A constitutive pitchfork at weak contact¶
The attracting sine registration has a positive constitutive tolerance at each fixed sufficiently small contact strength. That tolerance is not uniform as contact tends to zero. For the already declared potential family, a controlled joint expansion proves a supercritical pitchfork: the central registration loses its collective stability and two attracting registrations emerge, while every other relative phase direction stays positive.
At each fixed positive eta in the proved range, the same complete law also has a local scalar center dynamics with an inherited positive mobility. Its initialization depends on form as well as phase. Generic critical trajectories approach algebraically; a strong-stable exception approaches exponentially.
The quantitative domain proved here is \(0<\eta\le10^{-12}\). It is a narrower mathematical domain than the endpoint owners' \(\eta=10^{-5}\). This theorem does not classify the transition at that endpoint contact strength. It neither reruns a source nor refines an evaluated experimental protocol.
| Premise | Declaration |
|---|---|
| Constitutive family | \(V_\lambda(a)=1-\cos a+\lambda(1-\cos9a)^2/(6\,9^4)\), \(0\le\lambda\le1\) |
| Complete support | 54 labeled nodes of C9 x W6; 99 old internal/rim edges of weight one, 45 spokes of held weight eta |
| Complete law | Both weighted rows, held unit capacity, quarter factors and the original tau clock |
| Retained state | All signed forms, primitive circular phases and fine directions, on fixed-total-form and fixed-total-phase leaves |
| Comparison chart | \(\lambda=\kappa_*\mu\eta^3\), with independently supplied eta and lambda; this is not a parameter evolution law |
| Quantitative transition | \(0<\eta\le10^{-12}\), \(\mu\in[9/10,11/10]\), relative origins ( |
| Local dynamics | Fixed positive eta, finite smooth center manifold, full-state initialization and critical asymptotic relaxation; neighborhoods and transverse rates may depend on eta |
| Evidence | Analytic contraction, exact harmonic coefficients, Cauchy bounds and complete-state curvature; no trajectory, root search, sweep or source replay |
1. Complete law, coordinates and scope¶
Let c=0,...,4 label the host children, c=5=D the donor, and j=0,...,8 the internal slots. The primitive node order is 9*c+j. The graph, marked winding conventions and capacities are those of the weak-contact endpoint owners. Define
The support Laplacian L_eta is unnormalized and uses the same weights as the nonlinear current \(S_i=\sum_{b\sim i}w_{ib}j_\lambda(\theta_b-\theta_i)\). The complete, unforced equations and storage are
Primes in the evolution rows mean tau. Unit capacity is held, so the first row supplies the nodal pressure rather than reconstructing it from a response. Odd current and Laplacian row sums preserve total form and total lifted phase. Work on the zero-total-form leaf and a fixed-total-phase local chart, or equivalently remove common phase rotation. No laboratory units, later input or support event are introduced.
Put alpha=2pi/9, b=2pi/5 and \(b_c=b(c-2)\). Compatible lifts give
Here t is a relative origin, not the clock. The stationary tube is \(\|h\|_2\le6\eta\), \(\|q\|_\infty\le1/100\), with all t admitted before the local bifurcation restriction. It retains the six internal and nine rim winding-one classes and the 32 zero mixed host periods. It is not asserted to be an invariant source domain. Spoke angles have no acute restriction.
Eta and lambda are held, independently supplied model parameters. The substitution \(\lambda=\kappa\eta^3\) below is an analytical comparison chart near a balance of two torques. It does not derive a coupling between parameters, select contact strength, or create parameter dynamics. The endpoint source-capture theorems are not transferred to intermediate laws.
2. A joint complex contraction¶
Let D be the forward C5 incidence, \(L_5=D^TD\), and \(P=I-\mathbf1\mathbf1^T/5\). For this construction only, keep the sine linearization
The projected host stationarity equation, including every deformation term, is
The b inside a spoke vector means \((b_c)_{c=0}^4\); the b inside rim functions is the scalar gap. Constants disappear under D^T. Thus (5) is the actual deformed equation, not an unchanged tangent used as its nonlinear law.
The explicit row of G and bounds from the sine stationary proof give
These are exact properties of the sine matrix chosen in (4); all deformation terms remain in (5). Set
For real t, \(\|u^{(1)}\|_\infty<U_1\) and \(\|Du^{(1)}\|_\infty<D_1\). The complex ball therefore obeys \(\|u\|_\infty<U_0\), \(\|Du\|_\infty<D_0\). All shapes have zero mean.
