Free Count Patterns: Momentum Reduction and Bounded Shape Evolution

The moving count action admits a natural free material boundary. Removing a clamp changes the conserved momentum reduction and the wave response; its stability requires this separate calculation. For the same fourteen balanced real preparations, one common sufficient interval is

0 < r=s/c <= 1/100000,
M=100000, b=40, kappa=1/10.

Throughout this interval, the full linear dynamics at fixed zero material rigid charges is uniformly bounded after quotienting its stationary kernel. The quotient removes the stationary kernel of the linearized equations. Every nonconstant zonal preparation has exactly d-1 length-two neutral orientation chains; the full reduced evolution can grow linearly along those directions. The constant down preparation has no such orientation and is bounded without this additional quotient. Center motion is reconstructed from the conserved momentum, with complete boost inertia E0/c^2 as derived in the linked action.

This is a conditional linear theorem for the supplied active weighted ball and full classical component carrier. It retains the existing approximate count clocks, sector units and radius law R_d=r c/(b k_d). It does not select the observed branches, describe a complete lower-depth body in every ambient direction, or establish nonlinear persistence or observed lifetimes. Internal pattern orientation is distinct from the electron's deterministic six-dimensional position and orbital trajectory.

Physical coordinates are a=R_d y and the rest map is X=a. The material kinetic operator denoted M_r or M in the derivation is distinct from the chain ratio M=100000. Numbered section references below refer to the part in which they occur.

The first part establishes the actual free domain, stationary kernel and small-speed anchor. The second proves an explicit common interval using the whole free material resolvent. In particular, its endpoint proof does not assume that the auxiliary finite response matrix W_r stays positive.

The Free Action and Its Complete Stationary Shell

Action, Spaces and Reduction

Use the real wave space X_w with the actual full-form metric B_k=Lambda^k(I-yy^T), measure w dy with w=sqrt(1-|y|²), and polynomial-closure self-adjoint H from the complete finite-speed chain. V_w=Dom(H). J is the real temporal phase structure, J²=-1, J*=-J and [J,H]=0. The background is a real polynomial eigenprofile H psi=omega psi with omega>0, duplicated only into balanced temporal quadratures. Its density h=omega²|psi|² is smooth and bounded above and below by positive constants.

The material free form is the closure of smooth closed-ball fields in its weighted L2-plus-strain norm. It contains exactly the rigid kernel R_d={a+Omega y: Omega^T=-Omega}. The free Korn lift below proves compact form embedding and a positive gap on its weighted orthogonal complement. This is neither a clamped trace domain nor a thin sheet. Let E denote the normalized free material operator, T an orthonormal inclusion of its rigid kernel, Z=ker T*, and E_Z the restriction to Z. Then E_Z>=e>0. Write chi=T a+z, z in Z.

As in the independently audited action, put B=-JA, M=1-r²B*B, ||A||<=1 and

D_t=partial_t-J(H+omega), I=TM T, j=r TAD_t eta+TM z_t, L=TAJ(H+omega).

The rigid momentum is p=I a_t+j. At fixed p=0, the exact Routh action is L_red=L_0-j^T I^-1 j/2 and reconstruction is a_t=-I^-1j. Here L_0 means the full action with a_t=0; it still contains the nonrigid wave/material couplings. I>=1-r² and the reduced kinetic matrix is the positive Schur complement of the full matrix for r<1/sqrt(2).

The reduced static form is

V_red=diag(S+r² L*I^-1 L, r^-2 E_Z), S=H²-omega². (1)

All formulas first hold as graph forms. For these polynomial backgrounds, each A T e_j is a polynomial full-form field: the h factors cancel, leaving the principal symbol applied to psi times an affine rigid vector. Hence it belongs to Dom(H), and each row of L extends to a bounded wave-L2 functional by moving H across the inner product. Finite rank alone would not justify that assertion.

