Finite curvature from one additional reading¶

Question and conditional claim¶

The uncertain-clock inverse retains necessary bounds on geometry and the effective response scale \(J=G\rho\), where \(G\) is sensor gain and \(\tau=\rho s\) converts the observation clock to structural time. Its exact curvature witness disproves a universal gain/clock symmetry, but supplies no derivative measurement from noisy samples. Here a fourth scalar reading inside the first window yields a genuine finite-curvature constraint with full source and flow error bounds. It can narrow clock and gain separately under the same supplied complete law.

The result is a necessary interval enclosure. Local identifiability is proved separately on an ideal fixed-nuisance subfamily; it is not promoted to global recovery of the full uncertain state. No new reserved response is evaluated, and the earlier protocols and verdicts remain unchanged. The ontology retains the distinction between collective observation and physical identification.

F1-F2: one full source, two inputs and four readings¶

Retain the complete clocked model: eighteen nodes in two unit C9 rings, contacts (0,9) and (1,10), degree matrix \(M\), \(A=M^{-1}L\), and \(q=e_4-e_5\). Its structural rows are

\[ x_\tau=-Ax+\gamma f(\theta),\qquad \theta_\tau=\gamma Ax,\qquad \gamma=1/(1023\pi),\qquad f_i(\theta)=M_{ii}^{-1}\sum_{j\sim i}\sin(\theta_j-\theta_i). \tag{1} \]

In observation time \(s\), both rows multiply by one held unknown \(\rho\in[\rho_-,\rho_+]\), with \(\rho_->0\). The original full source has arbitrary common means and centered residuals

\[ x^-=m_x\mathbf1+u,\quad \theta^-=m_\theta\mathbf1+\theta^0(b,c)+v,\quad \|u\|_M\le X,\quad\|v\|_M\le Y. \tag{2} \]

The nominal geometry and source guard retain their original owners: \([b_-,b_+]\subseteq[11/8,3/2]\), \([c_-,c_+]\subseteq[2/3,1]\), continuous lifts and the full weighted centering. The actual original long-arc mean is still \(B_{\rm initial}=b-(v_1-v_0)/8\). No derivative of a reconstructed pressure, ideal target state or discarded intermediate state is supplied.

For \(0<a_1\le a_2\le1\), apply \(a_1q\) to phase at \(s=0\) and \((a_2-a_1)q\) at \(s=H>0\). Form is continuous at each jump; all thirty-six coordinates evolve between them. Require \(h_*:=\rho_+H\le1/2\). The additional reading is passive: it changes neither state nor events. The four observed times and true recorded values are

\[ (s_0,s_1,s_2,s_3)=(0,H/2,H,2H),\qquad r_i=Gq^Tx(s_i)+O+\eta_i,\quad |\eta_i|\le\delta, \tag{3} \]

with one held \(G\in[G_-,G_+]\), \(G_->0\), and one held arbitrary offset \(O\). Each primitive reading interval has exact midpoint \(c_i\) and half-width \(t_i\); its numerical width is separate from \(\delta\). The old endpoint inverse consumes indices \((0,2,3)\). It must not mistake the extra half-time reading for the old middle endpoint.

The finite contrast is

\[ C_r=r_2-2r_1+r_0,\qquad m_C=c_2-2c_1+c_0,\qquad \varepsilon_C=t_0+2t_1+t_2+4\delta. \tag{4} \]

Compute \(m_C\) exactly before outward interval arithmetic: its offset cancels. Then \(I_C=[m_C-\varepsilon_C,m_C+\varepsilon_C]\) encloses the noiseless sensor contrast. The fourth reading has zero coefficient in (4), but enters the old two-input bound on \(b,J\).

F3: initial curvature and a complete third-derivative bound¶

All norms below use \(M\). On this support, \(\|A\|\le2\), \(\|q\|_{M^{-1}}=1\), and \(\|Df(\theta)\|\le2\). The additional bound needed here is

\[ \|D^2f(\theta)[w,z]\|_M\le2\|w\|_M\|z\|_M. \tag{5} \]

For an oriented incidence matrix \(D\), \(\|M^{-1/2}D\|\le\sqrt2\). Every degree is at least two, so \(|w_i-w_j|\le\sqrt{1/d_i+1/d_j}\,\|w\|_M\le\|w\|_M\). Use this edgewise maximum in the incidence representation of the Hessian, whose sine weights have absolute value at most one, and bound the remaining edge difference norm by \(\sqrt2\|z\|_M\). This proves (5) globally, without an acute-trajectory assumption.

