Internal preparation modes and organized contact response¶
This owner answers the full-fine preparation question declared at b2801170
in the execution plan.
Every internal disturbance with zero mean in each child is invisible to the
donor mean at first variation along the entire evolving ideal carrier.
Opposite internal modes can nevertheless generate a second-order mean
response. Exact reflection protection is a separate, stronger condition.
The result concerns conditional perturbations of the exact limiting family in the dynamic cancellation theorem. It does not turn those perturbations into acquired residuals or supply a finite noisy response, transverse stability theorem or physical readout.
The constituent-memory theorem already establishes tangent decoupling, quadratic covariance feedback and closure collisions for the sine \(C_9\mathbin\square C_5\) model. The present contribution is the full \(C_9\mathbin\square W_6\) reciprocal contact under both retained laws: all-time internal-mode invisibility of its donor observable, the distinction between a linear kernel and exact reflection protection, and explicit phase/form disturbances that break the ideal onset behavior. Its coefficients and conclusions are derived on this support rather than transferred from that predecessor.
1. Full model and preparation directions¶
Retain exactly the two endpoint currents
and their potential \(V_\lambda\) from the prior owner. On all 54 nodes and 144 unit edges of \(C_9\mathbin\square W_6\), both complete rows are
Prime means \(\tau=(1023/1024)t\). Forms are signed, capacities are held at one, and phases occupy the full circle with declared lifts. There is no later input or event. The same quarter factor remains in both rows; the donor and host both react to contact.
Write \(\theta^0(\beta)\) for the ideal initial phases \(\psi_j+\phi_c\) on host child \(c=0,\ldots,4\), and \(\psi_j+\beta\) on the donor, where \(\psi_j=2\pi(j-4)/9\), \(\phi_c=2\pi(c-2)/5\). The perturbed initial state is
Both directions have all 54 coordinates. Their constraints fix total form and the phase gauge, so the complete histories have total form zero and lifted phase total \(9\beta\). The amplitude \(\epsilon\) scales specified form and phase directions in their respective structural units; no physical identification of these units is supplied.
Let \(\bar y_c=\frac19\sum_j y_{c,j}\). Each row splits uniquely into its six child means and eight zero-mean modes per child. Thus the constrained preparation space has ten collective and 96 internal real directions. The donor observation is the same linear functional \(d^Tx=\bar x_D=R_D\); it does not consume hidden state directly.
2. Variations along the evolving full carrier¶
Denote the unperturbed full solution by \((x_0,\theta_0)\). It remains on the twelve-coordinate carrier, but its forms and origins evolve under both rows of (2). Define the amplitude expansion on every fixed finite time interval by
The second coefficients are one half of the second amplitude derivatives. They are not additional preparations or frozen source coordinates. Smooth dependence follows from the smooth complete field. Put \(J(\tau)=DS_\lambda(\theta_0(\tau))\) and \(Q(\tau)=D^2S_\lambda(\theta_0(\tau))\). Explicitly,
With \(A=-L/4\), \(b=\gamma/4\) and \(C=\gamma L/4\), the complete variational equations are
All four variation rows have zero global mean. In particular, the second-order forcing retains equal opposite edge reactions; it creates no net form. The time dependence of \(J,Q\) is retained throughout.
Let \(T\) be the simultaneous cyclic slot shift in all six children. The combined map \((x,\theta)\mapsto(Tx,T\theta-\alpha\mathbf1)\), \(\alpha=2\pi/9\), fixes every carrier point circularly. It is a symmetry of the complete field because slot shift is a support permutation and common phase translation is a symmetry. Its tangent is \(T\) on both variation rows. Consequently \(J(\tau)T=TJ(\tau)\), for every time along the evolving carrier.
