{15}$$

Here $\delta q=\sum_a(\delta p_a q_a+p_a\delta q_a)$, so the support square includes both current/material and profile/material mixing. The positive kinetic metric makes the nonzero disturbance frequencies real in this retained reference problem. Rotations and same-shell profile changes with following support remain zero directions. The full fixed-charge kernel has dimension $45+4[\binom{K+d-1}{d-1}-1]$: four unit-profile spheres within the same shell, together with 45 material rotations. It is 153 at $d=3,n=4,K=6$. The 115-coordinate diagnostic below retains only one such tangent per profile, so its nullity is 49. No unique shape, observed cutoff, asymptotic damping or unrestricted spatial stability follows.

For the explicit profiles $\beta_{ij}=(n+1-i-j,i+j,1,\ldots,1,0,\ldots,0)$, the actual tensor is

$$q=\operatorname{diag}(n+3/2-l,l+1/2, 3/2\ [d-2],1/2\ [10-d]),\quad l=(1-k)/2+x^2/(1+x^2).$$

For $a_s=\chi=1$, the electron reference $(k,x)=(0,1/2)$ has $T_1/T_2=74/11$. This is a conditional shape ratio, not a measured electron size. The same construction admits all fourteen rows, including six and ten, without a special depth-five coefficient. Its profiles have decaying noncompact tails, not compact spatial support.

Using center proper time in (10) is an additional collective premise. It yields $E_{\rm lab}=\gamma E_{\rm rest}$ and $P=\gamma E_{\rm rest}v/c^2$ with every supporting term included. It does not turn the finite-band action into a local causal master PDE.

Protected Frequencies with Actual Spatial Response

A second construction tests whether a full zero and a nonzero geometric-mean frequency can survive a genuinely responding part of the same field. Its spatial source and normalization can be made explicit. Set

$$f(s)=\frac{\sqrt{15}}4(1-s^2)\quad(|s|\le1),$$

with zero outside, and take $h(s)=\sqrt{3/8}$ for $|s|\le1$, $h(s)=\sqrt{3/8}(2-|s|)$ for $1<|s|<2$, zero otherwise. Both have unit squared integral. The ten-dimensional product profile $u_d=\ell^{-5}\prod_{i\le d}f(X_i/\ell)\prod_{i>d}h(X_i/\ell)$ is in $H^1$. Its normalized density source is $s_d=u_d^2/\|u_d^2\|_2$. Direct integration gives

$$\int f^4=5/7,\quad\int f^2h^2=3/8,\quad\int h^4=27/80, \qquad \langle s_d,s_e\rangle=\left(\sqrt{7/12}\right)^{|d-e|}. \tag{16}$$

The amplitude overlap instead uses $\sqrt{5/8}$. Integrating a density over unused coordinates gives marginal weight one, but it does not set a finite-envelope contact to one. These are different operations.

Let $n,v,w$ be real orthonormal retained vectors, with orthogonal field components realizing their kinetic inclusion. A bounded finite-rank map $B$ sends retained coefficients into these actual response profiles, and $B^*$ is its spatial integral adjoint. Take $Bn=0,Bv\ne0$, $M=I/a$, $a,b,k>0$, $\lambda=ab$, and a positive self-adjoint response operator $L$. In normalized reference units define

$$R=(L-\lambda)^{-1},\quad K_{\rm base}=bP_v+kP_w, \quad K_0=K_{\rm base}+B^*RB,$$ $$\mathscr L=\frac{\dot q^TM\dot q+\|\dot y\|^2}{2} -\frac{q^TK_0q+\|L^{1/2}y\|^2}{2} +\epsilon\langle Bq,y\rangle,\qquad \epsilon=\pm1. \tag{17}$$

Vary both coordinates: $M\ddot q+K_0q-\epsilon B^*y=0$ and $\ddot y+Ly-\epsilon Bq=0$. The complete energy is

$$E=\frac{\dot q^TM\dot q+\|\dot y\|^2+q^TSq}{2} +\frac12\|L^{1/2}(y-\epsilon L^{-1}Bq)\|^2, \quad S=K_{\rm base}+B^*(R-L^{-1})B. \tag{18}$$

If $L\ge\ell_0I>\lambda I$, then $R-L^{-1}=\lambda L^{-1}(L-\lambda)^{-1}\ge0$. The full stiffness has exactly the zero $(n,0)$ and is positive on its complement. Every coupled row also admits the responding mode

$$h=\sqrt{ab},\qquad u_h=(v,\epsilon Y),\quad Y=RBv\ne0, \quad Z_h=1/a+\|Y\|^2. \tag{19}$$

The zero follows from a shift symmetry. The chosen nonzero frequency requires the stated inverse term in $K_0$. Static completion alone does not protect it. A dressed zero can instead have $y_0=\epsilon L^{-1}Bn$ with $(K_0-B^*L^{-1}B)n=0$; a pure zero is not the only possibility.

