Information from a small phase preparation¶

The two admitted constitutive laws can produce different finite observations without an initial mean-form contrast or a complete relative phase turn. For the two-C9 quarter-factor model, a supplied small relative phase \(d\) gives an endpoint mean-form separation proportional to \(\gamma d^3\). This owner bounds that separation from both sides and retains independent preparation and observation errors for the two candidate laws.

The question is a resource-information tradeoff for one specified observation, not the minimum resource for every possible experiment. The earlier organized nonlinear comparison already establishes finite constitutive discrimination on its different complete model. The new obligation is to relate a small declared phase resource to both distinguishable and indistinguishable endpoint records. Task status belongs to the research program.

1. Complete law and the endpoint observation¶

Keep the ordered eighteen nodes \(9c+k\), with \(c=0,1\) and \(k=0,\ldots,8\). Each marked child is a unit-weight C9. The nine matching edges have the fixed common weight \(a=\gamma^2/1000\), where \(\gamma=1/(1023\pi)\). Capacity is held at one; no input, impulse, capacity change or later support event is admitted. The state-preserving initial attachment and its work are supplied separately from the continuing flow.

With \(L=I_2\otimes L_9+aL_{\rm match}\), the complete structural law in clock \(\tau\) is

\[ \begin{split} V_\lambda(u)&=1-\cos u+ \lambda\frac{(1-\cos9u)^2}{6\,9^4},\\ j_\lambda(u)&=\sin u+\lambda h(u),\qquad h(u)=\frac{\sin9u(1-\cos9u)}{2187},\\ U_\lambda(\theta)&=\sum_e w_eV_\lambda(\theta_i-\theta_j),\\ x_\tau&=\frac{-Lx-\gamma\nabla U_\lambda}4, \qquad \theta_\tau=\frac{\gamma Lx}4, \qquad 0\le\lambda\le1. \end{split} \tag{1} \]

Both rows use the same weights and the unchanged canonical \(\gamma\). The pressure is specified by the form row, independently of the response. This is the retained comparison law, not a native operator word or a reciprocal-degree normalization. Smooth bounded currents and the linear form terms give global continuation in continuous phase lifts. The uniform current bounds, derived from (1), are

\[ |j_\lambda|\le J_0=\frac{1459}{1458},\qquad |j_\lambda'|\le J_1=\frac{245}{243},\qquad |j_\lambda''|\le J_2=\frac{10}{9}. \tag{2} \]

Write \(\alpha=2\pi/9\) and retain the marked closing-edge turn. The actual state decomposes as

\[ x_{c,k}=X_c+z_{c,k},\qquad \theta_{c,k}=\alpha(k-4)+\Phi_c+q_{c,k},\qquad \sum_k z_{c,k}=\sum_k q_{c,k}=0. \tag{3} \]

Thus \(r=X_A-X_B\) is a raw mean-form difference, \(v=\gamma r\), and \(\delta=\Phi_A-\Phi_B\) is a relative continuous phase lift. The time coordinate \(s=a\tau/2\) transforms both rows; the nominal readout \(s=1\) corresponds to \(\tau=2000/\gamma^2\), not to a claimed laboratory duration. Internal edge sums cancel exactly, giving

\[ v_s=-v-\gamma^2\overline J_\lambda, \qquad \delta_s=v, \qquad \overline J_\lambda= \frac19\sum_k j_\lambda(\delta+q_{A,k}-q_{B,k}). \tag{4} \]

The nominal preparation has \(x=0\), \(q=z=0\), \(\Phi_A=d/2\), \(\Phi_B=-d/2\), with

\[ 0<d\le\frac1{100}. \tag{5} \]

The carrier \(q=z=0\) is invariant: internal currents cancel and each matching current is constant over its child. On it, \(\overline J_\lambda=j_\lambda(\delta)\), \(v(0)=0\), and \(\delta(0)=d\). The observation is one endpoint raw contrast \(R_\lambda(d)=r_\lambda(1)\). It is neither a derivative estimate nor the harmonic integral used by the full-turn probe. Every nominal child retains its winding-one phase profile exactly.

This preparation supplies a relative phase. It does not derive the formation of either C9, autonomous attachment, an observation apparatus or a source of the required work. The NFR ontology distinguishes this conditional structural observation from physical identification.