The free configuration space is V_w direct_sum Dom(E_Z^(1/2)). Its compact embedding, bounded positive reduced kinetic matrix, closed lower-bounded stiffness and compact gyro map V to V* yield the same maximal combined-domain construction as the moving-action domain proof, now with this free space and the exact Routh terms. Explicitly the generator domain is

{(q,v) in V x V: V_red q+G_red v belongs to X}, (q,v) -> (v,-K_red^-1(V_red q+G_red v)). (2)

Adding a sufficiently large positive L2 stiffness makes the two real resolvents coercive; their onto property proves skew-adjoint generation in the shifted positive energy norm. Removing the shift is a bounded phase-space perturbation. Compactness of the gyro map follows from its H-graph/L2 factorization; Routh adds only bounded finite-rank maps and their form adjoints. The resolvent is compact by the compact V-to-X and X-to-V* embeddings. Thus (2) generates a C0 group and preserves the actual rotating quadratic energy. This verifies the free analogue of the domain hypotheses; it does not require V_red q and G_red v separately to belong to X.

2. An exact static kernel criterion despite excited negative directions

Let K=ker S, Q=K-perp, and S_Q=S|Q. Compact resolvent gives a bounded inverse S_Q^-1. Write L_K=L|K and L_Q=L|Q. For sufficiently small r the finite matrix

W_r=I+r² L_Q S_Q^-1 L_Q* (3)

is positive definite. A sufficient strict bound is r²||L_Q S_Q^-1 L_Q*||<lambda_min(I). The inverse on Q in the static block is also defined for sufficiently small r.

Eliminating Q from S+r²L*I^-1L gives exactly the K-Schur complement

F_K=r² L_K* W_r^-1 L_K. (4)

This follows either by multiplying the two block equations or by the finite-rank inverse identity. In particular it remains valid when S_Q is indefinite. Since W_r>0,

ker V_red=(K intersect ker L) direct_sum {0}, n_-(V_red)=n_-(S), rank(F_K)=rank(L_K). (5)

For a vector in ker L_K the eliminated Q component is exactly zero. The Q block keeps its inertia by small bounded perturbation; the K block (4) is nonnegative, proving the index assertion. Thus there are no additional cancellation zeros in this sufficiently small-r regime. Formula (5) is a reusable criterion, not an assumption that a negative and a positive form cannot cancel.

3. Translation and rotation rows on the complete zero shell

Write K=K_+ direct_sum K_-, with K_+=ker(H-omega), K_-=ker(H+omega), including both temporal quadratures. Every L row vanishes on K_- because its last factor is H+omega. This retains all protected negative-signed stationary modes.

For a translation v, the principal symbol is a commutator C(v)=[H,v dot x], with the appropriate fixed physical scaling. Since H psi=omega psi,

P_+ C(v)psi=P_+(H-omega)((v dot x)psi)=0. (6)

Multiplication by x preserves the domain, and for this profile every expression is polynomial. Consequently every translational row of L vanishes on all K. A center clamp is not used.

For rotations v=Omega y, one must actually compute the shell compression. Let lambda=N(N+d+1)>0. The selected scalar/exact-one-form shell has scalar basis u_alpha and gradient partners du_alpha/sqrt(lambda), with equal integrated norms. Keep its entire scalar multiplicity

m_N=binomial(N+d-1,d-1).

Define R_Omega u=(Omega y) dot grad u. Radial w and tangential v imply R_Omega*=-R_Omega on this finite scalar shell. Its action on one-forms is the induced rotation of forms, so it commutes with d. For the principal symbol epsilon_v-i_(A v), A v=v. Direct pairing gives

<u_alpha,-i_v du_beta/sqrt(lambda)> =-<u_alpha,R_Omega u_beta>/sqrt(lambda), <du_alpha/sqrt(lambda),epsilon_v u_beta> =-<u_alpha,R_Omega u_beta>/sqrt(lambda). (7)

The second equality uses the skew angular derivative. Contributions landing in two-form/coexact sectors are orthogonal to this scalar/exact shell and are not erased from H; they merely do not contribute to this compression.

On this shell the full chain matrix H(lambda) has weak bonds sqrt(lambda) sigma_x+sqrt(a_j) sigma_z, times k_d, and strong bonds M. Equation (7), slot by slot, therefore says

P_shell C(Omega y) P_shell=-2 R_Omega H'(lambda). (8)

The finite path-gauge construction gives each signed band an exact two-dimensional coefficient degeneracy for every scalar u_alpha. If f(lambda)>0 denotes its selected low frequency, differentiated spectral projection gives

P_+ H'(lambda)P_+=f'(lambda)P_+.