Below-gap response is sufficient, not necessary. For a target in the resolvent of uniformly positive $L$, the actual criterion is positivity of $S$ off the zero. Weak enough nonzero coupling can satisfy it even when response frequencies lie below the target. At a genuine resonance the inverse cannot be asserted; for $L=\lambda I,Bv\ne0$, the response row is impossible.

At off-pole squared frequency $z$, dynamic elimination gives

$$F(z)=K_0-zM-B^*(L-z)^{-1}B,\qquad -F'(z)=M+B^*(L-z)^{-2}B. \tag{20}$$

This is the complete lifted kinetic norm. It remains positive even if a self-energy contribution changes sign. A first-order canonical denominator cannot replace this wave denominator without transforming the full kinetic and source structure. The free zero pair permits drift at nonzero momentum; it is not a frozen zero Hamiltonian or an identified photon.

For an exact example use retained coordinates $(v,w)$, omitting the zero $n$ column, and the orthonormal response basis $(s_2,(12s_4-7s_2)/\sqrt{95})$. In those bases take

$$B=\begin{pmatrix}1&7/12\\0&\sqrt{95}/12\end{pmatrix},\quad L=\begin{pmatrix}9&2\\2&6\end{pmatrix},\quad a=1,\ b=4,\ k=9.$$

The eigenvalues of $L$ are 5 and 10, while $h^2=4$. Equation (19) gives $Y=(1/3,-1/3)$ and $Z_h=11/9$; the full active four-coordinate stiffness is positive. Another independently checked choice uses three times this source and $L=\left(\begin{smallmatrix}25&2\\2&36\end{smallmatrix}\right)$, giving $Y=(24/167,-3/334)$ and $Z_h=113869/111556$. These are different parameter examples, not competing predictions.

A Response with Spatial Gradient Energy

Take

$$L=\Omega_0^2-c_s^2\Delta_X\quad\hbox{on }L^2(\mathbb R^{10}), \qquad \Omega_0^2>h^2,\quad c_s>0. \tag{21}$$

Its operator domain is $H^2$, and its energy form domain is $H^1$. The response is not confined to the finite source span. In a unitary Fourier convention,

$$\widehat Y(p)=\frac{\widehat{Bv}(p)}{\Omega_0^2+c_s^2|p|^2-h^2}, \quad (B^*RB)_{ij}=\int\frac{\overline{\widehat B_i(p)}\widehat B_j(p)} {\Omega_0^2+c_s^2|p|^2-h^2}\,d^{10}p. \tag{22}$$

The exact coefficients are filtered spatial overlaps, not the old unfiltered Gram divided by a constant. Spectral calculus gives

$$\|Y\|\le\frac{\|Bv\|}{\Omega_0^2-h^2},\qquad \|L^{1/2}Y\|^2\le \frac{\Omega_0^2\|Bv\|^2}{(\Omega_0^2-h^2)^2}. \tag{23}$$

Thus the full mode has finite response kinetic, gradient and potential energy. For $a=1,b=4,k=9,\Omega_0=3,c_s=1$ and unit source strength, $1<Z_h\le26/25$. The completed nonnegative self-adjoint stiffness also defines linear energy evolution. Only the response subsystem is local: the retained profiles and target-dependent inverse term remain prescribed nonlocal data. No whole-system causal or autonomous localization theorem follows.

Doubling all real variables into two identical quadratures yields opposite phase charges and $E=h|Q|$ on circular modes, including the response norm. This is not yet an electric-charge law or a selected physical action unit. Likewise a geometric-mean frequency is a possible ingredient for the bottom exception, not its identified mechanism; a zero gap does not establish the photon's spanning reaction, target match, recoil or timing.

