Constitutive law and unknown collective initialization

Analytical scope and result

The unchanged joint form observations distinguish the sine law from its retained analytic countermodel throughout the declared initial class, even when initialization is independently unknown under each law. The whole-record theorem proves that their complete finite noisy record sets are disjoint. Its retained strict certificate includes both nonlinear histories and each law's fine-source error. This is a conditional information result; no measured record or physical law is selected. The earlier record-only forecast still predicts under its supplied sine law.

This owner establishes the alternative's own carrier, hidden-state error, finite identity and numerical growth bounds. It also proves a local obstruction: two distinct admitted initial phase gaps under the two endpoint laws have exactly the same two ideal frozen-current quadratures. An admitted adjacent gap can also match the three ideal receiver currents. Neither local test uniformly selects the law. The complete 150-by-35 formal sampled map is injective, but a direction with exact low-order cancellations obstructs every global scalar or positive block bootstrap using the retained unconstrained coefficient and remainder bounds. Those preliminary results alone establish neither separation nor overlap of the complete noisy nonlinear record sets.

A constrained continuation uses the actual shared harmonic phase. Two long-ramp contrasts cancel the fundamental without dividing by the short plateau duration, and their ideal joint signal has a uniform positive phase-circle margin. The middle ramp supplies stronger same-mode contrasts. Joint observations also force equal winding and bound the fundamental phasor difference. These restrictions, actual late-time heat decay and correlated phase transport close the nonlinear separation theorem. A separate heat bridge through the same readings retains its historical phase-increment error bound below 2.8e-8; the final theorem does not need to reconstruct an individual record's phase history.

Item	Retained analytical scope
Support/state	Unit C9 x C5, nodes 9*c+j, 90 edges, signed form and primitive circular phase; held unit capacities
Two laws	Endpoints lambda=0,1 of the existing potential/current family below; bounds hold uniformly for 0<=lambda<=1
Initial class	abs(x_i)<=X=1/100; theta=psi+R*Phi+z, psi_j=2*pi*(j-4)/9, child-zero hidden means, abs(z_i)<=1e-6, ||u||2<=31/10^9, ||z||2<=601/10^10; principal collective gaps at most 7/5, winding in {-1,0,1}
Input/clock	Original fifteen ramps and plateaus, exact command, tau=(1023/1024)*t, local time v=tau-200, horizon T=16
Actual observations	Original thirty five-form readings, arbitrary coordinate errors at most eta=1e-14; no primitive phase or initial-state label is observed
Information result	The two complete finite noisy record sets are disjoint under the fixed premises; local collisions and the unconstrained-bootstrap obstruction retain their smaller scopes
Retained limitations	No selection within the whole constitutive family, identification of a particular record, alternative-law acquisition, apparatus admission or physical law selection
Evidence	Analytical proofs, one fixed scalar-moment assessment and one strict interval certificate using its retained primitives; no fit, response trajectory, sampled coefficient design, physical data or earlier frozen producer is evaluated

The ontology requires retained constituent and boundary information. An invariant carrier or an identity-preserving containing class is not an acquisition theorem. Support and the input remain supplied; no operator occurrence, new event, physical constituent or uniquely fundamental law is inferred.

1. The complete alternative on the present support

Retain the current/potential owner:

\[ V_\lambda(a)=1-\cos a+\lambda\frac{(1-\cos9a)^2}{6\,9^4},\qquad j_\lambda(a)=\sin a+\frac\lambda{2187} (\sin9a-\tfrac12\sin18a). \tag{1} \]

With L_f=I5 tensor L9+L5 tensor I9, S_lambda,i=sum_(k~i) j_lambda(theta_k-theta_i) and gamma=1/(1023*pi), the full rows are

\[ x'=(-L_fx+\gamma S_\lambda(\theta))/4+Rf,\qquad \theta'=\gamma L_fx/4. \tag{2} \]

The signed nodal pressure is specified by the original-clock form row; both rows undergo the same clock conversion. The phase channel is not an independently supplied synchronization law. Odd periodic current preserves the global form and lifted-phase sums for neutral input. Its own storage is

\[ H_\lambda=\tfrac12x^TL_fx+\sum_{e}V_\lambda(\delta_e),\qquad H_\lambda'=-\|L_fx\|_2^2/4+(L_fx)^TRf. \tag{3} \]

Reciprocity cancels the current/phase work exactly. Storage values and preparation costs remain law-dependent; (3) transfers no earlier resource or acquisition verdict.

The shared derivative bounds give

\[ J_\lambda=1+\lambda/1458,\quad L_\lambda=1+2\lambda/243,\quad B_\lambda=1+\lambda/9 \tag{4} \]

as global bounds for |j|, |j'|, |j''|. The full lifted field is globally Lipschitz on the fixed finite support, and the prescribed piecewise-smooth input gives one continuous state history through its switches.

Write P=RR^T/9, Q=I-P, x=RU+u and theta=psi+R*Phi+z. Internal helical currents cancel by oddness and periodicity. Same-slot outer currents are constant within a child at z=0. Thus u=z=0 is an exact invariant carrier for both laws and every uniform-child input:

\[ U'=(-L_5U+\gamma S_{\lambda,5}(\Phi))/4+f,\qquad \Phi'=\gamma L_5U/4. \tag{5} \]

The full stiffness on this carrier is

\[ K_\lambda(\Phi)=j_\lambda'(2\pi/9)(I_5\otimes L_9) +L_5^{j_\lambda'(\Phi)}\otimes I_9. \tag{6} \]

It commutes with P and Q, so the collective/hidden tangent blocks still decouple. This is not a transverse-stability theorem. At internal gaps 2*pi/9 the deformation's first relevant derivatives vanish, but at the outer 2*pi/5 gaps they generally do not. The complete C9 x C5 tangent is therefore not matched by the old C9 target construction.

The original command removal remains exact: Y=U-z_cmd, Psi=Phi-H, with H'=gamma*L5*z_cmd/4, gives

\[ Y'=-L_5Y/4+\gamma S_{\lambda,5}(\Psi+H)/4,\qquad \Psi'=\gamma L_5Y/4. \tag{7} \]

This reuses the waveform and coordinates, not the sine-specific forecast gain.

2. Own-law hidden transport, identity and growth

Let g=1/3000, a0=31/10^9, b0=601/10^10, and s=2*g*T. The hidden form row contains Q S_lambda(psi+R*Phi+z), which vanishes at zero z and has Euclidean Lipschitz bound 8*L_lambda. The graph Laplacian has norm at most 8. Conservatively bound both coupling rows with 8*L_lambda and reuse the shared integral majorant:

\[ A_\lambda=\frac{a_0+sL_\lambda b_0}{1-(sL_\lambda)^2},\qquad Z_\lambda=\frac{b_0+sL_\lambda a_0}{1-(sL_\lambda)^2}. \tag{8} \]

All denominators are positive at the fixed corner. These are full nonlinear hidden bounds under each law, not transported sine-source states.

On an outer edge, the nine hidden gap differences have zero sum. Taylor expansion of the actual current cancels the linear correction and bounds the mean remainder by B_lambda/18 times their squared sum. External Laplacian norm at most 4 then gives a mean-form forcing bound

\[ \rho_\lambda=gB_\lambda Z_\lambda^2/9. \]

Compare each fine history with its own carrier starting at its actual initial means. Heat contraction and ||D S_lambda,5||2<=4*L_lambda give, with k=gT,

\[ E_U\le T\rho_\lambda+L_\lambda k E_\Phi,\quad E_\Phi\le k E_U,\qquad \boxed{E_\lambda=\frac{T\rho_\lambda}{1-L_\lambda k^2}}. \tag{9} \]

Consequently E_U<=E_lambda and E_Phi<=k*E_lambda on the entire horizon. Unknown means are proof initializations, not new observation inputs.

For finite identity, remove the form command at the fine level. With ||S_lambda||infinity<=4*J_lambda, the same exact integrated phase identity now yields

\[ M_\lambda=X+gJ_\lambda T,\qquad F_\lambda=2gX+2g^2J_\lambda T. \tag{10} \]

Substitute F_lambda for the drift in the original internal-positive, internal-acute, outer-acute and collar margins, with zeta=1e-6, A=7/5, epsilon=1/10. Their explicit expressions are

\[ 2\pi/9-2\zeta-2F_\lambda,\quad \pi/2-2\pi/9-2\zeta-2F_\lambda,\quad \pi/2-A-\epsilon-2\zeta-2F_\lambda,\quad \pi/2-2\zeta-4F_\lambda. \tag{11} \]

All are strictly positive. This preserves each initially admitted marked winding and collar throughout this word and tail, without proving that the alternative acquired the state from the earlier preparation.

For numerical reuse, the carrier Jacobian's quadratic form has cross term gamma*a^T(L5-K_lambda)*b/4 and negative form-diffusion term. Since ||L5-K_lambda||2<=4*(1+L_lambda), its identity-metric logarithmic norm is at most gamma*(1+L_lambda)/2. For every lambda in [0,1], using pi>3,

\[ \mu_2\le\frac{244}{745767}<\frac1{3000}. \tag{12} \]

The existing growth allowance is thus re-admitted. Actual-law Taylor derivatives, Picard tubes and source corrections must still be rebuilt; the frozen sine integrator consumer is unchanged.

