Observing outer winding with unknown child phase origins¶
Declared state class, probes and observation contract¶
The finite response comparison distinguishes two known preparations. This gate asks whether collective form readings can identify an unknown outer winding without supplying its phases to the inverse.
| Item | Fixed choice and scope |
|---|---|
| Complete law | Same C9-by-C5 support with nodes 9*c+j, 90 unit edges and held unit capacities; x_tau=(-Lx+gamma*S(theta))/4, theta_tau=gamma*Lx/4, gamma=1/(1023*pi), regular smooth-sine phase domain and no continuous input |
| Clock | tau=(1023/1024)*t; elapsed observation clock h begins at the unknown actual source, with both continuous rows retained |
| Admitted form | Every actual signed form satisfies abs(x_i)<=X=1/100; neither the fine forms nor their common or child means are supplied to the inverse |
| Admitted phase | theta[c,j]=2*pi*(j-4)/9+Phi_c+z[c,j], with unknown lifted child origins Phi, sum_j z[c,j]=0, and every abs(z[c,j])<=zeta=1e-6; global phase is arbitrary |
| Outer chart | Each oriented principal child-origin difference a_c=wrap(Phi_(c+1)-Phi_c) satisfies abs(a_c)<=A=7/5; the five angles are jointly compatible with actual origins, so their sum is 2*pi*w for unknown w in {-1,0,1} |
| Source scope | This is an admitted state class, not a newly derived preparation. Check inclusion of both earlier actual acquired families at tau=200 with all 80 original relative directions; a broader member is not thereby proved acquired |
| Probe collection | Ten separate branches from the same unknown actual state: for each marked child c and sign s in {-1,+1}, add s*epsilon*(delta[b,c]-1/5) to every phase of child b, with epsilon=1/10; forms, support and capacities are unchanged |
| Baseline | One unprobed continuation of that same source supplies five receiver readings; it is shared between both signed probes of each interface. Eleven matched mathematical branches are assumed, not sequential interventions or an authenticated laboratory cloning protocol |
| Readout | Fixed delay h=1/100000 in tau; for interface c read the arithmetic mean signed form of receiver child (c+1) mod 5. Three ordered vectors of length five contain baseline B, positive P and negative M, giving 15 raw scalar readings |
| Observation error | Each of the 15 raw readings has arbitrary additive error bounded by eta=1e-15. Preserve exact common-baseline cancellation: P-M has error at most 2*eta; P+M-2*B has error at most 4*eta; no noise independence is assumed |
| Inverse boundary | Consume only those 15 primitive readings and the frozen known model/protocol. Hidden phases, source reports, sector labels and evaluated response flags are not inputs. Unknown preparation is not reconstructed from an assigned answer |
| Required inference | Bound the actual mean boundary cosine/sine from finite nonlinear responses, infer child-origin angle intervals with the fine-error allowance, and intersect the complete cycle sum with 2*pi*w. Prove unique identification for every admitted source and allowed readout error, or report a scoped obstruction or unavailable sufficient certificate |
| Candidate meaning | Sector intersections are necessary compatibility conditions, not joint realizability witnesses. A singleton identifies the actual sector only under the declared same-source/state/law premises; empty candidates mean inconsistent records or premises, and unresolved alternatives remain explicit |
| Work | Each probe has its own actual signed jump W=H_plus-H_minus; require abs(W)<2 structural storage units per branch. No cumulative work budget, reservoir or funding law is inferred |
| Identity horizon | Preserve all five internal winding-one cycles and all nine respective outer slot-cycle windings at each jump and throughout 0<=t<=h. Generic all-future maintenance is outside this broader state-class contract |
| Numerical boundary | Exact rational admission and the existing shared 128-bit outward trigonometric arithmetic; finite nonlinear remainder retains the unknown background and both rows. No new trajectory, frozen-producer replay, amplitude scan or resolution refinement |
Freeze this declaration before evaluating the uniform inverse bounds. Reuse the complete finite-increment mechanism while preserving the earlier probe's fixed constants and evaluated result. A phase-origin chart or exact carrier alone cannot replace the actual fine-error or finite-time obligations.