Here this is differentiation of the finite matrix family, not differentiation of a discrete spatial eigenvalue. The lowest singular value has f'(lambda)>0: all nonzero entries of the sign-adjusted inverse triangular matrix strictly decrease with lambda, its Gram matrix is strictly positive, and its positive maximizing vectors give a strictly negative derivative of its largest singular value. Its reciprocal has strictly positive derivative.

Thus, on the complete selected eigenspace,

P_+ C(Omega y)psi=-2 f'(lambda) R_Omega psi. (9)

The parameter factors converting C to A T are nonzero common normalization factors and an invertible choice of rigid basis. They do not change the span or rank. No other form band or heavy band coincides with the selected frequency: the actual all-label extra-band separation and heavy-band certificate supply that fact, rather than a scalar truncation.

Because psi is real and J is the separate temporal structure, equations (6)–(9) show that the Riesz vectors of L_K span precisely J O, where

O=span_R{R_Omega psi: Omega skew} subset K_+.

For a nonconstant zonal Gegenbauer profile, rotations fixing its axis act trivially, while the d-1 rotations tilting that axis give independent derivatives. The full chain map from the scalar shell is injective. Hence dim O=d-1 for N>0. For N=0 the scalar profile is constant and L_K=0.

This proof keeps arbitrary angular perturbations in the entire shell. It computes only the rank of a finite collection of charge rows, not a truncation of admissible perturbations.

4. Exact kernel, phase and rotation structure

For N>0, decompose the real positive-signed eigenspace into O and its orthogonal complement, and append the temporal J copy. Equations (5) and (9) give

K_0=ker V_red=K_- direct_sum [(K_+) intersect (J O)^perp], codim_K K_0=d-1. (10)

Here K_+ already denotes both temporal quadratures; O is in its real quadrature. All real positive-signed shell tangents survive. The imaginary orientation directions J O are lifted by the positive Routh stiffness. Both the phase tangent J psi and amplitude tangent psi survive, because <psi,R_Omega psi>=0. The complete negative-signed stationary space survives too.

For N>0, each signed H frequency has real spatial multiplicity 2m_N; including J quadratures gives dim_R K=8m_N. Therefore dim_R K_0=8m_N-(d-1). The constant branch instead has dim_R K=dim_R K_0=4. These dimensions retain the full signed and angular multiplicities.

On K_0 every configuration row L vanishes. Therefore the Routh correction to the gyro compression vanishes on pairs of stationary vectors, and exactly

G_00=-2omega P_(K_0) J|_(K_0). (11)

Its radical is O for N>0, and zero for N=0. Indeed J maps O into the removed J O, while on the remaining paired real/imaginary directions it remains nondegenerate. The negative-signed shell is J invariant.

The standard zero-chain equation for (2) is V_red q_1=-G_red q_0. It is solvable precisely when G_00 q_0=0, since the self-adjoint stiffness has closed range and finite kernel. Consequently each nonzero q_0 in O has a generalized zero vector (q_1,q_0), and zero is not semisimple for every nonconstant zonal preparation. This conclusion is exact throughout the sufficiently small-r static regime, rather than a leading-order guess. For N=0, (11) is invertible and zero is semisimple.

The d-1 dimensions across the maintained nonconstant depths are 1,2,3,4,5,9; the down N=0 state is the constant exception. There is no special depth-five effect.

These reduced drifts are orientation tangents of the balanced family. They should not be identified automatically with arbitrary rigidly rotating exact nonlinear bodies: the counter-response, finite deformation and centrifugal terms are absent from this linear statement. They are also not decay rates. This identifies exactly why the clamped uniform-bound proof cannot be imported unchanged.

5. Negative-index exhaustion survives at small r

The free Routh action changes bounded-frequency wave pencils at O(r²). Elimination of the positive nonrigid block still contributes O(r⁴), with its positive free quotient gap e. The finite-rank Routh coefficients are bounded, and the remaining H terms are controlled on V_w. Thus the reduced analytic pencils converge on fixed contours about each nonzero bare wave cluster. The actual finite-M full-form resonance certificate excludes every opposite-energy nonzero collision at all fourteen labels, including the extra d=2 low family.

Track the complete finitely many negative-energy clusters, including their conjugates and full multiplicities. Their Riesz spaces converge, their energy Gram matrices remain negative definite, and their total real dimension equals n_-(S). Equation (5) says this exhausts the actual reduced energy's negative index. Denote this invariant negative-definite phase subspace by W. Since the generator is skew for its definite negative energy, its spectrum there is imaginary and semisimple.