The following rational majorants make the contraction quantitative:
The first three bound cosh(z)-1, sinh(z) and sinh(z)-z. For real a and complex v with \(|v|\le z<1/18\), \(|w(a+v)|\le H_0(z)\) and \(|w'(a+v)|\le H_1(z)\). These follow from \(|\sin(a+v)|,|\cos(a+v)|\le e^{|v|}\) and \(e^z\le(1-z)^{-1}\). Also take \(s_0=24/25>s_b\), \(c_0=31/100>c_b\). The map in (5) has derivative norm at most
Its displacement at the center is bounded by
All inequalities are rational arithmetic with the declarations above. The deformation contribution on a rim uses \(|w(b+d)-w(b)|\le|d|H_1(|d|)\); omitting it would not prove the joint result. The contraction theorem gives a unique solution, holomorphic jointly in eta and kappa on their disks, and real for real parameters. The same estimates work with eta and lambda independent on \(|\eta|\le R,|\lambda|\le1/25\), so coefficient extraction in those independent parameters is also justified.
The remaining wheel torque is
It equals the t derivative of the wheel potential after eliminating the projected shape. Uniformly on the complex parameter disks and real t,
3. Joint fifth-order torque and controlled derivatives¶
Cyclic host relabeling makes T periodic with period 2*pi/5. Phase inversion together with rim reflection makes it odd in t. These symmetries hold for complex parameters as identities of (5). Thus each Taylor coefficient \(T_n(t,\kappa)\) in \(T=\sum_{n\ge0}\eta^nT_n\) has only sine harmonics \(\sin(5kt)\).
The coefficient T_n is a trigonometric polynomial of degree at most 5n. To see the weighted-degree bound, a sine spoke forcing has eta order one and explicit harmonic degree one. A deformed spoke forcing has eta order four and explicit harmonic degree at most 18. A deformed rim term has eta order three and no explicit t harmonic. Products of lower shape coefficients add their orders and harmonic degrees. Induction in (5) therefore bounds shape and torque coefficients by five times their eta order. The bound does not assume the sine-only parity rule after lambda has become kappaeta^3.
For completeness, the fifth coefficient can be obtained in independent eta and lambda coordinates. Put
At eta=0, u=0. Its first eta coefficient is \(-G_\lambda f_\lambda\). The eta coefficient of the torque vanishes for every lambda, because the frequencies 1, 9 and 18 all sum to zero over five vertices. Its second coefficient is
Indeed the finite Fourier identity is
Its t derivative divided by -2 gives (14). The only surviving cross-harmonic is the pair of frequencies 1 and 9.
At lambda=0, the sine proof establishes that eta orders below five vanish and that the fifth coefficient is \(A\sin(5t)\), where
After lambda=kappa*eta^3, the only additional contribution through order five is the eta-squared, lambda-linear term of (14). The eta-linear term is identically zero for all lambda, and two lambda powers already cost six eta orders. Consequently
with no orders below five. This uses the established sine coefficient as an identity, while re-admitting every deformation contribution. It transfers no endpoint remainder or capture certificate.
Here is a uniform C3 remainder, with a mixed parameter derivative. Cauchy's estimate from (12) gives \(\|T_n(\cdot,\kappa)\|_\infty\le M/R^n\). Each sine coefficient has magnitude at most \(2M/R^n\), and its index k is at most n. For derivative order j=0,...,3,
Write \(h=\eta/R\). For \(0<\eta\le10^{-12}\), summing n>=6 yields
For the sum, use \((l+6)^{j+1}\le6^{j+1}(7/6)^{(j+1)l}\) and \((7/6)^4<2\). This proves absolute convergence after all stated derivatives. The estimates remain uniform for complex \(|\kappa|\le40000\). A kappa Cauchy circle of radius 20000 at each real \(|\kappa|\le20000\) therefore gives
Normalize by \(\kappa=\kappa_*\mu\) and set
For \(0\le\mu\le11/10\), (16) keeps kappa inside the required disk. Equations (19)--(20) imply, in particular,
The C0 and C2 bounds are also given in (19), divided by A. Thus this is a controlled C3 expansion with mixed-parameter evidence, not a derivative inferred from a value bound. Dividing the analytic series by eta^5 also extends f analytically to eta=0; this is an auxiliary parameter extension, not an assertion about dynamics on the disconnected eta=0 support.