In complex slot mode \(k\in\mathbb Z/9\mathbb Z\), write \(\mu_k=2-2\cos(2\pi k/9)\). The first variation retains the full six-form/six-phase system in that mode, with
Here \(K\) is the wheel Laplacian and \(\Phi\) comprises all six evolving origins. No mode is fed by another at first order. Real directions retain conjugate mode pairs. If \(\bar u_c=\bar v_c=0\) in every child, only \(k\ne0\) is present, and uniqueness in the zero mode gives
This is an all-time first-variation statement. It does not assert a uniform amplitude remainder over an infinite horizon or stability of internal modes. Collective directions instead obey the complete wheel tangent system and can affect the donor mean at first order. Equation (8) supplies a 96-real- dimensional invisible subspace; it does not classify every possible additional kernel of the full output history.
3. Quadratic selection and exact reflection protection¶
Slot equivariance gives \(TQ[z,w]=Q[Tz,Tw]\). Thus modes \(k,l\) force only mode \(k+l\pmod9\) at second order. A mean response receives the collective pair \((0,0)\) or internal pairs \((k,-k)\); a collective/internal pair cannot reach mode zero. In the internal case the mean second-order system is
Internal-cycle edge reactions cancel exactly in the last child average. They still affect \(y_1\) through (7). With the Fourier convention \(y_j=\sum_k\widehat y_ke^{2\pi ikj/9}\), the squared gap average is \(\sum_k\widehat{\Delta y}_k\widehat{\Delta y}_{-k}\). For real directions it is a sum of squared magnitudes, but its current coefficient \(j_\lambda''\) can have either sign. Opposite-mode pairing allows mean forcing; it does not guarantee a nonzero response for every internal direction.
Exact zero response has a stronger sufficient condition. At \(\beta=m\pi/5\), let the full-support reflection be \(P(c,j)=(4+m-c\bmod5,8-j)\), \(P(D,j)=(D,8-j)\). The complete-field conjugation fixes (3) for every amplitude whenever
Uniqueness then gives \(R_D(\tau;\epsilon)=0\) for all future time, not merely zero first or second variation. The reflection condition is a constraint on both rows and the complete fine state. It need not follow from child means, winding labels or the zero-mode kernel (8). Directions that satisfy neither this protection nor a nonzero coefficient below remain unclassified beyond the bounded amplitude and time orders; vanishing coefficients are not promoted to an invariance theorem.
4. Complete contact-time jets through order three¶
The following formulas retain all directions in (3), including mixtures of form, phase and collective perturbations. In this section evaluate \(J,Q\) at \(\theta^0\), let \(T_3=D^3S_\lambda(\theta^0)\), and put \(a=CbS_\lambda(\theta^0)=\theta_0''(0)\). Although \(\theta_0'(0)=0\), this acceleration is generally nonzero.
The linear coefficients of the form time derivatives are
The quadratic amplitude coefficients are
Apply \(d^T\) only after forming these full vectors. The terms containing \(a\), the diffusion terms and both reactions prevent a frozen-host or prescribed-donor interpretation. The factor one half in (12) follows the coefficient convention (4), not a change in the complete law.
For pure phase disturbances the first quadratic donor derivative is
For pure form disturbances \(v=0\), both quadratic time derivatives of orders one and two vanish. Phase feedback yields at order three
This term would be lost by suppressing the phase row. These are contact-time derivatives; the corresponding Taylor terms in time carry the usual factorials, independently of the amplitude convention.
5. Winding-admissible internal leakage witnesses¶
Set \(\beta=0\), where the unperturbed response is identically zero for both laws. Choose the rational zero-mean slot vector \(w=(1,-1,0,0,0,0,0,0,0)\), supported in host child \(c=1\) only. Use it first as \(v\), with \(u=0\), then as \(u\), with \(v=0\). Both directions belong to the all-time first-order kernel (8); neither satisfies the full reflection condition (10).
