Mixed interaction and the compatibility of localized formation¶

Scope: exact instantaneous mixed pressure under the existing cycle law, followed by a conditional two-carrier modulation calculation. The calculation identifies cubic interaction channels, distinguishes temporal resonance from spatial aliasing, and excludes one stationary bright-profile comparison. This pair is an analytical control with a separate static source-absence admission. It supplies no evaluated source, full-law envelope closure, formation event or physical identification. The research program owns admission of a preparation and any subsequent response evaluation.

1. Complete law, state and clock¶

Retain the complete localized model on C1024: unit edge conductances and lengths, held unit capacities, beta one, signed real form x and primitive circular phase with declared real lifts theta. Inputs and support events are absent. Keep e=2^-20, w=1-e, the normalized clock tau=w*t/pi and sigma=pi*e/w>0. Let D be the oriented cycle incidence, L=D^T D and A=L/2. Both real rows are

\[ x'=-\sigma Ax-\tfrac12D^T\sin(D\theta),\qquad \theta'=Ax. \tag{1} \]

Primes denote tau derivatives. Conversion to the original t clock multiplies both rows by w/pi; no laboratory-time identification is supplied. All coefficients and phases below use this fixed normalized model and clock. The bookkeeping variable z=x+i*theta retains two real channels; it does not make EPI complex or identify carrier phase with primitive phase. Writing F_sigma for (1), its nonlinear remainder is

\[ R(\theta)=-\tfrac12D^T[\sin(D\theta)-D\theta],\qquad R_3(\theta)=\tfrac1{12}D^T(D\theta)^3. \tag{2} \]

The original prepared encounter, its nominal interaction certificate and its strict-passage exclusion remain unchanged. The control below is selected analytically from velocity compatibility, not obtained by retuning that evaluated source.

2. Exact mixed pressure precedes a carrier approximation¶

Take two supplied full states on the same support, clock and lift convention. Put u=Dtheta_A, v=Dtheta_B, with edge products understood componentwise. The shared-state instantaneous defect is

\[ \mathcal M=F_\sigma(z_A+z_B)-F_\sigma(z_A)-F_\sigma(z_B) =\left(2D^T\left[ \sin\frac u2\sin\frac v2\sin\frac{u+v}2\right],\ 0\right). \tag{3} \]

Indeed sin(u+v)-sin(u)-sin(v) equals -4*sin(u/2)*sin(v/2)*sin((u+v)/2). The form-loss and phase rows cancel because they are linear on the supplied shared states. This does not cancel loss from their evolved histories: those histories and the transport of their nonlinear difference depend on the actual F_sigma. Equation (3) is not the endpoint superposition defect.

The leading mixed cubic form row and a global edgewise remainder are

\[ \mathcal M_{3,x}=\tfrac14D^T[uv(u+v)],\qquad \mathcal M_x-\mathcal M_{3,x}=D^T q, \]

\[ |q_j|\le \frac{|u_jv_j(u_j+v_j)|}{96} \left(u_j^2+v_j^2+(u_j+v_j)^2\right). \tag{4} \]

To prove the bound, telescope the product of three sines against the product of their arguments, using |sin a|<=|a| and |sin a-a|<=|a|^3/6. Multiplication by two and substitution a=u/2, b=v/2, c=(u+v)/2 give (4). There is no division by a phase gap, so zero gaps and cancellation u+v=0 retain their exact zero defect. Since ||D^T||_2<=2, the corresponding vector remainder is at most 2*||q||_2. Its form mean is zero because 1^T D^T=0.

The signed interaction depends on retained edge phase information. A positive local contribution is not a binding criterion, and summing contributions does not remove the regional boundary-work or formation obligations.

