Complete-law compatibility of three organized C9 constituents¶
The pairwise sine charts do not generally combine into one chart for a network. For the three prescribed oriented contacts, an unchanged normalized reciprocal sine law requires both equal contact amplitudes and a vanishing phase sum around the triangle. These conditions are necessary and sufficient in the positive nodewise affine form charts, constant phase-origin shifts and common clock specified below. Among the actual co-oriented nonnegative contacts, only the all-zero-offset control satisfies them.
The complete inherited dynamics still closes exactly on six registered coordinates for every admitted contact. Retaining the contact amplitudes and phase registrations gives that law without fitting or additional evolving variables. The obstruction concerns a more restrictive scalar network representation; it does not invalidate the supplied fine law, its storage balance or its conditional organized identity.
The pairwise owner supplies the interface family and its conditional sine-profile selection. This note fixes sine and tests whether those pair descriptions coexist on one cycle. The NFR ontology permits retained constituent and boundary information. It does not require the restrictive chart whose obstruction is proved here.
Supplied support, complete law and retained identity¶
There are three marked C9 constituents, with nodes \((r,\ell)\), \(r=0,1,2\), \(\ell=0,\ldots,8\), ordered as \(9r+\ell\). Each has its unit-weight internal cycle. Put \(\alpha=2\pi/9\) and orient the constituent triangle as
For each oriented pair \(e=(r,t)\), fix \(s_e\in[0,1]\) and add the undirected contacts
Indices are modulo nine. Zero-weight edges at the endpoints are absent; every positive edge acts at both ends with the same weight. Orientation records which constituent port is shifted, not a directed force. Each node has cross conductance two and total weighted degree four. There are 27 internal edges and between 27 and 54 positive contact edges.
The fine state retains signed real form \(x\), primitive circular phase with specified continuous lifts \(\theta\), held unit capacities, the marked support and its three fixed weights. Use the inherited structural clock \(\tau=(1023/1024)t\), a supplied constant \(\gamma>0\), and the actual weighted fine Laplacian \(L_f\). The complete law is
Primes denote \(\tau\)-derivatives. In original time both rows acquire the same factor \(1023/1024\). The form row specifies pressure in the unit-capacity nodal equation; it does not reconstruct pressure from an observed derivative. The common degree factor is neither a derived capacity nor an inverse mass. Structural units and any laboratory clock conversion remain those of the parameter foundation.
This is a detached supplied weighted reciprocal law. In particular, its weighted phase current differs from the native pressure rule's unweighted unique-neighbor phase channel. No operator word, support selector or native runtime is installed by (3). Odd exchange, symmetric weights and the supplied reciprocal row give
These identities concern continuous flow with no additional input.
The candidate registered organized family is
Each constituent keeps its winding-one internal geometry. The six coordinates include three signed forms and three lifted phase origins; the constituent labels, marked port embeddings and contact parameters remain part of the law. Independent changes of these phase origins without corresponding changes to the contacts are not symmetries of the fixed network. A common phase shift is a symmetry. A common signed form offset is also a symmetry of this supplied difference-only law.
The winding one's vanishing first phase resultant and registration-losing ninth moment cannot replace \(\Phi_r\) with its marked embedding. Equation (5) is a prescribed invariant family, not an acquired source. Arbitrary fine states outside it still retain their internal coordinates.
Exact six-coordinate dynamics and full storage¶
Let \(B\) be the three-by-three incidence matrix with a positive entry at an edge's tail and a negative entry at its head, in order (1):
Define \(\Delta=-B^T\Phi\), so \(\Delta_e=\Phi_t-\Phi_r\), and
The real part of \(m_e\) is positive. Thus its continuous argument is unambiguously chosen in \([0,\alpha]\), and
For a node at the tail of \(e\), the weighted cross current equals \(J_e\); at its head it equals \(-J_e\). Internal currents cancel pairwise. Form diffusion is constant within each constituent and equals \((LX)_r\). Consequently the exact restriction is
Both fine rows are tangent to (5). The fine vector field is globally Lipschitz in its lifted coordinates, so uniqueness proves invariance. This is exact nonlinear closure on (5), with fixed boundary coefficients; it makes no off-family approximation or transverse-attraction claim.
The internal potential baseline is \(B_0=27(1-\cos\alpha)\). Counting nine fine copies of every collective edge gives
Indeed the gradients of the bracket are \(LX\) and \(-BJ\); the reciprocal cross terms cancel exactly. Its potential remains nonnegative because each \(U_e\) is the actual convex combination of two fine sine potentials. The constants \(B_0\) and \(1-\rho_e\) are not removed from the support-work account.
Necessary and sufficient simultaneous chart conditions¶
The comparison target is the same unweighted three-edge triangle, with the normalized complete reciprocal sine law
Sine acts componentwise. The comparison uses every state on (5), not only an equilibrium, one trajectory or finitely many sampled responses. It does not allow edge-dependent diffusion, an independently adjustable phase-row support, nodewise clocks, state-dependent charts or an added forcing law. Those would be different comparison classes.
