A Moving Finite-Speed Count Action With Positive Inertia

The finite-speed count parent can be embedded in one explicit moving-material action. The same action retains its fourteen supplied frequency assignments, accounts for internal energy transport in the complete momentum, and has a positive joint wave/material kinetic response. Weighting the elastic energy by the wave's own local energy removes a separately imposed carrier mass and an uncontrolled stiffness-to-amplitude ratio.

This is an inverse construction. Its material geometry, component carrier, sector units and radius relation remain constitutive inputs. The frequency approximation is the existing finite-chain result, not a new fit or a forward selection of particle masses. The physical containing space is ordinary macroscopic space. The calculation below describes an active d-dimensional body for d=2,3,4,5,6,10; embedding its finite energy in all containing directions requires a further construction. Field components do not replace deterministic electron trajectories.

One Covariant First-Gradient Action

Let a be a physical material coordinate in a ball of radius R_d, y=a/R_d, w(a)=sqrt(1-|y|²), X(a,t) the physical material map and U(a,t) the retained real field components, including two real quadratures for complex notation. Set F=D_a X, v=X_t and gamma=(1-|v|²/c²)^(-1/2). The rest material derivative and rest strain are

D_A U=partial_A U+gamma²(F^T v)_A U_t/c²,
Cmat=F^T F+gamma² F^T v v^T F/c².

These formulas follow by taking spacetime material labels phi^A(x,t), defining B^{AB}=partial_mu phi^A partial^mu phi^B, and contracting the wave derivative with the rest-spatial dual E_A^mu=partial^mu phi^B(B^-1)_(BA). The material four-velocity is orthogonal to all label gradients. Thus D_A=E_A^mu partial_mu, Cmat=B^-1 and the invariant material measure is w da dt/gamma. Gradients must remain independent and spacelike; the material map must remain invertible.

Let H=k_d H_(M,b), k_d=mu_d M^(d-1)/d!, be the complete all-grade parent in y=a/R_d, with b_0=1 and b_k=b for k>=1. Its physical first-order expression H=sum_A C_A(a)partial_A+M0(a) is in its orthonormal component chart. Its physical derivative coefficients contain R_d, giving s=b k_d R_d. M, b, kappa and r are dimensionless; H, omega, k_d and mu_d have frequency units. Retain all of its weighted lower-order terms. Define H_D by replacing partial_A with D_A and put

W(C)=||C-I||_F²/4+[tr(C-I)]²/8,
S=int da dt (w/gamma){gamma²|U_t|²/2
                 -[1+2 kappa W(Cmat)]|H_D U|²/2}.       (1)

The dimensionless kappa is common across depths. All fields in (1) are retained parts of the proposed common field law, not exchanged particles. Equation (1) contains only first time derivatives. Covariantizing a high-order scalar count polynomial would not have this property; the retained first-order parent is essential to the present proof.

At X=a, H psi=omega psi with psi real gives the exact balanced solution

U=alpha psi exp(-i omega t),   X=a.

Here W(I)=DW(I)=0, the first material variation vanishes, and the internal flux is zero. With q=U_t, z=HU and h=(|q|²+|z|²)/2, this branch has |q|²=|z|²=h. Its rest energy and conserved phase action obey E0=Q omega. With p=L_(U_t), including the measure weight, the sign convention is Q=-int p.JU for U proportional to exp(-i omega t). Q is not identified with electric charge or fixed to an empirical action unit. No amplitude has been chosen to fit a mass.

Complete Kinetic Response and Local Momentum

At unstrained rest, the exact second velocity variation of (1), divided by w, is

delta²L/w=|delta q-C(vdelta)^T z/c²|²
            +h|vdelta|²/c²
            -[|C(vdelta)q|²+|C(vdelta)^T z|²]/c^4.

If ||C(b)||<=s|b| in physical units, the remaining Schur form is at least h(1-2s²/c²)|vdelta|²/c². It is strictly positive wherever h>0 and s<c/sqrt(2). This controls arbitrary spatially varying material velocities, not just rigid translation. Overall physical amplitude cancels from the sign condition.