At lambda=1, the exact bounds above imply

Quantity	Rationally controlled bound
Hidden form A_1	<3.17e-8
Hidden phase Z_1	<6.05e-8
Own-carrier mean-form error E_1	<2.42e-18
Fine endogenous phase drift F_1	<1.023e-5
Outer acute margin	>0.06997

If two fine histories under the two laws admit the same raw record, their own carriers satisfy, at every sample simultaneously,

\[ \|U_0(t_j)-U_1(t_j)\|_\infty\le \delta=2\eta+E_0+E_1. \tag{13} \]

This is a necessary joint condition. It proves neither record separation nor realizability of independently chosen sample boxes.

3. Exact local ambiguity after removing the unknown phase

The original finite observer uses three command levels to recover two quadratures. First consider its ideal frozen initial current references, before temporal and reading errors. For the interface c -> c+1, let a be its initial gap. The receiver's reference current at command level q is K-j(a-q), where the unmodulated neighbor contribution K cancels from the two contrasts. For epsilon=1/10, the normalized contrasts are

\[ C_j(a)=\frac{j(a+\epsilon)-j(a-\epsilon)}{2\sin\epsilon},\qquad S_j(a)=\frac{2j(a)-j(a+\epsilon)-j(a-\epsilon)}{2(1-\cos\epsilon)}. \tag{14} \]

These have exactly the orientation and scaling of the original O,V statistics. In particular (C_0(a),S_0(a))=(cos a,sin a). For the alternative,

\[ \begin{aligned} C_1(a)&=\cos a+ \frac{\sin(.9)\cos9a-\tfrac12\sin(1.8)\cos18a}{2187\sin(.1)},\\ S_1(a)&=\sin a+ \frac{(1-\cos(.9))\sin9a-\tfrac12(1-\cos(1.8))\sin18a} {2187(1-\cos(.1))}. \end{aligned} \tag{15} \]

There exist distinct gaps a and b, both of magnitude less than 1/2, with (C_1(a),S_1(a))=(cos b,sin b). Thus even exact ideal local quadratures need not distinguish these two laws when the phase is unknown.

To prove this, put F(a)=C_1(a)^2+S_1(a)^2-1, r=cos(.9)>0 and c9=sin(.9)/(2187*sin(.1))>0. At zero, S_1=0 and C_1=1+c9*(1-r)>1, so F(0)>0. At a=pi/9,

\[ S_1(a)=\sin a,\quad C_1(a)=\cos a-q,\quad q=c9(1+r)=\frac{\sin(.9)+\tfrac12\sin(1.8)}{2187\sin(.1)}. \]

The elementary bound sin(.1)>599/6000 gives 0<q<9000/(2187*599)<1/100. Hence F(pi/9)=q*(q-2*cos(pi/9))<0. Continuity supplies a_* in (0,pi/9) with F zero, without locating a root numerically.

For every a in that interval, C_1(a)>=cos(pi/9)-q>1-8/81-1/100>8/9, using pi<4. There is therefore a unique principal b with the matching unit phasor. Since cos(1/2)<=337/384<8/9, it satisfies |b|<1/2. Also a_*<pi/9<1/2, inside the original collective gap prior.

The matching phases are distinct. If b=a_*, both perturbations in (15) would vanish. Writing x=cos(9*a_*), the sine correction, with sin(9*a_*)>0, forces x=1/[2*(1+r)] in (0,1/2). But then the cosine correction is c9*(x+r-2*r*x^2)>0, a contradiction.

Each gap embeds in an admitted primitive initial state: choose U=u=z=0 and Phi=(0,d,3*d/4,d/2,d/4) for d equal to a_* or b. Its outer gaps are (d,-d/4,-d/4,-d/4,-d/4), with winding zero; the internal helical cycles retain winding one. The nuisance neighbor current cancels in the selected contrasts. The other interfaces and the absolute current baseline need not agree. Finite plateau increments also contain endogenous drift and integration error. This witness therefore excludes the local compressed statistic as a uniform selector; it is not a common complete record, an acquisition witness or a physical observation.

Retaining the local current baseline does not suffice either

The neighboring gap can realize the needed baseline, rather than treating K as a free unphysical offset. At a circle collision write kappa=S_1(a_*)-j_1(a_*)=sin b-j_1(a_*). Put s9=(1-cos(.9))/(2187*(1-cos(.1))). The bounds 1-cos(.9)<81/200 and 1-cos(.1)>1199/240000 give s9<1/26. Since 1+cos(.9)<2, (15) and (4) imply

\[ |\kappa|<3/26+1/1458<1/8,\qquad j_1(1/2)\ge\sin(1/2)-1/1458>1/8. \]

Continuity and oddness supply d in (-1/2,1/2) with j_1(d)=-kappa. Choose the initial outer gap cycles

\[ (b,0,-b/3,-b/3,-b/3),\qquad (a_*,d,-(a_*+d)/3,-(a_*+d)/3,-(a_*+d)/3) \]

for sine and the alternative, respectively. They sum to zero and obey the original gap bounds. Set all forms and hidden deviations to zero. Receiver 1 now has K_0=0 and K_1=-kappa. Equal O,V contrasts force the three edge-current differences to equal their baseline difference j_1(a_*)-sin b=-kappa. Therefore the complete receiver currents K_0-j_0(b-q) and K_1-j_1(a_*-q) agree at every q=0,+epsilon,-epsilon. This strengthens the local obstruction within admitted primitive states; other receivers and the continuous dynamics remain unconstrained by it.

4. The complete formal harmonic information remains distinct

For a cross-law carrier difference u=Y_0-Y_1, freeze each initial edge phasor in (7) while retaining the prescribed command h=B^T H. The formal heat response has five initial form coefficients and sine/cosine pairs on five edges for each frequency n in {1,9,18}: 35 real coefficients. Its equation is

\[ V'=-L_5V/4-B\sum_{n\in\{1,9,18\}} [b_{s,n}\odot\cos(nh)+b_{c,n}\odot\sin(nh)]. \tag{16} \]

The fundamental pair is the difference of the two initial phasors scaled by gamma/4. The ninth pair is the alternative's phasor scaled by -gamma/(4*2187), and the eighteenth by gamma/(8*2187). Their circle norms and shared underlying phase are constraints; treating these formal coefficients as free only enlarges the comparison.

The formal coefficient map is injective if the entire continuous V history is supplied. Indeed, a zero history forces zero initial form and zero divergence of the edge currents. On C5 all currents then equal one circulation. During a marked-child ramp three unactuated edges have constant command, so this circulation is constant. Each of its two actuated edges has a trigonometric polynomial of frequencies 1,9,18 equal to that constant on an open interval of command values. Independence of those sine/cosine functions and the constant forces all six coefficients on that edge to vanish. Every edge is actuated somewhere in the word, proving injectivity.

This theorem neither supplies continuous observations in the actual protocol nor proves rank or conditioning of its 150-row sampled matrix. It concerns the formal linear response, before nonlinear endogenous phase drift.

For the actual pair, let B_n=max(||b_s,n||infinity,||b_c,n||infinity) and D=4*g*X+2*g^2*J_1*T, bounding each endogenous edge-phase drift. Here the carrier has only two incident outer currents, so ||S_lambda,5||infinity<=2*J_1 and ||Y||infinity<=X+g*J_1*T/2. Integrating its live phase row gives coordinate drift at most 2*g*X+g^2*J_1*T, hence the stated edge bound D. This is separate from the fine-state drift F_lambda in (10). The fundamental difference remainder and the two one-law higher-harmonic remainders give

\[ u'=-L_5u/4-B\sum_n[b_{s,n}\cos(nh)+b_{c,n}\sin(nh)]+r,\quad \sum_c r_c=0, \]

\[ \|r(v)\|_\infty\le 3D(B_1+9B_9+18B_{18})+g^2\int_0^v\|u(s)\|_\infty\,ds. \tag{17} \]

For the fundamental term, use the original complex-phasor difference identity and |d_0-d_1|<=2*g*integral||u||infinity. For a higher harmonic, |exp(i*n*d)-1|<=n*|d|; its scaled phasor norm is at most sqrt(2)*B_n. Two incident edges and 2*sqrt(2)<3 yield (17). The zero-sum heat integral bound of 6 therefore implies

\[ \sup\|u-V\|_\infty\le 18D(B_1+9B_9+18B_{18})+6g^2T\sup\|u\|_\infty. \tag{18} \]

The old 15-column gain does not certify (16)-(18). A finite joint test must retain these harmonic blocks, their constitutive phase constraints, the two own-law source errors and every sample correlation. The finite-rank and relaxation results below distinguish information in the formal design from what this global remainder can certify about the nonlinear pair.

