Constitutive dependence and source selection under weak contact¶
The retained harmonic alternative changes the attracting relative registrations of the weakly contacted organization. At the fixed spoke weight eta=1/100000, a declared full-state neighborhood contains twenty stationary organizations modulo common phase rotation: ten strict relative minima and ten index-one saddles. All five origins that attract under the sine law become saddle origins under this alternative.
An own-law source theorem below also places ten explicit intervals of the original donor preparation inside the alternative's attracting basins, retaining all original errors and the complete observation history. On one declared common preparation interval, the two laws select asymptotic relative origins separated by more than pi/12. The alternatives retain the same support, capacities, clock and constituent profile; their collective registration geometry is nevertheless different. Neither reached sources nor the sine basin certificate are substituted for the alternative's own history and bounds.
| Premise | Fixed declaration |
|---|---|
| Constitutive law | Deformation-one potential and current already defined in the alternative contact owner |
| Support | The same 54 labeled nodes of C9 x W6; 99 old internal/rim edges of weight one, 45 donor spokes of weight eta=1/100000 |
| Complete dynamics | Both weighted form diffusion and weighted nonlinear phase current, held unit capacity, uniform quarter factors and the original tau clock |
| State and symmetry | All signed forms and primitive circular phases; zero total form, common phase rotation factored out; continuous compatible lifts only for the declared local coordinates |
| Stationary domain | Mean-zero host shape norm at most 6*eta and childwise mean-zero fine phase maximum norm at most 1/100; all donor origins admitted |
| Actual source | The original alternative-law outer-one acquisition, full sequential observation word and independent donor recovery; all twelve primitive allowances and all 96 relative directions |
| Certified preparation intervals | Donor origins within 1/10 of each odd multiple of pi/10 reach the unique attractive equilibrium in the opposite relative-origin window |
| Evidence | Exact finite Fourier identities, real contraction and explicit derivative/remainder bounds; no root search, trajectory, source replay, sweep or physical response |
1. Complete weighted law and the full-state domain¶
Write c=0,...,4 for host children and c=5=D for the donor; j=0,...,8 is the internal slot. The primitive node order is 9*c+j. Put
Let L_eta be the unnormalized weighted Laplacian, and let \(S_i(\theta)=\sum_{b\sim i}w_{ib}j(\theta_b-\theta_i)\). Write \(U(\theta)=\sum_{\{i,b\}}w_{ib}V(\theta_b-\theta_i)\) for its phase potential. The complete unforced law and its storage are
Prime means the declared tau clock in both rows. The pressure in the held-capacity nodal row is supplied by the first complete equation; it is not reconstructed from a response. Odd current and Laplacian row sums preserve total form and total lifted phase. All coefficients retain their structural units; no laboratory clock or storage unit is identified. Support, eta and the absence of later inputs/events remain premises. The balance follows from this alternative potential and current, not from copying a sine certificate.
Put alpha=2pi/9 and b=2pi/5. With compatible old marked-edge lifts, write
Here t is a relative origin, not the evolution clock. The domain is
Forms may be any finite signed values on that mean leaf. Internal closing edges and host-rim closing edges retain their original one-turn conventions. All old compatible gaps remain positive and below pi: internal gaps differ from alpha by at most 1/50; rim gaps differ from b by at most 12*eta+1/50. Thus (4) retains all six internal windings, all nine rim windings and all 32 zero mixed host periods of the complete marked class. Spoke phases have no added acute restriction.
The domain is a stationary classification neighborhood. It is not asserted to be positively invariant or to contain the initial actual source. Sections 6--8 retain that source in a larger energy trap and classify its limit using the stationary theorem. No finite settling time is supplied.
