A Local Ambient Count Action with Complete Inertia and Exterior Dynamics

This article gives an explicit changed local action on ordinary flat spacetime with ten spatial directions. Its conditional prepared branches reproduce all fourteen supplied count clocks, have uniform participating core energy and complete positive boost inertia, and retain nonnegative reference static forms while adding exterior phase and carrier dynamics. It is an inverse construction: the finite component carrier, occupation potentials, sector units and action normalization are premises. It does not complete the physical mass operator.

The action and its reference claims have independent mathematical review. This article incorporates the clarified full spacetime contraction in equation (6), with spacetime index b ranging over eleven coordinates and frame index i over ten. The positive perturbation form is always the co-rotating second variation specified below.

The IDOS energy and coherence derivation contains the preceding participation-action, occupation and persistence results. The action below changes the exterior law and its weighted domain. Its exact reference identities, full quadratic forms and principal bulk cones are derived here; the earlier action's nonlinear persistence and transmission theorems are not transferred to it. Both clock signs of the even action remain mathematical branches, not an antiparticle identification. Numerical check counts are verification coverage, not empirical evidence.

All formulas in the derivation use speed-one units. Thus $E_0=M_{\rm boost}=A_0\mu_d\binom{N+d}{d}$; restoring comparison speed $c$ gives $M_{\rm boost}=E_0/c^2$. The common $A_0$ is a phase-action normalization, not an adjustable physical field amplitude fitted to each particle.

1. Field, carrier and supplied preparations

The physical manifold is ordinary flat R^(1+10), with metric signature (-,+,...,+). Component pairs are real classical field coordinates, not extra spatial dimensions, particle position probabilities or mediator particles. The electron's literal deterministic R6 orbit and precession remain separate from these internal components.

The finite component set A is the union of B(2,94), B(4,71), B(6,34), B(10,22), where B(d,N) consists of nonnegative ten-component multiindices supported on the first d axes with total degree at most N. It contains M=69,117,447 labels. The actual preparations are:

The carrier target is C={b:sum_A|b_k|^2=1, Im b_0=0, b_0>0}. Its real tangent is <b,h>=0, Im h_0=0, with <u,v>=Re sum conjugate(u_k)v_k and orthogonal projection Pi_b z=z-b<b,z>-i e_0 Im z_0. The visible standing field remains S=exp(i theta)(b,conjugate(b))/sqrt(2). The visible-field map uses its local first-jet inverse on the stated chart. No new carrier component is added; the existing components acquire exterior dynamics.

For completeness, the visible first-jet inverse is local. With s_0=S_(+,0),

exp(i theta)=s_0/|s_0|,
b=sqrt(2) exp(-i theta)S_+,
dtheta=Im(ds_0/s_0),
db=sqrt(2)exp(-i theta)[dS_+-iS_+dtheta].

The condition b_0>0 makes s_0 nonzero. No Fourier transform or spatial averaging is required. The tensor support/frame/flow variables are declared components of this candidate field multiplet, not reconstructed from S. At a supplied prefix B(d,N), C_k=1/sqrt(D) on the prefix and zero elsewhere; permitted affine phases give the corresponding symmetry-related preparation.

All q in {2,4}, eta in {0,1}, both original entropy choices, the positive clock, both signs of the even clock, and the common-unit/supplied-six-unit alternatives are retained. These are 48 constitutive preparations per actual row, 672 in total. They are conditional variants of common laws, not parameters fitted separately to particles.

2. The same carrier in exterior coordinates

Keep the parent's support and frame variables W>0, Z orthogonal to a future unit timelike U, and an orthonormal spatial frame E_i orthogonal to U. Put

y_i=(E_i.Z) W^(-5/4),  xi_i=(L/ell)y_i,
c_k=exp(-i k.xi)b_k,  b_k=exp(i k.xi)c_k.                 (1)

This is an invertible pointwise map of the same constrained target at every finite point where W>0. It does not introduce a physical coordinate space. It preserves the origin constraint and the target inner product for variations at fixed xi. The first-jet map is explicit:

d c_k=exp(-i k.xi)[d b_k-i b_k k.dxi],
delta c_k=exp(-i k.xi)[delta b_k-i b_k k.delta xi],
delta xi_i=(L/ell)W^(-5/4)
  [delta(E_i.Z)-(5/4)(E_i.Z)delta W/W].                 (2)

For a metric variation, delta(E_i.Z)=delta g(E_i,Z)+g(delta E_i,Z)+g(E_i,delta Z). Derivatives of (2), with the ordinary product rule, give the complete varied first jet. Thus the action below is also a local first-jet action in the original visible-field and tensor coordinates.