5. Exact rank of the actual finite formal design

The 150-by-35 sampled map of (16), at the original thirty observation times, has column rank 35. This is an exact statement about the formal linear heat response. It supplies neither a quantitative inverse bound nor a noisy or nonlinear law discriminator. No sampled matrix is evaluated in the proof.

The recorded plateaus and first ramps determine scalar moments

Suppose a coefficient vector produces zero V at every recorded time. The first reading gives V(0)=0, eliminating all five initial-form coefficients. Write A=L5/4. On each plateau its forcing is a constant vector f, and the two recorded endpoints satisfy

\[ 0=e^{-hA}0+\left(\int_0^h e^{-sA}\,ds\right)f. \tag{19} \]

The integral matrix is positive definite: its eigenvalues are h for the constant mode and (1-exp(-h*mu))/mu>0 for every positive eigenvalue mu of A. Thus f is zero on each of the fifteen plateaus. At command zero, this means that all baseline edge currents sum_n b_s,n,e equal one circulation, because the kernel of the oriented C5 incidence matrix is the constant edge vector.

For an edge e define its command-dependent increments

\[ E_e(q)=\sum_n b_{s,n,e}(\cos(nq)-1),\qquad O_e(q)=\sum_n b_{c,n,e}\sin(nq),\qquad n\in\{1,9,18\}. \tag{20} \]

When a child is marked, only its two incident edges have nonzero commands, with opposite signs. The other three edges retain the baseline circulation. Zero divergence at the two nonzero plateau levels therefore gives E_e(epsilon)+O_e(epsilon)=E_e(epsilon)-O_e(epsilon)=0 on each incident edge. Every edge is incident to a marked child, so

\[ E_e(\epsilon)=O_e(\epsilon)=0\quad\hbox{for every edge e}. \tag{21} \]

Every child's first ramp starts at the recorded endpoint of its zero-level plateau and ends at the recorded beginning of its positive-level plateau. Both V values are zero. This includes child 4; its first ramp ends before the last reading. With local ramp time t in [0,1], put q(t)=epsilon*t^2*(3-2*t). After subtracting the zero-divergence baseline, the forcing is supported on the two incident incidence columns.

The two distinct nonzero eigenvalues of A are

\[ \mu_1=(5-\sqrt5)/8,\qquad \mu_2=(5+\sqrt5)/8. \tag{22} \]

Each has a real two-dimensional Fourier eigenspace. The projections of two adjacent incidence columns onto either eigenspace are linearly independent: in complex Fourier coordinates their ratio is exp(plus_or_minus 2*pi*i*k/5), for k equal to 1 or 2, which is not real. The zero ramp endpoint consequently forces each incident edge's weighted forcing integral to vanish separately for each mu.

Each edge is probed once from each endpoint in two child blocks. Their first ramps use the same q(t) and reverse that edge's command sign. Adding and subtracting the two integral equations therefore gives

\[ \int_0^1 e^{-\mu_j(1-t)}E_e(q(t))\,dt=0,\qquad \int_0^1 e^{-\mu_j(1-t)}O_e(q(t))\,dt=0, \quad j=1,2. \tag{23} \]

Only recorded endpoints and the known linear response between them were used. Neither derivatives nor extra within-ramp observations are supplied. Equations (21)-(23) reduce coefficient recovery on every edge to two three-by-three homogeneous systems. Their common matrix construction is

\[ M_f=\begin{pmatrix} f_1(\epsilon)&f_9(\epsilon)&f_{18}(\epsilon)\\ \int_0^1 w_1(t)f_1(q(t))dt&\int_0^1 w_1(t)f_9(q(t))dt& \int_0^1 w_1(t)f_{18}(q(t))dt\\ \int_0^1 w_2(t)f_1(q(t))dt&\int_0^1 w_2(t)f_9(q(t))dt& \int_0^1 w_2(t)f_{18}(q(t))dt \end{pmatrix},\qquad w_j(t)=e^{-\mu_j(1-t)}, \tag{24} \]

using either f_n(q)=sin(nq) or f_n(q)=1-cos(nq). The latter differs from (20) only by a common minus sign. Both matrices are nonsingular, as the following argument proves without computing their entries.

Strict independence of both trigonometric triples

For |z|<pi, the analytic function h(z)=z*cot(z) has expansion

\[ h(z)=1-\sum_{m\ge1}a_m z^{2m},\qquad a_1=1/3,\qquad (2m+1)a_m=\sum_{k=1}^{m-1}a_k a_{m-k}>0\quad(m\ge2). \tag{25} \]

The recurrence follows directly from z*h'=h-h^2-z^2; the first poles are at plus and minus pi. Thus for every positive d,

\[ H_d(\sigma)=\sqrt\sigma\cot(d\sqrt\sigma) =\frac1d-\sum_{m\ge1}a_m d^{2m-1}\sigma^m \quad\Longrightarrow\quad H_d'<0,\quad H_d''<0 \tag{26} \]

on 0<d*sqrt(sigma)<pi, where the derivatives are with respect to sigma. Termwise differentiation is valid inside that convergence domain.

First take f_n(q)=sin(nq), with 0<q<=epsilon=1/10 and n equal to 1, 9 or 18. Every f_n is positive because 18*epsilon=1.8<pi. For sigma=n^2, its logarithmic derivative and normalized second derivative are f_n'/f_n=H_q(sigma) and f_n''/f_n=-sigma. The first two-column Wronskian is negative by the strict decrease in (26). Factoring the positive functions from the three-column Wronskian leaves

\[ \det\begin{pmatrix} 1&1&1\\ H_q(1)&H_q(81)&H_q(324)\\ -1&-81&-324 \end{pmatrix}\ne0, \tag{27} \]

because strict concavity makes the three points on the graph of H_q noncollinear.

For g_n(q)=1-cos(nq)=2*sin(nq/2)^2, every g_n is also positive and

\[ g_n'/g_n=H_{q/2}(\sigma)=H,\qquad g_n''/g_n=(H^2-\sigma)/2. \tag{28} \]

Again its first two-column Wronskian is negative. Invert the strictly decreasing map in (26) locally as sigma=sigma(H). Throughout the interval between its three values,

\[ \sigma''(H)=-H_{q/2}''(\sigma)/(H_{q/2}'(\sigma))^3<0, \qquad \frac{d^2}{dH^2}(H^2-\sigma(H))>0. \tag{29} \]

The three-column Wronskian is therefore nonzero by strict convexity of the last row in (28) as a function of the middle row.

These Wronskian conditions imply that each triple has a nonzero evaluation determinant at any three distinct q values in (0,epsilon]. One direct Rolle argument is to divide a proposed three-zero linear combination by its positive first function. Put r=f_9/f_1 and s=f_18/f_1; r' never vanishes, and (s'/r')' is the three-column Wronskian divided by f_1^3*(r')^2, hence never zero. Applying Rolle twice forces the combination to be trivial. The evaluation determinant has a constant nonzero sign on the connected domain of ordered triples.

The two weighted moments retain that independence

Expanding (24) and interchanging its finite integrals gives exactly

\[ \det M_f=\int_{0<t<u<1} [w_1(t)w_2(u)-w_1(u)w_2(t)] \det\begin{pmatrix} f_1(\epsilon)&f_9(\epsilon)&f_{18}(\epsilon)\\ f_1(q(t))&f_9(q(t))&f_{18}(q(t))\\ f_1(q(u))&f_9(q(u))&f_{18}(q(u)) \end{pmatrix}\,dt\,du. \tag{30} \]

The weight bracket is strictly positive: mu2>mu1 and w2(t)/w1(t)=exp(-(mu2-mu1)*(1-t)) strictly increases. Also q(t) strictly increases on (0,1), so 0<q(t)<q(u)<epsilon on the integration domain. The evaluation determinant consequently has a constant nonzero sign. Thus (30) is nonzero for both bases. Equations (21)-(24) force all thirty edge coefficients to vanish. Together with the five initial forms this proves the claimed rank 35.

This finite theorem strengthens the continuous formal injectivity result; it does not make the local collision in section 3 into a complete-history collision or transfer formal linear identifiability to nonlinear dynamics. No determinant lower bound, inverse norm, finite-error separation or numerical budget verdict follows from strict nonvanishing alone.

6. Why the unconstrained global bootstrap cannot close the question

This obstruction concerns a specified proof method, not the actual laws' record sets. It needs no coefficient-matrix evaluation, proposed inverse, nonlinear history or choice of an unknown initial state.