4. A unique quantitative critical curve¶
The central registration t=0 is exact by oddness. Its normalized curvature is
G is positive at mu=9/10 and negative at mu=11/10. The intermediate value theorem and strict monotonicity give exactly one \(\mu_c(\eta)\) in this interval. The implicit function theorem, applied to the analytic extension at eta=0 and to (23), makes the curve analytic on the admitted range with \(\mu_c(0)=1\). Explicitly,
In particular the controlled shift is O(eta^4), rather than an unbounded formal error at the competing fifth order. For \(0\le\mu\le9/10\), (22) gives \(G>9/20\). Together with (23), the central registration therefore has positive collective curvature for every \(0\le\mu<\mu_c\), zero at mu_c, and negative curvature for \(\mu_c<\mu\le11/10\).
5. The emerging branches are a supercritical pitchfork¶
Work on \(|t|\le1/20\), \(\mu\in[9/10,11/10]\). Here \(\cos(10t)\ge7/8\), and (22) yields
Odd analyticity gives an exact factorization
For \(s\in[0,1/400]\), it has
This integral also proves the derivative bound at s=0 by continuity. It is not obtained by dividing a value-only remainder near t=0.
Now restrict \(|\mu-\mu_c|\le1/100\), which lies inside the preceding mu interval. Equation (23) gives \(|B(0)|<101/2000\), with the sign opposite to mu-mu_c. At the outer boundary,
Strict monotonicity in s now gives the complete local stationary classification:
- Below critical, t=0 is the only stationary origin in this interval, and its collective curvature is positive.
- At critical, t=0 is still the only stationary origin, with one zero collective curvature and a strictly positive quartic term.
- Above critical, t=0 has negative collective curvature, and there are exactly two additional origins \(t=\pm\sqrt{s(\mu,\eta)}\). They satisfy \(0<s<1/400\), and their curvature is positive because \(f_t=2sB_s>0\).
The implicit function theorem applies to B at its critical zero. For each fixed positive eta,
Indeed \(B_\mu\in(-101/20,-99/20)\), while \(\mu_c\in(99/100,101/100)\) and (22) imply \(55<B_s(0,\mu_c,\eta)<70\). At eta=0 the exact values are \(B_\mu=-5\) and \(B_s=125/2\). This proves the branch direction and square-root splitting, not just the existence of a central curvature crossing. Cyclic symmetry gives the same local theorem around each of the five origins \(2k\pi/5\). No global count of all stationary registrations for intermediate lambda is asserted.
6. Normal curvature and the complete two-row dynamics¶
All normal directions must be re-admitted for the intermediate law. For real \(0\le\lambda\le1\), the uniform current bounds are
The real host map with linearization \(a_\lambda L_5\) consequently has \(\|G_\lambda\|_2<5/2\). For \(0<\eta\le10^{-5}\), its image and derivative on \(\|u\|_2\le6\eta\) satisfy
Thus the real projected shape is unique in this ball. In the smaller joint domain it belongs to the complex ball of (7), since \(\|u-u^{(1)}\|_\infty<(6+47/20)\eta<r_0\). The real shape is therefore exactly the analytic branch used above, rather than a separately chosen solution.
Let \(C_6(v)=1-v^2/2+v^4/24-v^6/720\). On the stationary tube and on its segments to q=0, internal gaps have absolute value below 18/25, while rim gaps lie in (0,13/10). The inequality \(j_\lambda'(a)\ge\cos a-2/243\) gives the exact rational bounds
The 48 fine directions have internal Laplacian gap greater than 2/5. The spoke-star Laplacian has norm six. Even allowing negative spoke curvatures, their projected phase Hessian is bounded below by
At q=0 the fine gradient is exactly zero for every collective configuration: internal currents cancel and wheel currents are slot-constant. Integrating (33) therefore excludes any stationary q different from zero in this tube. Similarly, the four mean-zero host directions have wheel curvature \(L_5/4-\eta J_1I>I/5\). Their stationarity uniquely gives h=u(t). At a complete equilibrium, connected support and the phase row first give x=0; the form row then gives these phase stationarity equations. No fine coordinate was discarded before checking its equation.
The Schur complement of the four host directions is the reduced wheel curvature \(T'=A\eta^5 f_t\). In slot-first node order the full carrier Hessian is
Thus the relative phase Hessian has 53 positive directions below critical and on the two emerging branches; at critical it has 52 positive directions and one zero; on the central branch above critical it has 52 positive directions and one negative direction. These statements include every primitive phase direction apart from the exact common-rotation symmetry.