2. Own-law host-shape contraction¶
The real derivative bounds from (1) are
Let D be the forward C5 incidence, L5=D^T D and \(P=I-\mathbf1\mathbf1^T/5\). Define
G acts on the zero-mean host space. The elementary bounds \(3/10<c_b<1/3\) give \(\lambda_1>2/5\); also \(a<8/25\) and \(2-2c_b<7/5\) give \(\lambda_1<9/20\). The bound a<8/25 follows, for example, from \(c_b<5/16\) and \(\sqrt5/486<1/216\). Consequently \(\|G\|_2<5/2\), and \(\|D\|_2<2\).
On the internal carrier q=0, ground the donor phase at zero and seek host phases \(t+b_c+u_c\), with zero-mean u. The four projected stationary equations are exactly
The scalar b in N is the rim gap; the b inside the vector current is \((b_c)_{c=0}^4\). All current derivatives below are evaluated on real arguments. Since \(|N(d)|\le J_2d^2/2\), the map satisfies on \(\|u\|_2\le6\eta\)
The bounds use \(\|D^TN(Du)\|_2\le4J_2\|u\|_2^2\) and \(\|D^T\operatorname{diag}(N'(Du))D\|_2\le8J_2\|u\|_2\). Thus, for every real t, there is exactly one projected stationary shape u(t) in the declared host ball. It is smooth in t by the invertible projected derivative. This is an own-law real contraction; the complex growth bounds of the sine current are not reused.
Write \(f(t)=j(t+b)\), \(u_1(t)=-Gf(t)\). Each active frequency 1, 9 and 18 has zero sum on the five host vertices. Direct subtraction of (7) and differentiation in t give
In the last line, retain separately the rim derivative defect, \(\eta[j'(t+b+u)-j'(t+b)]\), and \(\eta\operatorname{diag}(j'(t+b+u))u_t\). These terms are what bound the derivative of the actual shape, rather than just its values.
3. Collective torque and its finite C1 remainder¶
Let W(t) be the alternative wheel potential evaluated at its projected stationary shape. The remaining stationary equation is
Cyclic relabeling makes W and T periodic with period 2*pi/5. Reflection and phase inversion make W even and T odd. Both claims use uniqueness of the shape in (8), as well as the actual symmetries of (1). Therefore \(T(m\pi/5)=0\) exactly for every integer m. Symmetry alone does not determine their curvature.
Put \(\varepsilon=1/2187\). The five-point Fourier identities give
Indeed the ninth harmonic is spatially mode minus one on this rim, whereas the eighteenth is mode minus two. Only frequencies 1 and 9 have a nonconstant cross term in f^TGf; their inner product is \(-5\cos(10t)/2\). The eighteenth-harmonic square contributes the constant last term. Equation (11) is an exact coefficient identity, not yet a finite-eta verdict.
Taylor's formula, (5) and (9) bound the actual torque and its derivative:
Here are explicit constants for both remainders:
The rational upper bound \(\sqrt5<161/72\), obtained from \(161^2-5\,72^2=1\), proves the displayed strict inequalities. For C0, subtract the exact linear term and use the actual current's quadratic J2 Taylor remainder. For C1, expand \(j'(t+b+u)(1+u_t)\): the two linear defects use 416 and 832, the second-order j' remainder contributes \(J_3\|u\|^2/2\), and the product \([j'(t+b+u)-j'(t+b)]u_t\) contributes \(J_2\|u\|\|u_t\|\). No unbounded derivative of a remainder is inferred from a value estimate.
4. Twenty stationary registrations in the declared domain¶
On the relative-origin circle, let
These intervals are disjoint. At their endpoints and everywhere outside their interiors, \(|\sin(10t)|\ge1/2\). The leading torque magnitude there exceeds \(\eta^2/80\), while its error is at most \(960\eta^3=0.0096\eta^2<\eta^2/80\). It has opposite signs at the two endpoints of every interval and no zeros outside the intervals.
Within I_k, \(|\cos(10t)|\ge\sqrt3/2>4/5\), and its sign is \((-1)^k\). The leading derivative magnitude exceeds \(\eta^2/5\), while its error is at most \(3001\eta^3=0.03001\eta^2<\eta^2/5\). Thus the actual torque is strictly monotone in each interval and has exactly one simple zero there.