One formal direction with a uniformly small response

Set the five initial-form coefficients and all b_c coefficients to zero. Repeat the same b_s coefficients on every edge:

\[ (a_1,a_9,a_{18})=(1,-323/19683,20/19683),\qquad b=(1,323/19683,20/19683). \tag{31} \]

The vector b lists their three block infinity norms. Direct rational algebra gives

\[ \sum_n a_n n^2=\sum_n a_n n^4=0,\qquad \sum_n|a_n|n^6=1+8721+34560=43282. \tag{32} \]

The command-zero current is a constant circulation and has zero divergence. At any instant only one child c is marked, with scalar command q satisfying |q|<=epsilon. The two changed edge commands are opposite; cosine is even. Consequently the complete formal forcing is exactly

\[ F(q)(e_{c-1}-e_{c+1}),\qquad F(q)=\sum_n a_n(\cos(nq)-1). \tag{33} \]

Taylor's theorem through degree five, with (32), gives for the entire command range

\[ |F(q)|\le f_*:=\frac{43282\epsilon^6}{720}. \tag{34} \]

The dipole in (33) has zero sum and Euclidean norm sqrt(2). The smallest nonzero eigenvalue of L5/4 is greater than 1/3. Variation of constants, including every change of marked child, therefore yields

\[ \sup_v\|V(v)\|_\infty \le\sqrt2 f_*\int_0^\infty e^{-s/3}\,ds <\frac92 f_* =:m=\frac{21641}{80000000}. \tag{35} \]

This bounds the complete formal history and hence every actual sample. It does not assert a zero response; it is consistent with rank 35. The retained elementary remainder coefficient is strictly larger:

\[ a=18D>m,\qquad D=4gX+2g^2J_1T. \tag{36} \]

All quantities in (31)-(36) are fixed rational bounds. In particular, m=0.0002705125, whereas 18D>0.000304.

Both scalar and block recovery inherit the obstruction

Let M denote the exact sampled formal map. Any proposed recovery K of all thirty harmonic coefficients has row norm k and harmonic residual block norm e for J=[0 I]-KM. Applying that identity to (31), whose initial forms vanish and harmonic infinity norm is one, gives

\[ 1\le e+k\|Mp\|_\infty\le e+km. \tag{37} \]

A scalar bootstrap with remainder coefficient at least a would require 1-e-k*a>0. Equations (36)-(37) exclude this for every K, even one obtained from an exact inverse. Taking upper residual bounds cannot improve it.

Keeping separate harmonic block norms also does not rescue this particular global envelope. Write B_n=max(||b_s,n||infinity,||b_c,n||infinity) for an arbitrary actual pair, and U_*=sup||u||infinity. Its last preceding sample is at most two units earlier, including the final unobserved ramp and tail. The static force and (17) then give

\[ cU_*\le\delta+6\sum_n(1+nD)B_n,\qquad c=1-2g^2T>0. \]

\[ R_*:=\sup\|u-V\|_\infty\le\sum_n A_nB_n+B_0\delta, \quad A_n=18nD+\frac{36g^2T(1+nD)}c,\quad B_0=\frac{6g^2T}c. \tag{38} \]

Here delta=2*eta+E_0+E_1 retains both own-law source errors. For arbitrary block recovery matrices K_n, let k_n be their row norms and E_nm the nonnegative row-norm bounds of their residuals on harmonic block m. Their proposed componentwise bootstrap has feedback matrix

\[ H=E+kA^T. \tag{39} \]

Using the formal witness instead of an actual pair in the exact recovery identities gives b<=E*b+k*m. But A^T*b>=A_1>18D>m, so

\[ Hb\ge b,\qquad b>0. \tag{40} \]

Thus I-H cannot have a nonnegative inverse: otherwise multiplying (I-H)b<=0 by that inverse would imply b<=0. This excludes the positive block bootstrap for every choice of dual weights and their certified residual bounds. Positive diagonal changes of coefficient units preserve the same obstruction. A more accurate matrix or a different inverse cannot repair (38)-(39); a sharper correlated remainder or constrained argument is a different method and is not excluded.

The obstruction is not an admitted cross-law history

An actual alternative has ninth and eighteenth phasor radii r9=gamma/(4*2187) and r18=r9/2. In complex notation z_n=b_c,n+i*b_s,n, its coefficients on each edge satisfy

\[ z_9=-r_9e^{9ia},\qquad z_{18}=(r_9/2)e^{18ia} =z_9^2/(2r_9). \tag{41} \]

Both are tied to the same phase a that occurs in the fundamental cross-law difference. On each edge the witness has phasor modulus ratio 323/20, instead of 2, so even rescaling it cannot produce these actual coefficients. This uses its zero b_c coordinates; general block infinity norms need not have the same ratio as phasor moduli. It also supplies no pair of initial cycle phases or nonlinear histories. Its role is to prove a limitation introduced by allowing independent formal coefficients and independent norm remainders. Finite formal rank, exact local ambiguity and this obstruction do not by themselves decide the joint question. The constrained theorem resolves it under the same word, noise and horizon using additional joint phase restrictions and stronger contrasts.

7. Constrained contrasts from recorded long-ramp endpoints

The following construction avoids dividing a form difference by the short plateau duration. It cancels the formal initial-form response, constant forcing and fundamental harmonic before using the alternative's shared phase. It supplies a conditional finite separation criterion. The complete certificate in sections 9-11 instead uses the middle ramp as well, while retaining this construction as a control and reusable conditional result.

An explicit functional of the unchanged readings

Orient each edge as e -> e+1, with incidence column b_e=e_(e+1)-e_e. Choose edge 1 -> 2 and its two marked-child blocks, c=1,2. Let h_p=1e-5 and s_c=3*c*(1+h_p). The first and third ramps have recorded endpoints

\[ a_{c,1}=s_c+h_p,\quad b_{c,1}=a_{c,1}+1,\qquad a_{c,3}=s_c+2+3h_p,\quad b_{c,3}=a_{c,3}+1. \tag{42} \]

The third endpoint is the next child's zero-level plateau beginning. All eight endpoints in this construction belong to the existing thirty times. Use command-removed form Y, or a difference u of two carrier forms, for which the common command already cancels.

For either positive eigenvalue mu in (22), let P_mu be its orthogonal spectral projector. Put Q_c,mu=[P_mu*b_(c-1),P_mu*b_c] and let R_c,mu,e be the row of its left inverse selecting edge e=1. Define

\[ \mathcal T_{c,\mu,e}(w)=-R_{c,\mu,e} [w(b_{c,1})-e^{-\mu}w(a_{c,1}) -w(b_{c,3})+e^{-\mu}w(a_{c,3})]. \tag{43} \]

The row already lies in the range of P_mu. Its norm is explicit: the Gram matrix of Q has diagonal d=8*mu/5 and off-diagonal d*(1-2*mu), with d*(1-(1-2*mu)^2)=1/2. Hence ||R_c,mu,e||2=sqrt(2) for both modes. Coordinate errors at most delta in the four inputs to (43) change its value by at most 2*sqrt(10)*(1+exp(-mu))*delta.

For the frozen formal equation (16), the first-minus-third ramp difference cancels every constant edge current. Only the marked child's two incident edges remain, so (43) extracts their separate weighted current contrasts. The command on edge 1 is positive when child 2 is marked and negative when child 1 is marked. Therefore define its two parity moments by

\[ \mathcal E_\mu(w)=-\frac{e^{\mu/2}}8 [\mathcal T_{2,\mu,1}(w)+\mathcal T_{1,\mu,1}(w)],\qquad \mathcal O_\mu(w)=\frac{e^{\mu/2}}8 [\mathcal T_{2,\mu,1}(w)-\mathcal T_{1,\mu,1}(w)]. \tag{44} \]

Each has observation error at most sqrt(10)*cosh(mu/2)*delta<4*delta. This bound uses the recorded forms themselves, without estimating a plateau derivative or an unknown initial form.

Positive kernels after removing the fundamental

Write the centered ramp coordinate as r=t-1/2 and put xi(r)=epsilon*(3*r/2-2*r^3). On 0<=r<=1/2, xi increases from zero to epsilon/2. The first and third commands are epsilon/2+xi(r) and -epsilon/2+xi(r). Their trigonometric difference gives the exact kernels

\[ C_{\mu,n}=\int_0^{1/2}\sinh(\mu r)\sin(n\xi(r))\,dr,\qquad D_{\mu,n}=\int_0^{1/2}\cosh(\mu r)\cos(n\xi(r))\,dr. \tag{45} \]

For the formal history, (44) equals respectively sum_n b_s,n,e*sin(n*epsilon/2)*C_mu,n and sum_n b_c,n,e*sin(n*epsilon/2)*D_mu,n. Set

\[ \alpha_E=C_{\mu_1,1}/C_{\mu_2,1},\qquad \alpha_O=D_{\mu_1,1}/D_{\mu_2,1},\qquad 0<\alpha_E,\alpha_O<1, \]

\[ \mathcal E=\mathcal E_{\mu_1}-\alpha_E\mathcal E_{\mu_2},\qquad \mathcal O=\mathcal O_{\mu_1}-\alpha_O\mathcal O_{\mu_2}. \tag{46} \]

The fundamental coefficients vanish exactly. The remaining coefficients are

\[ K_{E,n}=\sin(n\epsilon/2)(C_{\mu_1,n}-\alpha_E C_{\mu_2,n}),\qquad K_{O,n}=\sin(n\epsilon/2)(D_{\mu_1,n}-\alpha_O D_{\mu_2,n}), \quad n=9,18. \tag{47} \]

All four are strictly positive. Indeed, each numerator in (47) is an ordered double integral of a weight determinant times a function determinant. Both weight ratios sinh(mu2*r)/sinh(mu1*r) and cosh(mu2*r)/cosh(mu1*r) strictly increase for positive r. Both function ratios sin(n*x)/sin(x) and cos(n*x)/cos(x) strictly decrease for 0<x<=epsilon/2, where 18*x<=.9<pi/2. Consequently both determinants have the same positive orientation. This proves strict positivity without evaluating an integral.