The positive single-probe onset retains only one combination of sine and cosine; assess its ambiguity at the level actually proved, without inventing identical finite histories. The complete observation contract must retain all five interfaces and the compatibility of all nine fine slot cycles.
This tests observation of a conditional collective identity. It neither reduces the whole organization to one scalar EPI nor proves autonomous preparation, arbitrary-scale fractality or experimental physical identity.
The declaration above was committed as 52126ac0 before assessment. The
following argument uses the fixed protocol and exact rational outward
bounds; it evaluates neither a trajectory nor a frozen response producer.
1. State, marked cycles and the scope of the unknown source¶
Write \(\alpha=2\pi/9\), \(\psi_j=\alpha(j-4)\), and \(L=I_5\otimes L_9+L_5\otimes I_9\). All 45 signed forms and 45 primitive phase coordinates evolve under the declared complete rows. The origin \(\Phi_c\) is the arithmetic mean of the continuous lift \(\theta_{c,j}-\psi_j\) over child c. It is not the argument of that child's circular resultant, which vanishes for the exact C9 profile. Its residual \(z_{c,j}\) has zero child mean by definition; these means need not remain constant during the joined flow.
Let \(a_c\) be the admitted principal origin gap on the oriented interface from c to \(c+1\), cyclically modulo five. At slot j the corresponding actual lifted edge gap is
The fixed outer chart is unambiguous because \(A+2\zeta<\pi/2\). The residual differences telescope around each slot cycle, so
The allowed integer list follows from \(|\sum_c a_c|\le5A=7<4\pi\). Each internal C9 edge, including its declared closing turn, has gap \(\alpha+z_{c,j+1}-z_{c,j}\); its cycle sum is exactly \(2\pi\). Thus the admitted state has five internal winding-one identities and nine common outer winding identities before any observation. These are identities of the actual fine cycles, not a replacement by five independently assigned angle labels.
This wider class includes both actual previously acquired families at \(\tau=200\). The source theorem gives \(\|r\|_2<R<4\cdot10^{-70}\), \(\|z^{\mathrm{old}}\|_2<Z<7\cdot10^{-70}\), and the outer-one carrier bounds \(|X_c|<0.001394\) and \(|\Phi_c-v_c|<0.000022083\), with \(v_c=4(c-2)/\pi\). Subtracting each actual child's residual mean is an orthogonal projection, hence the resulting zero-mean fine residual still has norm at most Z. Incorporate the subtracted mean into the unknown origin; this does not reset or modify the reached state. The form bound is then below \(1/100\), the fine phase bound is below \(\zeta\), and the principal origin gaps remain below \(7/5\). For the outer-zero control the carrier is zero and the same residual argument applies, including its nonstationary actual residuals. All eighty original relative preparation directions are retained. This inclusion does not derive preparation of every other member of the larger class.
2. Finite response and the two boundary quadratures¶
For each marked interface c define its actual initial boundary moments
Neither moment nor any phase is supplied to the inverse. They are the quantities bounded from the finite mean-form readings. The positive or negative child-c probe changes the receiver's incident phase difference by \(s\epsilon\), where \(s=+1\) or \(-1\); the other initial incident differences at receiver child \(c+1\) are unchanged. Forms are unchanged across the event. Direct substitution into the full form row gives the exact initial probe-minus-baseline mean slope
The finite increment proof uses heat contraction, \(\|L\|_\infty=8\), degree four, \(\|S(\theta)\|_\infty\le4\), and the global sine derivative bound. These are the same full law and support here. Reapply its argument with the present \(\epsilon,h,X\), including the unknown background, rather than transferring the earlier evaluated response. With \(g\ge\gamma\) an outward rational upper bound and \(2g^2h^2<1/2\), the actual finite increment obeys
This bound retains both continuous rows, form diffusion, background phase motion and nonlinear probe feedback. In particular, the observed endpoint is not identified with the initial derivative. It applies to every actual fine state in the admitted class, with either probe sign and every receiver.