2. A two-sided nominal information bound¶

Let \(G=\gamma^2\). The canonical parameter satisfies \(G<10^{-7}\), using \(\pi>157/50\). On \(0<u\le d\), both \(\sin u\) and \(h(u)\) are positive, while \(j_\lambda(u)\le J_1u\). Variation of constants in (4), until a possible first exit from \(0<\delta_\lambda\le d\), gives

\[ \begin{split} -GJ_1d(1-e^{-s})&\le v_\lambda(s)\le0,\\ d-GJ_1d(s-1+e^{-s})&\le\delta_\lambda(s)\le d,\\ \delta_\lambda(s)&\ge d(1-GJ_1s^2/2)>\frac{99}{100}d \qquad(0\le s\le3/2). \end{split} \tag{6} \]

Here \(s-1+e^{-s}\le s^2/2\). The positive lower bound excludes exit through zero, and \(v_\lambda\le0\) excludes exit through the upper endpoint. This closes the corridor for every admitted \(\lambda\).

The perturbing current has a cubic bound obtained directly from its trigonometric factors:

\[ |h(u)|\le\frac{|u|^3}{6}\quad(u\in\mathbb R),\qquad h(u)\ge\frac{1081}{6912}u^3\quad(0<u\le1/18). \tag{7} \]

The upper bound uses \(|\sin9u|\le9|u|\) and \(1-\cos9u\le(9u)^2/2\). For the lower bound, put \(z=9u\le1/2\): \(\sin z\ge(23/24)z\) and \(1-\cos z\ge(47/96)z^2\) follow from the cubic and quartic Taylor bounds. In particular, (6) yields

\[ \frac{d^3}{7}<h(\delta_1(s))\le\frac{d^3}{6}. \tag{8} \]

For the endpoint comparison restrict again to \(0\le s\le1\). Put \(D(s)=\delta_1(s)-\delta_0(s)\). Subtract their exact Volterra equations, using the shared initial state:

\[ D(s)=-G\int_0^s(1-e^{-(s-t)}) \left[\sin\delta_1(t)-\sin\delta_0(t)+h(\delta_1(t))\right]dt. \tag{9} \]

Since sine is globally one-Lipschitz and the kernel lies in \([0,1]\), \(|D(s)|\le G\int_0^s(|D(t)|+d^3/6)\,dt\). The integral comparison and \(e^G-1\le2G\) give

\[ |D(s)|\le\frac{d^3}{6}(e^{Gs}-1)\le\frac{Gd^3}{3}. \tag{10} \]

Combining (8)--(10), with the strict canonical bound on \(G\), gives

\[ \frac{d^3}{8} <h(\delta_1(s))+\sin\delta_1(s)-\sin\delta_0(s) <\frac{d^3}{5}. \tag{11} \]

Both histories react freely. Their endpoint contrast difference, directly from the form row rather than separate approximations to each history, is

\[ \begin{split} R_0(d)-R_1(d) &=\gamma\int_0^1e^{t-1} \left[h(\delta_1(t))+\sin\delta_1(t)-\sin\delta_0(t)\right]dt,\\ \boxed{\frac{\gamma d^3}{20} <R_0(d)-R_1(d)<\frac{\gamma d^3}{6}.} \end{split} \tag{12} \]

For these deliberately loose constants it suffices to use \(3/5<1-e^{-1}<2/3\). The sine endpoint is larger, although both nominal form contrasts are negative. The proof compares the two actual solutions analytically; it evaluates no numerical trajectory. Subtracting independent endpoint approximations with an \(O(\gamma^3d)\) error would not prove a uniform cubic law as \(d\to0\). The direct difference (9) retains the shared linear response and controls its constitutive correction at order \(Gd^3\).

3. Full fine state and independent preparation errors¶

For each candidate separately, admit the primitive bounds

\[ |r(0)|\le\varepsilon_r,\qquad |\delta(0)-d|\le\varepsilon_\delta,\qquad \|z(0)\|\le\rho_x,\qquad \|q(0)\|\le\rho_\theta. \tag{13} \]

The last two norms combine all eighteen residuals in (3); their means vanish separately in each child. The relative phase error uses the declared initial lifts. A common form offset or phase rotation is immaterial by the complete law's exact symmetries. The candidates may choose different actual preparations anywhere in these bounds; no hidden pairing cancels their errors.