The same positive representation bounds their frequency ratios. For 0<x<y<=epsilon/2, the sine determinant sin(n*x)*sin(y)-sin(x)*sin(n*y) is (n^2-1)*sin(y) times the positive Dirichlet Green integral of sin(n*s) on [0,y], with operator -d^2/ds^2-1. Since sin(18*s)=2*sin(9*s)*cos(9*s), its n=9 to n=18 ratio is at most 40/(323*cos(.45)). For the cosine determinant use

\[ \left(\frac{\cos(ns)}{\cos s}\right)' =-\frac{n^2-1}{\cos^2s}\int_0^s\cos(nt)\cos t\,dt. \tag{48} \]

The n=9 to n=18 ratio is at most 80/(323*cos(.9)). Multiplying in each case by sin(.45)/sin(.9)=1/(2*cos(.45)) gives

\[ 0<\rho_E:=K_{E,9}/K_{E,18} \le\frac{20}{323\cos^2(.45)}<\frac1{12},\qquad 0<\rho_O:=K_{O,9}/K_{O,18} \le\frac{40}{323\cos(.9)\cos(.45)}<\frac14. \tag{49} \]

The strict rational comparisons already follow from cos(.45)>719/800 and cos(.9)>119/200. No sampled coefficient matrix or kernel magnitude is computed here.

A finite certificate with explicit remaining obligations

Let theta=9*a for the alternative's edge gap at one common reference time, and let r9=gamma/(4*2187). Retaining its actual coefficients (41) in the frozen contrasts gives

\[ \frac{2\mathcal E_{\rm fr}}{K_{E,18}} =r_9(\sin2\theta-2\rho_E\sin\theta),\qquad \frac{2\mathcal O_{\rm fr}}{K_{O,18}} =r_9(\cos2\theta-2\rho_O\cos\theta). \tag{50} \]

The Euclidean norm of this pair is at least r9*(1-2*max(rho_E,rho_O))>r9/2. This conclusion uses the shared phase and fixed radii, not independent bounds on two harmonic blocks.

For two histories sharing a noisy record, (13) bounds every input to (43) by delta. Their two final observation allowances are

\[ N_E=\sqrt{10}[\cosh(\mu_1/2)+\alpha_E\cosh(\mu_2/2)]\delta<8\delta, \quad N_O=\sqrt{10}[\cosh(\mu_1/2)+\alpha_O\cosh(\mu_2/2)]\delta<8\delta. \tag{51} \]

Suppose uniform, own-law arguments additionally establish |mathcal E(u)-mathcal E_fr|<=epsilon_E and |mathcal O(u)-mathcal O_fr|<=epsilon_O. A sufficient separation criterion for the actual record sets is then

\[ \sqrt{\left(\frac{2(N_E+\epsilon_E)}{K_{E,18}}\right)^2+ \left(\frac{2(N_O+\epsilon_O)}{K_{O,18}}\right)^2} <\frac{r_9}{2}. \tag{52} \]

This criterion alone supplies no passing verdict: it requires certified kernel magnitudes and actual-history departures from one shared reference phase. Reconstructing phase increments accurately does not make those increments zero: even known nonzero endogenous drift changes the frozen kernels. It must be incorporated into the contrasts or bounded, together with the uncertainty of the reconstruction and both laws' source errors. Local resetting must also retain transport between the two child blocks. The same-mode extension in sections 9-11 meets those obligations with stronger kernels and a common late-time reference. No trajectory, fit or reserved response is evaluated by either construction.

8. The same form readings constrain historical phase transport

The exact carrier phase row depends on the observed form history, not on an independently chosen phase velocity. This allows the initial phases to remain unknown while bounding their increments more tightly. This section uses only the historical interval 0<=v<=t_last=14+15/100000; it does not reconstruct the unobserved return ramp or the tail to time 16.

Put A=L5/4 and w_lambda=gamma*S_lambda,5/4 in (7). Then

\[ Y'=-AY+w_\lambda,\qquad \Psi'=\gamma AY,\qquad \|Y\|_\infty\le M_\lambda:=X+gJ_\lambda T/2. \tag{53} \]

At each reading the commanded form offset vanishes, so the raw fine-mean reading y differs from this law's carrier Y by at most epsilon_lambda=eta+E_lambda in each coordinate. Both compared laws use the same raw record if one is jointly compatible; this is a conditional argument for any such record, without evaluating one.

A bridge that retains arbitrary heat relaxation

On each interval of length h between successive readings, define

\[ Q(s)=\int_0^s e^{-rA}\,dr,\qquad V_h(s)=Q(s)Q(h)^{-1},\qquad \widehat Y(s)=(I-V_h(s))y_i+V_h(s)y_{i+1}. \tag{54} \]

The matrix Q(h) is positive definite, including its constant-mode eigenvalue h. The eigenvalues of V_h(s) lie in [0,1]. This bridge exactly reproduces every solution with constant forcing, including arbitrary initial heat relaxation. It is an observation-based interpolation, not a new supplied evolution law. It need not reproduce the actual current. Joining the intervals yields one continuous form interpolant. Define the common edge-phase increment reference

\[ \widehat d(v)=\gamma B^TA\int_0^v\widehat Y(s)\,ds. \tag{55} \]

The initial circular edge phase is not reconstructed. For each law its actual carrier increment d_lambda=B^T(Psi_lambda(v)-Psi_lambda(0)) is compared with this same reference.

With exact carrier endpoints, e=Y-hat Y obeys e''+Ae'=w_lambda', with zero values at both cell endpoints. For a spectral value 0<=mu<=1, let q_mu(s)=(1-exp(-mu*s))/mu, using s when mu is zero. The absolute scalar Dirichlet Green kernel is

\[ G_\mu(t,s)= \frac{q_\mu(\min(t,s))q_\mu(h-\max(t,s))}{q_\mu(h)} e^{-\mu\max(t-s,0)}. \]

For h at most one, q_mu(r)<=r and mu*h/(1-exp(-mu*h))<=1/(1-exp(-1))<8/5, with the continuous value one at mu zero. Consequently

\[ 0\le G_\mu(t,s)\le\frac85G_0(t,s),\qquad \int_0^hG_0(t,s)\,dt=\frac{s(h-s)}2. \tag{56} \]

The C5 geometry gives ||B||2<=2, ||A||2<=1, and each row of B^T A has squared Euclidean norm 5/4. The actual current derivative therefore satisfies, when the marked command level is q,

\[ \|w_\lambda'\|_2\le \frac{\gamma L_\lambda}{2} (\sqrt2|q'|+2\gamma\sqrt5 M_\lambda). \tag{57} \]

The bound uses B^T H', which has exactly two nonzero entries of opposite sign, and the complete phase row in (53). It is an own-law estimate, not an observed derivative.

Separate reading, command and endogenous errors

For a cubic ramp of amplitude a and duration h, q'=6*a*s*(h-s)/h^3, so integrating its absolute value against the weight in (56) gives |a|*h^2/10. Combining (56)-(57) with the edge-row norm gives at most sqrt(10)*gamma^2*L_lambda*|a|*h^2/25 in phase increment error. Use sqrt(10)<19/6. The endogenous contribution is at most gamma^3*L_lambda*M_lambda*h^3/3.

Endpoint perturbations have Euclidean norm at most sqrt(5)*epsilon_lambda. Both spectral bridge weights have norm at most one. After applying gamma*B^T A and integrating, their contribution is at most 5*g*epsilon_lambda*h per cell. No derivative estimate divides these errors by the short plateau duration.

Thus a uniform historical bound is

\[ \|d_\lambda(v)-\widehat d(v)\|_\infty\le R_\lambda, \]

\[ R_\lambda= 5g\epsilon_\lambda t_{last} +\frac{19}{150}g^2L_\lambda\sum_{\rm cells}|a|h^2 +\frac{g^3L_\lambda M_\lambda}{3}\sum_{\rm cells}h^3. \tag{58} \]

There are exactly fourteen observed ramps and fifteen short plateaus:

\[ \sum|a|h^2=19/10,\qquad \sum h^3=14+15\cdot10^{-15}. \tag{59} \]

The last return ramp has no right endpoint reading and is excluded. Summing only completed cells and the full nonnegative allowance for the current cell gives a valid bound also at interior times. In particular, for both endpoint laws (and uniformly through 0<=lambda<=1),

\[ R_\lambda<2.8\cdot10^{-8}. \tag{60} \]

The report retains the three error contributions and cumulative cell allowances separately. Unlike the old absolute drift D, the interpolation error has no term of order g*X; initial form enters only the much smaller endogenous term of order g^3*X. This uses additional information already present in the same record, without tightening its noise allowance.