For even k, the zero is exactly \(t=k\pi/10=m\pi/5\), by the symmetry already proved. Its derivative is negative. For odd k, the unique zero in I_k has positive derivative. The proof does not assign its exact position or assert that it differs from its interval center. There are exactly ten of each type.
The four host shape directions have positive curvature. In fact the rim weights in the domain satisfy the lower bound proved in section 5, so the host shape block exceeds \(L_5/4-\eta J_1I>I/5\). Eliminating that block gives the scalar Schur complement W''=T'. Consequently the wheel relative Hessian is positive definite at the ten odd-interval zeros, and has four positive directions and one negative direction at the ten exact even-interval zeros.
5. Full-state stationary exclusion and stability¶
The full fine directions cannot create additional stationary points inside (4). To prove this for the actual alternative, let \(C_6(v)=1-v^2/2+v^4/24-v^6/720\). Along the straight segment from the carrier phases to any q in (4), compatible internal gaps obey \(|\delta|<18/25\), and rim gaps obey \(0<\delta<13/10\). For example, \(\pi<22/7\) gives \(\alpha+1/50<18/25\) and \(b+12\eta+1/50<13/10\). Since \(j'(\delta)\ge\cos\delta-2/243\), exact rational cosine bounds give
Spoke curvatures may be negative, but are bounded below by -J1. Let P_f subtract each child's slot mean. On its 48-dimensional range, the internal C9 Laplacian has gap greater than 2/5, while the largest eigenvalue of the nine disjoint spoke stars is six. Every Hessian along that segment therefore satisfies
For any collective phases, the carrier has \(P_{\rm f}\nabla U=0\): internal currents cancel and wheel currents are constant in the slot. Integrating (16) consequently gives \(q^TP_{\rm f}\nabla U(\theta)>\|q\|^2/4\) for nonzero q.
At a full equilibrium, the phase row in (2), connected support and zero total form first force x=0. The form row then forces \(\nabla U=0\), which by (16) forces q=0. The projected host equation and contraction force h=u(t). Thus all full stationary states in (4) are exactly the twenty wheel equilibria already classified; noncarrier directions were retained and excluded by their own-law curvature, rather than discarded before testing stationarity.
At a carrier, \(j'(\alpha)=\cos\alpha\) exactly. In slot-first order the phase Hessian has the product form
The 48 fine directions remain positive by (16), while the five relative collective directions have the inertia from section 4. The ten minima therefore have 53 positive relative phase directions. The ten saddles have 52 positive directions and one negative direction, with no extra zero direction besides common phase rotation.
This phase curvature controls the actual complete two-row dynamics. On the relative space its own linearization has the symmetric pencil
For a positive phase Hessian, the scalar quadratic identity obtained by testing an eigenvector has all roots with negative real part. For an index-one Hessian, the pencil increases strictly for positive real s from one negative direction to a positive matrix, giving exactly one positive eigenvalue. The imaginary part of the same scalar identity excludes nonreal eigenvalues with nonnegative real part; invertibility of the relative Hessian excludes s=0. All remaining eigenvalues have negative real part. Thus the ten minima are locally exponentially attracting on fixed-mean leaves, and the other ten are hyperbolic index-one saddles of the full law.
6. The unchanged own-law source enters a full-state trap¶
Keep the original outer-one family from the retained alternative acquisition, its complete alternative-law observation history, and the independently recovered donor from that same owner. The twelve primitive form/phase error allowances stay in [0,1e-10], with all 96 relative directions and their original zero-sum constraints. The donor origin beta is supplied at primitive preparation. Its isolated rotation does not change its relative recovery trajectory.
All acquisition times, recovery dwells, commands and observation windows are unchanged. In particular the full fifteen-event word precedes contact; its final cumulative command vanishes, while every continuous drift and form change remains. Only the 45 new spoke weights are changed to eta in the subsequently supplied contact model (2). The earlier unit-contact observations and frozen responses are neither reused as weak-contact predictions nor re-evaluated.