3. An exact equal-velocity carrier control¶

The conservative comparison has lambda(k)=1-cos(k) and group velocity -sin(k) in the sign convention of the moving reference. Fix

\[ k_1=\pi/4,\quad k_2=3\pi/4,\qquad \lambda_1=1-\sqrt2/2,\quad\lambda_2=1+\sqrt2/2, \]

\[ v=\sin k_1=\sin k_2=\sqrt2/2,\qquad d_1=\tfrac12\cos k_1=\sqrt2/4,\quad d_2=-\sqrt2/4. \tag{5} \]

Both carriers are exact characters of C1024, at modes 128 and 384. They meet the leading relative-velocity condition without a carrier search. Their opposite curvature signs already prevent transferring a single focusing-profile argument to both channels. Equal velocity is only a kinematic admission; it does not establish retention or acquisition.

For a positive bookkeeping scale epsilon, put X=epsilon*(j+v*tau), S=epsilon^2*tau and phi_i=k_i*j+lambda_i*tau. The leading two-row reference is

\[ z_1=B_1(X,S)e^{i\phi_1}+B_2(X,S)e^{i\phi_2},\qquad x_{\rm lead}=\epsilon\operatorname{Re}z_1,\quad \theta_{\rm lead}=\epsilon\operatorname{Im}z_1. \tag{6} \]

No envelope, initial amplitude, source-error ball or finite horizon is selected by (6). Periodic envelopes require period epsilon*N in X. A line profile cannot be wrapped onto the cycle without its seam defect. The two carriers here are distinct channels; two lobes in a single carrier would instead require a justified observation of one total envelope.

4. Cubic labels, aliases and resonances¶

Freeze the envelope values when extracting the local carrier coefficients. For label (m,n), retain both

\[ K_{mn}=(m+3n)\pi/4\pmod{2\pi},\qquad \Omega_{mn}=m\lambda_1+n\lambda_2. \tag{7} \]

Spatial aliasing reduces K but never Omega. Write R_3(Im z_1)=sum c_mn*exp(i*(m*phi_1+n*phi_2)) for this frozen-envelope calculation. Define

\[ g_1=\frac{3-2\sqrt2}{16},\qquad g_2=\frac{3+2\sqrt2}{16},\qquad c=\frac18. \tag{8} \]

The eight labels below and their negatives give all sixteen. The negative coefficient is c_(-m,-n)=conj(c_mn) because this is a real form row.

Label	Coefficient `c_mn`
`(1,0)`	`-i(g1abs(B1)^2+cabs(B2)^2)B1`
`(0,1)`	`-i(g2abs(B2)^2+cabs(B1)^2)B2`
`(3,0)`	`-i(sqrt(2)-1)B1^3/48`
`(0,3)`	`i(sqrt(2)+1)B2^3/48`
`(2,1)`	`-iB1^2B2/16`
`(1,2)`	`-iB1B2^2/16`
`(2,-1)`	`i(sqrt(2)-1)B1^2*conj(B2)/16`
`(-1,2)`	`-i(sqrt(2)+1)conj(B1)*B2^2/16`

These follow directly by substituting theta=(B1*u-conj(B1)/u+B2*v-conj(B2)/v)/(2*i) into (2), with spatial shift multipliers u -> exp(i*k1)*u, v -> exp(i*k2)*v. For example, the resonant cross coefficient at the first carrier is -i*|1-exp(i*k1)|^2*|1-exp(i*k2)|^2/16=-i/8. The pure fundamental coefficients are -i*lambda_i^2/8. The mixed cubic defect has twelve labels: the four fundamentals and the eight signed mixed third-order labels. Only the four pure third harmonics are absent from that mixed defect; its fundamental coefficients retain only the cross terms, whereas the table displays full R_3. Omitting conjugate labels loses a real row.