The Full Phase-Resolved Source

Let $\xi$ change $a,b,B,L$ at fixed spatial measure and basis, with a common differentiable operator domain. Differentiate the actual inverse law:

$$R'=R(\lambda'I-L')R,\quad K_0'=b'P_v+B'^*RB+B^*RB'+B^*R(\lambda'I-L')RB, \quad\lambda'=a'b+ab'. \tag{24}$$

Evaluate the Hamiltonian derivative at fixed canonical coordinates before substituting the mode. For $q=Av\cos\theta,y=\epsilon AY\cos\theta$,

$$H_\xi=\frac{A^2}{2}\left[\frac{a'b}{a}\sin^2\theta +(b'+\lambda'\|Y\|^2)\cos^2\theta\right], \quad I=\frac{hA^2Z_h}{2},\quad \langle H_\xi\rangle=Ih'. \tag{25}$$

The explicit $B'$ and $L'$ terms cancel only because $K_0$ changes by equation (24). Holding $K_0$ fixed as the source deforms gives a different answer. The mean survives, while the twice-frequency source changes with $\|Y\|^2$. Moving measures, kinetic metrics, bases or protected vectors need their additional variations; arbitrary time-dependent deformations are not stationary solutions of this probe calculation.

A Corrected Clock in the Complete Local Ambient Action

The residual fit in this historical comparison changes the exact multiplier law. It is retained to document this action's mathematics, not as part of the current exact-count mass prescription or an active residual-fitting target.

The local ambient action supplies a different realization with full support, frame, flow, carrier and exterior dynamics. A common finite-chain correction can be inserted into its entire constitutive action, rather than merely attached to a mass table.

Let $C_d(n)$ be the lower bidiagonal matrix with diagonal $n+j$, $j=0,\ldots,d-1$, and subdiagonal $g_0=5000$. Define

$$S(n,d)=\binom{n+d-1}{d},\quad a_d=g_0^{d-1}/d!,\quad t_d(n)=\frac{a_d\,s_{\min}(C_d(n))}{S(n,d)},\quad z_d(n)=\frac{[n+(d-1)/2]^2}{g_0^2},$$ $$f_d(n)=\sqrt{t_d(n)^2+\eta z_d(n)},\qquad \eta=2.74181356771130384967671245187318169249\ldots . \tag{26}$$

The positive $\eta$ is fitted to the archived ratio 206.76828299: with $R=206.76828299/(3515/17)$, $\eta=[R^2t_6(13)^2-t_6(35)^2]/[z_6(35)-R^2z_6(13)]$. It is not a prediction of that ratio. The positive finite matrix $\mu_d^2[a_d^2C_d^TC_d+\eta S^2z_dI]$ supplies the scalar target $\Omega=\mu_dS f_d$. Its other matrix branches are not implicitly modes of the following local action.

For positive $x$, let $n_d(x)$ invert $\mu_dS(n,d)=x$, and set

$$F_d(x)=\mu_dS(n_d(x),d)f_d(n_d(x)),\quad H_c=\sum_d w_d(b)F_d(H),\quad m_c=\sum_d w_d(b)F_d(m). \tag{27}$$

Here $b$ denotes the ambient carrier, not the earlier current constant. The $w_d$ are exactly the ambient action's nonnegative sixth-power depth weights and flat origin switch. Their first two jets vanish at the actual prefixes; the down-origin neighborhood uses the depth-three branch before any undefined moment ratio is evaluated. Replace every material and exterior clock occurrence of $H$ by $H_c$, set $\alpha=H_c/2$, and use $(b_{\rm ent},\kappa)=(m_c/128,8/m_c)$ for branch entropy. The fixed entropy alternative remains $(\omega_*/128,8/\omega_*)$.

The distinction between $H_c$ and $m_c$ away from a reference is essential. For example $\delta H_c=F_x(b,H)\delta H+\sum_d\delta w_dF_d(H)$, with the analogous formula for $m_c$. All entropy, supporting, interface and metric sources must use these derivatives. In a generic metric variation of the saturation term, $X^2=g^{ab}A_aA_b$ obeys $\delta X^2=2g^{ab}A_a\delta A_b+\delta g^{ab}A_aA_b$; the second term cannot be discarded by using a fixed-metric source formula.

At every actual prefix $F_d'>0$, $\operatorname{Hess}H_c=F_d'\operatorname{Hess}H$, and the complete reference quantities become

$$T^{ab}=\chi\sigma\Omega V^aV^b,\quad Q_\theta=sA_0, \quad E_0=A_0\Omega,\quad M_{\rm boost}=E_0/c^2. \tag{28}$$

Both temporal signs belong to the same even-clock action. The full reference time matrix includes the positive material displacement contribution $(\sigma\Omega/2)Z_{\rm map}^TZ_{\rm map}$. Its principal spatial matrix obeys $0<K_{\rm space}\le M_{\rm base}/4\le M_{\rm full}/4$. The exterior change of variables gives the same congruence of both matrices. All bulk reference speeds therefore remain positive and at most one half in the parent's comparison-speed-one units. The nonnegative static form also uses the precise inherited weighted support-drift estimate; it is not a new proof of that older estimate or of nonlinear transmission and evolution.