This is a carrier edge-phase increment bound. A statement about the fine collective mean-phase increment must additionally retain twice the own-law mean-phase source error; individual hidden constituent phases remain unobserved. Neither version identifies the initial phase, its winding or the law. In particular, the nonzero common reference d_hat must be retained when using the contrasts in section 7: a small reconstruction error does not justify replacing the reconstructed motion by zero. The complete theorem in section 11 instead bounds actual motion about one late-time reference directly and retains relative transport from the same two form histories. This bridge remains a separate conditional reconstruction bound.

9. The observed middle ramp supplies a second command shape

The middle ramp also has recorded endpoints in blocks c=1,2 and c=2,3. For the continuation below select edge 2 and blocks c=2,3, whose complete ramps lie between local times 6.00007 and 12.00012. No final unobserved return ramp or additional reading is used. Translation of the child labels preserves the C5 projector and frozen-kernel algebra in section 7.

Write F_c,i(w)=w(b_c,i)-exp(-mu)w(a_c,i). Retain the first-minus-third functional T in (43), and define

\[ {\cal Q}_{c,\mu,e}(w)=-R_{c,\mu,e} [F_{c,1}(w)+F_{c,3}(w)-2F_{c,2}(w)]. \]

All three intervals have unit duration. The coefficients sum to zero, so constant forcing cancels. At centered ramp coordinate r, their commands are respectively epsilon/2+xi(r), -2xi(r), and -epsilon/2+xi(r). Combining the two marked-child orientations gives

\[ W_{E,\mu}=e^{\mu/2}({\cal Q}_{3,\mu,2}+{\cal Q}_{2,\mu,2})/8,\qquad W_{O,\mu}=e^{\mu/2}({\cal Q}_{3,\mu,2}-{\cal Q}_{2,\mu,2})/8. \]

Their frozen harmonic coefficients are respectively

\[ L_{D,n}=\cos(n\epsilon/2)D_{\mu,n}-D_{\mu,2n},\qquad L_{C,n}=\cos(n\epsilon/2)C_{\mu,n}+C_{\mu,2n}. \]

Thus E_mu-alpha_E W_E,mu and O_mu-alpha_O W_O,mu cancel the fundamental at a single mode when alpha_E and alpha_O take the ratios declared below. The signed high-harmonic coefficients still obey one common phase-circle relation. The new functional retains existing samples; it changes the analysis of their information, not the preparation or waveform. Actual phase motion and cross-law transport still require their own bounds.

Prospective scalar-moment evaluation

This is one analytical assessment of the scalar kernels, not a record, trajectory, fit, law selection or complete nonlinear separation. Commit this declaration, the helper and its exclusive driver before evaluation. Preserve the attempt and exact interval report; do not retry or change numerical budgets after observing it. The following choices are fixed:

Item	Declaration
Kernel set	Both exact C5 modes mu=(5-sqrt(5))/8,(5+sqrt(5))/8; harmonics 1,2,9,18,36; twenty scalar C,D integrals total
Arithmetic	Shared outward rational dyadic128 intervals; exact rational polynomial coefficients and antiderivatives; shared interval square root and trigonometry
Fixed series	Twenty terms in each hyperbolic and trigonometric series, giving odd degrees through 39 and even degrees through 38
Kernel budget	Every complete C,D enclosure must have radius at most 1e-28; zero-crossing elimination denominators give unavailable arithmetic
Derived quantities	Original across-mode contrasts and the three-ramp contrast at each mode, retaining signed coefficients and the full observation-noise multipliers
Selected continuation	Mode 2 is selected before evaluation; mode 1 and the old construction are retained controls, not candidates in a result-dependent search
No additional evaluation	No response, reconstructed phase history, nonlinear fit, sampled matrix, frozen source producer or physical observation is evaluated

Since xi(r)=r*(3/20-r^2/5), define exact rational polynomial moments

\[ J_{m,k}=\sum_{j=0}^{k}\binom{k}{j}(3/20)^{k-j}(-1/5)^j \frac{(1/2)^{m+k+2j+1}}{m+k+2j+1}. \]

Integrate the finite products by substituting J into

\[ C^{[20]}_{\mu,n}=\sum_{p,q=0}^{19} \frac{\mu^{2p+1}(-1)^qn^{2q+1}J_{2p+1,2q+1}} {(2p+1)!(2q+1)!}, \]

\[ D^{[20]}_{\mu,n}=\sum_{p,q=0}^{19} \frac{\mu^{2p}(-1)^qn^{2q}J_{2p,2q}}{(2p)!(2q)!}. \]

Combine each rational coefficient, including the exact J, before interval multiplication by a power of mu. The finite sums and every subsequent operation retain outward arithmetic rounding. On this domain abs(mu*r)<=1/2 and abs(n*xi)<=9/5. Uniform omitted-series bounds are

\[ E_{sh}=\frac{(1/2)^{41}}{41!}\frac1{1-1/(4\cdot42\cdot43)},\quad E_{ch}=\frac{(1/2)^{40}}{40!}\frac1{1-1/(4\cdot41\cdot42)}, \]

\[ E_s=(9/5)^{41}/41!,\qquad E_c=(9/5)^{40}/40!. \]

The hyperbolic tails use the decreasing positive-term ratio; the trigonometric bounds use Taylor's theorem with derivatives bounded by one. Every hyperbolic partial sum is below two and the actual sine/cosine has absolute value at most one. Consequently append the symmetric integrated product tails (E_sh+2*E_s)/2 and (E_ch+2*E_c)/2 to C and D respectively. These are product errors, not derivative or trajectory error estimates.

For a single mode put s_n=sin(n/20), c_n=cos(n/20) and

\[ \alpha_E=\frac{s_1C_{\mu,1}}{c_1D_{\mu,1}-D_{\mu,2}},\qquad \alpha_O=\frac{s_1D_{\mu,1}}{c_1C_{\mu,1}+C_{\mu,2}}. \]

The signed remaining coefficients are

\[ K_{E,n}=s_nC_{\mu,n}-\alpha_E(c_nD_{\mu,n}-D_{\mu,2n}),\qquad K_{O,n}=s_nD_{\mu,n}-\alpha_O(c_nC_{\mu,n}+C_{\mu,2n}), \quad n=9,18. \]

Their raw common-record uncertainty allowances are respectively sqrt(10)*cosh(mu/2)*(1+2*abs(alpha_E))*delta and the same expression with alpha_O. The factor two retains the doubled absolute ramp weights in the first+third-2*second functional. Evaluate cosh with the same fixed positive series and its admitted tail. Kernel arithmetic admission alone supplies no bound on the actual-history departure from these frozen coefficients.

benchmarks/organized_law_contrast_kernels.py uses the existing exclusive analytical runner: it requires a committed source/declaration, writes the source inventory, configuration, backend and pre-evaluation attempt, then writes the exact rational interval report. The reported scope is moment_kernel_arithmetic_only_not_a_record_verdict.

10. Necessary joint phase bounds from a shared complete record

These are necessary conditions for two admitted histories sharing one noisy record. No record is constructed or evaluated. Write r=gamma/4, r9=r/2187, g=1/3000, h=1e-5 and retain the own-law source errors in delta=2*eta+E0+E1<2.01e-14. Since pi<22/7, gamma>1/3300. For each endpoint law the command-removed carrier obeys

\[ Y'=-AY+rS_\lambda,\qquad \Psi'=\gamma AY, \qquad A=L_5/4. \]

The infinity norm of A is one and that of each row of B^T A is two. Thus on the entire horizon,

\[ \|Y\|_\infty\le M_\lambda=X+gJ_\lambda T/2<.013, \quad \|B^T(\Psi(v)-\Psi(0))\|_\infty \le D_\lambda=4gX+2g^2J_\lambda T<17\cdot10^{-6}. \]

On a plateau the command is constant. Therefore

\[ \|Y'\|_\infty\le M_\lambda+gJ_\lambda/2, \qquad \|Y''\|_\infty\le M_\lambda+gJ_\lambda/2+g^2L_\lambda M_\lambda =:M_{2,\lambda}<3/200. \]

The same winding follows before a detailed phase inverse

For one plateau with endpoint forms P,Q, use the original diffusion-corrected combination, now vector-valued,

\[ D(P,Q)=Q-P+\frac h2 A(P+Q). \]

For the difference of two carriers its receiver coordinate has absolute value at most (2+h/2)*delta: the coefficients are -1+h/4, 1+h/4 and four neighboring coefficients -h/8. Integrating the form equation gives

\[ D(Y(a),Y(a+h)) =r\int_a^{a+h}S_\lambda(v)\,dv+\mathcal R_\lambda, \qquad |\mathcal R_{\lambda,i}|\le M_{2,\lambda}h^3/12. \]

The error is the trapezoid error of the diffusion integral A int Y, not a trapezoid approximation of the current. Dividing by h gives an h^2/12 term per law, hence (M20+M21)*h^2/12, bounded by (3/200)*h^2/6, in the cross-law force estimate. No observed derivative or unknown intermediate state is inserted.

