Retained reservoir information and local organized equilibria¶
Declared comparison and mathematical question¶
The fixed reservoir experiment failed its prescribed disruption readout. This separate analytic question keeps that result and its protocol unchanged. It asks whether retained auxiliary state can change stable geometry, and what reservoir leakage changes.
| Item | Declared scope |
|---|---|
| Fine support | The same 36-node matching product C9 x K2 x K2, with 72 unit edges, node 18*g+9*a+j, and held unit network capacities |
| Complete law | x_tau=(-Lx+gamma*S(theta))/4+b*r, theta_tau=gamma*Lx/4, r_tau=-b^T*Lx-lambda*r, with gamma=1/(1023*pi) and unchanged b[g,a,j]=gamma*(2*a-1)/12 |
| Compared laws | Only lambda=0 and the original lambda=gamma^2/2; no coupling scan or change to the sine law |
| Auxiliary state and units | Signed real form r, held unit capacity in tau, independently supplied pressure -b^T*Lx-lambda*r, storage r^2/2 in the existing normalized units; no auxiliary phase is consumed |
| Clock | tau=(1023/1024)*t, s=gamma^2*tau/2; a local stability problem has no new finite readout or source dwell |
| Common modes | Fix sum(x)=0 and a continuous phase lift with sum(theta)=0; arbitrary common modes are outside this declared representative |
| Phase chart | Write theta[g,a,j]=psi[j]+delta[g,a,j], psi[j]=2*pi*(j-4)/9, with real mean-zero delta. On each oriented child edge j -> (j+1) mod 9, the adjusted gap is 2*pi/9+delta[next]-delta[j], including its fixed closing-edge turn. Require these gaps in (0,pi/2) and all matching contact gaps in (-pi/2,pi/2) |
| Retained-coordinate candidate | Define c=4*b/gamma and the lifted quantity J=r+c^T*theta. Determine its exact evolution before eliminating any state. For the zero-leakage classification restrict to abs(J)<=3 |
| Identity observations | Each marked child has winding one in this chart. A parent is registered when all its intraparent same-slot circular gaps have magnitude strictly below 1/4; equality is a boundary, not registration |
| Perturbations | Admit all fine form and phase directions compatible with the fixed means and chart, and independent auxiliary perturbations. State explicitly when stability is relative to a fixed J leaf rather than to the complete family |
| Required evidence | Full-state equilibrium classification, a valid local trapping neighborhood and attraction argument, or a scoped obstruction. A scalar reference root, positive Hessian or dissipative balance alone does not establish entry from an arbitrary preparation |
| Discriminating question | Can the same passive port support different locally stable registration geometries through retained initial auxiliary information, and which equilibrium/attraction conclusions survive the original positive leakage? |
Use the original joint storage H_joint=x^T*L*x/2+sum_edges(1-cos(gap))+r^2/2.
Derive every elimination, balance, domain and stability claim for the actual
complete law. Carry the phase lift used by J; a circular representative
change cannot silently change the supplied auxiliary state or its preparation.
This is a local conditional classification, not a new evaluation of the earlier acquired source. Any trapping condition is an explicit restriction on initial state; it does not prove that the original preparation enters it. No finite convergence time, trajectory campaign, response retuning, source refinement or replay of frozen producers is authorized by this question.
The auxiliary coordinate, its origin, reciprocal coupling and preparation remain supplied constitutive premises. The continuous path installs no operator, support event or occurrence selector. This comparison neither identifies a physical bath nor selects a unique TNFR law or material property.
The declaration above was committed as 53cc7b68 before this proof. The
zero-leakage law has a family of locally attracting equilibria indexed by
the retained lifted quantity. Attraction to a particular member is relative
to its invariant leaf. With the original positive leakage, only the
zero-reservoir equilibrium survives in the declared chart, and it has a
full-state local trapping neighborhood. Neither result admits the earlier
acquired preparation into that neighborhood.
1. Complete law, lift and retained quantity¶
Let \(\sigma_a[g,a,j]=2a-1\), \(\alpha=2\pi/9\), and \(\psi[g,a,j]=\alpha(j-4)\). Write
Every sum has the fixed 72 unit edges. Thus \(\nabla U=-S\), \(\mathbf1^Tc=0\), \(\|c\|_2=2\), and \(c^T\psi=0\). The declared chart is an open convex subset of the mean-zero lifted phase space: all its gap constraints are strict affine inequalities in \(\theta-\psi\). On a closing child edge, the fixed added turn is \(2\pi\). The cosine and its derivatives are unaffected by that turn, while the chart and winding statement depend on retaining it.
