Reciprocal form resources under arbitrary contacts¶
The matched-contact preparation bound does not directly cover unequal interface weights or additional reacting organizations. This owner derives a necessary form-resource bound for arbitrary reciprocal contacts on a fixed finite set of nodes. It retains every initial form, including donor means and internal deviations. Neither more contact time nor a different contact schedule removes this bound.
This is a conditional consequence of the supplied constitutive family, not a new TNFR axiom, a source mechanism or a sufficient preparation construction. The form foundation and parameter foundation retain their definitions. Task status belongs only to the research program.
1. Complete law and admitted contact histories¶
Fix an ordered set of \(N\ge2\) labeled nodes, held unit capacities, the nondimensional structural clock \(\tau\), and \(\gamma=1/(1023\pi)\). Forms \(x_i\) are signed real coordinates; phases \(\theta_i\) are circular, with continuous lifts used for evolution. For each unordered pair prescribe a nonnegative measurable weight \(w_{ij}(\tau)=w_{ji}(\tau)\), locally integrable in time. Zero weights are allowed. No connectivity, uniform degree or common interface weight is assumed. State-preserving changes of support are included in these weight histories; there are no form or phase jumps, inputs, node births, capacity changes or label changes.
Hold one \(\lambda\in[0,1]\) throughout the history and use
Let \(L(\tau)\) be the conductance Laplacian and \(U(\tau,\theta)=\sum_{i<j}w_{ij}(\tau)V_\lambda(\theta_i-\theta_j)\). The same weights occur in both complete quarter-factor rows:
These equations specify the pressure independently of the response. They are the comparison model's equations, not a native operator word or a reciprocal-degree normalization. A clock change transforms both rows. No laboratory-unit or physical-measurement interpretation is supplied.
Finite initial forms and phase lifts have a unique absolutely continuous solution on every finite time interval. Indeed, the current is globally bounded and Lipschitz, while the sum of the finitely many edge weights is locally integrable. The full vector field consequently has integrable Lipschitz and linear-growth bounds; its integral equation and Gronwall bound give continuation. Edge antisymmetry gives \(\sum_i x_i(\tau)=\sum_i x_i(0)\) and the analogous conserved sum of the chosen phase lifts. Neither conclusion requires connectivity. For state-dependent contact choices, the bound below applies along any separately admitted solution with locally integrable realized weights. This statement supplies neither a feedback selector nor its well-posedness.
The proof below uses the actual form row and a uniform bound on its phase currents. It does not hold phases fixed, project to a carrier, assume acute gaps, or replace the independently specified phase row.
2. An invariant envelope of sorted partial sums¶
Put \(\bar x=N^{-1}\sum_i x_i(0)\) and \(y=x-\bar x\mathbf1\). Sort only for the following observation, writing \(y_{(1)}\ge\cdots\ge y_{(N)}\). The physical labels are unchanged. Define
On the zero-sum subspace, \(q\ge0\), with equality only at zero. It is a finite maximum of linear functions, hence Lipschitz and absolutely continuous along the solution. The conditional theorem is
To prove it, choose any index \(k\) active in (3) at a state where \(q\ge c>0\). The neighboring partial sums satisfy their same \(qD_j\) bounds, including \(S_0=S_N=0\). Therefore
The top \(k\) set \(A\) is thus separated strictly from its complement; ties within either set cause no problem. Internal edges cancel in the derivative of its sum. At almost every time,
This includes a disconnected cut, whose sum is zero. Every active index has this property. The derivative rule for a finite maximum therefore gives \(q_\tau\le0\) almost everywhere where \(q\ge c\). More explicitly, all subset sums and their maximum are differentiable outside one null set; at those times the derivative of the maximum is an active derivative. No derivative of an edge weight is used. Where \(q<c\), the function \(\max(c,q)\) is locally constant; at its remaining differentiability points with \(q=c\), its derivative is also nonpositive. Integrating this absolutely continuous function proves (4), including arbitrary ties and measurable contact changes.
Equivalently, each polytope \(\{y:\sum_i y_i=0,\ S_k(y)\le\rho D_k\}\), with \(\rho\ge c\), is invariant. The endpoint inequalities give the useful corollary
3. Marked nine-node means and node-count resources¶
Now let \(N\ge18\), and retain any two disjoint marked sets \(A,B\), each containing nine nodes. They need not be entire connected components. Write \(r=\frac19\sum_Ax_i-\frac19\sum_Bx_i\) and \(v=\gamma r\), as in the free-probe preparation. At every time,
Indeed, a nine-node sum is no greater than the top-nine sum and no smaller than the bottom-nine sum \(-S_{N-9}\). Interchanging \(A,B\) gives the absolute-value bound. This observation retains all other nodes through the initial \(q\) and the fixed total \(N\).