Write
The retained own-law acquisition inequalities give host excess above 5V9+9V5 below 0.009588 and its product-cell boundary excess B above 0.01451. These are existing strict rational summaries of the preserved source evidence, not newly evaluated endpoints. The same cycle-boundary formula applies after adding the donor C9: the old cycles now comprise six C9 and nine C5 cycles, with unchanged old edge weights.
6.1 Observation drift and new contact work remain explicit¶
For a fresh analytical upper bound \(\bar\gamma=1/3000>\gamma\), put X=1/100 and T_o=1/5000. The own-law current bound J0 in (5) gives
Heat contraction bounds every host form during the word by B_o; its degree-four form row then bounds each net form change by d_x. The phase row bounds each continuous phase drift by m_o after removing only the explicit command. Every one of the 90 host edges has net form-gap change at most 2d_x and net phase-gap change at most 2m_o. The square-difference identity and \(|V'|\le J_0\) therefore give
This endpoint comparison does not cancel the paid work of intermediate events or identify the reached endpoint with the pre-word state.
The donor retains form and relative-phase norms at most \(q_D=10^{-70}\). Its target phase gradient vanishes, its C9 Laplacian norm is below four, and \(|V''|\le J_1\), so \(H_D\le V_9+2(1+J_1)q_D^2<V_9+5q_D^2\). The weighted support event changes no coordinate and has work
The extra 2/19683 is the global upper bound on the alternative's nonnegative added potential. This is supplied attachment work; past loss does not fund the event or select its occurrence.
Combining these own-law inequalities proves, for every original beta and every admitted original error member,
Time zero is the actual final contact, not a reset. The pre-contact phase inclusions from the own-law source and complete observation word place every old edge in its original acute branch. The donor has the same winding-one C9 branch. No energy inequality is used to invent this initial branch information.
6.2 The old identity is retained for every future time¶
The own-law acute convexity proof establishes \(V''>0\) on the entire closed interval [-pi/2,pi/2]. Thus Jensen's inequality on the actual old cycles gives the boundary excess B from that owner. The 198 signed old-edge faces consist of 54 positive and 54 negative internal faces, 45 positive rim faces and 45 empty negative rim faces. Negative internal faces have larger cost; negative rim faces cannot accommodate winding one. The added donor cycle changes their number, not the minimum cycle-face formula.
Form storage and the weighted spoke potential are nonnegative. Hence every old-cell boundary has \(H\ge V_*+B\), independently of beta. Equations (2) and (23) exclude the first such boundary encounter. All six internal windings, nine rim windings and 32 zero mixed host periods persist for the whole future trajectory. The retained sublevel stays a positive distance from the old acute faces.
At fixed positive eta, connected support and zero total form make the form coordinates bounded by their storage. The two old support components have bounded relative phase lifts inside the old cell; their remaining relative origin is a circle. Thus the full trapped state is precompact modulo common phase rotation. This argument keeps all fine coordinates and does not assert invariance of the smaller stationary tube (4).
7. Own-law fine relaxation and classification of source limits¶
First improve the phase domains using the alternative potential itself. For a winding-one C9 cycle with one gap fixed at s, write \(F_9(s)=V(s)+8V((2\pi-s)/8)\). The added potential is nonnegative and vanishes at alpha. Therefore the existing rational cosine inequality gives \(F_9(\pi/3)-V_9>1/25\) for this law as well. The negative face has larger cost by even acute convexity. Since the full excess is below 3/250, every actual internal gap has magnitude below pi/3 for all future time.
