Dynamical protection and regeneration of organized contact¶
This owner answers the bounded mechanism question declared at 173265a3 in
the execution plan.
The ideal fivefold host cancels the initial aggregate contact current for
both retained constitutive laws. That cancellation persists at particular
symmetry origins; at every other origin reciprocal dynamics regenerates a
nonzero donor mean. The first nonzero derivative depends on the supplied law.
These are exact conditional results about the declared limiting profiles. The sine contact and alternative contact retain their actual acquired sources, finite error bounds and observation histories. No finite source is reset to the profiles studied here, and no response derivative is identified with an experimental measurement.
1. Complete model, initial state and balances¶
Retain the constitutive family at its two declared endpoints, \(\lambda=0,1\):
The support is \(C_9\mathbin\square W_6\): six internal nine-cycles, five host rim edges and five donor spokes at each slot. Its 54 nodes are \((c,j)\), where \(c=0,\ldots,4\) labels the host, \(c=D\) the donor, and \(j=0,\ldots,8\). All 144 edges have weight one. On that full support,
Prime denotes the declared clock \(\tau=(1023/1024)t\). Forms are signed, capacities are held at one, and phases occupy the full circle. Both rows have the same supplied quarter factor; neither row uses inverse degree. There is no continuous input or later event. The pressure in the form row is supplied independently of the response. Smooth periodic currents and the linear terms give global existence and uniqueness.
The conditional initial family is exactly
The symbolic registration \(\beta\) ranges modulo a turn. The observation is \(R_D(\tau;\beta)=\frac19\sum_jx_{D,j}(\tau)\). The initial zero total form remains zero; the declared phase lifts have constant total \(9\beta\). This is the exact limiting organization supplied to a new conditional question, with no observation word or residual replacement.
Reciprocity retains both contact reactions and the full storage identity:
The attachment in (3) supplies the separate work \(W=9\sum_cV_\lambda(\beta-\phi_c)=45+5\lambda/2916\), independent of registration. The root-of-unity sums of harmonics 1, 9 and 18 vanish. Thus initial storage is \(6V_9+9V_{5,\lambda}+W\), where \(V_9=9V_\lambda(2\pi/9)\) and \(V_{5,\lambda}=5V_\lambda(2\pi/5)\). Independence of \(\beta\) does not make the positive attachment cost passive or select its occurrence.
2. Exact carrier and complete-field symmetries¶
The twelve-coordinate carrier \(x_{c,j}=X_c\), \(\theta_{c,j}=\psi_j+\Phi_c\) is invariant for each law in (1): the internal clockwise and counterclockwise currents cancel, as do the internal form differences. With the wheel Laplacian
the complete restriction is \(X'=(-KX+\gamma S_\lambda(\Phi))/4\), \(\Phi'=\gamma KX/4\), and \(R_D=X_D\). All six forms and six phase coordinates evolve. The carrier storage is \(6V_9+(9/2)X^TKX+9\sum_{\text{wheel edges}}V_\lambda\), with loss \(-9\|KX\|^2/4\). This restriction preserves the full fine-state embedding; it is not a closed equation for an instantaneous contact moment.
The eigenvalues of \(B\) on the nonconstant rim Fourier modes are
Support permutations and common phase translations are exact symmetries of (2). Since each current is odd, every graph automorphism \(P\) also gives the complete-field conjugation
Indeed both right-hand sides acquire the same required minus sign and permutation. Equality of phases here is circular; constant integer turns can be supplied to the transformed lifts to match initial representatives.
For \(\beta=m\pi/5\), take \(b=\beta\) and the full-support reflection
Here \(\psi_{8-j}=-\psi_j\) and \(\phi_c+\phi_{r-c}\equiv2\beta\pmod{2\pi}\). The state (3) is fixed by (7). Uniqueness therefore preserves this fixed sector for all future time, and \(x_{D,j}=-x_{D,8-j}\) gives
This proof uses the actual complete field, including slot reflection and form sign reversal. A stationary reflection constraint alone would not establish invariant dynamics.
For general registration, a common phase shift by \(2\pi/5\) followed by a rim relabeling restores the host family. Conjugation with \(b=0\), \(r=4\) reverses its donor origin. Consequently
These restrictions will remove impossible response harmonics. They also reduce the odd multiples of \(\pi/10\) to the two representatives \(\pm\pi/10\), related by sign.