For the conservative comparison, set Delta_mn=Omega_mn-lambda(K_mn). The complete gaps are

Label pair	`Delta_mn`	`Delta_(-m,-n)`
`(1,0)` and `(-1,0)`	`0`	`-2+sqrt(2)`
`(0,1)` and `(0,-1)`	`0`	`-2-sqrt(2)`
`(3,0)` and `(-3,0)`	`2-2*sqrt(2)`	`-4+sqrt(2)`
`(0,3)` and `(0,-3)`	`2+2*sqrt(2)`	`-4-sqrt(2)`
`(2,1)` and `(-2,-1)`	`2-sqrt(2)`	`-4`
`(1,2)` and `(-1,-2)`	`2+sqrt(2)`	`-4`
`(2,-1)` and `(-2,1)`	`-sqrt(2)`	`-2+2*sqrt(2)`
`(-1,2)` and `(1,-2)`	`sqrt(2)`	`-2-2*sqrt(2)`

Only (1,0) and (0,1) are resonant. One direct check is that every cubic label has odd m+n; lambda(K) is either 1-sqrt(2)/2 or 1+sqrt(2)/2. Equality with (m+n)+(n-m)*sqrt(2)/2 forces one of those two labels. Although 3*k1=k2, the temporal gap is 2-2*sqrt(2), not zero. All fourteen nonresonant gaps have magnitude at least 2-sqrt(2). These are carrier gaps for the stated comparison, not full nonlinear spectral gaps or positive-loss eigenfrequencies.

5. Leading equations retain cross interaction and actual loss¶

Cancellation of the two resonant cubic terms, together with the common transport in (5), gives the conditional leading system

\[ i\partial_S B_i=d_i\partial_X^2B_i+ (g_i|B_i|^2+c|B_j|^2)B_i -i\frac{\gamma\lambda_i}{2}B_i,\qquad j\ne i, \quad\gamma=\sigma/\epsilon^2. \tag{9} \]

The damping term comes from both carrier directions in -sigma*A*Re(z). A corresponding counterrotating correction is -i*gamma*conj(B_i)*exp(-i*phi_i)/4; its conservative divisor is -2*i*lambda_i. Keeping only the positive carrier loss would discard part of the real form row. The interpretation of damping at this order requires the declared scaling of gamma; fixed positive sigma cannot be removed by sending epsilon to zero. Equation (9) is neither an installed TNFR law nor a certified closure of (1).

For smooth periodic envelopes, or on the line with vanishing boundary flux, the individual modulation masses satisfy

\[ M_i=\int|B_i|^2\,dX,\qquad M_i'=-\gamma\lambda_i M_i. \tag{10} \]

Multiply (9) by conj(B_i) and take the appropriate real part after solving for B_i,S. The real cubic multipliers contribute zero to M_i'; the dispersion term is the boundary derivative 2*d_i*partial_X Im(conj(B_i)*B_i,X). Thus the leading mixed term shifts phase and can affect spatial evolution, but supplies no direct exchange between these two masses.

Equation (9) is invariant under independent constant rotations B_i -> exp(i*alpha_i)*B_i. Its resonant coupling therefore does not select a relative uniform carrier phase or supply a phase-dependent exchange term. These carrier rotations are not primitive common-phase gauge. The full sine law, its nonresonant reconstruction and higher-order errors have not been proved to inherit this independent symmetry or these separate mass laws. Their absence at leading order does not exclude higher-order phase effects, profile reorganization or all finite-lived collective motion.

6. Nonresonant reconstruction and the remaining residual¶

Let L_nr be the fourteen nonresonant labels and define

\[ C_3=\sum_{(m,n)\in L_{\rm nr}} \frac{c_{mn}}{i\Delta_{mn}}e^{i(m\phi_1+n\phi_2)}. \tag{11} \]

For constant envelopes, partial_tau-i*A multiplies each term by its displayed divisor, so (11) cancels exactly the nonresonant cubic form row. If S_3=sum_(L_nr)|c_mn|, then pointwise |C_3|<=S_3/(2-sqrt(2)). Its frozen fast derivative is bounded by sum_(L_nr)|Omega_mn*c_mn|/(2-sqrt(2)). These finite bounds retain all generated cubic labels; they are not bounds for later generated sidebands.