Let p_k=|b_k|^2=|c_k|^2, extended by zero outside A; r=sum p_k^2, K=sum p_k k, M2=sum p_k kk^T. The parent's three occupation potentials are

V0=sum p_k(p_0-p_k)^2,
Vminus=sum_(k,i:k_i>0) p_k(p_(k-e_i)-p_k)^2,
Vexchange=sum_(i!=j)K_i K_j
    sum_(k in N0^10:k_i>0)(p_k-p_(k-e_i+e_j))^2,
Vocc=V0+Vminus+Vexchange,
Vphase=sum_(k,i:k_i>0)p_k|b_0 b_k-b_(e_i)b_(k-e_i)|^2,
H=mu/r+2mu r^(-3)[Vocc+(1+eta)Vocc^q+eta Vphase].        (3)

The full zero-extended directed exchange sum includes outside-A source endpoints. Under (1), each phase residual is multiplied by exp(i k.xi). Consequently

H(b)=H(c),  m(b)=m(c)=mu/r,
beta(b)=beta(c)=omega_*ell^2/[16(1+tr M2)]              (4)

exactly, for arbitrary xi and all target points, not just at a preparation. Both the common positive mu and the supplied smooth six-unit function depend only on p, so they obey this identity. The supplied unit function is the parent's sixth-power depth interpolation with the flat small-degree switch. Its first two jets vanish at each actual prefix and it equals mu_3 near the origin. Take 0<omega_*<=inf mu as in the parent.

Here is the explicit optional six-unit prescription, so its supplied character is reproducible without a separate interpolation choice. For I={2,3,4,5,6,10}, define

z1=sum_i K_i, z2=sum_i K_i^2,
d_eff=z1^2/z2, n_eff=z1+z2/z1,
L_d(t)=product_(j in I,j!=d)(t-j)^6,
w_d(t)=L_d(t)/sum_(j in I)L_j(t),
mu=(1-B(2n_eff-1))mu_3+B(2n_eff-1)sum_d w_d(d_eff)mu_d,
B(t)=h(t)/[h(t)+h(1-t)],
h(t)=0 for t<=0, h(t)=exp(-1/t) for t>0.

Use mu=mu_3 wherever the switch is zero, including the full origin neighborhood; no singular ratio is used there. At every positive supplied preparation n_eff=N>=2 and the interpolation equals mu_d with vanishing first two jets. This is a common inverse constitutive function using the six supplied positive units, not a dynamical derivation of them.

The two entropy prescriptions are (b_ent,kappa)=(omega_/128,8/omega_) or (m/128,8/m). They are retained without replacing m by H. Set K_c=4kappa and

F_plus(t,H)=t-H,
F_even(t,H,kappa)=
   [log cosh(K_c t)-log cosh(K_c H)]/[K_c tanh(K_c H)]. (5)

The even sign is a property of the same action. It does not require reversing the future material current.

3. The complete changed action

Write p_s(W)=min(1,256W^2), sigma=A0/(V10 L^10), V10=pi^5/120, chi=Theta(Psi), Psi=1-|y|^2, and P_U^(ab)=g^(ab)+U^a U^b. The parent material current and density remain

J=(A0/V10) star(dy_1 wedge ... wedge dy_10),
n=sqrt(-J.J), V=J/n,
A_theta=P_U dtheta,
h_kappa(A)=[sqrt(1+4kappa^2 A.A)-1]/(4kappa),
F_m=V.dtheta-H-h_kappa(A_theta)                         (positive),
F_m=F_even(V.dtheta,H,kappa)-h_kappa(A_theta)             (even),
alpha=H/2,
D_a b_k=(P_U)_a^b partial_b b_k-i b_k k_i E_(i,a)/ell,
L_car=sigma beta[|U.db|^2-|D b|^2/4]/2,
L_m=rho F_m-b_ent[rho log(rho/n)-rho+n]
        -n alpha[(U.V)^2-1]/2+L_car.                   (6)

The material map must be regular, future timelike and oriented on a neighborhood of Psi>=0; V is not used in the new exterior term. The support/frame/flow density is unchanged:

mu_0=L^2 omega_*sigma/(212W^2),
A_s=P_U dW+8Z/L,
C_s=P_U P_U nabla Z-W^(5/4)P_U/L+10ZZ/(LW),
G_s=Z.Z-W^(9/4)+W^(5/2),
Omega_(a,ij)=E_i.nabla_a E_j,
L_E=(kappa_E sigma/2)sum_(i<j)[(U.Omega_ij)^2
                                  -c_E^2|P_U Omega_ij|^2],
L_U=(kappa_U sigma/2)[|nabla_U U|^2
                                  -c_U^2|P_U nabla U|^2],
L_e=mu_0[(UW)^2+|P_U nabla_U Z|^2]/2
       -mu_0[|A_s|^2+|C_s|^2]/8-kappa_G G_s^2/2+L_E+L_U.       (7)