On the energy-orthogonal invariant complement U=W^[perp], the conserved energy is nonnegative. Its radical is exactly K_0 embedded as (q,0). To justify coercivity on the quotient, use the reference norm made from |V_red|+P_(K_0) on positions and K_red on velocities. The energy represents identity minus a finite-rank negative/zero correction. Removing W leaves an identity-minus-finite-rank positive operator with radical K_0. Therefore it has a strictly positive lower bound on the Hilbert quotient U/K_0. The induced C0 group preserves an equivalent positive norm and is uniformly bounded there.

This proves that the full reduced generator has only imaginary spectrum, including zero: a nonzero nonimaginary eigenvector in U would have zero conserved energy and hence lie in the stationary radical, contradicting its eigenvalue. Compact resolvent excludes an unexamined finite continuous spectrum. This closes the high-frequency issue without extrapolating the small-r Schur series to material frequencies of order 1/r.

The unquotiented group has at most linear growth. One way to see this without assuming a bounded off-diagonal generator block is to choose any bounded complement of the finite stationary kernel. In that decomposition the C0 group is an upper triangular extension of its uniformly bounded quotient group by the identity on K_0. The cocycle identity and its bounded norm on 0<=t<=1 give a linear bound by telescoping unit time steps. It follows also that zero Jordan chains have length at most two. Section 4 supplies d-1 independent length-two chains for N>0; no longer chains or exponential modes are hidden there. On N=0 the invertible compression (11) supplies a bounded stationary projection and the entire reduced group is uniformly bounded.

Taking the minimum of the finitely many positive static, contour and kinetic thresholds gives one common r_*>0 for the fourteen preparations. This part supplies the small-speed anchor; the quantitative continuation below supplies an explicit threshold. The very small W detuning remains a quantitative cost; neither the material gap nor r=1/3 certifies its contour bounds.

6. Reconstruction and limits

At fixed p=0 reconstruct a_t=-I^-1j. At other fixed linear momenta the equation is a_t=I^-1(p-j). A bounded reduced oscillation can have a nonzero mean reconstructed velocity, yielding linear center or orientation motion. The independently checked exact boost tangent includes its simultaneity phase and gives complete positive translational inertia E0/c². That motion must not be labelled loss of shape stability simply because absolute position grows.

The additional stationary quotient in section 5 is necessary for the present excited zonal family even after the material rigid momenta have been fixed. It removes neutral wave-shell directions, not physical field components or angular multiplicity. Calling it nonlinear orbital stability would require identifying and controlling the actual nearby nonlinear family, its orientation evolution and support. The theorem is linear spectral/quotient stability only.

No observed particle lifetime, selected occupancy, preferred physical radius, full finite-energy ambient support, or preservation of the nonlinear material map is supplied. Finite persistence remains a separate property, especially for unstable observed branches. Classical field modes and their phase coefficients do not replace deterministic electron positions.

An Explicit Common Free Interval

Elementary Explicit Free Korn Constant

Use normalized uniform measure mu on the ordinary unit ball B^n. Let v be smooth on its closure and subtract its L2(mu) orthogonal projection onto {a+Omega x: Omega skew}. The remainder, still called v, satisfies

integral v dmu=0,
M_ij=integral v_i x_j dmu is symmetric.

Indeed translation orthogonality gives the first statement, while rotation orthogonality annihilates the antisymmetric part of M. Isotropy gives integral x_i x_j dmu=delta_ij/(n+2). Define

D(v)=integral integral [(v(x)-v(y)) dot (x-y)]² dmu(x)dmu(y).

Expanding the four scalar terms v(x) dot x, -v(x) dot y, -v(y) dot x, v(y) dot y yields exactly

D(v)=2 integral (v dot x)² dmu
     +2/(n+2) integral |v|² dmu
     +2(tr M)²+2tr(M²).

All other cross terms vanish by integral v=0 and integral x=0. Since M is symmetric, tr(M²)>=0. Thus D(v)>=2||v||²/(n+2). Subtracting any rigid field leaves D and eps(v) unchanged, so using the projection does not alter the upper estimate.

Convexity keeps z_t=(1-t)y+t x in B^n. The fundamental theorem of calculus and cancellation of skew strain give

(v(x)-v(y)) dot (x-y)
   =integral_0^1 (x-y)^T eps(v)(z_t)(x-y) dt.