The two fixed means are conserved by the complete law. Its vector field is globally Lipschitz on the finite-dimensional real lifted state, so solutions exist uniquely for every finite time. The joint balance and retained quantity follow directly, without restricting to child-constant states:
For the slow clock, put \(\ell=2\lambda/\gamma^2\), which is either zero or one. Every row becomes
The positive clock changes do not change equilibrium or local attraction. They do not supply a laboratory time scale.
The conserved quantity for zero leakage is lift dependent. Replacing \(\theta\) by \(\theta+2\pi k\) while retaining the same physical auxiliary state changes its numerical label by \(2\pi c^Tk\). Such a change cannot be used to compare preparations in different charts without transporting the chart and its retained data. The following classification uses exactly the lift and means in the declaration.
2. Full equilibrium classification in the declared chart¶
First take \(\lambda=0\) and a fixed \(J\). Eliminating the auxiliary coordinate on its invariant leaf gives
The complete restricted rows and their storage are
In the acute chart, the Hessian of \(U\) is the graph Laplacian whose edge weights are the positive cosines of the adjusted gaps. The graph is connected. Consequently \(U\), and also \(U_J\) with Hessian \(\nabla^2U+cc^T\), are strictly convex on the mean-zero chart. This is a statement about every fine phase direction, including directions outside the four-child invariant family.
Any equilibrium has \(Lx=0\), hence \(x=0\) under the fixed form mean. Its remaining equation is \(\nabla U_J=0\). A candidate can be constructed without assuming that the equilibrium must inherit a particular symmetry:
The scalar left-hand side is strictly increasing; its derivative is \(6+3\cos B\ge3\). For \(|J|\le3\), its unique root satisfies \(|B_*|<1/2\): its value at \(1/2\) is strictly greater than three. For nonnegative \(J\), useful enclosing inequalities are \(J/9\le B_*\le J/6\); negative roots follow by odd symmetry. At this candidate all child gaps equal \(\alpha\), all interparent gaps vanish, and all intraparent gaps equal \(B_*\). It lies inside the declared chart. The internal sine contributions cancel, and \(S(\theta_*)=-\sigma_a\sin B_*=-c r_*\), so it is an exact equilibrium. Strict convexity then proves that it is the only equilibrium in that chart on this \(J\) leaf. This uniqueness argument does not discard transverse coordinates or infer equilibrium from a scalar root alone.
Both parents are registered at equilibrium precisely when
At equality their gaps equal the strict identity boundary. For \(J_{\rm reg}<|J|\le3\), both equilibrated parents are unregistered while all four children retain winding one. In particular \(J=0\) gives registered parents, whereas \(J=3\) gives unregistered parents, since \(J_{\rm reg}<9/4<3\). These are different preparations of the same zero-leakage law, not different fitted couplings.
For \(\lambda=\gamma^2/2>0\), equilibrium again requires \(Lx=0\). The reservoir row then requires \(r=0\), and the form row requires \(S(\theta)=0\). Strict convexity of \(U\) in the declared chart, together with the explicit critical point \(\psi\), gives the unique equilibrium
The variable \(J\) is not a constraint on this positive-leakage equilibrium: equation (2) changes it during evolution. Removing \(r\) as though \(J\) were conserved would change this complete model.
3. Uniform local trapping bounds¶
The fixed product graph has \(\lambda_2(L)>1/3\) and \(\|L\|_2\le8\). Indeed its Laplacian spectrum is the sum of the C9 spectrum and two copies of \(\{0,2\}\); the C9 gap is \(2-2\cos\alpha>1/3\). The acquisition proof establishes these same fixed-support bounds. No graph coefficient from a different support is used here.
Set \(\rho=1/8\) and write \(\eta=\theta-\theta_*\), with zero mean. In the closed phase ball \(\|\eta\|_2\le\rho\), every edge-gap perturbation has magnitude at most \(2\rho=1/4\). The adjusted internal gaps lie in \((\alpha-1/4,\alpha+1/4)\subset(0,\pi/3)\); the absolute matching gaps are at most \(|B_*|+1/4<3/4<\pi/3\). The positive-leakage target is the case \(B_*=0\). Thus the entire closed ball stays strictly inside the declared chart, retains all four child windings, and has every edge cosine greater than \(1/2\). In particular,
For zero leakage, use the storage excess on the actual fixed \(J\) leaf,
The second equality uses both \(\nabla U(\theta_*)=c r_*\) and \(r-r_*=-c^T\eta\). It is the affine cancellation that makes this excess positive near the target. The separate differences \(U(\theta)-U(\theta_*)\) and \((r^2-r_*^2)/2\) need not be positive. Nor is this a positive excess relative to an arbitrarily selected target on a different \(J\) leaf.