All original positive constants are common; 0<c_E,c_U<=1/2 and kappa_U c_U^2>=20omega_L^2 remain sufficient. Introduce a single common lambda>0, for example lambda=1/omega_, and common 0<c_theta,c_b<=1/2. Define

t=U.dtheta, X=P_U dtheta, B_a=partial_a c,
T=U^a B_a, B_perp=P_U B,
K_tail=lambda[F(t,H,kappa)^2-c_theta^2 X.X]/2
             +beta[|T|^2-c_b^2|B_perp|^2]/2,
L_tail=sigma p_s K_tail,
S_new=integral sqrt(-g)[chi L_m+p_s L_e+(1-chi)L_tail] d^11x.    (8)

Use F_plus or F_even consistently with the material clock. All variables in (8) are the same field components as in (6)-(7), related by (1). This is a changed constitutive law, not an equivalence to the gated parent. The exterior now has the full 2M+65 intrinsic dynamical coordinates, rather than the parent's 66 tensor coordinates alone. Its large finite field content remains a supplied assumption.

4. Complete local first variation of the added term

It is clearest to use independent c coordinates, with all tensor constraints imposed in a regular chart or with algebraic multipliers. At fixed metric, put

delta t=a_a delta U^a+U^a partial_a delta theta,
delta X=delta P_U a+P_U d(delta theta),
delta T=delta U^a B_a+U^a partial_a delta c,
delta B_perp=delta P_U B+P_U d(delta c), a=dtheta.

The exact variation is

delta L_tail=sigma p_s' K_tail delta W
  +sigma p_s {lambda F[F_t delta t+F_H delta H+F_kappa delta kappa]
    -lambda c_theta^2 <X,delta X>
    +(delta beta/2)[|T|^2-c_b^2|B_perp|^2]
    +beta[<T,delta T>-c_b^2<B_perp,delta B_perp>]}.       (9)

For general constrained metric variations an equivalent expanded formula is

delta K_tail = lambda F(F_H delta H+F_kappa delta kappa)

• Jtheta^a partial_a delta theta + <Jc^a,partial_a delta c>
• {lambda(F F_t-c_theta^2 t)a_a +beta(1-c_b^2)<T,B_a>}delta U^a +(delta beta/2)(|T|^2-c_b^2|B_perp|^2) -(lambda c_theta^2/2)delta g^(ab) a_a a_b -(beta c_b^2/2)delta g^(ab)<B_a,B_b>, Jtheta^a=lambda[F F_t U^a-c_theta^2 P_U^(ab)a_b], Jc^a=beta[U^a T-c_b^2 P_U^(ab)B_b]. (10)

Add K_tail g^(ab)delta g_ab/2 for the volume variation, p_s' as in (9), and the algebraic constraint variations. Here delta g^(ab)=-g^(ac)g^(bd)delta g_cd. In c coordinates no tail derivatives of W,Z,E,U occur. There are algebraic U, W, H, kappa and beta forces; omitting them would change the action.

For the positive clock F_t=1, F_H=-1, F_kappa=0. For the even clock, writing f=F_even and K=4kappa,

F_t=tanh(Kt)/tanh(KH),
F_H=-1-[2K/sinh(2KH)]f,
F_kappa=[t tanh(Kt)-H tanh(KH)]/[kappa tanh(KH)]
            -f[1/kappa+4H sech^2(KH)/tanh(KH)].         (11)

At t=eH with e=+1 or -1, these are (e,-1,0), and F=0. At the positive reference e=+1 the same first derivatives apply.

For completeness every algebraic variation in (9)-(10) is determined locally as follows:

delta p_k=2Re(conjugate(c_k)delta c_k),
delta r=2sum p_k delta p_k, delta K=sum k delta p_k,
delta M2=sum kk^T delta p_k,
delta beta=-beta tr(delta M2)/(1+tr M2),
delta m=delta mu/r-m delta r/r.

For any directed difference Delta=p_k-p_l, its square varies as 2Delta(delta p_k-delta p_l). Thus

delta V0=sum[delta p_k(p_0-p_k)^2
                      +2p_k(p_0-p_k)(delta p_0-delta p_k)],
delta Vminus=sum[delta p_k(p_prev-p_k)^2
                      +2p_k(p_prev-p_k)(delta p_prev-delta p_k)],
delta Vexchange=sum_(i!=j){(delta K_i K_j+K_i delta K_j)sum Delta^2
                   +2K_i K_j sum Delta(delta p_k-delta p_shift)}.

