Observing collective winding with finite form inputs

Prospective protocol and decision

The single finite pulse admitted one approximate phase intervention. This question admits a complete continuing observation history and derives its inverse anew. Known inputs and collective form measurements must suffice; actual phases, source labels and hypothetical reference readings are not inverse inputs.

Item	Fixed choice and scope
Source	Both actual outer-zero and outer-one acquired five-child hosts, immediately after outer formation. Retain primitive component allowances at most 1e-6, quarter formation 200, recovery dwell 8e10, recovered norm q=1e-8, decay power 80, outer formation 200, all fine state and conserved means
Larger inference class	Unknown initial signed forms abs(x_i)<=X=1/100; theta[c,j]=psi_j+Phi_c+z[c,j], psi_j=2*pi*(j-4)/9, sum_j z[c,j]=0, abs(z[c,j])<=zeta=1e-6; all principal mean gaps have magnitude at most A=7/5. Initial outer winding is in {-1,0,1}. Inclusion of both acquired families must be proved; acquisition of every member of this larger class is not claimed
Support and complete law	Fixed C9-by-C5, nodes 9*c+j, 90 unit edges, held unit capacities; x'=(-L*x+gamma*S(theta))/4+T*f, theta'=gamma*L*x/4, gamma=1/(1023*pi), full-circle pairwise sine
Clock	Prime denotes tau=(1023/1024)*t. Observation begins at the actual formed source; no earlier word, donor or contact is consumed
Input program	Mark children c=0,1,2,3,4 in that order. In each block interleave three unforced plateaus of duration h=1e-5 with three cubic form pulses of duration d=1 and phase-command increments epsilon,-2*epsilon,epsilon, epsilon=1/10. Each plateau precedes its pulse, at cumulative command 0,+epsilon,-epsilon. All inputs are fixed before readings
Actual duration	One block is 3*(d+h) and the complete word is Tobs=15+15*h. Release input at the final pulse endpoint, retaining the reached state
Readings	All five actual mean signed forms at both endpoints of every plateau: ordered shape 5 blocks x 3 plateaus x 2 endpoints x 5 children, 150 scalar readings. No phases, derivatives, reset, copies or reference branch are observed
Uncertainty	Each raw scalar has arbitrary additive error at most eta_raw=1e-14; correlations are unrestricted. Input waveform, clock, support, gain and sampling times are ideal stipulated quantities in this mathematical protocol
Inference	Use the diffusion-corrected increments (8), then the two quadratures and complete cycle constraint. Identify the actual initial winding for every member of the admitted class and every allowed error, or report a scoped unavailable certificate. Necessary compatibility of arbitrary records is not their joint realizability
State and work	Preserve all five internal winding-one cycles, all nine respective outer windings and the within-child phase collar throughout the complete history. Bound release errors, full storage, continuous loss and positive supplied work
Resource decision	Peak full storage below 5e8, integrated loss below 2e9, absolute supplied work below 2e10, in structural units. Report necessary cost floors separately. Old jump-work and single-pulse budgets do not transfer
Evidence	Freeze declaration, consuming source and driver before one rational/outward dyadic128 analytical assessment at the stated source/noise corner. No trajectory, raw physical data, adaptive input, scan or frozen-producer replay

The supplied input is an intervention used to obtain information. Its autonomous occurrence and a physical measurement apparatus remain open. The question concerns collective observation of an already formed identity, not acquisition by measurement or identification of a material particle.

1. One continuous history and the global phase bound

Let R repeat a child vector over nine slots (called T in the complete row in the declaration), and use L=I5 tensor L9+L5 tensor I9. For marked child c set k_c=e_c-1/5 and v_c=L5^+ k_c. On a ramp with signed increment a and local fraction r=u/d, the reused waveform is

\[ s(r)=3r^2-2r^3,\quad z(u)=\frac{4a}{\gamma d}v_c s'(r),\quad f(u)=z'(u)+L_5z(u)/4. \tag{1} \]

The command H evolves from its current value by a*k_c*s; it is constant on the plateaus. Both z endpoints vanish. The complete H is continuous, with H'=gamma*L5*z/4, starts and ends at zero, and never changes any edge by more than epsilon. The middle increment has magnitude 2*epsilon, but crosses from +epsilon to -epsilon; this is not an excursion of 2*epsilon. The fine input Rf is zero on plateaus and after release. Bounded switches in f jump no state. Endpoint readings are state values; the plateau estimates use the unforced one-sided dynamics.