3. Alternative-law generic onset¶
Put \(\delta_c=\beta-\phi_c\), \(f_c=j_\lambda(\delta_c)\), and \(p_c=j_\lambda'(\delta_c)\). Root-of-unity filtering gives \(\sum_cf_c=0\). The exact hub equation, using zero total form, is
Initially the phase velocity is zero. Thus \(R_D(0)=R_D'(0)=R_D''(0)=0\), \(\delta''=-\gamma^2Bf/16\), and
Write the three current coefficients as \(b_1=1\), \(b_9=\lambda/2187\), \(b_{18}=-\lambda/4374\). In \(f^TBf\), only the pair \(1+9=10\) contributes registration dependence: every other nonconstant sum or difference fails the fivefold filter. Modes 1 and 9 have the same eigenvalue \(\ell_1\), so
For \(\lambda=1\) and \(\beta\notin(\pi/10)\mathbb Z\), this is the first nonzero derivative, with a strictly positive prefactor. For the sine law it vanishes identically, requiring the separate argument below. Matching initial cancellation does not transfer the later response.
4. Alternative-law exceptional origins¶
Even multiples of \(\pi/10\) are already protected by (9). At \(\beta=\pi/10\), put \(u=1/2187\). On the five initial gaps, \(10\delta_c\equiv\pi\pmod{2\pi}\), so the actual current and its derivatives, differentiated before this substitution, are
The third derivative is zero by (13). The fourth contains only bilinear terms of these discrete modes with powers of \(B\); their sums are zero at this origin. More explicitly, \(\delta'''=\gamma^2B^2f/64\) and \(p^TB^2f=0\), so (11) gives \(R_D^{(4)}(0)=0\).
For the fifth derivative set \(v=Bf\). Differentiating both complete rows, with the first four donor derivatives zero, gives
Here \(pv\) is componentwise multiplication. To check that the rim reaction has not been omitted, let \(J=DS_1(\Phi(0))\). On \((v,0)\) its rim row is \(-wL_{C_5}v-pv\), where \(w=j_1'(2\pi/5)\), and its hub row is \(p^Tv=0\). The fourth gap derivative therefore contains \(-\gamma^2B^3f/256-\gamma^4(KJ(v,0))_{\rm rim}/256\). The terms \(p^TB^3f\) and \(p^TBL_{C_5}Bf\) vanish by the same bilinear filter; the remaining terms give (15). The factor 3 comes from the second phase derivative in the fourth derivative of the current.
For an explicit sign calculation, write \(U=\ell_1(1+u)\), \(V=\ell_2u/2\), \(D=1-9u\), \(E=-9u\), \(F=-(1+81u)\), \(G=-162u\). The only surviving cubic products have total discrete harmonic five. The sums of \(\cos\delta\cos2\delta\sin2\delta\) and \(\cos^22\delta\sin\delta\) are \(5/4\); the sum of \(\sin\delta\sin^22\delta\) is \(-5/4\). Thus
Positivity follows already from \(\ell_2<2\ell_1\): the bracket exceeds \(2250\ell_1-15\ell_2>0\). Substitution simplifies (15) to
The coefficient is strictly negative. Equation (10) extends the calculation at \(\pi/10\) to every odd exceptional origin, where \(\sin5\beta=\pm1\). The fifth derivative is therefore the first nonzero one throughout this exceptional set.
5. Sine-law onset and the fivefold filter¶
For \(\lambda=0\), use the temporary algebraic clock \(s=\tau/4\) in both carrier rows. This removes their quarter factors; derivatives in the declared clock will be restored explicitly. Every contact derivative is a finite Fourier polynomial in \(z=e^{i\beta}\). Induction in the complete rows gives
For phase, the bound concerns its displacement from the initial phase. The displacement has no constant or first-order term. A contact current has one initial factor \(z^{\pm1}\); a rim current has none. Expanding its phase exponential, each displacement coefficient of time order \(n\) contributes at most \(\lfloor n/2\rfloor\) to Fourier degree. Multiplication, one integration in the form row, and the next integration in the phase row preserve (18). Linear damping adds no degree. By (10), the donor coefficients are odd and contain only multiples of harmonic five. Their degree is at most four through order eight, so all those donor derivatives vanish exactly.