For R_ki=c_0 c_k-c_ei c_prev,

delta R_ki=delta c_0 c_k+c_0 delta c_k-delta c_ei c_prev-c_ei delta c_prev,
delta Vphase=sum[delta p_k|R_ki|^2+2p_k Re(conjugate(R_ki)delta R_ki)].

With Vstar=Vocc+(1+eta)Vocc^q+eta Vphase,

delta H=delta m+2r^(-3)Vstar delta mu-6mu r^(-4)Vstar delta r +2mu r^(-3){[1+(1+eta)q Vocc^(q-1)]delta Vocc+eta delta Vphase}. (12)

For fixed entropy delta kappa=0; for branch entropy delta kappa=-8delta m/m^2. For the supplied unit interpolation, put z1=sum K_i, z2=sum K_i^2, d_eff=z1^2/z2, n_eff=z1+z2/z1. Where its switch is nonzero,

delta d_eff=2z1 delta z1/z2-z1^2 delta z2/z2^2,
delta n_eff=(1-z2/z1^2)delta z1+delta z2/z1,
delta mu=2B'(2n_eff-1)(sum_d w_d mu_d-mu_3)delta n_eff
                          +B(2n_eff-1)sum_d w_d' mu_d delta d_eff.

Use the parent's exact weights w_d(t)=product_(j!=d)(t-j)^6 divided by their sum, and set delta mu=0 in the flat origin neighborhood. These formulas cover zero components without polar-angle coordinates.

The tail Euler expressions in c coordinates are -div(sigma p_s Jtheta) for theta, and the tangent projection of

sigma p_s[lambda F F_H grad_C H+lambda F F_kappa grad_C kappa
    +( |T|^2-c_b^2|B_perp|^2 )grad_C beta/2]
              -div(sigma p_s Jc)                              (13)

for c, plus the explicit algebraic tensor terms in (9)-(10). The geometric constraints and carrier normals determine multipliers; they do not add dynamical variables.

To pull this first variation back to b and the tensors, substitute (2) everywhere, including d(delta c). Let R_c(s)_k=i c_k k.s. With D=diag(exp(-ik.xi)), the derivative momentum one-form pulls back as

<Pi_c^a,delta c>=<D^*Pi_c^a,delta b>-<R_c^*Pi_c^a,delta xi>,
Pi_b^a=D^*Pi_c^a,
Pi_q^a=-(partial_q xi)^T R_c^*Pi_c^a                   (14)

for tensor coordinates q=W,Z,E at fixed metric, in addition to their parent momenta. The derivative of partial_q xi contributes the corresponding zeroth-order terms automatically in (2). For metric variation use its full delta xi. This is the complete chain rule; the original-coordinate support/frame derivative terms are not omitted.

The complete first variation of S_new is the accepted local variation of chi L_m+p_s L_e, plus (9)-(14), with the material surface replacement in the next section. In particular the unchanged parent variation retains its material-current cofactor delta J, delta n=-V.delta J, delta V=P_V delta J/n, all entropy and alignment variations, delta beta, the support residual variations, frame and flow connection variations, and delta Gamma. These unchanged material-current variations follow by the ordinary product rule from equations (6)-(7), on their stated regular timelike chart; no exterior V is introduced. This gives a fully specified local weak variation of the complete changed action.

5. Moving interfaces and global conormals

Before integration by parts the full density variation is

chi delta L_m+delta(p_s L_e)+(1-chi)delta L_tail
   +delta(Psi)(L_m-L_tail)delta Psi,

with the full volume and constraint terms just stated. In a solved real chart q^A, the explicit material-interface distribution is

delta(Psi){Psi_A(L_m-L_tail)
              -(partial_a Psi)(Pi_m,A^a-Pi_tail,A^a)}.          (15)

The remaining divergence is that of p_s Pi_e and the regular piecewise fluxes. With the parent's outward normal convention N=-dPsi/|dPsi|, the regular weak conormal rule is

N_a[Pi_m,A^a+p_s Pi_e,A^{a,-}-p_s Pi_e,A^{a,+}-Pi_tail,A^a]
       +Psi_A(L_m-L_tail)/|dPsi|=0.                            (16)

Use the projected target conormal and the chain rule (14) when working in original coordinates. For a spacetime material surface this is the spacetime conormal, so it includes a moving surface's temporal flux. Formula (16) is the regular weak variational convention of the parent; strong transmission solutions need appropriate traces and regularity, not an undefined product of distributions. It is not a general free-boundary existence theorem.