For the one actual solution throughout the entire program, put

\[ y=x-Rz,\qquad e=\theta-\theta(0)-RH. \]

Every switching boundary retains y and e continuously. The exact rows are

\[ y'=-Ly/4+\gamma S(\theta)/4,\quad e'=\gamma Ly/4=-\gamma y'+\gamma^2S(\theta)/4. \tag{2} \]

With g=gamma_upper, heat contraction and ||S||infinity<=4 give

\[ \|y(t)\|_\infty\le M=X+gT_{obs},\qquad e(t)=\gamma\{x(0)-y(t)\} +\frac{\gamma^2}{4}\int_0^t S(\theta(u))\,du, \]

\[ \boxed{\|e(t)\|_\infty\le E=2gX+2g^2T_{obs}}. \tag{3} \]

This exact integration uses the live phase row and form diffusion. It avoids treating a transient large prescribed form offset as uncontrolled phase drift. No stationary background, phase feedback or reset is used. Global form and lifted-phase sums are conserved because every input is neutral and the edge currents sum to zero. Common means need not vanish in the larger inference class; the actual acquired hosts retain their original zero totals.

At every sampling endpoint z vanishes, so abs(X_c)<=M. At final release both z and H vanish, yielding ||x_end||infinity<=M and ||theta_end-theta_initial||infinity<=E. For example ||x_end-x_initial||infinity<=M+X; neither state is restored exactly. The subsequent unforced flow must start at this actual endpoint.

2. Complete identity and actual-source inclusion

Initial internal fine gaps are 2*pi/9 plus differences of z; outer fine gaps are the admitted principal mean gaps plus those residual differences. Using (3), sufficient whole-history identity margins are

\[ \begin{split} m_{internal,+}&=2\pi/9-2\zeta-2E,\\ m_{internal,acute}&=\pi/2-2\pi/9-2\zeta-2E,\\ m_{outer,acute}&=\pi/2-A-\epsilon-2\zeta-2E. \end{split} \tag{4} \]

Strict positivity prevents a cut crossing on each of the five internal and nine outer marked cycles. Every initial outer identity is retained, including negative winding in the larger admitted class. Removing the current child mean from e changes a within-child residual by at most 2E, so a sufficient strict circular-collar margin is

\[ m_{collar}=\pi/2-2\zeta-4E. \tag{5} \]

The actual source is rebuilt from the recovery and resource primitives and outer acquisition, not from the old observation or contact reports. If the latter proof gives nominal form bound M, nominal outer-gap bound A, and full Euclidean departures R,Z, the inclusion checks are

\[ M_*+R<X,\qquad Z<\zeta,\qquad A_*+2Z<A. \tag{6} \]

Within-child mean removal is an orthogonal projection, so Z still bounds its residual; there is no state change in this decomposition. These bounds also cover the outer-zero control with stationary nominal carrier. Independent child acquisition/recovery, matching law and clock, support, capture, and historical storage/work premises remain necessary. Equality of their five resource bounds does not equate five actual preparations. The first plateau supplies the unforced first window; no extra prefix is inserted before it.

3. Form readings isolate the interaction current

Let U be the five actual child mean forms. Internal sine currents cancel exactly within a child, including off the carrier, leaving two external currents per child. On an unforced plateau,

\[ U'=-L_5U/4+\gamma\bar S/4. \tag{7} \]

The observer supplies two ordered vectors P,Q of measured mean forms at that plateau's endpoints. For the receiver r=(c+1) mod 5, form the exact primitive combination

\[ D=Q_r-P_r+\frac h8\{L_5(P+Q)\}_r. \tag{8} \]

All terms are in form units. The second term compensates diffusion rather than equating the observed increment to a sine-current response. Integrating (7) shows that its true-value counterpart equals gamma/4 * integral(bar S_r) plus the trapezoid error in the diffusion term. The fine unforced rows bound each mean's second derivative by

\[ M_2=4M+2g+4g^2M. \tag{9} \]

Since ||L5||infinity=4, the trapezoid error in D is at most M2*h^3/12. During any plateau the command is fixed. Each of the two external phase differences differs from its initial shifted counterpart by at most 2E, so the mean current differs by at most 4E. The plateau integral may therefore be compared to the initial shifted current with error at most g*h*E, irrespective of its block's position.