For the complete material density L(F,v,U,q,G;a), including w, define

p=L_q, P_i=L_(v_i), R_A=L_(G_A), S_iA=L_(F_iA),
e=q.p+v.P-L, K_A=q.R_A+v_i S_iA.

The full Euler equations imply

partial_t P_i+partial_A S_iA=0,
partial_t e+partial_A K_A=0.

With J_X=det F>0, the physical energy density, momentum and energy flux are respectively e/J_X, P/J_X and (e v+F K)/J_X. Covariance gives the exact identity

e v+F K=c² P,

and a symmetric physical momentum flux. At unstrained rest these reduce to

e=w h, J_i=-w z.C_i q, P_i=J_i/c², S_iA=0.

Internal wave energy transport therefore contributes to complete momentum. A mixed wave with changing flux cannot consistently keep its material momentum fixed. These are local conservation identities for solutions of the coupled equations, without a time average or a common clock imposed on distinct patterns. The zero total rest stress is a consequence of this particular isospectral material law, not a universal field postulate.

Positive Translational Inertia for the Free Symmetry Family

A uniform boost of the stationary solution contracts the material map and adds its simultaneity phase. At infinitesimal velocity V,

xi_V(a,t)=Vt,
eta_V(a)=omega J psi(a)(V.a)/c².

The phase structure J is multiplication by i in complex notation. The wave phase cannot be omitted while claiming the boosted solution. A direct check uses [H,f]=C(grad f) with f=V.a/c². Substitution into the complete quadratic wave and material equations gives

delta P(a)=w(a)h(a)V/c²,
delta P_total=E0 V/c².

Thus the complete translational inertia is positive E0/c², without an added carrier mass. The canonical multiplier-domain result and the actual polynomial profiles justify this tangent. The boosted family uses natural free boundary conditions; a clamp does not permit the same translation symmetry.

Ballistic center displacement is consequently an exact generalized zero mode of the free linear system. A norm measuring absolute position cannot stay uniformly bounded for a moving particle. Free stability must compare shape relative to its moving symmetry family and retain conserved momentum. A thin d-dimensional sheet in R10 is not automatically a complete ambient body: strain depending only on F^T F leaves arbitrary normal deformations without quadratic bending stiffness. This is a neutral quadratic kernel, not evidence of instability or decay; generic finite normal graphs have positive quartic strain energy.

The Supplied Profile Assignments

The spatial polynomial degree is N=n-1 relative to the maintained mass-law index n. These assignments are inputs to this action test.

Nodeless Profiles Without a Separate Physical Amplitude Fit

The full scalar/exact-one-form pair removes zeros of a scalar zonal mode. For u(t)=C_N^a(t), a=(d+1)/2, t=y1 and lambda=N(N+d+1), define

E_N(t)=u(t)²+(1-t²)u'(t)²/lambda.

The Gegenbauer equation gives E_N'=2(d+1)t u'²/lambda. Parity then implies 0<E_N(0)<=E_N(t)<=E_N(1)=binomial(N+d,d)². The constant mode uses E_0=1 and its scalar profile directly. The exact central value is

E_(2m)(0)=[(a)_m/m!]²,
E_(2m+1)(0)=[2a(a+1)_m/m!]²/lambda.

A real eigenprofile of the signed chain can balance its scalar and gradient weights, giving pointwise |psi|² proportional to E_N/2. This is one profile, not an extra duplicated spatial field. The balancing follows from choosing a unit vector with expectation 1/2 in a real two-by-two positive matrix of trace 1 formed from the exact chain rotations. It is a preparation choice within the degenerate eigenspace, not a dynamical occupation rule.

After spatial normalization on the unit y-ball, let W_d=int_B w. All fourteen profiles obey

W_d min|psi|² >= 2103280464675/8796093022208.