Theta and c are now global fields with matching two-sided traces, as are the tensor variables. The old inside-only carrier trace/zero-action exterior quotient is inapplicable. At the separate constitutive corner W=1/16, p_s is continuous, while p_s' has a finite jump. There is no delta from its first derivative; the total weighted conormal matches. If the two surfaces intersect, (15) and the total flux give their weak rule. No extra codimension-two term is inserted.

6. Exact preparations, field equations, metric stress and charges

For B=B(d,N), C_k=1_B(k)/sqrt(D), let H0=mu_d D, or the common-unit value mu D; the down state has H0=mu_3 in the supplied-unit law. In physical rest coordinates,

bbar_k=C_k exp(i k.x/ell), theta=eH0 t,
W0=(1+|x|^2/L^2)^(-4),
Z0=(x/L)(1+|x|^2/L^2)^(-5),
E_i=e_i, U=V=(1,0), y=x/L, rho=n=sigma.               (17)

The material surface and weight corner coincide at |x|=L. Every actual prefix lies inside b_0>0. The occupation proof gives dH=dm=0 tangentially at each branch, for both unit prescriptions. From (1), cbar=C everywhere, so dcbar=0. Moreover F=0 and P_U dtheta=0. Hence the tail density and its full constrained first field variation vanish pointwise. Its conormal and material shape term also vanish. The parent reference weak equations therefore remain exact for every row and every stated conditional variant.

The constrained metric variation is equally zero. This statement uses the unit-U constraint. For example, at a=-eH0 U_flat the apparent direct phase metric term in (10) is canceled by its U variation, since

2g(U,delta U)+delta g(U,U)=0,
delta[g^(ab)a_a a_b+(U.a)^2]=0.                        (18)

One must not hold an unconstrained U fixed while varying the metric and then mistake the incomplete term for a stress. In a multiplier formulation the associated change of the U-normal multiplier cancels it. The carrier metric terms vanish because dcbar=0, and F times any clock variation vanishes. The measure, gate and weight variations multiply K_tail=0. Thus delta_g S_tail=0 exactly on the constrained reference.

Consequently the complete reference stress, participating energy and charges are unchanged:

T^(ab)=chi sigma H0 V^a V^b,
Q_theta=e A0, E0=A0 H0, P=0, M_boost=A0 H0>0.          (19)

The proper participating energy density is sigma H0 throughout the core; the tail has zero reference stress and energy. The equal-mode allocation sigma mu_d remains available, under the same parent assumptions. A full Lorentz transform of every field and both surfaces still has constant c, F=0 and P_Udtheta=0. Thus E=gamma E0 and P=gamma E0 v for the entire boosted preparation. No partial carrier norm replaces complete inertia.

The tail phase-current addition is

j_tail^a=(1-chi)sigma p_s lambda
                  [F F_t U^a-c_theta^2 P_U^(ab)partial_b theta]. (20)

It is zero on every preparation and boost. The same global phase symmetry remains. The charge magnitude A0 is preserved on this family, not dynamically selected by the extension. Opposite even-clock signs do not establish observed electric charge or antiparticles.

For generic data the action's energy, momentum and phase current change. Define the new full canonical charges from the new total momenta, including (14), or the complete constrained Hilbert stress. It would be false to claim that the parent's E and P are still separately conserved under the new coupled equations. Translation and phase symmetry conserve the NEW totals. Their physical reciprocal-response consequences remain to be calculated.

7. Full added quadratic form, including tensor mixing

Use the parent's constrained perturbations at a fixed actual reference:

h_k=exp(-ik.x/ell)delta b_k, <C,h>=0, Im h_0=0,
phi=e delta theta, Y=(w,z)=(delta W,delta Z_spatial),
F_ji=delta E_i^j, F^T=-F, v=delta U_spatial,
zeta_i=L delta y_i
   =L W0^(-5/4)[z_i+sum_j F_ji Z0_j-(5/4)Z0_i w/W0],
R(s)_k=i C_k k.s/ell,
g=delta c=h-R(zeta).                                  (21)

The time/space constraint components delta E_i^0=v_i and delta Z^0=v.Z0 produce no first-order contribution to delta(E_i.Z); zeta has no v term. Both h and g obey the same constant carrier tangent constraints, including the origin and all unoccupied components.