Write \(s_1=\sin(2\pi/5)>0\), \(s_2=\sin(4\pi/5)>0\), and \(q_1=j_\lambda''(2\pi/5)\), \(q_0=j_\lambda''(4\pi/5)\). The actual endpoint laws give
For the phase witness, \(\overline{w^2}=2/9\), so (13) gives
For the form witness, the spoke gap in \(Lu\) is \(-(L_{C_9}+4I)w\) at child 1, zero at its two rim neighbors, and \(-w\) at the two remaining host children. Direct integer arithmetic gives \(\|w\|^2=2\) and \(\|(L_{C_9}+4I)w\|^2=100\). The full set of five contact gaps in (14), rather than the disturbed child alone, therefore gives
Both are strictly positive: \(s_2<s_1\), and the alternative expression has two positive terms. This is the first nonzero quadratic time derivative for the form witness. The linear donor variation is zero at all times in both examples, not just through this jet budget.
An explicit common initial winding allowance is \(0<|\epsilon|\le1/10\). In the phase witness internal edge deviations are at most \(2|\epsilon|\le1/5\), and host outer edge deviations at most \(|\epsilon|\le1/10\). Since \(\alpha-1/5>0\), \(\alpha+1/5<\pi/2\), and \(2\pi/5\pm1/10\in(0,\pi/2)\), every marked internal and outer cycle retains its initial winding-one acute chart. Undisturbed cycles are unchanged. The form witness leaves all initial phases unchanged. No winding is assigned to the new contact loops, and initial admission proves no finite-time maintenance interval.
Nonzero quadratic coefficients imply leakage for sufficiently small nonzero amplitudes inside this admitted range; the range itself is a winding allowance, not a uniform truncated-series error certificate. For the pure form witness the third contact-time derivative is exactly quadratic in \(\epsilon\): differentiating (2) three times introduces at most two initial form factors when initial phases are fixed. Its constant and linear donor terms vanish here. Neither statement identifies an acquired source or requires a trajectory evaluation.
6. Work, storage and the scope of robustness¶
Attachment preserves the prepared coordinates and adds the actual contact storage. With \(\delta_c=\beta-\phi_c\), \(\Delta u_{c,j}=u_{D,j}-u_{c,j}\), and similarly \(\Delta v\),
Here \(W_r\) are amplitude coefficients. Internal zero-mean phase directions have \(W_1=0\). The quadratic correction need not be nonnegative because the contacts need not be acute. The positive baseline attachment cost remains supplied work; continuous loss does not fund it without a separate reservoir and occurrence law.
The full storage remains \(H=\frac12x^TLx+\sum_{\text{all edges}}V_\lambda\), with \(H'=-\|Lx\|^2/4\). At preparation its quadratic coefficient is
This is not automatically a coercive perturbation energy. Along the evolving base the quadratic coefficient of the storage-loss identity is \(-[\|Lx_1\|^2+2\langle Lx_0,Lx_2\rangle]/4\), which need not be nonpositive by itself. First-order invisibility therefore implies neither internal damping nor long-time transverse stability.
The mechanism separates three statements: the complete slot symmetry hides all internal modes at first order; opposite modes can feed a collective response at second order; full reflection conditions can protect exact zero response. The leakage witnesses show that zero response at a protected ideal origin is not a property of its full open preparation neighborhood. The ideal law-dependent onset orders consequently require their exact preparation scope. This does not alter the earlier finite acquired-source contact bounds, classify every zero quadratic direction, establish a noisy discriminator, select a unique law or identify a physical property.
7. Detached coefficient observer¶
The full-fine observer admits both complete primitive direction vectors, the endpoint law and a supplied registration before forming the derivatives in (11)--(12). It reuses the shared contact geometry, unit Laplacian, edge law and outward rational interval arithmetic. The report distinguishes amplitude coefficients from time derivatives and retains the evolving base's initial phase acceleration, work/storage coefficients and initial winding allowance.
Exact child-mean and reflection relations justify their corresponding symmetry flags through (8) and (10). An interval containing zero does not establish a vanishing derivative or invariant output. The symbolic registration and mode statements belong to this proof; the observer at a supplied origin is not a registration scan, source certificate or finite trajectory. Its fixed limits are preparation amplitude order two and contact-time derivative order three. The contract tests differentiate the complete fine-state time rows independently of the amplitude recurrence, and check all form/phase coefficients, work, storage, explicit witnesses, primitive admission and bounded stopping. They evaluate no acquired source or trajectory.