The uniform fine-retention proof uses neither a large mean form nor a particular mean phase. Its Hessian bounds hold over the entire relative phase circle. Consequently it applies here, and through the extended timing horizon, if

\[ F_0=\frac12(\rho_\theta+\gamma\rho_x)^2 +\frac12\gamma^2\rho_x^2<\frac1{40000}. \tag{14} \]

Both fine identities are retained, and the true current defect satisfies

\[ |\overline J_\lambda-j_\lambda(\delta)|\le B e^{-Ks}, \qquad B=\frac{4J_2F_0}{9},\qquad K=\frac{125}{2}. \tag{15} \]

This is the existing own-law estimate on every retained fine direction. It does not set the reached state to its carrier, infer the preparation from an operator label, or transfer a different law's source certificate.

Compare an actual history with the nominal history of its own law. Let \(\Delta r=r-r_{\lambda,\mathrm{nom}}\) and \(\Delta\delta=\delta-\delta_{\lambda,\mathrm{nom}}\). Variation of constants in the two actual mean rows gives

\[ \begin{split} \Delta r(s)&=e^{-s}r(0)-\gamma\int_0^s e^{-(s-t)} \left[j_\lambda(\delta(t))-j_\lambda(\delta_{\lambda,\mathrm{nom}}(t)) +e_f(t)\right]dt,\\ \Delta\delta(s)&=\delta(0)-d+\gamma(1-e^{-s})r(0) -G\int_0^s(1-e^{-(s-t)}) \left[j_\lambda(\delta(t))-j_\lambda(\delta_{\lambda,\mathrm{nom}}(t)) +e_f(t)\right]dt, \qquad |e_f(t)|\le Be^{-Kt}. \end{split} \tag{16} \]

For \(0\le t\le3/2\), define the following rational bounds:

\[ A=\varepsilon_\delta+\gamma\varepsilon_r+\frac{GB}{K}, \qquad P_t=\frac{A}{1-GJ_1t}. \tag{17} \]

The denominator is positive. Taking a supremum in the second row of (16) gives \(M_t\le A+GJ_1tM_t\), hence \(|\Delta\delta(s)|\le P_t\) for every \(s\le t\). At the nominal readout, the exact fine convolution is \((e^{-1}-e^{-K})/(K-1)\). Using \(e^{-1}<2/5\) and \(1-e^{-1}<2/3\) in the first row of (16) therefore gives

\[ |r(1)-R_\lambda(d)|\le E, \qquad E=\frac25\varepsilon_r+ \frac23\gamma J_1P_1+\frac25\frac{\gamma B}{K-1}. \tag{18} \]

The factor involving \(K\) keeps the fast decay of the fine state. A persistent fine forcing was not inserted in its place. This bound is applied once to each candidate, without assuming equal hidden states.

4. Reading sets, separation and a true overlap witness¶

Fix nonnegative exposure, gain and additive reading allowances \(\eta\le1/2\), \(g\), and \(\varepsilon_y\). The sole observation is

\[ Y_\lambda=(1+\kappa)r_\lambda(1+\xi)+b, \qquad |\xi|\le\eta,\quad |\kappa|\le g,\quad |b|\le\varepsilon_y. \tag{19} \]

Here \(\xi\) is an error in the independently declared \(s\) clock, and \(\kappa\) changes the readout, not a kinetic coefficient. An unknown clock rate or physical sensing law is not supplied by these allowances. A gain allowance exceeding one is permitted but may destroy useful information.

Put \(T=1+\eta\). The nominal corridor (6) gives \(|r_{\lambda,\mathrm{nom}}(s)|\le\gamma J_1d\) through \(T\). Also, \(\int_0^s e^{-(s-t)}e^{-Kt}dt\le1/K\). Equations (16)--(17) and the actual mean form row imply

\[ \begin{split} |r(s)|&\le R=\varepsilon_r+\gamma J_1(d+P_T)+\frac{\gamma B}{K},\\ |r_s(s)|&\le L=\varepsilon_r+2\gamma J_1(d+P_T) +\gamma B(1+1/K),\qquad 0\le s\le T. \end{split} \tag{20} \]

\[ |Y_\lambda-R_\lambda(d)|\le E_{\rm obs}=E+\eta L+gR+\varepsilon_y. \tag{21} \]