Allow any fixed positive nodewise affine form chart, constant phase origin shifts and one positive constant common clock:
All chart constants may depend on the supplied contact weights, but not on state or measured response. We retain the transformations of the marked ports, inputs and observations below.
Theorem. A chart (12) turns both rows (9) into (11) if and only if
Every such chart has
The common form offset \(b_*\) and common phase origin are arbitrary symmetries, not fit parameters that remove the compatibility conditions.
To prove necessity, compare the terms linear in \(Z\) in the form row. The transformed diffusion matrix is \((4c)^{-1}A^{-1}LA\). Its diagonal entries must equal those of \(L\); since they are two, \(c=1/4\). The off-diagonal entries then force \(a_t/a_r=1\) on every edge. Hence \(A=aI\). The phase row independently requires \(Lb=0\) and \(\Gamma=\gamma a\). Connectedness gives \(b=b_*\mathbf1\).
The remaining form-row equality is
The three oriented edge characters on the node-phase torus are distinct up to sign. In a given node row the two incident nonconstant Fourier characters are also distinct. Uniqueness of their Fourier coefficients therefore implies, for each edge separately,
This step uses the full network vector field: merely assuming individual edge equality would miss the need to exclude cancellation of net nodal currents. The identity \(B\mathbf1=0\) permits constant circulating edge currents, but the nonconstant characters in (15) cannot produce such an unobserved circulation for every independent node phase.
Taking magnitudes gives \(\rho_e=a^2\); summing the argument differences around the triangle gives \(\Theta=0\) modulo \(2\pi\). Conversely, when (13) holds, set \(\beta_0=0\), \(\beta_1=\phi_{01}\), \(\beta_2=\phi_{01}+\phi_{12}\). The closing edge then satisfies (16) as well. Equations (14) transform both rows directly to (11), proving sufficiency.
This theorem classifies the declared affine/phase/clock maps. It is not a theorem ruling out every nonlinear state conjugacy, observation equivalence or model with additional boundary variables.
The co-oriented family and the two fixed controls¶
For the actual contacts (2), \(0\le\phi_e\le\alpha\), with \(\phi_e=0\) if and only if \(s_e=0\). Therefore
Vanishing modulo \(2\pi\) is possible only when every \(s_e=0\). Then every \(\rho_e=1\), and the complete target law (11) follows with \(\Gamma=\gamma\) and \(\sigma=\tau/4\). All other contacts in this specified co-oriented cube fail at least the cycle condition. Changing an interface orientation changes the supplied support and must be treated as a different model, not used to erase this result.
For the equal-weight control \(s=(1/2,1/2,1/2)\),
All pairwise form and reciprocal scales agree, but their phase-origin charts cannot coexist around the cycle. This isolates the phase obstruction independently of amplitude mismatch.
For the unequal control \(s=(1/4,1/2,3/4)\), the identity \(m(1-s)=e^{i\alpha}\overline{m(s)}\) yields \(\phi(1/4)+\phi(3/4)=\alpha\), so again \(\Theta=\pi/3\). Its magnitudes satisfy
Thus this fixed control also fails the common positive form/reciprocal scale condition. Choosing a different form unit for each pair would assign incompatible values to a constituent shared by two pairs; the nodewise argument leading to (14) makes that failure explicit.
Failure of (13) does not imply absence of an equilibrium. At equal \(s_e=s\), take \(X=X_*\mathbf1\), \(\Phi=\Phi_*\mathbf1\). Every edge current is \(s\sin\alpha\), so \(BJ=0\) and both rows vanish. The same cancellation holds at every fine node in (3), making this an exact fine equilibrium. For \(s>0\) the structural edge-current terms form a nonzero cycle with zero net nodal current. This algebraic statement assigns no transported physical quantity. The static control makes no source-acquisition or neighborhood-stability assertion. At unequal weights the same aligned state instead has \(X'=\gamma\sin\alpha\,Bs/4\), which is nonzero for (19).
A sufficient retained boundary law when the chart fails¶
For all contacts, fix the same spanning-tree phase chart
Use \(Z=X\), \(\sigma=\tau/4\), and retain the transformed contact phases
The complete law becomes
It retains exactly the amplitudes and the closing-edge phase that the unmodified scalar-node law (11) cannot absorb. All quantities are determined by the supplied interfaces. No new evolving coordinate, memory approximation, controller or hidden force is added on (5).
A different spanning tree redistributes the residual phases among edges. A constant change of node phase origins changes each edge phase by a node difference, whose sum around the cycle is zero. The product \(\prod_e m_e/\rho_e=e^{i\Theta}\) is therefore unchanged by this coordinate change. This is a property of the marked contact law. It is not evidence of a physical gauge field, a charge or the separate auxiliary gauge construction.