At the reference dH=dm=dkappa=0, delta t=e phi_t, delta F=phi_t, and

delta(P_U dtheta)=e(grad phi+H0 v),
delta(U.dc)=g_t, delta(P_U dc)_j=partial_j g,
delta(P_U dc)_0=0.                                    (22)

There is no delta U.dcbar term because dcbar=0, and no derivative of a nonconstant cbar. The entire new quadratic action is therefore

Delta S2=(sigma/2)integral dt integral_(|x|>L) p_s(W0) {lambda[phi_t^2-c_theta^2|grad phi+H0 v|^2] +beta0[|g_t|^2-c_b^2|grad g|^2]} dx. (23)

No second derivative of the even clock enters (23), since it is multiplied by the zero background F. Both even signs give the same form. First variations of beta, p_s and the surface multiply a quadratic residual and start above second order. Since K_tail and its first variation vanish, the moving interface adds no independent quadratic tail term.

In original variables the carrier squares in (23) are explicitly

|h_t-R(zeta_t)|^2
  =|h_t|^2-2<h_t,R(zeta_t)>+zeta_t^T M2 zeta_t/ell^2,
sum_j|partial_j h-R(partial_j zeta)|^2
  =|grad h|^2-2sum_j<partial_j h,R(partial_j zeta)>
                     +sum_j(partial_j zeta)^T M2(partial_j zeta)/ell^2. (24)

The derivatives of zeta include the x-dependent coefficient derivatives of W0 and Z0, as well as derivatives of w,z,F. Equations (21),(24) retain all support/frame mixing. The phase-flow cross term is 2H0 v.grad phi, with positive H0^2|v|^2 in the static square. There is no exterior carrier R(v) drift and no added velocity-linear gyro term in these variables.

In c coordinates g is independent. Within the core, h=g+R(zeta), so the parent carrier residuals become

h_t+R(v)=g_t+R(zeta_t+v),
grad h-T(F)=grad g+R(grad zeta)-T(F),
T(F)_(j,k)=i C_k sum_i k_i F_ji/ell.                   (25)

These are genuine core support/frame/flow mixing terms and are retained. Since affine phase directions R(s) lie in the exact symmetry kernel of H and dH=0, the positive semidefinite Hessian satisfies Q_H(g+R(zeta))=Q_H(g). This also follows directly from exact identity (4). It holds for eta=0 and eta=1 and at the origin R=0.

8. Full time and static positivity at every actual frequency

Let mu_s=p_s(W0)mu_0(W0), alpha=H0/2 and D_s v=(v.grad W0,v.grad Z0). The accepted complete parent time form is

M_old(q,q)=sigma integral_B[(1/b_ent+a2)phi^2+beta0|h|^2 +(H0-alpha)|zeta|^2] +integral_R10[mu_s|Y|^2+sigma p_s kappa_E|F|^2 +sigma p_s kappa_U|v|^2], (26)

where a2=0 for the positive clock and a2=8kappa/sinh(8kappa H0)>0 for either even sign. The exact new time form adds

Delta M=sigma integral_ext p_s[lambda phi^2+beta0|h-R(zeta)|^2]. (27)

Every independent coordinate now has positive time weight at each finite point. The tensor baseline is positive; the core material rank update has H0-alpha=H0/2>0; beta0>0 on all carrier tangents. Formula (27) is positive semidefinite in original coordinates and diagonal positive in the exterior c coordinates. It cannot create a time-negative direction. Positivity is reference-local, not a statement about every off-reference even-clock jet.

For the unchanged parent define

A1_i=partial_i w+8z_i/L,
C1_ij=partial_i z_j-(5/4L)W0^(1/4)w delta_ij
  +10(z_i Z0_j+Z0_i z_j)/(LW0)-10Z0_i Z0_j w/(LW0^2),
G1=2Z0.z-[(9/4)W0^(5/4)-(5/2)W0^(3/2)]w.

Its full static form is

a_old=sigma integral_B[beta0|grad h-T(F)|^2/4+2Q_H(h) +kappa|grad phi+H0v|^2+alpha|v|^2-beta0|R(v)|^2] +integral_R10{mu_s[|A1|^2+|C1|^2]/4+p_s kappa_G G1^2 +sigma p_s kappa_E c_E^2|grad F|^2 +sigma p_s kappa_U c_U^2|grad v|^2-mu_s|D_s v|^2}. (28)

The new static addition is

Delta a=sigma integral_ext p_s[lambda c_theta^2|grad phi+H0v|^2
                                         +beta0 c_b^2|grad g|^2]. (29)

The full zero-extended occupation theorem gives Q_H>=0 for every actual row, both q/eta choices and both unit laws. The parent's complete flow estimate uses the raw second moment, not covariance:

beta0 M2/ell^2<=omega_* I/16,
integral mu_s|D_s v|^2/sigma
  <=2I0 omega_*Vol(B)|mean_B v|^2
       +(1024/53)omega_*L^2 integral p_s|grad v|^2,
I0=[529-760log 2+11017/693]/212<11/128.                 (30)

Under kappa_U c_U^2>=20omega_L^2, the parent leaves the positive margins 17sigma omega_/64 on integral_B|v|^2 and 36sigma omega_*L^2/53 on integral p_s|grad v|^2, in addition to its nonnegative residual and occupation terms. Adding (29) preserves these margins exactly. This proves full static nonnegativity, including responsive support, frame, flow, newly occupied carrier directions and exterior perturbations, within this quadratic reference problem. No coefficient singles out depth five.