Physical da normalization instead uses volume R_d^d W_d and density multiplied by R_d^-d; the volume-density product and h_min/h_max stay unchanged. This lower bound concerns density relative to w. The weight and the form metric still degenerate at the ball boundary. On a fixed nonzero-amplitude profile h is nevertheless smooth, positive and bounded on the closed ball.

A Common Positive Causal Principal Regime

At unstrained rest normalize material displacement by chi=sqrt(h)xi/c and write r=s/c. The full directional acoustic pencil, including mixed wave and material derivatives, is positive on every covector at or outside speed c if

Acone=[[1-r²,-r(1+r)],[-r(1+r),1-r²-3 kappa]]>0.

A sufficient common range is r<1/2 and kappa<(1+r)(1-2r)/[3(1-r)]. For example kappa=1/10 and r<=1/3 satisfy this strictly. Positive kinetic and directional spatial forms supply real semisimple characteristic roots. This is a local principal statement; lower-order spectral stability requires the separate calculation below.

The audited finite-strain extension admits an open regular neighborhood: ||Cmat-I||_op<1/10, abs(|q|²-|z|²)/(|q|²+|z|²)<1/100 and normalized unweighted flux |J0|/(c h0)<1/200, where J0_i=-z.Cphys_i q and h0=(|q|²+|z|²)/2, with the same kappa and reference symbol bound s0<=c/3. The strained physical symbol Cphys(b)=Cref(F^T b) can reach sqrt(1.1)s0; that change is included in the bound. These are regular interior statements, not uniform boundary coercivity. It retains all mixed strain/flux terms. This does not prove that nonlinear evolution stays in that neighborhood or reaches the vacuum boundary regularly.

Clamped Linear Evolution and the Quantitative Spectral Test

The clamped control has the canonical full-form wave domain Dom(H), and material domain obtained by completing compact-interior displacements in its actual weighted strain norm. These closures are different and must not be interchanged. In dimensionless material coordinates put

chi=sqrt(h)xi/c, B=-J A, ||A||<=1,
r A chi=C(c chi/sqrt(h))z0/c², z0=omega psi,
E[chi]=kappa(b k_d)² int w h[2|eps(chi/sqrt(h))|²
                                  +div(chi/sqrt(h))²].

Here k_d=mu_d M^(d-1)/d!, r=b k_d R_d/c, and the full normalized matrices are

K_r=[[I,r A],[r A*,I-r²B*B]],
G_r=[[-2omega J,r(H+omega)B],[-rB*(H+omega),0]],
V_r=diag(H²-omega²,r^-2 E).

The mixed H terms are weak forms; B need not map the material domain into Dom(H). K_r is bounded positive for r<1/sqrt(2). The exact weighted elastic identity with f=sqrt(1-|y|²) gives

int w[2|eps(v)|²+(div v)²] >= (d+2)int w|v|²/(1-|y|²).

Density comparison proves E>=eI with

e=kappa(b k_d)²(d+2)h_min/h_max>0.

The stronger boundary-tail estimate supplies compact material embedding; interior Korn and Rellich handle the remaining compact region. The wave embedding is compact by the complete polynomial-block spectrum. The gyro is compact from the configuration form domain to its dual.

The actual phase generator is

L_r(q,p)=(p,-K_r^-1(V_r q+G_r p)),
Dom(L_r)={(q,p) in V x V: V_r q+G_r p belongs to X}.

A positive stiffness shift gives coercive resolvents at both signs of real lambda and a skew-adjoint shifted generator. Removing that bounded shift gives a global strongly continuous evolution on V x X, preserving the rotating energy. Compactness of the mixed forms gives compact phase resolvent. Separate strong membership of V_r q and G_r p is not imposed.

For the nonzero spectral calculation define

D(lambda)=lambda-J(H+omega), F(lambda)=lambda-J(omega-H),
T_r(lambda)=r^-2E+lambda²(I-r²B*B).

The exact wave Schur expression is

D(lambda)[F(lambda)-r²lambda² A T_r(lambda)^-1 A*D(lambda)].

