Same-band source admission and regional work

Scope: a bounded two-lobe preparation on one carrier band, a geometric screen absent throughout that source class, an exact necessary screen-work condition, and a strictly signed nominal nonlinear rate distinction between resource-matched initial controls. A lobe is a region of concentrated full-state amplitude; a chosen sum does not supply independent dynamical constituents or an NFR identity. This is analytical admission, not an evaluated trajectory, formation result or physical identification. The research program owns this branch's status and any later response evaluation.

1. Complete state and a bounded one-carrier preparation

Retain the complete C1024 law, with unit conductances and lengths, held unit capacities, beta one, no input or support event, and the supplied real lift of primitive circular phase. Write x=EPI, z=x+i*theta, (Dq)_j=q_(j+1)-q_j for cycle incidence, L=D^T D and A=L/2. The bookkeeping complex field retains two real rows:

\[ x'=-\sigma Ax-\tfrac12D^T\sin(D\theta),\qquad \theta'=Ax,\qquad \tau=wt/\pi,\quad w=1-2^{-20},\quad \sigma=\pi2^{-20}/w>0. \tag{1} \]

The prime denotes the fixed normalized clock. Carrier phase is not primitive phase, and |z|^2=x^2+theta^2 is a localization diagnostic rather than local structural storage. The full state, including generated modes and error means, remains retained. Neither this preparation nor a Fourier observation closes the dynamics on one band.

Set N=1024, Q=16, k=pi/4, a=2^-12 and delta=a/1000. Reuse the Fejer normalization and tail bound only, without transferring the earlier carrier's transport certificate:

\[ b_m=1-|m|/16\quad (|m|<16),\qquad S_2=171/16, \qquad f(j)=\frac{\sum_{|m|<16}b_m e^{2\pi imj/N}}{\sqrt{NS_2}}. \tag{2} \]

The profile is real, nonnegative, even, periodic and unit-normalized. For each supplied psi in [-pi,pi], define the preparation class

\[ z_\psi^0(j)=a e^{ikj}[f(j)+e^{i\psi}f(j-512)],\qquad z(0)=z_\psi^0+r,\qquad \|r\|_2\le\delta. \tag{3} \]

The primitive form is the real part of (3), and its imaginary part is the supplied real phase lift. Explicitly, their nominal values are a[f(j)cos(kj)+f(j-512)cos(kj+psi)] and a[f(j)sin(kj)+f(j-512)sin(kj+psi)]. The two terms are preparation labels on the same band, modes 113 through 143. There are no independent band projections that recover them from an evolved total field.

The exact initial overlap and norm are

\[ s=\langle f,f(\cdot-512)\rangle =\frac{\sum_m b_m^2(-1)^m}{S_2}=\frac1{171},\qquad \|z_\psi^0\|_2^2=2a^2(1+s\cos\psi). \tag{4} \]

Indeed the numerator of the overlap is 1/16, while the squared-weight sum is 171/16. Thus changing relative phase at fixed component amplitude does not generally hold total norm fixed. Arbitrary errors are retained, not projected onto the nominal band. The triangle inequality gives

\[ a\sqrt{340/171}-\delta\le\|z(0)\|_2 \le a\sqrt{344/171}+\delta, \qquad \frac{1409}{1000}a<\|z(0)\|_2<\frac32a. \tag{5} \]

Since ||f||_infinity=2/sqrt(171), primitive admission also gives

\[ \|z(0)\|_\infty\le a(4/\sqrt{171}+1/1000)<a/3, \qquad |D\theta(0)|<2a/3=1/6144<\pi/2. \tag{6} \]

The nominal mean vanishes because the band omits zero. Error means remain: |mean z(0)|<=delta/sqrt(N)=a/32000. Equation (1) conserves both real means; it does not set them to zero.

The exact structural storage and initial bounds are

\[ H=\tfrac12\|Dx\|_2^2+\sum_j[1-\cos(D\theta)_j],\qquad H'=-\tfrac\sigma2\|Lx\|_2^2, \qquad 0\le H(0)\le\tfrac12\|Dz(0)\|_2^2<\tfrac92a^2<1. \tag{7} \]

The bound uses 1-cos u<=u^2/2 and ||D||_2<=2. Together with the initially acute lift, the storage balance prevents a first edge crossing of +/-pi/2, which would contribute one to H. This familiar chart bound does not preserve the initial localization or full-state norm, and does not prove organization.