If every initial form is the same, \(q(y(0))=0\) regardless of the actual initial phases. For the original eighteen nodes, (8) yields
The strict numerical inequality uses \(\pi>157/50\) in the declared \(\gamma\), not an evaluated response. Arbitrary unequal interfaces and support switching on these nodes cannot prepare the target. The value \(v=12\) is a sufficient choice for the declared full-turn probe, not a necessary preparation for every possible discrimination. For additional nodes also starting at that same common form, a necessary node-count condition for \(|v|=12\) is
This is not a sufficient construction, a proposed reservoir, a minimum number of physically realized constituents, or proof that the bound is dynamically attainable. Donors with other initial forms instead contribute to \(q(y(0))\); they cannot be counted as zero-resource additions.
The observation bound is sharp as a statement about the polytope alone. Its arithmetic-progression vertices \(y_{(i)}=\rho(N+1-2i)/2\) attain every partial-sum cap and the top-nine versus bottom-nine bound in (8). No claim follows that the complete phase/form dynamics reaches those vertices from a given source.
4. Primitive preparation information cannot be discarded¶
The function \(q(x-\bar x\mathbf1)\) is unchanged by a common form offset and by a simultaneous permutation of node labels. It is not determined by child means or by donor count. To admit uncertain primitive forms, let \(\widehat x\) be an ordered record with \(|x_i(0)-\widehat x_i|\le\varepsilon\) for every node. For any subset \(A\) of size \(k\), its centered sum has coefficients \((N-k)/N\) on \(A\) and \(-k/N\) on its complement. Hence
Each normalized subset functional has exactly that error bound; taking their finite maximum proves the first inequality. The upper endpoint is attained for every record: choose an active maximizing subset of \(\widehat x\), add \(+\varepsilon\) on it and \(-\varepsilon\) on its complement. That functional increases by \(4\varepsilon/N\). Thus the exact box supremum of the true \(q\) is the recorded \(q\) plus \(4\varepsilon/N\). In particular, \(|x_i(0)-m_0|\le\varepsilon\) about any supplied common origin gives \(q(y(0))\le4\varepsilon/N\). Two nonempty groups at the opposite endpoints \(m_0\pm\varepsilon\) attain this constant. Uncertain forms must be admitted before sorting or centering.
A simple full-state witness explains why zero child means are insufficient. Take two marked winding-one C9 phase carriers with equal phase origins. In A give two distinct slots forms \((-H,+H)\), \(H>0\), with all other forms zero in both children. Both child means and the global mean are zero. Keep the internal cycles and add only one cross edge of weight \(w>0\), from the \(-H\) slot to its matching zero-form slot in B. The phase current of this edge is zero at this state. Nevertheless,
Internal terms cancel in each mean, and the two endpoint form rates from that cross edge are \(wH/4\) and \(-wH/4\). Thus child means alone miss an arbitrarily large initial boundary rate; the full resource in (3) retains it. This is an instantaneous analytic control, not a trajectory, maintained-identity theorem or preparation of \(v=12\).
5. Contact work and scope of the exclusion¶
Nonnegative reciprocal weights need not be storage-passive when they change. If they are absolutely continuous between finitely many support jumps, the same complete rows give
A state-preserving weight jump changes \(H\) by the corresponding sum with \(\Delta w_{ij}\). Arbitrary locally integrable weights need not have a finite-variation work ledger. The form envelope requires neither that ledger nor event passivity; the preparation work owner retains their separate obligations.
The proof remains valid even when constituent winding or an acute chart is lost. It therefore excludes a preparation without having to assume identity maintenance, but supplies no maintenance certificate when a resource bound fails to exclude. Contact occurrence, source origin and physical identification remain open. Asymmetric contacts, independently weighted phase currents, changed capacities or normalization, form jumps, inputs and node creation lie outside (2); no verdict transfers to them.
The detached calculation in
_reciprocal_form_resources.py
re-admits ordered primitive forms and their declared uncertainty before
computing a sufficient upper envelope. Its
static controls
exercise cut cancellation, resource admission and counterexamples.
These calculations do not evaluate a trajectory, select a support history,
replay a formation source or certify physical preparation.