On the fixed-mean relative state space, the actual 106-dimensional linearized two-row dynamics has the symmetric quadratic pencil
Testing an eigenvector gives a scalar quadratic with positive coefficients for s-squared and s. If M is positive definite, every eigenvalue has negative real part. If M has one negative direction, P(s) increases strictly for real s>0 from that one negative direction to a positive matrix, giving exactly one positive eigenvalue. The imaginary part of the scalar identity excludes nonreal eigenvalues with nonnegative real part.
At critical, M is positive semidefinite with a one-dimensional kernel. The Schur complement of its positive subspace is \(s/4+O(s^2)\), so the zero eigenvalue of the complete first-order system is algebraically simple. The other 105 eigenvalues have negative real part. Consequently the attracting branches are locally exponentially stable, and the central branch above critical has one unstable direction. Normal stability has been checked for the whole law, not inferred from a wheel-only eigenvalue.
7. The critical state is attracting, with quartic restoration¶
The zero linear eigenvalue does not by itself decide nonlinear stability. At mu_c, (27) and B(0)=0 give \(f(t)/t=B(t^2)>(100/3)t^2\) for t different from zero. For the reduced wheel potential \(\mathcal W\), this implies
The complete phase potential contains nine wheel copies. Integrating the fine and host normal bounds on the same fixed-mean chart gives
The fixed internal carrier potentials cancel in this difference. Adding the positive form storage in (2) produces a strict local minimum at the critical equilibrium. For each fixed positive eta, a sufficiently small compact sublevel remains inside the local chart and stationary tube. Storage is nonincreasing there. In the invariant part of H'=0, connected support and zero total form give x=0, and the form row then requires a stationary phase state. Sections 5--6 leave only the central critical equilibrium in this neighborhood. The elementary compact LaSalle argument proves Lyapunov stability and asymptotic attraction.
The critical state is not locally exponentially stable: its complete linearization has the simple zero eigenvalue just proved. No uniform settling time or decay rate as eta tends to zero is claimed.
8. The inherited local dynamics of the soft registration¶
Fix one positive eta in the proved interval. This section derives a local dynamical consequence of the same complete law in the original tau clock. Every neighborhood, remainder constant and transverse tracking rate below may depend on that fixed eta. There is no uniform-in-eta center manifold, source-entry theorem, or finite-time approximation claim.
Write \(\sigma=\mu-\mu_c(\eta)\). Let \(\Theta(t,\sigma)\) be the stationary carrier shape after eliminating the four host shape coordinates, with the total phase fixed. It is a curve of projected stationary shapes, not a curve of complete equilibria: its remaining torque can be nonzero. Its phases are
At criticality put \(\theta_*=\Theta(0,0)\), \(w=\Theta_t(0,0)\), and \(N=w^Tw\). Then
Differentiating the real contraction in (31) gives
This derivative belongs to the actual constitutive shape at the critical parameter.
In the state order \((\theta-\theta_*,x)\), the critical Jacobian and its right and normalized left zero vectors are
Here \(M_{54}w=0\) follows from the critical Schur complement; the common-rotation zero mode has already been removed. For the linear coordinate \(\zeta=w^T(\theta-\theta_*+\gamma x)/N\), the full nonlinear equations have the exact projected row
This row still depends on hidden phase coordinates. Projection is not yet a scalar closure.
8.1 An invariant center manifold, with matched hypotheses¶
The vector field is analytic on this finite-dimensional relative chart. Its critical spectrum consists of one simple zero and 105 strictly stable eigenvalues. The local center-manifold theorem therefore gives a finite \(C^k\) invariant curve for any prescribed finite k, chosen here at least five. Applying the same theorem after adjoining \(\sigma'=0\) gives the local parameter family. These are the hypotheses of the local theorem and parameter construction in Nagata, Lecture 25, Theorem 3.7 and the final parameter extension.
The reflection symmetry acts as \(t\mapsto-t\). An invariant, reflection-symmetric cutoff in a local center/stable construction gives an equivariant center manifold; its scalar row is odd in t. The exact carrier subspace, in which fine phases and fine forms vanish, is itself invariant: internal currents cancel, and the wheel rows are constant in each slot. The center manifold can first be constructed in this ten-dimensional carrier system and is then an invariant manifold of the full system. Fine perturbations retain their stable directions; they are not assumed absent from nearby initial states.