Cauchy-Schwarz in t and the Frobenius matrix bound, with |x-y|<=2, imply

D(v)<=16 integral_0^1 integral integral |eps(v)(z_t)|² dmu(x)dmu(y) dt.

The x/y symmetry changes the t integral to twice the interval [0,1/2]. At fixed x, y -> z_t has Jacobian (1-t)^n. Its image is contained in the ball, so replacing that image by the whole ball increases the nonnegative integral. The normalized y integral is bounded by (1-t)^(-n) times the normalized z integral; integrating x then leaves the same bound. Hence, for n>1,

D(v)<=32 (2^(n-1)-1)/(n-1) integral |eps(v)|² dmu,

inf_rigid ||v-rigid||²_L2(mu)
   <= C_n integral |eps(v)|² dmu,
C_n=16(n+2)(2^(n-1)-1)/(n-1).

This is a conservative direct proof, not a quoted unspecified Korn constant.

For the actual weight w(y)=sqrt(1-|y|²) on B^d, extend a vector field to V(y,z)=(v(y),0) on B^(d+1). This extra integration coordinate is solely a mathematical device. Integrating z gives twice the weighted norm and strain norm. The first d components of an ordinary rigid approximation are a+Omega y+beta z. Integrating the odd z cross term vanishes; dropping its nonnegative beta² z² term and the last component square proves

inf_r in R_d integral w|v-r|²
   <= C_d integral w|eps(v)|²,
C_d=16(d+3)(2^d-1)/d.

For d=2,3,4,5,6,10 the constants are 120,224,420,3968/5,1512,106392/5. The same proof covers each depth without a depth-five coefficient. It extends to the natural free strain closure by continuity. The independently accepted second-Korn lift supplies compact form embedding; this first-Korn calculation supplies its explicit quotient gap.

For rho=w h, with 0<h_min<=h<=h_max, minimize over the same rigid set after density comparison:

inf_r integral rho|v-r|²
   <= C_d (h_max/h_min) integral rho |eps(v)|².

Thus the free elastic form a_h(v)=integral rho[|eps(v)|²+(div v)²/2] has quotient gap

K_h >= h_min/(C_d h_max).

The nodal audit gives h_min/h_max=E_N(0)/binomial(N+d,d)² for its prepared full profiles; N=0 gives one. With a=(d+1)/2,

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

Physical amplitude cancels from this ratio. In normalized displacement variables the actual free material operator E has a rigid kernel and nonzero spectrum bounded below by

e=2 kappa (b k_d)² K_h,
k_d/mu_d=M^(d-1)/d!.

All subsequent arithmetic divides frequencies by the same supplied mu_d within each depth.

2. The complete material resolvent retains rigid motion

Keep the exact unreduced quadratic pencil of the moving-action quadratic derivation:

P_r(lambda)=[[D(lambda)F(lambda), r lambda D(lambda) A],
             [r lambda A* D(lambda), T_r(lambda)]],
D=lambda-J(H+omega), F=lambda-J(omega-H),
T_r=r^(-2)E+lambda² M_r,
M_r=I-r² B*B, B=-JA, ||A||<=1.

Here E is FREE, and (1-r²)I<=M_r<=I. The full kinetic matrix is positive for r<1/sqrt(2). For fixed positive r define the nonnegative self-adjoint operator Etilde=M_r^(-1/2) E M_r^(-1/2) through its closed form. Its compact form embedding persists, and its kernel has the complete rigid multiplicity d(d+1)/2. Its first positive eigenvalue is >=e. To justify the latter without a mistaken orthogonality claim, use the generalized minmax quotient <u,E u>/<u,M_r u>. Its denominator is no larger than ||u||², so every ordered eigenvalue, including the zero multiplicities, is no smaller than the corresponding E eigenvalue. The transformed kernel need not equal the original kernel.

For lambda!=0, |lambda|<=L and r²L²<e, spectral calculus gives

lambda² T_r(lambda)^(-1)
  =M_r^(-1/2) lambda²[r^(-2)Etilde+lambda²]^(-1)M_r^(-1/2),

||lambda² T_r(lambda)^(-1)||
  <=1/(1-r²) max{1, L²/(e/r²-L²)}.