2. A total-field concentration screen initially absent

For a real observation center c on the unit-length cycle, let W(c,96) be the radius-96 ball. Let W_1=W(0,96) and W_2=W(512,96). The reused spatial-tail inequality gives, for each shifted unit profile,

\[ \|\mathbf1_{W_i^c}f_i\|_2^2\le b:=\frac{1024^3}{16\cdot16^3\cdot95^3} =\frac{16384}{857375}<\frac1{49}. \tag{8} \]

The nominal components are not disjoint, but both initial lobes have a nonvanishing regional signal even with arbitrary allowed error and phase:

\[ \|\mathbf1_{W_i}z(0)\|_2 \ge a[\sqrt{1-b}-\sqrt b]-\delta>\frac45a. \tag{9} \]

Here the other lobe contributes at most a*sqrt(b) because the two source windows are disjoint. Outside their union, ||1_(W_1 union W_2)^c z(0)||_2<=2a*sqrt(b)+delta<3a/10. These inequalities describe two separated concentrations, without assigning them unique constituent identities under future overlap or deformation.

The proposed common-window screen requires ||z||_2>=a and one radius-96 window containing at least 3/4 of the total squared norm. It is a geometric prerequisite for a narrower possible organization, not a complete relational identity. It does not consume independent component projections.

For every c, at least one source center is at cycle distance at least 256. Since 256>96+96, W(c,96) misses that source window. Its component contribution is therefore at most a*sqrt(b); the other component's full norm is a. Equations (5) and (8) imply

\[ \frac{\|\mathbf1_{W(c,96)}z(0)\|_2^2}{\|z(0)\|_2^2} \le\left(\frac{1+\sqrt b+1/1000} {\sqrt{340/171}-1/1000}\right)^2 <\left(\frac{8007}{9863}\right)^2<\frac23. \tag{10} \]

Thus the 3/4 concentration requirement fails with margin greater than 1/12 for every center, phase and member of the source-error ball. The signal floor holds independently by (5), and the cycle triangle inequality also covers fractional centers. This is a fresh same-band total-field argument; the previous disjoint-band absence test has not been reused.

Later entry into this screen could occur by ordinary interference or dispersion, and by itself would not establish acquisition, retention or a collective response. A two-lobed organization wider than this screen is not excluded by its failure. The formation contract still requires a retained relation, a justified lifetime and a discriminating response on the continued full state.

3. Exactly matched initial resources without deleting phase

Equal component amplitudes do not match total norm across arbitrary psi, by (4). They do not match exact H either. The quadratic storage comparison is H_2=||Dz||_2^2/2, and

\[ 0\le H_2-H\le\frac1{24}\sum_j(D\theta)_j^4. \tag{11} \]

Replacing exact H by this comparison would discard a phase-dependent remainder. Rescaling amplitudes to match norm alone supplies another preparation and does not certify matched storage or initial loss.

A useful matched pair is instead +psi and -psi. Let (Jq)_j=q_(-j) and define the real-linear isometry

\[ \mathcal S(x,\theta)=(Jx,-J\theta),\qquad (\mathcal S z)_j=\overline{z_{-j}}. \tag{12} \]

Evenness of f and its antipodal shift gives S z_psi^0=z_(-psi)^0. Pair errors by r_(-psi)=S r_psi; independently chosen errors of equal norm need not match resources. For forward edge index j, put (J_e q)_j=q_(-j-1), y=Dx and u=Dtheta. Then

\[ y(\mathcal S z)=-J_e y(z),\qquad u(\mathcal S z)=J_e u(z),\qquad h_j=\tfrac12y_j^2+1-\cos u_j. \tag{13} \]

Consequently the paired full preparations have exactly equal norm, total H and ||Lx||_2^2, hence equal initial global loss. Pointwise localization and edge storage are reflected. For an edge subset E, H_E(Sz)=H_(J_e E)(z); a same-region comparison requires J_e E=E, otherwise its reflected region must be paired explicitly. Means are conjugated, including the retained error means, rather than erased. No fitted amplitude or hidden reservoir is involved.

The map S is not a positive-time symmetry of (1). It matches initial resources while changing the allocation of signed form and phase, whose actual subsequent rows must be evaluated separately. Reflection J alone is a graph symmetry; adding phase reversal changes this conclusion. The paired source classes establish a valid initial control, not equality of their future trajectories or a sign for future regional work.