There is a substantive changed premise. Equal participating energy allocation within each prefix survives, but exact band-independent energy per counted mode does not: $\Omega/D=\mu_df_d(n)$. For example $f_6(13)\simeq1.000003314458$ and $f_6(35)\simeq1.000020614894$. After electron anchoring, the muon shift is about 17.300378 ppm; the fitted ratio is restored by that shared function.

For the following numerical comparison, frequencies are quoted in energy-equivalent units: $\hbar$ times angular frequency is expressed in MeV, with $\hbar=c=1$ in the numerical prescription. Use the pinned archived electron reference $m_ec^2=0.51099895$ MeV and

$$\mu_2=m_e\sqrt{2890},\quad \mu_3=m_e\sqrt{32\sqrt7},\quad \mu_4=\frac{m_e}{15}\sqrt{48/\sqrt7},\quad \mu_6=\mu_{10}=m_e/18564, \quad \Omega_{n,d}=\frac{\mu_d}{f_6(13)}S(n,d)f_d(n). \tag{29}$$

These are supplied numerical scale relations, not newly derived couplings. Rounded examples must not be compared as if they evaluated this prescription exactly. The neutrino scale has two legitimate branches: retain the earlier unit calibrated to the archived solar splitting $7.49\times10^{-5}$ eV², or refit that one depth-five unit after (26). The latter unit is 0.9999958065550636 times the former. Holding the old unit changes the solar splitting by about +8.38694 ppm. Both give the same scale-free corrected splitting ratio, about 0.0303495874; refitting the scale does not repair that ratio. These values use dated stored targets, not newly checked measurements.

What the Constructions Establish Together

The spatial construction shows that a real count operator, nonaffine current, responding anisotropy and a positive joint reference Hessian can coexist in one finite-band action. The response construction shows that a protected zero and a chosen massive frequency can coexist with actual reciprocal spatial response and its full energy. The local ambient construction shows how a corrected scalar frequency can retain complete reference inertia and bulk cone bounds when every affected constitutive term is changed consistently.

These are useful compatibility results at the maintained higher depths, including both six and ten. They do not prefer those depths over all others. The choice of physical layer, observed occupied mode, sector unit, common action magnitude and underlying field content remains to be derived. The finite-band model's material connection and the response model's inverse term remain designed structures. The local ambient model still has its large finite carrier and nonempty weighted exterior, rather than a derived regular fundamental vacuum. No physical photon, bottom assignment or measured pair interaction is supplied by joining the names of these constructions.

Reproducible Evidence

The original spatial pair was checked independently and replayed unchanged with 341 and 59574 assertions. Its combined spatial calculation tests the original action away from reference, all fourteen assignments, both signs, the actual kinetic embedding and a 115-dimensional current/profile/material Hessian. The variable-coefficient calculation has 5770 assertions and the expected 49 zero directions in that finite diagnostic: 45 rotations and four same-shell shape-following directions. The exact all-cutoff statement rests on (8), not on numerical eigenvalue tolerances. Omitting $\eta_d'$ changes the tested force, even though it leaves the pure-shell Hessian unchanged.

The original response pair has 128 and 185 independently reproduced checks. The non-flat response examples and changed-source calculation have another 22-check construction and a separate 42-check comparison. Exact source integrals, full coupled matrices and normalization are tested. The gradient response uses the analytic resolvent and energy bounds (22)–(23), not a claimed ten-dimensional grid simulation.

The corrected-clock pair includes 84 reference clock/entropy/sign cases, 474 independent checks, and 123 comparison checks resolving the calibration conventions and full metric derivative. Rational Collatz bounds enclose the small-matrix spectral values and fitted squared ratio. Positive derivative, stress and full principal-form conclusions also require the analytic proofs. Check counts describe coverage, not experimental evidence.

The spatial checker, response checker and clock checker are exact copies of their reviewed executable sources. Their adjacent results.json files retain the executed results. The provenance record identifies the original proofs, independent comparisons and code hashes. Exact proofs, source versions and dispositions remain in the research catalog. The common conditional constructions are an advance in consistency and calculability; their remaining physical inputs define the next mass-operator derivation problem.