The center curve need not be unique or analytic. The claims below concern its finite-order coefficients and the tracking of nearby states, with physical t as its coordinate. The derivative of that coordinate along v is one, so it is a valid local parameter. The graph \((\Theta(t,\sigma),0)\) is not invariant away from its torque zeros: its phase row is zero while its form row is nonzero. It cannot replace the center manifold.
8.2 The normal-form coefficients come from the actual two rows¶
Let \(t'=g(t,\sigma)\) be the center row. At criticality odd symmetry and the zero eigenvalue give \(g(t,0)=O(t^3)\). Write the center phase and form curves as \(\Theta_{\rm c}(t)\), \(x_{\rm c}(t)\). The phase row implies
Invariance in the form row then forces the normal phase gradient to vanish through degree two. The invertible normal Hessian from section 6 consequently gives \(\Theta_{\rm c}(t)-\Theta(t,0)=O(t^3)\), at the same physical t. It is this agreement of finite-order expansions, not invariance of the stationary graph, that permits use of its reduced potential.
Eliminating x from the full equations gives the exact second-order phase equation on the relative space,
On the critical center curve its inertial term is O(t^5). The cubic normal displacement changes the cubic gradient by \(M_{54}\) times that displacement; contraction with w annihilates this term. On the stationary curve, \(w^T\nabla U_\lambda(\Theta(t,0))=9T(t)\): the fine and projected host gradients vanish, and the host-shape part of w is orthogonal to the remaining mean gradient. Projection of (44) at cubic order therefore gives
where the inherited mobility is
The positive coefficient is fixed by the complete damping row and the physical coordinate geometry. It is neither a new primitive capacity nor an independently assigned time rescaling. Equation (45) does not assert the exact scalar identity \(t'=-m_\eta T(t)\) at all orders.
For the parameter-linear term, differentiate the simple critical eigenvalue. At the central equilibrium the stationary tangent satisfies \(w(\mu)^TM(\mu)w(\mu)=9T_t(0,\mu)\). Since \(M(\mu_c)w=0\), differentiation at criticality gives \(w^TM_\mu w=9T_{t\mu}\). The left/right vectors in (41) therefore yield
The resulting local row in the unchanged tau clock is
The subscript on the remainder records fixed-eta local constants. The dynamic branch ratio agrees with (29), an independent check of the pressure, damping and coordinate factors. The slow eigenvalue is \(a_\eta\sigma+O_\eta(\sigma^2)\) at the central equilibrium and \(-2a_\eta\sigma+O_\eta(\sigma^2)\) on the emerging branches. These are local spectral statements, not uniform relaxation times.
9. Hidden initialization and critical relaxation¶
At criticality, the stable spectrum has a positive separation from zero for each fixed eta. In an adapted norm, choose \(\omega_\eta>0\) such that the stable semigroup is bounded by \(e^{-\omega_\eta\tau}\). After the spectral change of coordinates, the nonlinear remainder and its derivative vanish at the origin. A smooth local cutoff can therefore make its Lipschitz constant arbitrarily small compared with this separation, while leaving the actual equations unchanged on a smaller neighborhood.
Taking the stable block as A, the zero center block as B, and \(\alpha=-\omega_\eta,\ \beta=0\), this satisfies \(2\operatorname{Lip}(H)<\beta-\alpha\) in the foliation and exponential-tracking theorem of Chung and Jolly, Theorem 1.1 and Proposition 1.2. For the finite \(C^k\) center curve, choose the cutoff still smaller so the center growth weights required through derivative order k remain strictly below the stable decay. Differentiating the Lyapunov--Perron fixed-point equation then has convergent stable integrals and invertible contraction derivatives through that order. This is the finite-smoothness center-manifold construction used in section 8.1; the cited Lipschitz foliation theorem by itself is not being used to assert smoothness or analyticity of the tracking map. Section 7 supplies a smaller invariant neighborhood of the actual critical law, so its forward solutions never use the modified equations outside the cutoff. This is a proof extension, not a new physical or runtime law.
Every sufficiently close full initial state consequently tracks a center-manifold trajectory exponentially in the actual tau clock. Denote the physical t coordinate of that trajectory at time zero by \(\Pi_\eta(\theta_0,x_0)\). It is a local initialization map from the full state, not generally the observed t_0.
The uniform graph slopes also give the needed linear-size initial bound, without requiring a smooth projection theorem. Write p for the stable spectral coordinates and c for the center coordinate. If the stable leaves and center graph have Lipschitz slopes at most \(q<1\), their intersection \((p_+,c_+)\) satisfies
Thus the tracking initial state is O(norm(z_0)). On the critical center curve, (45) makes its scalar magnitude nonincreasing in a sufficiently small interval. Shrinking the initial neighborhood consequently keeps both the tracking center orbit and the actual orbit inside the region where the cutoff equals one.