4. A necessary prefix-work condition for the absent screen

The concentration diagnostic and structural storage are different observations. A positive local storage rate alone cannot establish entry into (10). Fix a node window W, put P_W=diag(1_W), alpha=3/4 and B_W=P_W-alpha*I, and retain the quadratic screen gap

\[ V_W(z)=x^TB_Wx+\theta^TB_W\theta =\|P_Wz\|_2^2-\alpha\|z\|_2^2. \tag{14} \]

Screen entry in that window requires V_W>=0 and the separate signal floor ||z||_2>=a. For every admitted source, (10) gives V_W(z(0))<-||z(0)||_2^2/12. No regional-storage bound was substituted for that initial diagnostic margin.

Differentiate (14) using both actual rows in (1). With u=Dtheta,

\[ \begin{split} V_W'&=\mathcal L_W(z)+\mathcal N_W(z),\\ \mathcal L_W(z) &=-2\sigma x^TB_WAx+2\theta^T(B_WA-AB_W)x,\\ \mathcal N_W(z)&=(DB_Wx)^T(u-\sin u). \end{split} \tag{15} \]

Here L_W is the rate under the actual damped linear tangent evaluated at the same full state, not a nonnegative loss estimate. A restricted quadratic observation has boundary transport and can grow even while global structural storage decreases. N_W is the exact nonlinear rate insertion relative to that tangent. Its cubic-sine comparison is

\[ \mathcal N_{W,3}=\tfrac16(DB_Wx)^Tu^{\circ3},\qquad |\mathcal N_W-\mathcal N_{W,3}| \le\tfrac1{120}\sum_j |(DB_Wx)_j|\,|u_j|^5. \tag{16} \]

The elementwise cube is not an envelope equation. Under joint amplitude scaling this leading rate is fourth order; the remainder is sixth order. Both still consume full form, primitive phase and the selected boundary. The sine Taylor remainder proves (16) globally on real lifts.

If a solution first satisfies the screen in W at a time T, necessarily

\[ \int_0^T [\mathcal L_W(z(\tau))+\mathcal N_W(z(\tau))]\,d\tau \ge -V_W(z(0))>\frac{\|z(0)\|_2^2}{12}. \tag{17} \]

This is a source-dependent necessary condition, not a sufficient entry criterion. For entry in any radius-96 window, apply (17) to the successful window held fixed over the prefix; its initial absence holds for every center. A moving-window rate or continued moving identity requires its own membership transport. The norm floor and later maintenance remain separate obligations.

If W is invariant under node reflection, B_W commutes with J. Equation (13) then gives

\[ V_W(\mathcal Sz)=V_W(z),\qquad \mathcal N_W(\mathcal Sz)=-\mathcal N_W(z). \tag{18} \]

The dissipative contribution in L_W is unchanged and its conservative commutator contribution changes sign. Thus relative-phase allocation can change the linear part of the same criterion as well. Equation (18) does not prove that N_W is nonzero on a particular source or stays signed along its future evolution.

5. Structural regional work and its nonlinear rate insertion

The existing exact local storage balance uses L_e=DD^T, y=Dx, u=Dtheta and

\[ F_{jk}=\tfrac12(\sin u_j\,y_k-y_j\sin u_k) +\tfrac\sigma4(y_j^2-y_k^2)=-F_{kj}. \]

For a fixed edge subset E define

\[ \begin{split} H_E&=\sum_{j\in E}(y_j^2/2+1-\cos u_j),\\ I_E&=-\sum_{j\in E,\,k\notin E,\,k\sim j}F_{jk},\\ D_E&=\tfrac\sigma4\sum_{j\in E}\sum_{k\sim j}(y_j-y_k)^2. \end{split} \qquad H_E'=I_E-D_E. \tag{19} \]

An internal edge pair appears twice in the local-loss sum, as required by the local balance. Taking E to be the full edge set gives the global loss in (7). The inward current is signed; neither it nor a global loss budget is an event reserve or a guarantee of acquisition.

For comparison, use the same structural H_E and the damped tangent

\[ x'_{\rm lin}=-\sigma Ax-A\theta,\qquad \theta'_{\rm lin}=Ax. \tag{20} \]

This tangent is real-linear in the two state channels. Its positive-loss term acts on form, so a uniform complex carrier rotation does not generally commute with its propagator. Each phase preparation must retain both rows; rotating one precomputed history is not an exact control for another.