The term 1 is the whole rigid kernel contribution. For positive eigenvalues use |a+lambda²|>=a-|lambda|². Therefore

r²L²<=e/2  implies  ||lambda² T_r(lambda)^(-1)||<=1/(1-r²).

This holds on complex circles, not just the imaginary axis. It includes all nonrigid material frequencies and all rigid mixing caused by M_r.

Eliminating material gives the exact form Schur expression

D [F-r²lambda² A T_r^(-1) A*D].

Since D and F commute,

D F^(-1)=2(lambda-J omega)F^(-1)-I.

If the contour distance from every bare F frequency is >=delta, then

||D F^(-1)||<=1+2(L+omega)/delta,

||r²lambda² A T_r^(-1)A*D F^(-1)||
    <=q_free(r)=r²/(1-r²)[1+2(L+omega)/delta].

Thus q_free(r)<1 excludes roots on the contour. The full signed chain spectrum is symmetric under H -> -H, so D and F have the same spectral set; both outer factors are invertible there. Realize D:X_w -> V_w* and F:V_w -> X_w as in the audited form-resolvent argument. The bounded middle inverse gives the weak Schur inverse, and material reconstruction belongs to its form domain. The combined strong generator condition follows from the resolvent equation. No unbounded product is being asserted on arbitrary L2 vectors.

The bound explicitly controls the order-r² rigid/Routh correction. Bounding the whole free inverse is simpler and stronger for this purpose than bounding every finite charge row separately. It does not silently omit their finite-rank contribution.

3. Equivalence to the fixed-charge reduction away from zero

Write chi=T a+z with T orthonormal into ker E and Tz=0, and I_r=TM_r T. Projecting the material pencil onto T gives

lambda² I_r a+lambda² T*M_r z+r lambda T*A*D eta=0.

For lambda!=0 this solves exactly as

lambda a=-I_r^(-1)[r T*A*D eta+lambda T*M_r z].

The bracket is the linear momentum j. This is precisely the fixed-zero-momentum Routh reconstruction. Eliminating the a block, whose coefficient lambda² I_r is invertible, yields the already derived reduced pencil. This block elimination works equally for forced equations and supplies resolvent equivalence on nonzero contours, including algebraic multiplicities. No condition fixing the center position is imposed.

Equivalently, in first-order variables the conserved charge p has p_t=0, so every nonzero root subspace lies at p=0. Restoring cyclic positions gives the corresponding full root subspace. The unreduced static form is diag(S,r^(-2)E), with S=H²-omega², and has negative index n_-(S). Its rigid positions add zero directions, while kinetic energy is positive. The Routh energy on p=0 agrees with the full energy after reconstruction and is independent of the cyclic positions. Its static form is

V_red=diag(S+r² L*I_r^(-1)L, r^(-2)E_Z),
L=T*A*J(H+omega), E_Z>=e.

Consequently n_-(V_red)<=n_-(S) for every r<1/sqrt(2). No extra negative index is contributed by the discarded charge coordinates.

4. Continuation and the static kernel without an extra tiny charge-row bound

The accepted free small-r theorem supplies a negative-definite Riesz space of dimension n_-(S) for all sufficiently small positive r. Use the fixed contours from the exact spectral certificate, their conjugates, and the estimates above along the whole interval up to the proposed endpoint. Both q_free(r) and r²L²/e increase with r, so checking the endpoint suffices.

For any positive r in this interval, the reduced conserved Hermitian energy is nondegenerate on each nonzero reflection-invariant contour range. This statement does not require knowing its sign or its static kernel in advance. The energy-adjoint resolvent identity makes the Riesz projection energy-self-adjoint. A vector in its range orthogonal to that range is then orthogonal to the complementary range as well, hence lies in the full energy radical. Positive kinetic energy makes every radical vector (q,0) with V_red q=0; these are stationary zero eigenvectors. A contour excluding zero annihilates every such vector. Therefore the restricted energy is nondegenerate. Local norm continuity of the form resolvents and finite-dimensional Gram matrices makes its inertia constant. This is the audited continuation argument, now applied to the actual free Routh system with the nonzero contour equivalence established above.

The continued negative Riesz space therefore still has dimension n_-(S). The preceding upper bound forces

n_-(V_red)=n_-(S).

An elementary useful lemma now removes the need for a separate numerical bound on L_Q S_Q^(-1)L_Q*.

Lemma. Let V=S+B be lower-bounded self-adjoint forms with finite equal negative index, B>=0. Then ker V=ker S intersect ker B.