Only the maximal positive Fourier degree can contribute harmonic five at order nine. At these minimal time orders, damping and the rim current cannot attain that degree. Retain the maximal-degree part of the host phase displacement in the form
Here the projection retains only \(s^{2n}z^n\), \(1\le n\le4\); the conjugate negative modes remain in the real phase but cannot contribute to this maximal positive-degree coefficient. The rim Laplacian remains present through its eigenvalues \(\ell_n\) in the phase row. The donor phase cannot contribute at these orders: \(\Phi_{D,s}=6\gamma R_D\), and its displacement starts at order ten or later. Equating maximal-degree terms gives the finite recurrence
In particular,
For example \(\sqrt5<9/4\) already proves the last strict inequality. The resulting donor current term is \(-5b_4\gamma^8s^8\sin5\beta\). Its form damping only affects later orders, so integration and restoration of the tau clock give
The prefactor is strictly negative. For every \(\beta\notin(\pi/5)\mathbb Z\), the ninth derivative is the first nonzero one. Every remaining origin is protected for all time by (9), rather than inferred invariant from a zero finite jet.
6. Exact implementation and outcome¶
The exact coefficient kernel retains all twelve carrier coordinates and all ten reciprocal wheel edges. It expands each edge exponential by its differential recurrence in the exact coefficient algebra \(\mathbb Q[\zeta_{20}][\gamma,z,z^{-1}]\), where \(\Phi_{20}(a)=a^8-a^6+a^4-a^2+1\), \(i=\zeta_{20}^5\), and \(z=e^{i\beta}\). This preserves exact cancellation without floating phase substitutions. The generic rows and the one symmetry-distinct exception at \(\beta=\pi/10\) stop at their first nonzero donor coefficient, within the frozen ceiling of twelve. Derivatives are factorial times the separately retained Taylor coefficients.
The independent root sums and maximal-degree calculation above determine the signs and the origin domains; the exact recurrence checks all preceding coefficients and the complete carrier wiring. No zero jet alone certifies an invariant sector, and no trajectory or acquired-source producer is used. The kernel shares the exact Fourier coefficients with the constitutive owner and existing contact observer. Its contract tests check the cyclotomic algebra, analytical coefficients, complete 54-node field residuals, conserved sums, reflection and bounded admission/stopping.
| Supplied law | Registration domain | First nonzero donor derivative |
|---|---|---|
| Either endpoint | \(\beta\in(\pi/5)\mathbb Z\) | None: complete-field symmetry proves zero response for all future time |
| Sine, \(\lambda=0\) | Every other origin | Order 9, equation (22) |
| Alternative, \(\lambda=1\) | \(\beta\notin(\pi/10)\mathbb Z\) | Order 3, equation (13) |
| Alternative, \(\lambda=1\) | Odd multiples of \(\pi/10\) | Order 5, equation (17) |
Thus the initial fivefold cancellation is not an origin-independent interaction constraint. Reciprocal motion generates a collective response at a law-dependent order, while an exact symmetry protects a discrete set of origins. All registrations are classified within the original budget. A nonzero leading derivative guarantees a nonzero response for sufficiently small positive time at that fixed origin; it supplies no noise-resolved horizon or actual-source detection threshold.
At common generic origins, where \(\sin10\beta\ne0\), the onset orders nine and three cannot be made equal by a regular clock change or regular scalar re-expression of this observation. Indeed, if \(R(\tau)=a\tau^n+o(\tau^n)\), \(a\ne0\), a clock \(\tau=f(s)\) with \(f(0)=0\), \(f'(0)>0\) preserves order \(n\). So does \(h(R)-h(0)\) with \(h'(0)\ne0\). This is a local obstruction for the same matched preparation and donor observable, not an exclusion of arbitrary state transformations or observation maps consuming hidden coordinates. A common formed identity does not force a common effective interaction law.
All-future zero donor mean does not imply zero local currents, stationary fine state or permanent retention of the marked windings. Nor does the difference of onset orders select a fundamental law, establish a particle property, provide autonomous attachment or complete a physical measurement bridge. The prior finite-source contact bounds and the separate unresolved maintenance problem retain their original scopes.
The full-fine preparation analysis separates the robustness obligations: internal disturbances can be invisible to every first-order mean response while generating quadratic leakage. Exact reflection-preserving preparations remain protected; winding and child means alone do not preserve this owner's ideal onset orders.