With the damping correction included in C, a candidate z_app=epsilon*z_1+epsilon^3*C has the exact two-row residual

\[ r_x=x_{\rm app}'+\sigma Ax_{\rm app} +\tfrac12D^T\sin(D\theta_{\rm app}),\qquad r_\theta=\theta_{\rm app}'-Ax_{\rm app}. \tag{12} \]

Slow derivatives of C, envelope variation across edges, actual loss and generated cubic cross terms with C remain in (12). They cannot be removed by the frozen-carrier cancellation. The sine tail has a separate bound: if |Dtheta_app|<=H on every edge, then

\[ \|R(\theta_{\rm app})-R_3(\theta_{\rm app})\|_2 \le\frac{\sqrt N H^5}{120}. \tag{13} \]

This follows from |sin a-a+a^3/6|<=|a|^5/120 and ||D^T||_2<=2. For a split edge gap a+b, the exact correction in the cubic row is D^T[3*a^2*b+3*a*b^2+b^3]/12; no generated term is discarded. Together with declared envelope derivative bounds these identities give inputs to a full residual estimate, not an evaluated estimate themselves. The carrier calculation itself supplies no profile, source-error ball, horizon or maintenance margin; section 8 admits a separate static source. The complete-law comparison owner provides the methodology; its single-envelope constants do not transfer.

7. A scoped stationary bright-pair obstruction¶

Consider only the conservative comparison (9) with gamma zero on the real line, and a stationary real-profile ansatz B_i(X,S)=exp(-i*Omega_i*S)*f_i(X). Assume sufficient regularity, integrable squares and fourth powers, vanishing integration-by-parts boundary terms, and nonzero exponentially decaying tails on the usual branch Omega_i/d_i>0. The second profile equation implies

\[ \Omega_2\int f_2^2 =-d_2\int(f_2')^2+g_2\int f_2^4+c\int f_1^2f_2^2>0 \quad\hbox{if }f_2\not\equiv0. \tag{14} \]

Every right-hand term is nonnegative, and the fourth-power term is strictly positive. But d2<0 makes the stated exponential-tail branch require Omega2<0. This contradiction excludes that stationary bright-bright template. The tail-sign premise is explicit; (14) is not a theorem about every possible decaying spectral solution.

At positive gamma, (10) already excludes a nonzero fixed-amplitude stationary modulation profile under these boundary conditions. The conservative obstruction is a comparison control, not permission to delete actual loss. Neither result excludes periodic profiles with a nonzero background, dark profiles, breathing or decaying organizations, different preparations, or finite-lived acquisition under the complete C1024 law. In particular, the line-tail premise does not transfer to the finite cycle.

8. Source admission and formation boundary¶

The carrier calculation selects no localized source and proves no entry into a common identity. A prospective source must define its initial pattern identities, absence of the proposed common relation with a margin, retained relative and boundary state, and a discriminator against separate evolution and linear overlap. The formation contract owns these obligations. Source-specific admission remains separate from the carrier control and from the deferred prepared-envelope certificate.

A bounded source with a common-window screen initially absent¶

This static class reuses only the Fejer normalization and spatial-tail proof from the transport owner. It does not inherit that owner's pi/2 transport or encounter certificate. Set Q=16, a=2^-12, delta=a/1000, and define

\[ b_m=1-|m|/Q,\quad S_2=\sum_{|m|<Q}b_m^2=171/16,\qquad f_Q(j)=\frac{\sum_{|m|<Q}b_m e^{2\pi imj/1024}} {\sqrt{1024S_2}}. \]

For fixed supplied phases psi1 and psi2, with zero as one explicit choice, let

\[ v_{1,j}=a e^{i(\psi_1+\pi j/4)}f_Q(j),\qquad v_{2,j}=a e^{i(\psi_2+3\pi j/4)}f_Q(j-512), \qquad z(0)=v_1+v_2+r,\quad\|r\|_2\le\delta. \]