For positive leakage use the full-state excess
Taylor's formula along the line segment in the ball, the critical-point identities, and (9) give the lower bounds
Every point on the phase-ball boundary therefore has excess at least
A sufficient initial-state condition is
where \(E=E_J\) on the specified zero-leakage leaf, or \(E=E_+\) for the positive-leakage model. The phase-ball admission is required in addition to the energy bound; a low periodic storage elsewhere is not an entry certificate. The storage derivative is nonpositive. If a solution first reached the phase-ball boundary, (13) would contradict \(E\le e\). It remains in the ball for all future time.
For a simple rational sufficient admission from norm allowances, let \(\|x(0)\|_2\le X\), \(\|\eta(0)\|_2\le\Theta<1/8\). The global Hessian upper bounds \(\nabla^2U\preceq L\preceq8I\) and \(\nabla^2U_J\preceq L+cc^T\preceq12I\) imply
These are sufficient bounds, not the exact basin. On a fixed zero-leakage leaf the auxiliary deviation is constrained by \(r-r_*=-c^T\eta\). An independent auxiliary perturbation must instead be included when determining the state's actual \(J\) and target. It cannot be dropped while invoking the first row of (15).
4. Full-state attraction and the remaining neutral information¶
For each \(e<1/768\), the portion of the closed phase ball with \(E\le e\) is compact and positively invariant under the corresponding complete law. Compactness follows from the form bound in (12), the bounded phase ball, and either \(r=J-c^T\theta\) or the reservoir bound in (12). Invariance was proved by the boundary barrier, not assumed from passivity.
In the zero-leakage case, zero storage loss requires \(Lx=0\), hence \(x=0\). To remain in this zero-loss set, the form row in (5) also requires \(\nabla U_J=0\). The unique equilibrium in (6) is therefore the largest invariant subset of the zero-loss set. The finite-dimensional LaSalle invariance argument gives
This controls every fine transverse form and phase direction on the invariant leaf: with the two means fixed, the full joint state has dimension 71 and the fixed-\(J\) leaf has dimension 70. It is not only stability in the scalar \(B\) direction or in the four-child invariant family. The positive energy bounds also establish Lyapunov stability relative to that leaf.
In the positive-leakage case, zero joint loss requires both \(x=0\) and \(r=0\). Invariance additionally requires \(S(\theta)=0\), so the largest invariant zero-loss set is the single equilibrium (8). The same argument proves local asymptotic stability in all 71 mean-fixed joint directions, including independent auxiliary perturbations. This conclusion does not give global attraction from the full acute chart or entry from the previous finite-reservoir source.
For zero leakage, a particular equilibrium cannot attract arbitrary nearby states on different \(J\) leaves: nearby members of (6) are exact equilibria themselves. The locally attracting object without fixing \(J\) is the family, with each sufficiently close state subject to its own conditions (14). Its target varies continuously, since
This is retained initial information, not an unforced periodic motion. For a trapped state whose \(J\) lies strictly below the threshold in (7), convergence implies eventual registration. Strictly above that threshold, it implies eventual nonregistration of both parents. At the threshold, convergence to the identity boundary alone settles neither strict finite-time alternative. The uniform phase ball by itself preserves children, not registration of every parent throughout the transient.
The extra Hessian term \(cc^T\) is positive semidefinite at a fixed phase state. It does not show that every transverse stiffness increases when comparing different equilibria: the equilibrium shift also changes the edge cosines. Neither this rank-one term nor dissipation selects the port law, the auxiliary coordinate or its initial value.
5. Evidence boundary¶
This proof classifies the supplied joint law on one declared lifted chart. It derives an exact information constraint for zero leakage, distinct registered and unregistered equilibrium geometries, and full-state local attraction under explicit trapping premises. The positive-leakage model has a different local equilibrium selection because it changes that constraint. No finite deadline or convergence rate is asserted.
The static implementation reads admitted primitive states and rebuilds their retained quantity, equilibrium, chart and local-energy evidence. The norm-allowance conditions in (15) are additional mathematical sufficient conditions, not a required implementation API. The owner tests check the static identities and admission boundaries without trajectories or source replay.
Root and threshold enclosures must retain unresolved strict boundaries, and local sufficient admission must remain distinct from an actual source-capture result. Cached verdicts are not premises. This owner supplies no second research queue.