The two receiver-reading coefficients in (8) are -1+h/4 and 1+h/4; the four neighboring endpoint coefficients are -h/8. With h<4, their absolute sum is exactly 2+h/2. Correlations need not be discarded. For every measured corrected increment,

\[ D=\frac{\gamma h}{4}\bar S_r(\theta(0)+RH_q)+\delta, \quad |\delta|\le b=g h E+M_2h^3/12+(2+h/2)\eta_{raw}. \tag{10} \]

No derivative or phase is measured. Reads from different plateaus belong to one carried state history; the initial state in (10) is a proof reference, never a copied or observed branch.

4. Quadratures, circular geometry and the initial winding

For interface c define the initial fine edge phasor by

\[ C_c+iS_c=\frac19\sum_j e^{i d_{c,j}},\quad d_{c,j}=a_c+z_{c+1,j}-z_{c,j},\quad |a_c|\le A. \tag{11} \]

The three plateau levels of block c affect just its receiver's boundary with marked child c. Their initial currents satisfy

\[ \bar S_r^{\pm}-\bar S_r^0 =\mathord\pm\sin\epsilon\,C_c +(1-\cos\epsilon)S_c. \]

Use the three new corrected increments to compute

\[ O_c=D_+-D_-,\quad V_c=D_++D_--2D_0. \]

\[ O_c=\frac{\gamma h\sin\epsilon}{2}C_c+\delta_C, \quad V_c=\frac{\gamma h(1-\cos\epsilon)}2 S_c+\delta_S, \quad |\delta_C|\le2b,\quad |\delta_S|\le4b. \tag{12} \]

The coefficients and initial class in (12) exactly match the shared quadrature-to-cycle inverse. Consequently its existing geometry and circular kernel apply unchanged after the new primitive reading combinations and new complete-history errors have been established. This transfers only that algebraic inverse, not the earlier matched-branch or instantaneous sequential theorem.

In particular C_c>=cos(A+2*zeta)>0, the phasor modulus is at least cos(2*zeta), and its circular argument differs from a_c by at most 2*zeta. The kernel intersects those constraints, bounds each mean gap, and tests their sum against 2*pi*w, w in {-1,0,1}. It retains shared outward rounding and the existing angle-arithmetic allowance. A uniform returned cycle width strictly smaller than 2*pi admits at most one winding; an actual valid history supplies its included winding. This proves conditional identification for every admitted state and error record.

Malformed primitive readings reject before arithmetic. Impossible raw magnitudes, incompatible phasor annuli or absent cycle candidates remain explicit. A candidate or even a singleton for arbitrary supplied records does not authenticate their origin, their law, their error bounds or their joint realizability. Noise that defeats the uniform certificate leaves protocol admission unavailable, even if particular data look decisive.

5. Complete continuous resource account

The storage and balance remain

\[ H=\tfrac12x^TLx+\sum_e(1-\cos\delta_e),\qquad H'=-\|Lx\|_2^2/4+(Lx)^TRf. \tag{13} \]

Let ell=gamma_lower. The largest ramp has amplitude 2*epsilon and duration one. With the full-history residual M, the pulse extrema give

\[ Z_m=\frac{24\epsilon}{5\ell},\quad J=\frac{48\epsilon}{5\ell}+8M,\quad F_m=\frac{108\epsilon}{5\ell}. \]

\[ \sup H\le180(Z_m+M)^2+180,\quad Loss\le45T_{obs}J^2/4,\quad W_{abs}\le45\cdot15 JF_m. \tag{14} \]

The input is present only over the fifteen ramps. The last bound also bounds total positive supplied work, including the sum of positive individual channel contributions. It uses absolute powers before summing, so simultaneous supplying and absorbing channels do not conceal costs. Storage change, loss and supplied work are distinct ledger quantities; old event work and signed hidden storage supply no reservoir.