To prove the nontrivial inclusion, let Vu=0 and let Y be a maximal V-negative subspace, whose dimension is n_-(S). If B[u]>0, then S=V-B is strictly negative on every nonzero vector y+t u: if y!=0 the V part is negative; if y=0 and t!=0 the B part is strictly positive before subtraction. This produces an S-negative subspace of dimension n_-(S)+1, contradiction. Hence B[u]=0; positivity gives B(u,v)=0 for every form-domain v, and Vu=0 implies Su=0. The reverse inclusion is immediate. The argument is a form argument and does not assume simultaneous diagonalization.

Apply this lemma to S on the wave space and B=r²L*I_r^(-1)L. The positive E_Z block contributes no kernel. It yields exactly

ker V_red=ker(H²-omega²) intersect ker L.

The previously audited angular compression now applies at every certified r: for N>0 its codimension in the full signed stationary shell is d-1, and the gyro compression has radical O=span{R_Omega psi} of dimension d-1. At N=0, L annihilates the stationary shell and its gyro compression is invertible. There is no circularity: nonzero signature continuation used only the fact that any energy radical is stationary, not the precise kernel formula now obtained.

Negative-index exhaustion and the identity-minus-finite-rank energy argument from the free spectrum report give positive coercive energy on the stationary quotient. Its induced group is uniformly bounded. The upper-triangular group extension by the finite stationary space grows at most linearly. The angular compression proves exactly d-1 length-two zero chains for nonconstant profiles; no longer chain is possible. These are neutral linear orientation drifts, not a decay rate or failure of physical ballistic motion.

5. Exact arithmetic and scope

The executable standalone free checker recomputes all 319 full-form low-frequency intervals with exact integer Sturm recurrences, rather than reading rounded gaps. It reconstructs each contour and verifies its distance from all signed scalar and extra-form frequencies, from the omitted high spatial families, from heavy bands, and from zero. It uses the same supplied full finite-M parent as the audited existence theorem. Every target interval fits strictly inside its rational contour; conjugate contours have identical bounds.

At r=1/100000 the maximum q_free per label is:

Table entries are rounded upward summaries; exact fractions are authoritative in complete exact result. The certificate has 456 positive contours and 1836 recorded assertions, including twelve small exact affine moment/Korn controls. These controls check algebra, not an approximation to the infinite-dimensional spectrum. The report provides the infinite-dimensional estimates.

The scale law remains R=r c/(b k_d), with supplied M,b,kappa and sector units. Reducing r retains the finite-M clocks and their existing errors, while reducing physical radius and symbol speed together. This is a conservative sufficient interval, not an optimized threshold or a measured particle speed. Physical amplitudes remain unspecified. All six maintained depths use one rule; no depth-five exceptional suppression appears.

The result concerns the square active weighted ball with natural free boundary and fixed zero linear charges. It is not a completed finite-energy body in all ten ordinary spatial directions, a nonlinear existence theorem, a bound on material-map invertibility, empirical particle identification, or a universal requirement that finite-lived particles be infinitely stable. Free center reconstruction and the previously derived positive boost inertia remain separate physical consequences. Internal field coefficients do not replace deterministic electron positions.

Derived Orientation Response

The complete orientation calculation solves the generalized zero modes, including their wave correction and material counter-motion. It gives positive orientation inertia and a certified lower bound throughout the same interval. A supplied rotating perturbation-energy fraction then bounds the linear angular coefficient per carrier cycle. No disturbance or obligatory drift is imposed; finite nonlinear rotation and observer-space reconstruction remain distinct.

Reproducibility

The standalone standard-library checker recomputes the complete spectral window, excludes omitted spatial and heavy bands, retains all signed and angular multiplicities, and verifies 456 positive contours and their conjugates. It also checks twelve exact moment identities. All acceptance decisions use integers and fractions. The 1,836 recorded free assertions supplement the root isolation and full-domain proofs above; they do not replace those proofs with a finite simulation.

The finite-time bound applies in the weighted form norm for each fixed allowed parameter set. It does not give a numerical amplification constant or pointwise control of the material Jacobian. The ratio r is a material characteristic-speed parameter, not the particle center's translational speed. Its conservative bound is sufficient, not an optimized threshold. No force strength or physical amplitude was fitted to obtain it.