For one interface write the three noisy raw readings as \(B,P,M\). The same unprobed reading B occurs in both differences. Define
Equations (4)-(5) and the arbitrary raw error bounds imply
The odd combination cancels baseline motion and its actual reading error exactly. The even combination carries twice that error. No independence, zero-mean noise, stationary baseline or equality of different branches' later fine states is assumed. All eleven branches share the same initial actual state; measurements from different unknown preparations do not satisfy this contract merely because they have equal winding labels.
Let outward intervals enclose \(\gamma\), \(\sin\epsilon\) and \(1-\cos\epsilon\), and form the positive scale intervals
Then the moment rectangles are
The lower cosine prior follows from (1), not from an assigned answer. Empty intersections are inconsistent with the declared premises; they do not prove that no mathematical state under any model could give the data.
Two further necessary checks reject direct contradictions of the state class. Heat contraction gives \(\|x(h)\|_\infty\le X+gh\) on every branch, so each raw reading must have magnitude at most \(X+gh+\eta\). Rotating the mean boundary phasor by \(-a_c\) puts each constituent phasor in \([-2\zeta,2\zeta]\). Its real part is at least \(\cos(2\zeta)>0\), hence \(\cos(2\zeta)\le |C_c+iS_c|\le1\). Each computed moment rectangle must therefore intersect this annulus. These filters retain all actual admissible data. Passing them still does not prove that one full trajectory realizes an arbitrary supplied record.
3. Angle inclusion, cycle consistency and uniform identification¶
Every phasor in (3) has its argument in \([a_c-2\zeta,a_c+2\zeta]\), an acute interval. Positive averaging therefore keeps its nonzero resultant in the same angular cone. If \(\beta_c=\arg(C_c+iS_c)\), then
This argument uses the actual fine boundary deviations. It does not set them to zero or equate the argument of an average phasor with an arithmetic mean phase. Since the entire rectangle (9) has positive real part, the shared outward principal-argument calculation is regular. Define
Each true \(a_c\) belongs to its interval. Keep precisely those \(w\in\{-1,0,1\}\) for which the outward enclosure of \(2\pi w\) intersects \(\mathcal T\). These are necessary sector candidates. Independent interval corners need not correspond to one jointly realizable source, so the retained set is not a collection of constructed witnesses. Nevertheless, under the declared actual-source premises it contains the true sector; a singleton then identifies that sector, simultaneously on all nine fine cycles by (2).
A uniform bound shows that every admitted source and every allowed reading error yields a singleton. For a scale interval K, write \(K^+\) for its upper endpoint and \(\operatorname{wid}(K)\) for its width. Set
Since \(|C_c|,|S_c|\le1\), (7) gives \(|D_C|\le F_C+b_C\) and \(|D_S|\le F_S+b_S\). The product-width inequality therefore bounds the two pre-intersection moment widths by \(w_C,w_S\), including the outward uncertainty in the scale intervals. Materializing \(D+[-b,b]\) adds at most u at each endpoint. Its width is therefore at most \(2b+2u\), and its maximum magnitude at most \(F+2b+u\). Multiplication by K followed by endpoint rounding adds the remaining \(2u\), giving exactly the additional allowance \(\rho_K\). Clipping cannot increase either width. On \(C\ge c_0\), \(|S|\le1\), \(\arg(C+iS)=\arctan(S/C)\), and the elementary derivative bounds give
The last term covers the shared argument evaluation and subsequent interval arithmetic uniformly, not a sampled rounding error. Here \(c_0>1/6\) and \(|S|\le1\), so \(|S/C|<6\); division adds fewer than ten grid units to its endpoint enclosure. The shared arctangent reduces point arguments to magnitude at most \(1/2\). Its 64 alternating terms have omitted tail at most \((1/2)^{128}/129\); interval term multiplication contracts by less than \(0.251\), and bounding each term and accumulated sum rounding gives fewer than \(10^4u\) of point-enclosure width. At most two pi-based argument transformations, both endpoints and the remaining additions/clips/cycle sum stay below the looser total \(10^6u<3\cdot10^{-33}\). The \(10^{-20}\) allowance in each angle is consequently sufficient. The transcendental owners and their precision are unchanged.