At any of the three levels q in a marked-child block, each of the two receiver edge gaps departs from its initial gap plus the stipulated command by at most D_lambda. Its current consequently departs by at most 2*L_lambda*D_lambda. A shared record therefore requires the two ideal initial receiver currents at that q to differ by at most

\[ \kappa=\frac{4}{\gamma_{lo}} \left\{\frac{(2+h/2)\delta}{h} +\frac{M_{2,0}+M_{2,1}}{12}h^2 +\frac g2(L_0D_0+L_1D_1)\right\}<\frac1{7500}, \qquad \gamma_{lo}=1/3300. \]

Taking the second three-level contrast cancels the adjacent baseline. If a0,a1 are the two initial gaps of that edge, its ideal sine quadratures obey

\[ |S_0(a_0)-S_1(a_1)| \le\frac{2\kappa}{1-\cos(.1)}<3/50. \]

The alternative quadrature deformation has magnitude less than 3/26: write s9=(1-cos(.9))/(2187*(1-cos(.1)))<1/26; the eighteenth coefficient is (1+cos(.9))*s9<2*s9. Thus

\[ |\sin a_0-\sin a_1|<3/50+3/26. \]

On the admitted interval [-7/5,7/5], cosine is larger than 1/6. For example its alternating Taylor lower bound 1-A^2/2+A^4/24-A^6/720 at A=7/5 is larger than 1/6. The mean-value theorem gives |a0-a1|<342/325. Both initial gap cycles sum to 2*pi*w, so

\[ 2\pi|w_0-w_1|\le\sum_e|a_{0,e}-a_{1,e}| <342/65<6<2\pi. \]

Hence the two histories must have the same initial outer winding. This is a necessary compatibility statement, not a reconstruction of it.

Initial current balance bounds the fundamental phasor

At the first plateau there is no preceding command. Replace D_lambda in the preceding force estimate by the tighter direct increment bound 2*g*M_lambda*h. Then, with

\[ b_{s,e}=r(\sin a_{0,e}-\sin a_{1,e}),\qquad f_e=b_{s,e}-r_9\{\sin(9a_{1,e})-\tfrac12\sin(18a_{1,e})\}, \]

the necessary initial force estimate is

\[ \|Bf\|_\infty\le\zeta:={ (2+h/2)\delta\over h} +{M_{2,0}+M_{2,1}\over12}h^2 +g^2h(L_0M_0+L_1M_1)<4.1\cdot10^{-9}. \]

Adjacent f values differ by at most zeta. The five-cycle has diameter two, so max f-min f<=2*zeta. The deformation's range is at most 3*r9. Equal winding and monotonicity of sine on the initial gap domain imply min b_s<=0<=max b_s. Consequently

\[ \|b_s\|_\infty\le2\zeta+3r_9<\frac{13}{4}r_9. \]

The last strict inequality follows using r9>1/(3300*4*2187). For m=(a0+a1)/2, the fundamental phasor identity gives

\[ r|e^{ia_0}-e^{ia_1}|= {|b_s|\over\cos m}<6|b_s|<\frac{39}{2}r_9. \]

At any later common reference time in the admitted horizon, removing the common command, the change in this phasor difference is at most r*(D0+D1). Hence its magnitude P_ref satisfies

\[ \boxed{P_{ref}<\{39/2+2187\cdot34\cdot10^{-6}\}r_9<20r_9.} \]

This bound uses all sampled blocks to establish equal winding and the initial plateau to constrain the fundamental. The 9th and 18th harmonic radii retain their exact common-phase relation. No nonlinear record separation verdict follows from these bounds alone.

11. Late-block whole-record separation theorem

This proof keeps the complete laws, initial class, 30 five-form readings, noise, clock, command and horizon in ORGANIZED_LAW_OBSERVATION unchanged. The kernel calculation is the retained single assessment. No further moment, trajectory, record, inverse or fitted state is evaluated.

Frozen final assessment declaration

Consume only the five retained mode-two primitive C,D enclosures at harmonics 1,2,9,18,36. Re-admit their labels and common C5 eigenvalue, rebuild the same-mode three-ramp contrasts, including alpha and K, and rebuild all own-law and common-record analytical bounds below. Use children 2 and 3, edge 2 -> 3, and the shared reference s3=9.00009. No selection or retry is allowed. The final decision is the shared circle check's STRICT squared inequality, including both observation and nonlinear-history errors. An unsafe denominator, nonpositive margin or failing strict inequality is unavailable, not an overlap witness. A passing comparison establishes disjointness of the two complete finite noisy record sets under the declared premises, not physical identification or uniqueness of the family.

The input is the immutable mode-two kernel evidence declared at 0b21a040, in docs/assets/organized_law_contrast_kernels/. The read-only input owner checks its byte receipts; the theorem consumes primitive C,D intervals, not cached coefficients, margins or verdicts. Arithmetic remains outward dyadic128. Exponentials use the shared exact-time rational series, with sixteen equal argument subdivisions for decay on [0,16]. There is one assessment through benchmarks/organized_law_record_separation.py, archived in a fresh docs/assets/organized_law_record_separation/ directory after this declaration and all consumed source are committed. No precision escalation, source replay or additional response calculation is admitted.

Endpoint functional and ideal coefficients

Put \(h=10^{-5}\), \(s_c=3c(1+h)\) and \(\mu=\mu_2=(5+\sqrt5)/8\). Use all three unit ramps in each of children \(c=2,3\); every endpoint is an existing reading. With the spectral left-inverse row \(R_c\) selecting edge \(2\to3\), define

\[ F_{c,i}(w)=w(b_{c,i})-e^{-\mu}w(a_{c,i}),\qquad \mathcal T_c=-R_c(F_{c,1}-F_{c,3}),\qquad \mathcal Q_c=-R_c(F_{c,1}+F_{c,3}-2F_{c,2}). \]

The two parity pairs are

\[ \mathcal E_0=-\frac{e^{\mu/2}}8(\mathcal T_3+\mathcal T_2),\qquad \mathcal O_0=\frac{e^{\mu/2}}8(\mathcal T_3-\mathcal T_2), \]

\[ W_E=\frac{e^{\mu/2}}8(\mathcal Q_3+\mathcal Q_2),\qquad W_O=\frac{e^{\mu/2}}8(\mathcal Q_3-\mathcal Q_2). \]

The target command is positive in child 3 and negative in child 2. The centered ramp profiles are \(\epsilon/2+\xi,-2\xi,-\epsilon/2+\xi\). Suppressing the fixed mode index, the frozen harmonic coefficients of \(\mathcal E_0,\mathcal O_0\) are \(\sin(n\epsilon/2)C_n,\sin(n\epsilon/2)D_n\), while those of \(W_E,W_O\) are

\[ L_{E,n}=\cos(n\epsilon/2)D_n-D_{2n},\qquad L_{O,n}=\cos(n\epsilon/2)C_n+C_{2n}. \]

The ratios declared in section 9 therefore cancel the fundamental in

\[ \mathcal E=\mathcal E_0-\alpha_EW_E,\qquad \mathcal O=\mathcal O_0-\alpha_OW_O, \]

\[ K_{E,n}=\sin(n\epsilon/2)C_n-\alpha_EL_{E,n},\qquad K_{O,n}=\sin(n\epsilon/2)D_n-\alpha_OL_{O,n}. \]

Both higher harmonics retain the same alternative-law phase \(\theta=9a_1(s_3)\) on the selected edge. The coefficients are rebuilt from the primitive intervals; their signs and magnitudes are not assumed. With \(\delta=2\eta+E_0+E_1\) from (13), the observation allowances are

\[ N_E=\sqrt{10}\cosh(\mu/2)(1+2|\alpha_E|)\delta,\qquad N_O=\sqrt{10}\cosh(\mu/2)(1+2|\alpha_O|)\delta. \]

Uniform actual phase motion on the late blocks

Set \(\mathcal I=[s_2,s_4]\), \(t_*=s_3\) and \(L=3(1+h)=3.00003\). For each edge, let \(a_\lambda(v)=(B^T\Psi_\lambda(v))_e\). Section 10 proves

\[ P(v):=\frac\gamma4\max_e |e^{ia_{0,e}(v)}-e^{ia_{1,e}(v)}|<20r_9, \qquad r_9=\frac\gamma{4\cdot2187}, \]

at every historical time. Equal winding, the first plateau and the acute initial gap prior first bound the initial amplitude; the absolute phase drift bound then preserves this containing bound.

