Collective state and a matrix-response comparison¶

Status and responsibility¶

This owner specifies one complete collective comparison for the foundational question raised by the physical feasibility review. The law below is a supplied constitutive choice. Its balance and coordinate identities are conditional algebra. The prepared-reference theorem adds own-law identity retention, a finite response separation and a precise state-reduction obstruction. Acquisition of this preparation, fine-source inheritance and physical realization remain unestablished. The research program alone selects work and acceptance conditions.

The question is how retained state, response and storage must change together when an organized object has an additional structural coupling. It does not assume a new primitive capacity or a uniquely selected fundamental law. The NFR ontology requires any claim about constituents to retain their influence in the same configuration.

1. Complete comparison and units¶

Use the fixed, unit-weight, undirected C5 Laplacian L and five signed forms U with five primitive circular phases Phi. These are the collective reference coordinates in the quarter-law dictionary. They do not include the 45-node source's internal deviations. Retain the same normalized structural clock tau=(1023/1024)t and units, gamma=1/(1023 pi), and held unit scalar capacities. Primes below denote derivatives in tau. Laboratory calibration is a separate obligation. Write m for the conserved mean form; r below always denotes the response parameter, not the mean denoted r in the earlier electrical protocol.

Let r be a supplied finite constant with r>=0, and define

\[ W_r=I+rL^2,\qquad M_r=W_r^{-1},\qquad S_i(\Phi)=\sum_{j\sim i}\sin(\Phi_j-\Phi_i). \tag{1} \]

The dimensionless parameter r changes the response in this declared chart; it is not fixed by the nodal row. The complete driven equations are

\[ W_r U'=(-LU+\gamma S(\Phi))/4+f(\tau),\qquad \Phi'=\gamma LU/4. \tag{2} \]

Input f is an independently specified, neutral, locally integrable five-vector; no phase feedback, support event or input selector is implicit. There are no jumps. Thus the solved form row has input M_r f, not f. The unforced pressure is the independently specified map p=M_r*(-LU+gamma*S)/4, with U'=p at unit capacity. Circular differences enter S without choosing a principal-phase average. Means of phase lifts are used only after declaring those lifts.

W_r is symmetric positive definite, commutes with L and satisfies W_r 1=1. The full equations preserve the form mean under neutral input; on any continuous phase lift they also preserve its common mean. One may fix the neutral form leaf and quotient a common phase origin, while retaining four relative forms and four relative phases. These are symmetry reductions, not removal of response-relevant internal state. An admitted initial set, its lifted chart where needed and every proposed identity must still be specified before a finite claim.

At r=0, (2) recovers the nominal collective quarter law. For r>0 it changes only the form row in these coordinates. This differs from the phase-dependent diagonal reciprocal mobility, which changes both rows and keeps another storage. Its proofs do not transfer.

The electrical audit motivates (1) through an additional positive ring capacitance, with r=c_J/c. That motivation does not force this law on arbitrary NFRs or provide a fine-node electrical realization. In particular, replacing L by L45 would be a further constitutive choice requiring its own admission.

2. Storage and supplied work¶

Let V(Phi)=sum_edges[1-cos(Phi_j-Phi_i)], so grad V=-S. The comparison's own storage is

\[ H_r(U,\Phi)=\tfrac12 U^TLW_rU+V(\Phi). \tag{3} \]

Since L and W_r commute, LW_r is symmetric and positive semidefinite. Differentiating along (2) gives, almost everywhere for integrable f,

\[ H_r'=(LU)^T W_rU'-S^T\Phi' =-\|LU\|^2/4+(LU)^Tf. \tag{4} \]

The cross terms cancel. This is a storage/work balance, not a formation, stability, recurrence or observation theorem. The old H0 is not the storage in (4); the difference is r*U^T*L^3*U/2. Consequently equal initial states need not have equal storage across r. A comparison must declare which preparation quantities are matched instead of assuming both equalities.

There is an explicit obstruction to reusing the old decay claim. Set f=0, q=LU and H0=H_r at r=0. Along the positive-r law,

\[ H0'=-q^TM_rq/4+\gamma q^T(M_r-I)S/4. \]

Choose Phi=(0,pi/2,0,0,0), so S=(1,-2,1,0,0), and U=-(gamma*r/2)LS. Then q=(gamma/2)(I-W_r)S and, for every r>0,

\[ H0'=\frac{\gamma^2}{16}S^T(M_r-I)W_r(M_r-I)S>0, \qquad H_r'=-\|q\|^2/4<0. \]

The strict inequalities follow because S is a nonzero neutral vector and L is positive definite on that subspace. This example uses the full circular-phase domain of (2); it is not an admitted acute winding source. It disproves a global transfer of the old storage monotonicity, without claiming loss of an acquired identity.