Let \(a_c=\frac19\sum_jx_{c,j}\), \(z_{c,j}=x_{c,j}-a_c\), and keep the fine phase q from (3). Let P_f subtract each child's mean and \(L_f=L_\eta|_{\operatorname{ran}P_f}\). The graph gives \(2I/5<L_f\le10I\). The segment from the actual phases to their carrier at the same child means keeps internal gaps in (-pi/3,pi/3) and rim gaps acute. Its own-law projected phase Hessian satisfies
because its lower bound is \((1/2-2/243)(2/5)-6\eta J_1>19/100\), and its upper norm is at most \(10J_1<11\). The rim contribution is nonnegative by own-law acute convexity. The projected gradient at the carrier is exactly zero, for every set of origins, as in section 5.
The exact fine equations and a fixed quadratic proof function are
Young's inequality bounds the absolute cross term by \((m_f/2)\|q\|^2+(242\gamma^2/m_f)\|z\|^2\). Since \(2/5-242\gamma^2/m_f>m_f/2\), \(F\le\|q\|^2+\|z\|^2\), \(\|q\|^2\le4F\), and \(\gamma^2\|z\|^2\le2F\),
These are own-law bounds: the Hessian norm eleven and cross-term constant 242 replace the sine values. A matching constituent tangent alone would not justify (24)--(26).
7.1 The actual collective equations keep their fine remainder¶
Let L_W be the weighted wheel Laplacian and S_W its current under j. Averaging the actual rows gives
On each wheel edge, the mean-zero fine gaps cancel the linear Taylor term. The own-law constant J2 bounds the quadratic remainder, giving
The donor bound retains the spoke factor eta. Neither its force nor the host remainder is set to zero during the actual transient.
Jensen across the nine rim slot cycles gives, at every future time, \(V_{\rm rim}(\Phi_H)-V_5<(H-V_*)/9<1/750\). Every averaged acute winding-one rim gap is positive. To bound its upper endpoint, put
The own-law constrained face cost exceeds its sine counterpart minus \(5\max(V-(1-\cos))=10/19683\). Exact rational arithmetic gives \(R_5-10/19683>1/750\). Hence every actual averaged rim gap lies in (0,29/20). With C6 from section 5, \(C_6(29/20)-2/243>1/10\). The segment from the actual host shape h to the stationary shape u(t) therefore has projected host Hessian
where g_H is the projected host potential gradient at the actual mean phases. Its lower bound is \(L_5/10-\eta J_1I>9I/100\). This controls the actual shape outside the small stationary ball (4).
7.2 Every retained source converges to one classified equilibrium¶
An equilibrium in the larger retained sublevel first has x=0 by (2). Equation (24) then forces q=0; equation (29) forces h=u(t). Its last stationary equation is T(t)=0. Thus every stationary point in the actual trap belongs to the twenty-target classification of section 4. This is a full-state conclusion within the admitted old-cell sublevel, not a classification of distant or nonacute states.
The precompact orbit has a nonempty compact connected omega-limit set. Storage has one limiting value, constant along every orbit in that invariant limit set. Equation (2) then forces L_eta*x=0, hence x=0; invariance of the form row forces S=0. The omega-limit is consequently contained in the finite classified equilibrium set modulo common rotation. Connectedness makes it a singleton. Every retained source therefore converges to one of those twenty relative equilibria.
On the full phase torus, the conserved product \(\prod_i e^{i\theta_i}=e^{i9\beta}\) leaves only 54 common-rotation representatives of each relative equilibrium. The same connected-limit argument therefore gives one full circular equilibrium. Once each primitive phase stays in a proper arc about its limit, its continuous lift stays in one corresponding real chart and converges there as well. No finite-total-variation bound is asserted by this argument.
This uses the actual law, trap and finite equilibrium classification, without importing a global analytic-convergence theorem from sine. It does not yet choose a destination or exclude saddle capture. The following source restriction supplies that additional conclusion.