For either law, decompose the command-removed edge-phase increment into homogeneous heat and forced contributions. The row \(Y'=-AY-(\gamma/4)Bj\) gives

\[ d_{\mathrm{heat}}(v)=\gamma B^T(I-e^{-vA})Y(0),\qquad d_{\mathrm{forced}}'(v)=-\gamma^2\int_0^v A_{\mathrm{edge}}^2e^{-(v-s)A_{\mathrm{edge}}}j(s)\,ds. \]

Here \(A_{\mathrm{edge}}=B^TB/4\) is the same C5 generator in edge order; incidence intertwining gives the second identity. Its positive-mode projectors satisfy \(\|P_\mu\|_\infty=(2+2\sqrt5)/5<13/10\), and \(\mu_1+\mu_2=5/4\). Consequently

\[ \int_0^\infty\|A_{\mathrm{edge}}^2e^{-sA_{\mathrm{edge}}}\|_\infty\,ds <\frac{13}{8},\qquad \|d_{\mathrm{forced}}'(v)\|_\infty\le\frac{13}{8}\gamma^2J_1. \]

The exact homogeneous edge-row norms are

\[ \|\operatorname{row}(B^TP_{\mu_1})\|_1 =1+\frac1{\sqrt5}<\frac{29}{20},\qquad \|\operatorname{row}(B^TP_{\mu_2})\|_1 =1+\frac3{\sqrt5}<\frac{47}{20}. \]

Each spectral exponential difference is monotone away from \(t_*\). Thus the larger endpoint sum of these row norms times \(|e^{-\mu t}-e^{-\mu t_*}|\), multiplied by \(\gamma X\), bounds the homogeneous motion throughout \(\mathcal I\). Adding the forced allowance \((13/8)\gamma^2J_1L\) gives

\[ \sup_{v\in\mathcal I} \|d_\lambda(v)-d_\lambda(t_*)\|_\infty \le D_{\mathrm{late}}<10^{-6},\qquad \lambda=0,1. \]

The shared rational exponential owner verifies this without a nonlinear response. An elementary coarse bound uses \(\bar\gamma=1/3200>\gamma\), from \(\pi>3.14\). The left heat row is below \(1.45(0.127-0.044)+2.35(0.005)=0.1321<0.14\); the right is below \(1.45(0.045)+2.35(0.001)=0.0676<0.14\). Hence

\[ D_{\mathrm{late}} <\frac{0.14X}{3200}+\frac{13J_1L}{8\cdot3200^2}<10^{-6}. \]

Both laws retain their actual phase motion about the common reference; no reconstructed increment is set to zero.

Joint difference and nonlinear current remainder

Let \(u=Y_0-Y_1\) and \(U=\sup_{\mathcal I}\|u\|_\infty\). The actual fundamental phasor difference is below \(20r_9\), and the additional harmonics total at most \(3r_9/2\) per edge. Each node has two incident edges, so the exact pair forcing has infinity norm below \(43r_9\). Every historical cell has length at most one and two observed endpoints. Heat contraction from its preceding reading yields

\[ U\le\delta+43r_9<44r_9. \]

Since \(\|B^TA\|_\infty=2\), the same pair of histories satisfies

\[ \|[d_0(v)-d_0(t_*)]-[d_1(v)-d_1(t_*)]\|_\infty \le2\gamma LU. \]

Freezing both laws at \(t_*\), the complex-phasor difference identity bounds the fundamental current error by \(20r_9D_{\mathrm{late}}+(\gamma/4)2\gamma LU\) per edge. The two higher-harmonic errors total at most \((9r_9+18r_9/2)D_{\mathrm{late}}=18r_9D_{\mathrm{late}}\). The resulting uniform current remainder is therefore bounded by

\[ \mathcal R :=r_9\bigl(38\cdot10^{-6}+22\bar\gamma^2L\bigr) <46\cdot10^{-6}r_9. \]

This bound retains the correlation between relative phase motion and form differences. It does not use the unconstrained remainder excluded in section 6.

Exact contrast operator bounds

At mode two, \(R_cB\) has entries, up to cyclic order and reflection, \((1,0,-1,\varphi,-\varphi)\), where \(\varphi=(1+\sqrt5)/2\). Its row norm is \(3+\sqrt5\). Put \(q_\mu=(1-e^{-\mu})/\mu\). The final ramp weights at each child are, up to the overall and child signs in the odd contrast,

\[ w_E=(1+\alpha_E,-2\alpha_E,\alpha_E-1),\qquad w_O=(1-\alpha_O,2\alpha_O,-1-\alpha_O). \]

Writing \(S_E=\|w_E\|_1\), \(S_O=\|w_O\|_1\), the two child blocks and normalization \(e^{\mu/2}/8\) give

\[ H_E=\frac{e^{\mu/2}q_\mu(3+\sqrt5)}4S_E,\qquad H_O=\frac{e^{\mu/2}q_\mu(3+\sqrt5)}4S_O. \]

These bounds allow signed interval coefficients. The retained inequalities \(0<\alpha_E<1<\alpha_O\) simplify them to \(S_E=2+2\alpha_E\), \(S_O=4\alpha_O\); the consumer instead rebuilds the full absolute-value sums. The inclusive error budgets are

\[ T_E=N_E+H_E\mathcal R,\qquad T_O=N_O+H_O\mathcal R. \]

For nonzero high-harmonic denominators define

\[ m=1-2\max\left\{ \left|\frac{K_{E,9}}{K_{E,18}}\right|, \left|\frac{K_{O,9}}{K_{O,18}}\right|\right\},\qquad e^2=\left(\frac{2T_E}{r_9|K_{E,18}|}\right)^2 +\left(\frac{2T_O}{r_9|K_{O,18}|}\right)^2. \]

The shared phase-circle theorem excludes a common record whenever the outward bounds satisfy \(m_{\mathrm{lower}}>0\) and \(e^2_{\mathrm{upper}}<m_{\mathrm{lower}}^2\). Equality does not certify separation. Both hidden source errors are already included in \(\delta\).

Result: the two complete noisy record sets are disjoint

The single final assessment, declared at b9b2e49a8146d6544407eed55a395377d91504c4, passed the strict squared comparison. Its attempt and exact result retain the source, input receipts, primitive moment intervals and rebuilt history/error bounds. Conservative outward summaries are

\[ D_{\mathrm{late}}<8.737\cdot10^{-7},\qquad e^2_{\mathrm{upper}}<(0.337)^2<(0.716)^2<m_{\mathrm{lower}}^2. \]

The decision uses the retained rational squared inequality, not rounded display values. This closes the stated distinction: under the unchanged initial class, complete laws, waveform, clock and noise allowance, no 150-coordinate raw record is compatible with both endpoint laws, even when each law may start from a different admitted state and have its own hidden constituent deviations.

The conclusion concerns these two supplied models and this protocol. It does not identify a law of nature, select a unique member of the wider constitutive family, reconstruct all initial phases, or prove acquisition under the alternative law. No raw record, nonlinear trajectory, physical observation or frozen source producer was evaluated in the assessment.

12. Shared implementation and verification

The detached consumer reuses the canonical potential/current, product geometry, hidden transport, interval/jet arithmetic and waveform. It supplies own-law analytical bounds, an actual carrier field, ideal local quadratures and the exact analytical relaxation bounds. Its conditional phase-circle checker re-admits coefficient and total-error primitives and excludes simultaneous membership in two zero-centered error bands only under the declared shared-phase contrast premises. Unsafe denominators and nonpositive circle margins are unavailable; equality at the squared error threshold does not certify exclusion. The phase-transport bound rebuilds the historical grid, current variation and both own-law source budgets. It consumes no raw record and does not implement an individual phase reconstruction. Neither consumer installs a runtime law or evaluates a trajectory. The shared prescribed-time validator remains the execution owner for any separately declared future numerical test.

The joint phase owner rebuilds the necessary shared-record conditions from the same source and plateau premises. The complete certificate re-admits primitive moments and rebuilds the late grid, decay, contrast and strict exclusion. The read-only input owner binds those moments to the retained kernel assessment without consuming its cached contrasts or verdict. The current consumer additionally rejects incorrect plateau levels; the evaluated source and original result remain recoverable at the frozen declaration revision.

Admission and field controls check invalid domains, the fine/carrier projection, both clock rows, derivative bounds, source denominators and identity/growth margins. Information controls compare actual-current quadratures with independently expanded harmonics and verify the interval sign witnesses used by the analytical obstruction. Finite-rank controls check the actual endpoint schedule, reversed edge probes and C5 spectral independence used in section 5. Relaxation controls check moment cancellation, actual forcing orientation, the dipole heat bound, own-law remainder constants and the obstruction's constitutive exclusion. Constrained-contrast controls check shared-phase synthetic cases, strict margins and primitive rejection. Phase-transport controls check the actual historical domain, exact C5 norms, independent scalar Green and constant-forcing cases, and separate reading/source/command budgets. Joint phase controls check the trapezoid, cycle and phasor arguments. Complete theorem controls independently check C5 projector/dual norms, exponential bounds, three-ramp parities and primitive admission. The evidence controls rebuild the strict inequality and source associations without producer replay; input controls separate fixed-byte admission from scientific inference. These controls exercise proof premises; the proofs establish their stated mathematical conclusions. No root search, fit, sampled coefficient assessment or frozen producer is run.