Because W_r is invertible, unforced equilibria of (2) agree with those of the r=0 complete law. Their tangent rates and finite response need not agree. Matching equilibria does not transfer attraction, an initial-state capture bound or an observation certificate. A finite positive-r study must declare its parameter range and horizon before requesting uniform bounds.

3. Coordinate, locality and memory boundaries¶

The invertible charge-like chart Z=W_r U changes both equations:

\[ Z'=(-LM_rZ+\gamma S(\Phi))/4+f,\qquad \Phi'=\gamma LM_rZ/4, \tag{5} \]

with storage Z^T*L*M_r*Z/2+V(Phi) and work pairing (LM_r Z)^T f. Moving W_r out of the form row while retaining Phi'=gamma*L*Z/4 would therefore change the model. A regular chart change is distinct from discarding state and from choosing a different pressure law.

Although W_r is positive definite, M_r is not generally entrywise nonnegative. For example, at r=1/100 its first row is (2181,81,-19,-19,81)/2305. It is not a convex averaging map or a vector of nonnegative nodal capacities. Its coupling need not have the one-edge locality of the original sine-pressure row; locality must be justified for the actual map, including the phase and input channels.

This fixed matrix is instantaneous. It introduces no temporal memory by itself. Derived memory in the cited reductions comes from eliminating dynamical state under a specified observation, retaining hidden initialization and forcing, as in the memory owner. The inherited-metric result already distinguishes coordinate response factors from primitive capacity. Neither construction justifies replacing nu_f by M_r in the native runtime.

Holding r constant is part of (2). On the neutral form leaf, a time-dependent charge law q proportional to W_r U differentiates to an expression proportional to W_r U'+r' L^2 U; off that leaf the charge map uses W_r(U-m*1). Substituting W_r(tau) into (2) alone would be a different model. Parameter changes or support events also need their storage/work and occurrence laws.

4. Identity, scale and implementation boundary¶

The state in (2) is sufficient for this declared ten-coordinate ODE. The finite theorem below uses a bounded preparation of these same coordinates, including the relative phases. It is not a proof that ten coordinates predict the full acquired source. The same-slot source remainder, its hidden transport, constituent identities and acquisition use another complete law. They remain valid there. Neither r=0 recovery nor a shared equilibrium establishes those claims at positive r.

A claim about a parent NFR must define whether it concerns this prepared collective reference or a finer organization with retained constituent state. The latter needs a separately declared fine law, a state/clock/input map, exact closure or controlled error, and own-law identity bounds. Static re-encoding or formal associativity of an exact elimination is insufficient as a new capability; finite nonlinear prediction must retain the consumed boundary information. Events additionally require the existing operator descent criterion.

This comparison installs no execution law, SDK method or operator. The independent algebra controls check (1)-(5), the r=0 boundary and deliberate wrong-chart/input controls. They evaluate no reserved response. Exact matrix calculations reuse shared linear algebra. The coordinate-memory owner can split an admitted fixed linear/tangent generator or rational spatial blocks. It does not close the nonlinear sine law. Such reuse must retain nonlinear residuals, hidden initialization and forcing where consumed; rationalizing gamma would specify a different coefficient, not certify the exact comparison above. A nonlinear reduction needs its own proof. Generic runtime integration is justified only by a proved consumer contract.

5. Declared finite prepared-reference question¶

The analytical comparison fixes the following domain before evaluating any response. It concerns (2) alone, without a fine-node extension, source replay, physical sensor or numerical trajectory. The two models match their complete initial coordinates and prescribed input; their initial storages need not match.

Primitive	Declaration
Response	Fixed `r in [1/100,1/10]`; separate reference `r=0`
Prepared identity	Either C5 winding `w=+1` or `w=-1`
Phase chart	`Phi_i=2piwi/5+theta_i`, with a continuous perturbation lift and `sum(theta)=0`; the closing oriented edge retains its `2pi*w` offset
Full preparation	`sum(U)=0`, `\|\|U(0)\|\|_2+\|\|theta(0)\|\|_2 <= delta=2^-26`
Input	`f=a*v` for `0<=tau<=T`, zero afterwards; `a=1/16`, `v=(1,-1,0,0,0)`
Clock and horizon	The existing structural tau; `T=1/1024`
Observation	`y=v^T*U(T)` with separate additive error bounded by `eta=2^-26`
Identity criterion	Every oriented phase gap stays within `rho=1/10` of `2piw/5` during the pulse and its unforced continuation
Finite prediction criterion	Whole-preparation observed record intervals for the positive-r family and r=0 are disjoint, with separation strictly greater than `2^-21`
Calculation	Exact rational inequalities and certified transcendental bounds; no integration, parameter scan, precision search or response fitting

The prediction must retain both evolution rows, the full preparation bound, the input channel M_r*f, and the law's own storage and supplied work. A separate state-sufficiency control tests whether identical forms, winding, conserved means and storage can still conceal different form derivatives. Neither success condition would derive acquisition of this preparation, constitutive selection, a laboratory clock or a physical measurement map.