At an identical supplied full state their derivative difference is

\[ G_E(z):=\left.\frac{dH_E}{d\tau}\right|_{F_\sigma(z)} -\left.\frac{dH_E}{d\tau}\right|_{J_\sigma z} =\tfrac12(P_Ey)^TL_e(u-\sin u),\qquad P_E=\operatorname{diag}(1_E). \tag{21} \]

Only the form-rate difference contributes because the phase rows agree. This exact instantaneous insertion is not H_E(joint)-H_E(A)-H_E(B) and not an endpoint difference between evolved histories. Its leading term and remainder are

\[ G_{E,3}=\tfrac1{12}(L_eP_Ey)^Tu^{\circ3},\qquad |G_E-G_{E,3}|\le\tfrac1{240}\sum_j|(L_eP_Ey)_j|\,|u_j|^5. \tag{22} \]

Writing q=u-sin u, the exact incidence decomposition is

\[ G_E=\tfrac12\sum_{\{j,k\}\subset E,\,j\sim k} (y_j-y_k)(q_j-q_k) +\tfrac12\sum_{j\in E,\,k\notin E,\,j\sim k} y_j(q_j-q_k). \tag{23} \]

Internal pairs in the first sum are unordered and counted once. Thus G_E is not solely boundary inflow: it is the change of the structural storage rate caused by replacing the tangent pressure by actual sine pressure. The linear comparison still satisfies an exact structural ledger, H_E'=I_E-D_E-G_E, when all terms are evaluated on its own linear state.

The matched preparation map gives G_E(Sz)=-G_(J_e E)(z). For J_e E=E this is an exact sign reversal, while H_E and the local loss match initially. It does not preserve the two full trajectories or their later current.

6. A strictly signed nominal phase control

Sign reversal alone permits a zero insertion. A finite exact witness within (3) avoids this ambiguity. Take the nominal source r=0, psi=pi/2 and the two-edge region E={-1,0} modulo N, which is invariant under j -> -j-1. This small region measures a local rate; it is distinct from the radius-96 acquisition screen.

Put f_j=f(j), g_j=f(j-512) and c=1/sqrt(2). Define

\[ A_0=f_0-cf_1>0,\qquad B_0=cg_1>0,\qquad U=cf_1,\qquad V=f_2-cf_1<U. \tag{24} \]

These inequalities follow from the source, without a numerical fit. The positive Fourier weights give f_1<=f_0, so A_0>=(1-c)f_0>0. Their nonzero frequencies satisfy |2*pi*m/N|<=15*pi/512<pi/4, hence f_1>cf_0 and 2cf_1>f_0>=f_2, proving V<U. The Fejer formula gives g_0=0 and g_1>0: its numerator at the antipodal offset is proportional to sin(pi/64)^2, and its denominator is nonzero.

Evenness and the exact carrier values at nodes 0, +/-1 and +/-2 give

\[ \begin{array}{ll} y_{-1}=a(A_0-B_0),&y_0=-a(A_0+B_0),\\ u_{-1}=a(U-B_0),&u_0=a(U+B_0),\\ u_{-2}=a(V+B_0),&u_1=a(V-B_0). \end{array} \tag{25} \]

Let q(t)=t-sin(t), q_(U,+/-)=q(a(U+/-B_0)) and likewise for V. Substitution into (21) yields

\[ \begin{split} G_E(z_{\pi/2}^0)=\frac a2\{& A_0[3(q_{U,-}-q_{U,+})+(q_{V,-}-q_{V,+})]\\ &-B_0[(q_{U,-}+q_{U,+})-(q_{V,-}+q_{V,+})]\}<0. \end{split} \tag{26} \]

Indeed q is strictly increasing: its derivative 1-cos(t) is nonnegative and vanishes only at isolated points, with no constant interval. Because a,B_0>0, each difference in the first bracket is negative. Since U>V, each q value in the second U sum exceeds its corresponding V value, so the second bracket is positive. Therefore

\[ G_E(z_{-\pi/2}^0)=-G_E(z_{\pi/2}^0)>0. \tag{27} \]

The two preparations have exactly equal initial total norm, exact storage, global loss and reflected regional storage. Nevertheless the actual sine pressure changes this structural regional storage rate above the same-state linear value in one preparation and below it in the other. This is a strict local, initial nonlinear distinction. It does not imply positive total regional growth, an attractive force, screen entry, restoration, lifetime or acquired NFR identity. No robust sign over the nonzero error ball, or sign of the screen insertion N_W, has been proved by this witness. In particular, B_0>0 comes from the remote Fejer lobe's nonzero tail: the witness already has local overlap initially. A strict nominal sign need not dominate the admitted arbitrary source error or the linear contribution. It does not establish newly acquired contact.