8. Ten explicit intervals of actual preparation reach strict minima¶
For k=0,...,9, define the primitive donor-origin center \(\beta_k=(2k+1)\pi/10\). At the fixed eta in (1), every original own-law preparation with
converges to the unique attractive equilibrium whose relative origin lies in the interval of radius pi/60 centered at \(-\beta_k\). The inequality uses the lift nearest the chosen center. All twelve original allowances, all 96 relative directions and the complete own-law history are retained. No initial residual or observed endpoint is replaced by a stationary or exactly reflected source.
8.1 A coupled comparison bounds shape and origin transport¶
Let bars with subscript H denote the average over five host children. Keep h and t from (3), and put
The actual donor current is T(t)+delta. The two complete mean rows (27) give exactly
The corrected origin y retains the relative mean form through chi. It is an auxiliary coordinate, not a new phase measurement or a claim of instantaneous form relaxation.
Define \(\mathcal E^2=\|e+\gamma p\|^2/2+\gamma^2\|p\|^2/2\). Then \(\|e\|\le2\mathcal E\), and (28) gives
Here \(2\sqrt5J_1<5\). The fixed-shape terms in the derivative of \(\mathcal E^2\) are \(-\gamma^2(e^Tg_H+p^TK_Hp+2\gamma p^Tg_H)/4\). Use (29), \(I<K_H<5I\), and Young's inequality with m_H=9/100. Since \(1-50\gamma^2/m_H>m_H/2\), these terms yield decay \(\mathcal E'\le-9\gamma^2\mathcal E/1600\). The moving-shape term uses the actual u_t from (9), and is bounded by \(28\eta^2|\chi|\); the fine forcing contributes at most \(\gamma^2\|Pr_H\|\).
Also \(\lambda_1>2/5\) in (6) gives K<3/100 in (11). The finite torque bound (12) consequently gives \(|T|<(3/100+960\eta)\eta^2<\eta^2/20\). With upper right derivatives at vanishing norms, the complete comparison is
8.2 The reached preparation controls the entire transient¶
The host and donor form totals are separately zero before contact. The host lifted phase total is zero; each prescribed observation jump has zero global phase increment. The donor total is 9*beta. Therefore the actual contacted endpoint satisfies exactly \(\chi(0)=0\) and \(t(0)=y(0)=-\beta\).
The retained own-law fine error before observation is below 1e-6, a coarse container of its much smaller preserved bound. Using (20) and subtracting child means gives \(|q_{c,j}(0)|<1/400000\), \(|z_{c,j}(0)|<21/1000\) and \(|\gamma z_{c,j}(0)|<7/10^6\). The donor errors are smaller. Consequently
For the second inequality, own-law rim convexity in the domain from section 7 gives \(\|h(0)\|^2<2/75\), hence \(\|h(0)\|<1/6\). Equations (8) and (20) give \(\|u\|<6\eta\) and \(\|p(0)\|<\sqrt5B_o<1/40\). Substitution into the definition of E proves (35). These estimates use the reached state rather than narrowing any primitive allowance.
Set \(\mathcal X=\mathcal E+20\eta|\chi|\) and \(\lambda=\gamma^2/200\). At the fixed eta, \(9/1600-30\eta^2>1/200\), \(\eta>\gamma^2/20\), and \(80/27+(80/9)\eta^2<3\). Thus (34) gives
The transient coefficient is \(3\gamma^2/(\rho-\lambda)=160\); the nondecaying shape bound is \(((3/10)\gamma^2\eta^3)/\lambda=60\eta^3\). The last line follows by integrating the second row of (34) using \(\chi(0)=0\). The nonzero shape remainder is retained; it does not assert exact tracking of the stationary shape during motion.
8.3 The inward torque separates each selected source from saddles¶
Fix a center t_c=-beta_k, and first consider the upper strip \(1/10\le y-t_c\le1/5\). Equation (36) puts \(10(t-t_c)\) strictly between 0.99 and 2.01. This interval is inside \((0.99,\pi-0.99)\), and \(\sin(0.99)\ge0.99-0.99^3/6>4/5\). Since t_c is an odd multiple of pi/10, the leading torque in (12) has the restoring sign here. Its full finite remainder gives
Equations (33) and (36) bound the nondecaying part of |delta| by \(300\eta^4\), smaller than \(\eta^2/100\). The corresponding lower strip has the opposite restoring sign by the same leading-torque and absolute-remainder bounds. No exact reflection symmetry about an odd interval center is assumed.