6. Own-law identity and finite prediction¶

Declaration revision a205a8fe fixes Section 5. The result is analytical: no measured or simulated response is used to set its constants or decide between the laws. Write kappa=2*pi*w/5, let D take the five oriented cyclic differences, and use the declared closing-edge offset. All norms below are Euclidean except the explicitly marked maximum norm. The mixed form/phase norm uses numerical coordinates in the fixed normalized chart of Section 1; it is not an invariant metric or a laboratory calibration.

6.1 Identity uses the actual storage¶

On the centered form leaf, define excess storage relative to the prepared twist by

\[ E_r=H_r-V_* =\tfrac12 U^TLW_rU+ V(\Phi_w^*+\theta)-V_*,\qquad V_*=5[1-\cos(2\pi/5)]. \tag{6} \]

The full preparation cost includes V_*; E_r is not total storage or a budget that supplies the external input. Consider the convex relative phase cell ||D theta||_infinity <= rho=1/10, sum(theta)=0. Here cos(kappa+d)>1/5 for every |d|<=rho: cosine is 1-Lipschitz and cos(2*pi/5)=(sqrt(5)-1)/4>3/10. The first variation of V at the twist vanishes because sum(D theta)=0. Taylor's integral remainder and LW_r>=0 therefore give

\[ E_r\ge\tfrac1{10}\|D\theta\|^2. \tag{7} \]

Every first contact with this cell's boundary requires E_r>=1/1000. Also ||L||<=4, as follows directly from the edge-square bound sum_edges(z_j-z_i)^2<=4||z||^2. The initial preparation consequently obeys

\[ E_r(0)\le2(1+16r_{\max})\|U(0)\|^2+2\|\theta(0)\|^2 \le(4+32r_{\max})\delta^2=\tfrac{36}{5}\,2^{-52}. \tag{8} \]

The bound also holds for the r=0 reference. Initial cell admission follows from ||D theta(0)||_infinity<=2 delta<rho. Completing the square in the own-law balance (4), rather than transferring H0 decay, yields

\[ E_r'=-\tfrac14\|LU-2f\|^2+\|f\|^2\le\|f\|^2. \tag{9} \]

During the declared pulse, ||f||^2=2a^2. Thus

\[ E_r(\tau)\le\overline E=\tfrac{36}{5}\,2^{-52}+2^{-17} <2^{-16}<\tfrac1{1000}\quad(0\le\tau\le T). \tag{10} \]

Continuity excludes a first boundary contact. After T the input is zero and E_r is nonincreasing, so the same identity survives for all future times. In particular the gaps remain strictly acute with the original winding. The smooth periodic sine term and constant positive-definite W_r give global continuation; there is no hidden event at pulse release.

On the fixed-mean lift, the intersection of E_r<=overline E with the closed relative cell is compact and contained in its interior. Zero dissipation requires LU=0, hence U=0; remaining invariant trajectories also require S=0. The potential is strictly convex on this relative cell by the cosine Hessian bound used in (7), so its only critical point is the prepared twist. The invariant-set argument therefore gives convergence to that twist after the pulse for each fixed r. It supplies no uniform settling time, acquisition from another winding sector or indefinite external activity.

6.2 A finite response with both rows live¶

Let z(tau)=||U(tau)||+||theta(tau)||. For every phase state, the sine-source Jacobian is an incidence matrix times a diagonal cosine matrix times its transpose; hence ||S(Phi_w^*+theta)||<=4||theta||. Also ||M_r||<=1 and the exact gamma=1/(1023*pi)<1. Both rows of (2) give

\[ D^+z\le2z+2a,\qquad \sup_{0\le\tau\le T}z(\tau)\le Z=\frac{\delta+2aT}{1-2T}. \tag{11} \]

The rational bound follows by taking the supremum in the integrated inequality, using T<1/2; it is not a numerical exponential or a frozen phase approximation. Integrating the form row and taking its v projection gives the finite prediction

\[ y=aT h(r)+R,\quad h(r)=v^TM_rv, \quad |R|\le\mathcal R=2\delta+2TZ. \tag{12} \]

The remainder retains initial form, unknown initial phase shape, damping and the phase response throughout the pulse. Here ||v||=sqrt(2)<2 and ||M_r(-LU+gamma S)/4||<=z supply (12). It is not a tangent trajectory or a finite difference at time zero.