The complete velocity-linear form remains

G_old=sigma integral_B[beta0<h_t,R(v)>-alpha v.zeta_t]
                                 +integral_R10 mu_s Y_t.D_s v. (31)

Use h=g+R(zeta) when changing coordinates. The quadratic action is (M_new/2)+G_old-(a_new/2), evaluated on velocities where appropriate. Its conserved canonical perturbation energy is [M_new(q_t,q_t)+a_new(q,q)]/2. Because the background has theta=eH0t, this is the stationary/co-rotating second variation of E_new-eH0 Q_new. It is not an assertion of positivity of the raw unconstrained energy Hessian at arbitrary phase charge. Along fixed-charge paths, including exact boosts, it is the corresponding complete energy variation.

For the parent boost tangent h=-tR(v), zeta=-tv, Y=-tD_s v, phi=-H0 v.x and F=0, equation (21) gives g=0 and grad phi+H0v=0. All added quadratic terms vanish. The parent cancellation of parent carrier/support drifts therefore still gives E2=A0 H0|v|^2/2, consistently with exact (19).

9. Principal cones, including the full mixed system

At a reference exterior point use independent c coordinates. The principal time matrix is diagonal with weights sigma p_s lambda, sigma p_s beta0, mu_s, sigma p_s kappa_E, sigma p_s kappa_U on phi,g,Y,F,v. Its spatial counterpart has respective ratios c_theta^2,c_b^2,1/4,c_E^2,c_U^2. The v.grad phi term in (29) has only one derivative and is lower order. The exterior principal mixed time-space block is zero. Thus the full exterior bulk speeds are

c_theta; c_b on every real carrier tangent;
1/2 on the eleven support variables;
c_E on the 45 frame rotations; c_U on the ten flow variables.   (32)

They are all strictly positive and at most 1/2. In the original variables the principal map (h,Y,F)->(g,Y,F) is invertible at each finite point, so the mass and spatial matrices undergo the same congruence. Its rank-at-most-ten support/carrier mixing cannot change the characteristic roots. This explicitly includes the mixed tensor directions rather than testing theta with frozen support.

The interior action is unchanged. Its exact reference inequality 0<K_space<=M_base/4<=M_old/4 remains true, including the material support/frame rank update and the separate phase speed 1/4 or 1/[4sqrt(1+b_ent a2)]. Hence all core reference bulk speeds remain positive and at most 1/2. The global derivative form likewise satisfies the principal quadratic inequality K_space<=M_new/4: add the tail residual contributions in (23), each with speed at most 1/2. Zeroth-order coefficient derivatives do not alter this assertion.

There is no claim of a uniform unweighted energy bound at infinity, a nonlinear cone theorem for arbitrary detuning, or a general characteristic interface theorem. All fourteen actual clocks are retained. Hypothetical off-reference values are not used to exclude any of them.

10. Changed weighted domain and linear trace rules

At (17), put p=p_s(W0). It equals 1 for r<=L and 256(1+r^2/L^2)^(-8) outside. It is positive at finite r and behaves as r^(-16) at infinity. The support weight mu_s is bounded above and below by positive constants, and is exactly 256L^2omega_*sigma/212 outside the core.

In independent c coordinates a natural mass space is the completion under M_new of

phi,g in L2(R10,p dx), Y in L2(R10,dx), F,v in L2(R10,p dx),     (33)

with the constant target tangent for g and skew frame tangent for F. Each fixed-branch M_new is equivalent to this product norm: core coefficients and the displacement map are bounded on the closed core, and the positive tensor baseline controls the finite core cross terms. A natural closed form domain is

phi,g in H1(R10,p dx), Y in H1(R10,dx), F,v in H1(R10,p dx),     (34)

defined by closure of globally smooth test fields in the displayed norm, with the same tangent constraints. The nonnegative form plus mass norm is equivalent to (34), by the positive baseline derivative terms, bounded core mixing coefficients, and (30). Lower-order tail phase mixing is controlled by the p-weighted v mass. The finite carrier Hessian is bounded on the core for each fixed branch. This proves the relevant form closure, rather than assuming the old quotient domain remains valid.