Its leading term is fixed by (41):
One can obtain the quadratic remainder without assuming an analytic foliation. Let z and z_c be a full orbit and its exponentially tracking center orbit. Their projected difference satisfies
where \(\mathcal N\) is the nonlinear remainder after the critical linearization. The center orbit decreases in magnitude by (45); (48a), its smooth graph and exponential tracking keep both orbits O(norm(z(0))). Since \(D\mathcal N=O(\|z\|)\), the exponentially decaying difference makes (50) O(norm(z(0))^2). Finally \(\ell z_{\rm c}(0)=t_{\rm c}(0)+O(t_{\rm c}(0)^2)\). These facts prove (49).
The set \(\Pi_\eta=0\) is the local 105-dimensional strong-stable manifold of the equilibrium. It is tangent to the kernel of the linear functional in (49); it is not asserted equal to that linear kernel. On this exceptional set the full state approaches the equilibrium exponentially.
If \(\Pi_\eta\ne0\), its center trajectory eventually obeys \(t'=-b_\eta t^3+O_\eta(t^5)\), remains on the corresponding side of zero and approaches zero. Therefore
The exponentially small tracking difference transfers this asymptotic formula to the physical relative origin of the full trajectory. Equivalently \(2b_\eta=m_\eta T_{ttt}(0,\mu_c)/3\). The limit is as tau tends to infinity at fixed positive eta. It is not a joint eta/time estimate or a finite experimental settling-time prediction.
9.1 Equal phase observations can have opposite subsequent approaches¶
There is a concrete local initialization witness. Keep every phase equal to the critical equilibrium and choose the two signed form states
They obey the zero-total-form constraint and share all initial phase observations, including t_0=0 and the marked identity. For sufficiently small epsilon both lie in the preceding local domain. Equation (49) gives
Thus their asymptotic approach signs are opposite. Even their initial physical relative-origin rates already differ:
To check (54) directly, the host mean of w is 1/6, the donor mean is -5/6, and the exact mean phase row gives \(t'=(3/2)\gamma\eta(\bar x_H-\bar x_D)\). No selected trajectory was integrated to construct this witness.
The scalar center law is therefore a justified local description with a full-state initialization and an exponentially decaying transverse discrepancy. It is not an exact instantaneous quotient of all nearby states onto their phase origin. This distinguishes inherited dynamics from stationary elimination and preserves the hidden-initial-state obligation in the ontology.
10. Meaning and validation boundary¶
For every eta in the proved positive interval, the central registration attracts for a strictly positive interval \(0\le\lambda<\lambda_c(\eta)\), and the critical state remains asymptotically attracting with quartic restoration. Immediately above lambda_c, the central state becomes a saddle and the two new registrations attract. The explicit curve \(\lambda_c=\kappa_*\eta^3+O(\eta^4)\) tends to zero. There can therefore be no positive deformation tolerance independent of eta that preserves central attraction throughout all sufficiently weak contacts. Explicitly, for any proposed tolerance lambda_0>0, choose eta in the proved interval small enough that \(\lambda_+(\eta)=\kappa_*[\mu_c(\eta)+1/100]\eta^3<\lambda_0\). The central state at this supplied deformation is a saddle. This countersequence proves nonuniform tolerance; it does not assert a fixed-lambda asymptotic classification outside the bounded mu chart.
This is a constitutive dependence result for a conditional formed organization. It does not select a unique TNFR law, identify a physical force or species, derive support occurrence or scale selection, or determine the basins of the original sources under intermediate laws. Fixed-contact continuity and a nonuniform small-contact limit are distinct statements; neither licenses interchanging stationary sets at eta=0 with those at positive eta.
The proof range \(0<\eta\le10^{-12}\) is explicit. The existing sine and deformation-one theorems at eta=1e-5 remain valid in their own domains, but do not extend this quantitative transition theorem to that larger contact strength.
Static controls in the transition research tests check the finite Fourier coefficient, rational contraction and remainder inequalities, derivative margins and full-state curvature. The local-dynamics tests independently check the two-row projection, inherited mobility, invariant-manifold coefficients, initialization and critical decay identities. No source producer, frozen response, trajectory, parameter scan or numerical root search is used. The execution plan owns task status and subsequent research.