On the neutral C5 space, L^2-5L+5I=0. With d(r)=1+15r+25r^2,

\[ M_r=\frac{(1+20r)I-5rL}{d(r)},\qquad h(r)=\frac{2(1+5r)}{d(r)},\qquad h'(r)=-\frac{20+100r+250r^2}{d(r)^2}<0. \tag{13} \]

The formula for M_r here is restricted to the neutral space; its common mode remains one. Since v^Tv=2 and v^TLv=6, it determines h exactly throughout the parameter interval, not merely at sampled endpoints. Write b=mathcal R+eta. The observed record sets are contained in

\[ \mathcal Y_+=\left[aT\tfrac{12}{11}-b,\; aT\tfrac{840}{461}+b\right],\qquad \mathcal Y_0=[2aT-b,\;2aT+b]. \tag{14} \]

For the fixed declaration, exact arithmetic gives

\[ Z=\frac{8193}{66977792},\qquad \mathcal R=\frac{9215}{34292629504},\qquad \inf\mathcal Y_0-\sup\mathcal Y_+ =\frac{40665805}{3952225550336}>2^{-21}. \tag{15} \]

An independent, coarser spectral bound also suffices: L>=I on the neutral space implies h(r)<=2/(1+r), so the separation is at least 2aT/101-2b=555365/865888894976>2^-21. Thus the conclusion does not depend on fitting the exact response gain or on the last digits of a trajectory.

The pulse's actual supplied work and continuous loss are separately bounded:

\[ \int_0^T |(LU)^Tf|\,d\tau\le8aTZ =\frac{8193}{137170518016},\qquad \int_0^T\tfrac14\|LU\|^2\,d\tau\le4TZ^2. \tag{16} \]

Equation (4) still retains the signed work. The 2a^2 T allowance in (10) is an energy-growth inequality, not an equality for delivered work or an autonomous reservoir. No hardware or observation mechanism is inferred from the bounded additive error in (14).

7. Why form, winding and storage do not close the state¶

This comparison also admits a constructive obstruction inside its prepared domain. Fix w=+1, one node i, s=delta/2>0, and two states

\[ U_+=U_-=0,\qquad \theta_\pm=\pm s(e_i-\mathbf1/5). \tag{17} \]

They have the same complete form vector, both conserved means, winding and own storage V_*+2*cos(2*pi/5)*(1-cos(s)). Their changed edge gaps are the same pair kappa-s,kappa+s in opposite positions; all other gaps equal kappa. Both states satisfy the full preparation radius.

At the same independently specified input, direct edge summation gives

\[ (U_i')_\pm=(M_rf)_i\mp\tfrac\gamma2 (M_{ii}-M_{i,i+1})\cos(2\pi/5)\sin s,\qquad M_{ii}-M_{i,i+1}=\frac{1+5r}{d(r)}>0. \tag{18} \]

Reflection symmetry makes the two adjacent entries of M equal; their even terms cancel in this node's response. The two rates are strictly different. Therefore form, even augmented by winding, means and storage, is not an exact autonomous state on this class. The finite envelope (12) remains valid because it retains a bound on the discarded phase information. This obstruction does not exclude a controlled approximation or a memory description retaining hidden initialization and the supplied input.

8. Executable certificate and remaining scope¶

The detached arithmetic owner rebuilds (8), (10)-(16) from admitted primitive budgets. It reports identity and record separation separately: a failed sufficient bound is not a trajectory failure or an impossibility theorem. Its variable-budget interface supports arithmetic controls, not a new search or evaluation campaign. The shared C5 preparation owner evaluates (8) and supplies this collar's radius and barrier to the unloaded, loaded-sensor and recovery consumers. It admits primitive budgets without inferring that a supplied state belongs to the preparation. The independent controls use edge-built matrices, symbolic polynomial identities, direct sine sums, boundary coercivity and deliberate loss of admission. They preserve the existing frozen evidence and execute no reserved response.

This closes the declared prepared-reference identity/prediction question. It does not select r, derive this law from the nodal identity, inherit it from the 45-node source, or supply acquisition, operational preparation, physical identification or a sensor/clock model. The exact state and work requirements now constrain the separate finite-window observation question; only the execution plan activates that work.