The branch D(partial_t)eta=0, chi=0 is protected exactly. The other branch has a bounded-frequency correction of order r^4. Neither statement depends on making the physical amplitude small: it cancels in this normalized action.

On a contour with |lambda|<=L and bare spectral distance at least delta, right composition by F^-1 and the identity D F^-1=2(lambda-J omega)F^-1-I give the full-operator sufficient bound

q(r)=r^4 L²/(e-r²L²) [1+2(L+omega)/delta] <1.       (2)

All high frequencies enter the resolvent norm; (2) is not a finite spatial truncation. The material denominator must remain positive. The underlying exact Sturm certificate includes every full-form family below the required window and excludes all remaining spatial and heavy chain bands analytically. It finds no nonzero opposite-energy collision across the fourteen labels.

For M=100000, b=40, kappa=1/10 and the single common r=1/100, exact rational arithmetic verifies (2) on 456 positive-frequency negative-energy contours and their conjugates. The maximum is 0.235340209758762, strictly below 1/4. The ground assignment has no negative-energy contour. The circles are not counts of negative dimensions: every coincident angular, form and signed eigenspace is retained at its full multiplicity. A positive spatial family of dimension 2h has signed-chain multiplicity 2h; the constant has multiplicity 1, and temporal quadratures are retained. Contour arithmetic divides frequencies by mu_d and e by mu_d². These numbers use h_min/h_max from the nodeless profiles and the conservative Korn bound above. The very small former W collision is included, not rounded to zero.

The signed symmetry of H makes the spectra of D and F identical: both contain all values plus/minus i(omega plus/minus nu). The same distance certificate therefore excludes protected D roots. The weak factorization uses D:X_w to Dom(H)* and F:Dom(H) to X_w; its bounded middle inverse reconstructs the combined generator domain. For fixed contours q(r) increases with positive r while its denominator stays positive. The certificate consequently covers every 0<r<=1/100, not just one endpoint.

The spectral continuation argument is as follows. At small r, rescaling the material displacement by r gives phase resolvents converging on these fixed contours to the bare wave resolvent with zero material block. Their contour integrals are norm-close finite-rank projections, preserving full algebraic multiplicity. The pulled-back energy forms converge too, so their complete negative-energy subspaces remain negative definite. For positive r, a spectral contour invariant under lambda -> -conjugate(lambda) has an energy-self-adjoint Riesz projection. Its restricted energy cannot be degenerate: a radical vector would be in the full stationary kernel, which the contour excludes. Its finite energy signature therefore cannot change while (2) keeps the contour free of spectrum. This continues negativity to the certified endpoint.

The phase-space comparison is explicit. On the fixed form phase space use U_r(eta,z,v,u)=((eta,r z),(v,u)); material velocity u is not rescaled. The pulled-back energy converges in bounded-form norm to bare wave energy plus e(z,z)+||u||². For forcing (f,f_m,g,g_m), the weak Schur equations give, uniformly on each fixed contour, eta-eta0=O(r) in Dom(H), chi=O(r²) in Dom(E^(1/2)), chi/r=O(r), and both velocity differences O(r) in their L2 spaces. Here eta0 solves the bare wave resolvent. Thus

||(lambda-U_r^-1 L_r U_r)^-1
     -i_w(lambda-L_w)^-1 pi_w|| <= C_Gamma r.

The limiting expression vanishes on material inputs; it is not a resolvent on the entire comparison space. Its contour integral is nevertheless the bare finite-rank wave projection. Projections at norm distance less than 1 have equal rank by mutual injectivity on their ranges. The simultaneous energy-form convergence makes the full nearby Riesz range negative definite.

For continuation, represent the energy by the bounded Hermitian operator Jenergy on a fixed positive phase norm. Its conservation gives

[(lambda-L_r)^-1]* Jenergy
      =-Jenergy(-conjugate(lambda)-L_r)^-1.