Only the decaying part can drive y outward in either strip. Its entire possible adverse displacement is
The prepared interval (30) gives \(|y(0)-t_c|\le1/10\). If y first reached t_c+1/5, take its last prior crossing of t_c+1/10. Integrating (32) between those times would give \(1/10\le J<1/1000\), a contradiction. The lower strip gives the same conclusion. Therefore
The source already converges to one classified equilibrium by section 7.2. The only stationary origin allowed by (39) is the unique minimum in the central odd interval (14): its radius is pi/60, while the nearest exact saddle is pi/10 from t_c. Hence this is the source's destination. Its local exponential attraction eventually controls the complete state on the conserved-mean leaf. Continuous phase lifts then converge to the corresponding lifted equilibrium. No uniform entrance time or finite settling deadline has been proved.
The ten certified intervals are disjoint. Fix any one original allowance tuple and every primitive error and relative source member, retaining the full prescribed history; vary only the primitive donor origin beta. Preparation and the own-law finite-time flow depend continuously on beta. On the zero-total-form state space with common phase rotation removed, the distinct attracting equilibria have disjoint open basins. Every gap between successive certified intervals contains at least one saddle-converging origin for this fixed member: otherwise the connected interval joining its two certified endpoints would be covered by those disjoint open basins, although the endpoints have different destinations. The own-law convergence and finite classification justify the saddle conclusion. Other gap destinations, exceptional counts and measures are not determined.
8.4 A common primitive preparation interval separates the two laws¶
Take the common primitive preparation interval \(|\beta-\pi/10|\le1/10\), with the original primitive allowances, commands and durations under each complete law. This entire interval lies inside the sine theorem's \(|\beta|\le3/5\) source basin. The sine law therefore selects t_infinity=0. The alternative instead selects its unique minimum in
The wrapped asymptotic relative-origin difference exceeds pi/12 for every allowed pair of own-law source members. This compares the same declared primitive preparation class and commands, not identical reached endpoints or transplanted source bounds. Each law carries its own acquisition, observation backaction, storage and complete subsequent evolution. The two own-law certificates establish the contrast without generating another trajectory or re-evaluating a frozen response.
The conclusion is an asymptotic mathematical discriminator. It has no finite reading time, laboratory clock, independently admitted sensor or observation-error budget. It does not by itself select which law describes nature.
9. Meaning and implementation boundary¶
The sine law at this same eta has attracting origins t=2kpi/5. Every one is among the alternative's exact saddle origins. The ten alternative attracting origins lie instead in the odd intervals (14). This finite-strength comparison resolves a constitutive dependence of the collective registration while preserving the declared constituent profile and old marked identity. It does not follow from graph symmetry alone.
The stationary proof is local to (4); the source proof separately extends its use to the larger retained energy sublevel. Neither result classifies all nonacute or distant organizations. Ten actual-source intervals now reach alternative minima, while the prototype common primitive interval has distinct destinations under the two laws. Own-law acquisition, observation history, attachment work and transient fine-state transport are part of that conclusion. Matching the isolated constituent tangent did not transfer the sine source basin.
These labeled registrations are not physical species. Their difference neither selects a unique fundamental law nor supplies a physical clock, measurement bridge, autonomous contact occurrence or collective operator. The ontology retains those separate obligations.
The static controls check the derivative bounds, weighted full-state identities, fivefold Fourier coefficient, contraction and C1 margins, full-state curvature, actual-source budgets, moving-shape transport and inward-torque inequalities. Those controls support the written proof; they do not sample roots or evaluate a trajectory. This analytical result installs no new runtime law or source producer. The execution plan owns its status and any later admission.