The fixed outward calculation gives the strict conservative bounds
Adjacent exact sector sums are separated by \(2\pi>6.28\). The shared 128-bit outward \(2\pi\) intervals retain a separation greater than 6.28 as well; this checks the actual numerical intersection rule, not only ideal real arithmetic. Thus (14) precludes two candidates. True-source inclusion precludes zero candidates. The protocol identifies the unknown outer winding uniformly on the complete admitted class, including both signs of winding one and winding zero. It recovers neither all hidden fine coordinates nor absolute phase origins. A common phase translation leaves the complete law and every reading unchanged.
For arbitrary supplied records outside the assumed family, the inverse must retain empty, multiple or otherwise unavailable outcomes explicitly. An empty set signals inconsistency of records or premises; multiple necessary candidates are unresolved by this sufficient calculation, not proof of identical complete histories. The frozen protocol has no provision for smaller noise, a longer horizon or an amplitude retry after assessment.
4. What one positive onset loses¶
The positive slope alone is proportional to \(\cos(a-\epsilon/2)\) on a uniform boundary. It is unchanged by \(a\mapsto\epsilon-a\). This ambiguity occurs within the admitted state class and can span different cycle sectors. Take zero forms and zero fine residuals. One state has
A second has
All gaps satisfy \(|a_c|\le7/5\); choose successive lifted origins to realize them, using the respective closing turn. The two marked opening gaps obey \(-7/10=\epsilon-4/5\), so the positive initial receiver slopes are exactly equal. This disproves identification of winding from that local onset alone. It does not assert equal finite histories or equal noisy records under every finite observation protocol. The two signed probes recover both boundary quadratures, while the five marked interfaces supply the cycle information absent from one local current. Winding itself is not a sufficient state for predicting all those responses.
5. Actual work and the finite identity horizon¶
For a marked child c, the supplied jump is constant within each child and has zero global phase sum. It preserves forms, capacities, support, every internal gap and the phase mean. Exactly eighteen outer edges change, each by \(+\epsilon\) or \(-\epsilon\). The actual signed work is
By the global cosine Lipschitz bound, \(|W|\le18\epsilon=9/5<2\), independently of its sign. This accounts for each branch's own work. It supplies no reservoir, cumulative budget or occurrence law, and credits no prior continuous loss. The same eleven-branch preparation assumption used in the inference is retained; this is not a theorem about ten sequential probes on one evolving object.
Across a jump the outer gaps have magnitude at most \(A+2\zeta+\epsilon\), and each slot-cycle sum is unchanged. For every probed or unprobed branch, heat contraction and the complete phase row give, throughout \(0\le t\le h\),
The frozen bounds imply
Every internal edge stays positive and acute, and every outer edge stays strictly acute. No branch crossing can occur. Consequently all five internal winding-one cycles and all nine respective outer cycles retain their identities through every declared observation. This is a finite identity theorem for the larger state class. The earlier acquired sources' unprobed capture is a separate result; their earlier smaller-probe work and capture certificates do not transfer to the present stronger probes. Generic all-future maintenance or recovery after repeated observations is not established here.
6. Result and computational boundary¶
Fifteen fixed finite mean-form readings identify the unknown outer winding under the admitted state, complete law, marked support, known clock and matched-source assumptions. The bound includes hidden child origins, every allowed fine phase deviation, arbitrary bounded fine forms, full nonlinear finite motion and all permitted reading errors. Both previously acquired organizations lie in this class without replacing their actual states by limiting profiles. Work and identity have separate finite guarantees.
The observation implementation keeps the public-data inverse separate from hidden-state checks and source inclusion. The independent tests exercise complete-row coefficients, the one-onset counterexample, signed work, all marked cycles, raw-error cancellation, invalid domains and the uniform interval argument. Exact rational/outward arithmetic is shared with the existing proof owners; no numerical trajectory or frozen producer is replayed. The earlier finite-response protocol remains unchanged.
This is observation of a conditional collective identity, not derivation of a laboratory measurement bridge. It neither makes winding a complete state nor selects the supplied law, support, preparation or probe event. The organization can be treated as a composite NFR only with its declared regional identity and retained constituent/boundary state; the result does not justify collapsing those coordinates into one scalar nodal variable.