Integrating on the reflected contours yields Q_r* Jenergy=Jenergy Q_r. The Riesz range is energy-orthogonal to its complete projection complement, without assuming an eigenbasis. Any vector in the restricted radical would then lie in the full radical; the contour excludes its zero eigenvalue. Continuous finite-dimensional nonsingular energy Gram matrices have constant inertia. This proves the continuation step for the entire certified interval.

These finite subspaces exhaust the negative index of V_r. On the stationary kernel, the compressed gyro is exactly -2omega J and invertible. Its explicit stationary projection removes that finite radical. Relative to the natural positive form norm the energy is identity minus a finite-rank operator; the infinite complementary subspace is consequently coercive. Reversing the sign on the finite negative subspace and adding the squared stationary projection produces a conserved positive norm equivalent to the complete phase norm. Hence all linear disturbances remain bounded for all time in this clamped constitutive regime. No eigenbasis completeness assumption or finite Galerkin calculation replaces these steps.

Writing N=ker V_r and using its finite-dimensional orthogonal projection, the stationary projection is exactly

G_N=P_N G_r|N=-2omega J,
P0(q,p)=(G_N^-1 P_N(G_r q+K_r p),0).

The projection on dual forms means restriction to N. Differentiation gives P0 T(t)=T(t)P0=P0. If P_W is the energy-orthogonal projection onto the full negative invariant range, an explicit conserved positive norm is

||z||_sharp²=Henergy(z,z)-2Henergy(P_W z,P_W z)+||P0 z||_*².

Its equivalence to the reference form norm follows because the remaining energy operator is identity minus finite rank, positive off its finite stationary radical. The stationary projection gives a bounded complement.

The norm bound is for each fixed profile and parameter set. Its numerical amplification constant has not been calibrated, and it does not control the pointwise material Jacobian needed for nonlinear persistence. The free body's rigid momentum reduction changes the wave correction at order r²; its separate complete proof is linked below. Neither result alone determines observed lifetimes.

Natural Free Boundary Extension

The free momentum reduction and complete linear proof now certify a common interval 0<r<=1/100000 for all fourteen backgrounds. The stationary quotient is uniformly bounded; nonconstant zonal preparations retain exactly d-1 neutral orientation chains. This uses the actual free elastic resolvent and its rigid kernel, with an explicit weighted Korn constant. It is a separate proof from the clamped r<=1/100 result above. Center reconstruction preserves the complete boost inertia. Nonlinear orientation evolution and the full ambient particle remain unresolved.

Physical Scope

Equation (1) connects retained count frequencies, local momentum balance, positive inertia, causal principal propagation and a controlled clamped linear regime in one common construction. The finite chain still has extra modes and an explicit radius cost R_d=c/[100 b k_d] in the certified example. Its small radius is a consequence of this parent, not a measured size or a universal restriction on alternative constructions. The sector frequency units, support, component origin and observed occupation remain inputs.

A reciprocal pair interaction can preserve these isolated clocks, but its full domain and stability must be checked after adding it. A local causal Hessian bound alone is not that combined evolution theorem. The physical mass-frequency conversion is already known; this work addresses a candidate spatial mechanism and its consistency, not the conversion itself.

Reproducibility

The standalone exact checker reconstructs the full-form spectral windows, isolates 319 lowest squared roots with integer Sturm recurrences, excludes omitted spatial and heavy bands, and verifies all 456 positive-frequency contours and their conjugates. Its complete result retains exact rational intervals, multiplicities, elastic bounds and every contour margin. All decisions use exact fractions; displayed decimals are summaries. The accompanying domain and energy arguments establish the infinite-dimensional implications; arithmetic alone does not establish them.

The group-generation step uses skew-adjoint generation and bounded perturbation as stated in Engel and Nagel, One-Parameter Semigroups for Linear Evolution Equations, Chapters II–III. The actual weighted forms, closures, compactness and energy projections are specified above. Finite propagation for the fixed first-order parent is proved with its full domain in the linked count-chain article.