The original fine storage in this chart is
Each potential term splits into \((1-\rho_e)+ \rho_e[1-\cos(\Psi_t-\Psi_r+\chi_e)]\). If \(\Theta\ne0\) modulo \(2\pi\), all three second terms cannot vanish simultaneously: their effective angles sum to \(\Theta\). Thus the sum of isolated pair minima is unattainable on this triangle. This conclusion does not identify the minimizing network state or prove how it is reached.
More generally one may retain a common positive scale \(X=aZ+b_*\mathbf1\) in (20). Then the phase coefficient is \(\Gamma=\gamma a\) and the form-current coefficients in its reciprocal normalization are \(\rho_e/a^2\). Retaining these edge coefficients is legitimate; declaring them all equal to one without (13) changes the model.
Inputs, support work and measured coordinates¶
Constituent-uniform fine inputs \(f_r(\tau)\) in the form row and \(g_r(\tau)\) in the phase row preserve (5). Their complete balance is
For \(X=aZ+b_*\mathbf1\), \(\Psi=\Phi+\beta\) and \(\sigma=\tau/4\), their transformed rows receive \(4f(4\sigma)/a\) and \(4g(4\sigma)\), respectively. The support and charts stay fixed. Applying these transformations to the full storage gives exactly four times the work rate in (24). Original coordinate observations retain
A form sensor therefore does not generally read \(Z_r\), and a phase sensor does not discard \(\beta_r\). Other observations must compose with this same embedding. No physical sensor calibration is derived.
When (13) is satisfied, the full storage in the target chart (11) is
This explicitly retains the multiplicative scale and additive boundary cost, rather than equating two differently normalized storages.
Starting with the three isolated cycles, attach (2) without changing any fine form or phase. The support-work jump is exactly
On (5) it equals
At the aligned state with constant forms and phase origins, this is \(9(1-\cos\alpha)\sum_e s_e\), equal to \(27(1-\cos\alpha)/2>0\) for both fixed controls. A positive jump requires supplied work; the continuous loss in (4) is not an available reservoir or a reason for the contact to occur.
At fixed support, a uniform form jump \(h_r\) becomes \(h_r/a\), while a uniform phase jump keeps its angle and the fixed origin offsets. Its work is the difference of the full storage. These coordinate facts do not admit or select a registered hybrid operator. Its live preconditions, atomic targets, collective descent and occurrence need their own contracts. Time-varying weights would add explicit support-work and chart-derivative terms absent from (3).
Consequence for inheritance and the remaining scope¶
Pairwise same-profile representations require compatibility at every shared constituent and every network cycle. Even the sine profile selected by the additional pairwise premise has a nontrivial boundary law on the prescribed triangle. The sufficient inherited description is (9), or its explicit tree chart (22), with marked boundary information retained. The collective interaction owner already distinguishes such information from a universal scalar-node representation; this is an exact organized example of that distinction.
The winding-one geometry is preserved for every trajectory initially on (5). This is conditional invariant-family identity, with no assertion that arbitrary nearby states remain there or that previously acquired two-object error budgets survive this different support and degree. Matching target geometry and local coefficients does not transfer nonlinear recovery or acquisition certificates. The separate acquired-assembly proof establishes full-state capture and winding retention for the two fixed controls from their actual independently recovered sources, with supplied attachment work retained. Other mixtures, later inputs and event selection require their own admissions. No trajectory, source producer, physical dataset, topology search or new numerical protocol was evaluated here.
This result neither identifies a fundamental physical interaction nor rules out a hierarchy carrying its actual boundary laws. The execution plan owns the next admission decision.
Implementation and independent controls¶
The detached static owner _organized_network_compatibility.py rebuilds the actual fine support, degree, contact coefficients, complete collective rows and full storage from admitted primitive data. Its fine reciprocal coefficient is fixed to \(1/(1023\pi)\); the proof permits any positive fixed \(\gamma\). The shared contact factors in _oriented_interface_constitutive.py carry no pair-model degree, source certificate or network verdict.
The static observation admits three joint node-phase phasors, then derives the three edge differences from that assignment. Three independently supplied edge differences need not satisfy the cycle identity and are not substituted for joint state. These circular observations do not construct continuous lifts or run the fine law. Exact primitive conditions decide chart compatibility: all weights vanish for the phase condition, and all \(s_e(1-s_e)\) agree for the amplitude condition. Overlapping rounded intervals cannot substitute for these equalities. A compatible common amplitude scale is reported separately even if the phase condition fails; it then belongs to a law retaining its phase lags, not to (11).
The independent controls compare these reports with the full 27-node edge construction and cover the fixed equal and unequal contacts, the compatible zero-offset control, the tree chart, storage/loss and admitted-state boundaries. No trajectory or source-acquisition producer is part of these static checks.