The weights are bounded and bounded away from zero on each finite collar, so global fields in (34) have matching traces across the material surface and weight corner. Constants in the p-weighted fields are permitted when their constraints hold; p has finite integral. Their presence does not imply a positive static gap. Far-out broad carrier/phase wave packets still have Rayleigh quotient tending to zero.

In original perturbation variables the equivalent statement is h inside H1(B), g=h-R(zeta) outside H1(p dx), together with (34) for tensors and matching g/phi traces. At the material boundary,

g_out|boundary=h_in|boundary-R(zeta|boundary).          (35)

One must NOT separately impose exterior h in H1(p dx). The coefficients of zeta grow polynomially at infinity: for x/L=u e_1, its support coefficients include -(5/4)L u(1+u^2)^4 and L(1+u^2)^5. Thus the global change is locally smooth but not a uniformly bounded map on the naive old exterior h norm. Equation (34) is the energy domain selected by the new action. A nonlinear Banach-chart/evolution theorem at infinity requires separate work.

For a fixed reference interface let N be the outward spatial unit normal, D_N=partial_N h-T(F)N, and define the baseline support flux J_Y=(mu_s/4)(A1.N,C1(N,j)), frame flux J_F=sigma p kappa_E c_E^2 partial_N F, and flow flux J_v=sigma p kappa_U c_U^2 partial_N v. The full positive stiffness conormal one-forms in original coordinates are

J^- = sigma kappa(partial_N phi+H0v.N)delta phi +(sigma beta0/4)<D_N,delta h> +J_Y^-.delta Y+J_F^-.delta F+J_v^-.delta v, J^+ = sigma p lambda c_theta^2(partial_N phi+H0v.N)delta phi +sigma p beta0 c_b^2<partial_N g,delta h-R(delta zeta)> +J_Y^+.delta Y+J_F^+.delta F+J_v^+.delta v. (36)

The linear natural condition is J^-=J^+ for every allowed common trace variation. In c coordinates replace delta h=delta g+R(delta zeta); the carrier-induced support/frame flux then appears on the core side instead of the exterior side. This is the same condition, not an independent boundary choice. The parent null-divergence cancellation includes the material moving-surface contribution; the new term adds none at quadratic order. Formula (16) remains the exact nonlinear rule. The corner uses the analogous total flux match. General nonlinear transmission and long-time persistence remain open.

11. Why extending the old carrier kinetic term is a different option

For comparison ONLY, replace the new carrier part by the parent's L_car multiplied by (1-chi)p_s. This alternative also vanishes on the actual prepared references, but its exact quadratic addition is

(sigma beta0/2)integral_ext p
      [|h_t+R(v)|^2-|grad h-T(F)|^2/4].                (37)

Its time form is positive, its gyro acquires sigma beta0 integral_ext p<h_t,R(v)>, and its static form acquires sigma beta0 integral_ext p[|grad h-T(F)|^2/4-|R(v)|^2]. This is an actual-reference negative flow load; it is not a detuned-clock example.

If the same phase tail is retained, choose smooth compactly supported f on a physical annulus, v_R(x)=grad_x[R f(x/R)], phi_R=-H0 R f(x/R), h=Y=F=0. The phase square vanishes exactly. The negative carrier contribution scales as -const R^(-6), with a strictly positive constant for any actual N>0 and f whose gradient has an active-axis component. The positive flow-gradient term scales as O(R^(-8)); the unchanged support drift is nonpositive. Thus sufficiently large R gives negative static energy in this alternative for each of the thirteen nonorigin actual rows. All such tests have finite energy and compact support. At down, M2=0 and this specific load is absent. This proves a limitation of (37), not of the gated parent or the c-tail action (8). No observed instability timescale, decay or force follows from this static witness.

12. Scope of the construction

The changed action (8) gives the same actual prepared reference fields, uniform participating core energy, full reference stress, common phase-charge magnitude and complete boost inertia, while adding nondegenerate exterior phase and carrier dynamics. Its complete reference time form is positive, its static form is nonnegative under the parent's common flow bound, and its full bulk characteristic speeds are at most 1/2. The c-tail variable avoids the old exterior carrier flow load exactly. The global weighted domain and interface traces change, and generic states have the new action's total energy and current.

The construction does not select fourteen particles from the many allowed supports, derive the supplied sector units or A0, calibrate the field amplitude, identify a photon, infer a common frequency for unlike particles, establish an observed reciprocal force, or complete the mass operator. Depth five obeys the identical law and has no special suppression from these formulas. General nonlinear transmission, persistence and dynamical index selection remain open. The phase-current and reciprocal-response follow-up remains open.