7. Matched histories retain transport and comparison state

Let z be a full-law solution and ell the tangent solution (20), with the identical admitted primitive preparation z(0)=ell(0). The same screen observation has the exact difference

\[ V_W(z(T))-V_W(\ell(T)) =\int_0^T[\mathcal L_W(z)-\mathcal L_W(\ell) +\mathcal N_W(z)]\,d\tau. \tag{28} \]

The linear transport terms are evaluated on different evolving states and do not cancel. Integrating only the initial or future N_W omits this feedback. Conversely, the structural regional-storage difference is

\[ H_E(z(T))-H_E(\ell(T)) =\int_0^T[I_E(z)-I_E(\ell)-D_E(z)+D_E(\ell)+G_E(\ell)]\,d\tau. \tag{29} \]

Here G_E is evaluated on ell because its structural ledger in (23) differs from the full-law ledger. Equations (28) and (29) concern different observations and cannot substitute for each other.

A separate-history control requires an explicit primitive split, including an allocation of the admitted source error. If z_A and z_B evolve independently under (1), their sum s=z_A+z_B is an observation on the common support. It need not solve (1). With u_A=Dtheta_A, u_B=Dtheta_B, define

\[ \mathcal N_W^{\rm sep}(z_A,z_B) =(DB_W(x_A+x_B))^T [u_A-\sin u_A+u_B-\sin u_B]. \tag{30} \]

One declared split covering the whole source class is z_A(0)=a*exp(i*k*j)*f(j)+r/2 and z_B(0)=a*exp(i*(k*j+psi))*f(j-512)+r/2. It gives the exact total preparation (3), with each assigned error at most delta/2. This equal assignment is comparison metadata, not an inferred physical partition. Other splits require their own retained initial metadata and bounds.

For a matching initial sum, the actual and separate observed screen gaps obey

\[ V_W(z(T))-V_W(s(T)) =\int_0^T[\mathcal L_W(z)-\mathcal L_W(s) +\mathcal N_W(z)-\mathcal N_W^{\rm sep}(z_A,z_B)]\,d\tau. \tag{31} \]

At one supplied shared state, the nonlinear insertion difference is

\[ \mathcal N_W(z_A+z_B)-\mathcal N_W^{\rm sep}(z_A,z_B) =4(DB_W(x_A+x_B))^T \left[\sin\frac{u_A}{2}\sin\frac{u_B}{2} \sin\frac{u_A+u_B}{2}\right]. \tag{32} \]

This reuses the exact mixed sine identity. The cross terms in V_W(s) are retained; replacing it by V_W(z_A)+V_W(z_B) would change the observable. Likewise the initial structural cross storage is generally nonzero:

\[ H_E(z_A+z_B)-H_E(z_A)-H_E(z_B) =\sum_{j\in E}\left[y_{A,j}y_{B,j} +4\sin\frac{u_{A,j}}2\sin\frac{u_{B,j}}2 \cos\frac{u_{A,j}+u_{B,j}}2\right]. \tag{33} \]

Its evolved difference must retain each history's current and loss. The split is supplied preparation information, not a unique nonlinear decomposition or proof of autonomous constituent identities.

8. Bounded result and remaining formation obligations

The admitted source supplies two separated concentrations with an initially absent total-field screen. Equations (15)-(17) give the exact necessary prefix work for that screen; (21)-(27) prove a strictly signed nonlinear local rate distinction for two exactly resource-matched nominal phases. The static controls check these algebraic and source statements without evaluating a prepared trajectory or reserved response.

This establishes a bounded conditional compatibility result and its comparison boundary. It does not establish that the positive local insertion can overcome the screen deficit, that the full-law history passes a screen which the linear or separate controls fail, or that any resulting concentration maintains a new relation. A future formation discriminator still needs those actual-state transport estimates, a separate signal floor, a justified retention interval and a continued-response condition. The exact balances identify that missing signed work; they do not admit a trajectory, parameter sweep or long-horizon calculation by themselves.