Form and the supplied real phase lift are Re z and Im z; arbitrary fine-state error modes remain. The full field is sufficient state. Let P1 and P2 be unitary Fourier projections onto modes 113 through 143 and 369 through 399. Their bands are disjoint, omit zero, and give ||v_i||_2=a, P_i v_i=v_i, P_i v_(3-i)=0. Consequently

\[ \|P_i z(0)\|_2\ge a-\delta=999a/1000>a/2. \]

These projections are observations; retain their complement and both real rows. The nonlinear dynamics has not been proved to close on these bands, their intensities or their centers.

Let W(c,96) denote the cycle ball centered at a real observation position c. The common-window screen asks for one such ball containing at least 3/4 of each projected squared norm, with both projected norms at least a/2. It is a geometric prerequisite for a possible identity, not a sufficient definition of organization or a formation discriminator by itself.

The source centers are 0 and 512. The existing spatial-tail inequality gives

\[ \frac{\|\mathbf1_{W(c_i,96)^c}v_i\|_2^2}{a^2} \le b:=\frac{1024^3}{16\cdot16^3\cdot95^3} =\frac{16384}{857375}<\frac1{49}. \]

For every c, at least one distance to the antipodal centers is at least 256, exceeding 96+96. Its common window is disjoint from at least one source window. Contractivity of P_i and the triangle inequality then imply, for that i,

\[ \frac{\|\mathbf1_{W(c,96)}P_i z(0)\|_2^2}{\|P_i z(0)\|_2^2} \le\left(\frac{\sqrt b+1/1000}{1-1/1000}\right)^2 <\left(\frac{1007}{6993}\right)^2<\frac1{36}. \]

Thus throughout this source ball and for every proposed common center, at least one required fraction fails by more than 3/4-1/36=13/18. Both norm floors hold, so absence is not inferred from a vanishing signal. The proof covers fractional centers by the cycle triangle inequality.

Primitive phase admission is also explicit. Since ||f_Q||_infinity=2/sqrt(171)<2/13,

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

Orthogonality gives ||z(0)||_2<3a/2. Nominal means vanish, but arbitrary error means remain: |mean z(0)|<=delta/sqrt(1024)=a/32000. No projection onto zero conserved means is allowed. These are initial bounds, not continuing phase, norm or localization certificates.

This is a bounded mathematical preparation and absence result, not a materialized source, frozen response protocol or demonstrated path to the target. Entry into the screen would still require a retained relation, nontrivial lifetime and a continued-response control beyond linear overlap. A reorganization outside these fixed bands may require another identity; failure of this screen would not exclude all common organizations.

Information retained by a single total envelope¶

For a single carrier, the total-envelope cubic retains different information. Writing N(B)=|B|^2*B, any chosen sum satisfies the exact algebraic identity

\[ N(B_1+B_2)-N(B_1)-N(B_2) =2|B_1|^2B_2+2|B_2|^2B_1+B_1^2\overline B_2+B_2^2\overline B_1. \tag{15} \]

Here all terms share the same carrier, so relative phase remains in the total cubic response. Equation (15) does not assign unique lobe identities or closed equations to B1 and B2. Deriving a regional interaction or an acquisition discriminator still requires retained state, boundary work and the complete-law residual; the two-carrier control supplies none of them.

The same-band source and work owner admits a separate total-field preparation and derives exact regional rates directly from both real rows. Its initial resource matching and necessary concentration-prefix condition do not require an envelope closure or transfer the two-band screen; a maintained common organization remains unproved.

The static controls check exact incidence coefficients, retained labels and aliases, mixed pressure, leading loss, initial-absence margins and the scoped sign obstruction independently. They evaluate no prepared trajectory or reserved response. The reusable regional storage balance and relative-state observation boundary remain necessary when a later source and common identity are selected.