Returning the command to zero does not erase the cost. On each ramp, the actual collective phase increment is its command plus at most 2E componentwise. The marked-minus-adjacent increment has magnitude at least delta_a=max(0,abs(a)-4E). The previously proved C5 contrast inequalities then imply, on disjoint ramp intervals,

\[ Loss\ge\frac{90}{g^2} (2\delta_{\epsilon}^2+\delta_{2\epsilon}^2),\qquad \sup\tfrac12x^TLx\ge12\delta_{2\epsilon}^2/g^2. \tag{15} \]

Plateau loss is nonnegative. Since initial storage is at most 180X^2+180, positive supplied work is at least max(0,Loss_lower-180X^2-180). These structural bounds prove neither calibrated joules nor optimal cost. Finite bandwidth, tracking uncertainty and physical sampling remain additional premises before interpreting this protocol as a realizable measurement.

6. Instrument and handoff boundary

The pure protocol, word and inverse admit only their primitive time, noise and ordered form readings. The source assessor separately rebuilds actual acquisition and inclusion, then assesses the fixed corner through the exclusive driver. The independent controls exercise synthetic algebra and source failures without generating a reserved response. No existing frozen driver or inverse is rewritten.

The final state belongs to this new finite-input history. Neither the old post-observation contact response nor its source-specific electrical error certificate transfers merely because the same winding was retained. Their full source, means, support, law and clock must match the new endpoint. The execution plan alone selects the next dependency. No physical dataset, autonomous actuator, laboratory calibration or material identity is admitted by the inverse.

7. Frozen source-inclusive decision

The declaration, consuming source and driver were committed at 0d46c82f46e722618920b6b8ebce0db475f13c64 before the single assessment. The preserved attempt and result retain verdict finite_input_observation_admitted, fourteen true source flags, eleven positive source margins and nine positive protocol margins. No actual-source trajectory, raw response record or physical experiment was generated by this assessment: it evaluated the uniform analytical bounds. The read-only evidence controls reconstruct the receipt, source inclusion, observation errors, cycle resolution and resource margins without importing or rerunning the assessor.

Quantity	Outward summary of the archived bound
Whole-history phase departure from the prescribed command	less than 9.128e-6 radians
Mean form magnitude at every reading and release	less than 0.014668
Returned initial cosine / sine interval widths	less than 0.001247 / 0.049824
Each returned initial mean-gap interval width	less than 0.336298 radians
Returned cycle-sum interval width	less than 1.681488 radians
Winding-resolution margin below 2*pi	greater than 4.60169 radians
Minimum outer acute-gap margin throughout the word	greater than 0.070776 radians
Within-child circular-collar margin	greater than 1.5707578 radians
Peak full storage	less than 4.284e8 structural units
Full integrated loss	greater than 5.5748e7 and less than 1.607e9 structural units
Positive supplied work	greater than 5.5748e7 structural units
Absolute supplied work and summed positive channel work	less than 1.446e10 structural units

The cycle width is strictly smaller than the separation 2*pi between adjacent possible winding sums. Hence every valid record within the admitted class and error allowance identifies its actual initial winding. Both acquired families belong to that class; the broader inference also covers winding minus one without asserting its acquisition by the supplied preparation. All five internal and nine respective outer identities survive throughout the 15.00015 structural-time history. Large transient form, supplied work and residual release state remain explicit consequences of the finite inputs.

This closes the finite-input observation gate. Reusing the same prior and quadrature kernel preserved one algebraic owner while the new current correction and whole-history proof replaced the inapplicable instantaneous reading formula. The earlier frozen evidence and producers remain unchanged. The result supplies an observable collective identity under a complete conditional protocol; it establishes neither a physical sensor nor an autonomous force or material constituent.