The gain term already bounds the entire actual endpoint, so another gain factor on the preparation or timing error would count that uncertainty twice. Let \(\mathcal Y_\lambda\) be the set of all readings (19) from all allowed preparations of that candidate. The positive gap (12) gives the sufficient full-state conclusion

\[ 2E_{\rm obs}<\frac{\gamma d^3}{20} \quad\Longrightarrow\quad \mathcal Y_0\cap\mathcal Y_1=\varnothing. \tag{22} \]

Both candidates can choose their nuisance values independently; the factor two is essential. A failed inequality establishes no intersection by itself. There is, however, a separate sufficient ambiguity condition:

\[ \varepsilon_y\ge\frac{\gamma d^3}{12} \quad\Longrightarrow\quad \mathcal Y_0\cap\mathcal Y_1\ne\varnothing. \tag{23} \]

To prove it, use the exact nominal preparations, zero timing and gain errors, and the common reading \(Y_*=(R_0(d)+R_1(d))/2\). Under each law choose the additive error \(b_\lambda=Y_*-R_\lambda(d)\). Equation (12) places both magnitudes below the allowance. This is an actual common-reading witness, not overlap of two conservative enclosures. It establishes failure of uniform identification in this observation class; it does not make every possible record ambiguous or rule out richer observations.

5. Attachment resource and what the scaling means¶

At the supplied state-preserving initial attachment, the exact new storage is

\[ W=a\sum_{k=0}^8\left[ \frac12(r(0)+z_{A,k}-z_{B,k})^2+ V_\lambda(\delta(0)+q_{A,k}-q_{B,k})\right]\ge0. \tag{24} \]

The linear fine terms cancel because their means are zero. Using the global curvature bound and \(V_\lambda(u)\le J_1u^2/2\) gives

\[ W\le a\left[\frac92\varepsilon_r^2+\rho_x^2+ \frac92J_1(d+\varepsilon_\delta)^2+J_1\rho_\theta^2\right]. \tag{25} \]

For the exact nominal preparations, \(W_\lambda=9aV_\lambda(d)\). The Taylor bounds \(d^2/2-d^4/24\le V_\lambda(d)\le d^2/2+d^4/24\) imply

\[ 4ad^2<W_\lambda<5ad^2. \tag{26} \]

Thus this fixed model and observation have a nominal signal of order \(\gamma d^3\) for nominal attachment storage of order \(ad^2\). At fixed canonical parameters, the signal is of order \(\gamma(W_\lambda/a)^{3/2}\). Reducing preparation reduces the signal faster than this attachment cost. This is not a universal thermodynamic measurement bound; formation, phase preparation, sensing and timing costs are separate, and \(W\) is the declared structural storage jump.

For sufficiently small constants, (22) admits nonzero errors with \(\varepsilon_r,\varepsilon_y\) of order \(\gamma d^3\), \(\varepsilon_\delta,F_0\) of order \(d^3\), and \(\eta,g\) of order \(d^2\). These are sufficient joint scalings, not separately proved necessary ones. The explicit additive-noise obstruction (23) is retained independently. The horizon remains \(\tau=2000/\gamma^2\); reducing \(d\) does not shorten it.

6. Detached calculation and scope¶

The shared calculator re-admits all primitive error allowances and the nominal phase before rebuilding (14)--(26). It reuses the current family, canonical parameter enclosures and fine-energy owner from the free probe. Exact rational arithmetic retains arbitrarily small nonzero preparations; outward gamma endpoints give conservative bounds without a floating-point cutoff.

Its statuses distinguish guaranteed_separation, observation_class_nonidentifiable and not_certified. The first is a prospective guarantee, the second uses the nominal midpoint witness, and the third establishes neither conclusion. No status selects a law from data. Bounds outside the fine-retention domain reject admission instead of claiming that an identity was lost. Supplied errors and source premises are not authenticated by the calculation.

The static controls check direct difference algebra, fine convolution, independent uncertainty, work cancellation, precision boundaries and a strictly positive budget with all error allowances nonzero. No trajectory, source producer or physical response is evaluated. A phase preparation, a clock and a sensor remain independent obligations before comparing with measured physics.