Set \(a=a_1\) and \(\phi=\theta^0(b,c)+aq\). At the original post-event state, differentiating both rows of (1) gives

\[ x_{\tau\tau}(0^+)=A^2u-\gamma Af(\phi+v) +\gamma^2Df(\phi+v)Au. \tag{6} \]

The nominal scalar curvature is

\[ \begin{aligned} K(b,a)&=-\gamma q^TAf(\phi)\\ &=\frac\gamma2[4\sin(b+a)-3\sin(b-2a)-\sin b]. \end{aligned}\tag{7} \]

Only rows three through six enter \(q^TAf\); other rows retain their actual noncritical values. The initial full-state error satisfies

\[ |q^Tx_{\tau\tau}(0^+)-K(b,a)|\le \varepsilon_2:=4(1+g^2)X+4gY,\qquad g=1/3069>\gamma. \tag{8} \]

Indeed, the form terms in (6) contribute at most \(4X+4g^2X\), and the sine-map difference followed by \(A\) contributes at most \(4gY\). A nonzero initial form residual cannot be dropped because the nominal constant-form case has zero initial phase rate.

For stable positive coefficient bounds, rewrite (7) as

\[ \begin{aligned} P(a)&=2\sin^2(a/2)(1+3\cos a),\\ R(a)&=\sin a(2+3\cos a),\\ K(b,a)&=\gamma[P(a)\sin b+R(a)\cos b]>0. \end{aligned}\tag{9} \]

Both coefficients and both trigonometric coordinates are positive on the admitted domain. This avoids subtracting nearly equal sine values when \(a\) is small. Numerical positivity still needs an outward lower bound strictly above zero; an unresolved bound is not a zero physical curvature.

Reuse the freshly rebuilt maximum-history form bound

\[ Q=\frac{X+2gh_*(4+4a_2+2Y)}{1-16g^2h_*^2}. \tag{10} \]

It retains the complete first-to-second-window history. On the first window, the actual phase is \(\phi+v+d(\tau)\), with \(\|d\|_M\le4gh_*Q\). Therefore a valid first-window forcing bound is

\[ S=\min\{7,\ 4+4a_1+2Y+8gh_*Q\}. \tag{11} \]

The second bound follows from the nominal forcing bound four and global Lipschitz constant two. The first follows independently from \(|f_i|\le1\) and \(\sum_i d_i=40<49\).

Writing \(U=2Q+gS\) gives

\[ \|x_\tau\|\le U,\quad \|\theta_\tau\|\le2gQ,\quad \|x_{\tau\tau}\|\le2U+4g^2Q,\quad \|\theta_{\tau\tau}\|\le2gU. \]

Differentiating once more,

\[ x_{\tau\tau\tau}=-Ax_{\tau\tau} +\gamma D^2f[\theta_\tau,\theta_\tau] +\gamma Df\,\theta_{\tau\tau}. \]

Consequently

\[ \boxed{\|x_{\tau\tau\tau}\|_M\le M_3:=8(1+2g^2)Q+4g(1+g^2)S+8g^3Q^2.}\tag{12} \]

These are complete-law derivatives on the first continuous segment. No Taylor expansion crosses a phase jump. The endpoint \(x(H^-)\) equals \(x(H^+)\), so the reading at the second event remains admissible in (4).

A finite difference, not assumed access to derivatives¶

For any twice differentiable scalar \(y\), the exact identity

\[ y(h)-2y(h/2)+y(0) =\int_0^{h/2}\int_0^{h/2}y''(u+v)\,du\,dv \tag{13} \]

has a positive kernel of area \(h^2/4\). If \(|y'''|\le M_3\), its difference from \(h^2y''(0)/4\) has absolute value at most \(M_3h^3/8\), since the double integral of \(u+v\) is \(h^3/8\). Apply this to \(q^Tx\), use (8), and set \(h=\rho H\). The actual noiseless sensor contrast obeys

\[ \left|C-\frac{G\rho^2H^2}{4}K(b,a_1)\right| \le G\left[\frac{\rho^2H^2}{4}\varepsilon_2 +\frac{\rho^3H^3}{8}M_3\right]. \tag{14} \]

Dividing by the actual positive \(J=G\rho\) yields

\[ \left|\frac{4C}{H^2J}-\rho K(b,a_1)\right| \le D_C:=\rho_+\varepsilon_2+\frac{\rho_+^2H}{2}M_3. \tag{15} \]

The source term does not vanish merely by shortening \(H\). Meanwhile the observation error after normalization scales as \(H^{-2}\). A shorter window reduces the smooth-flow error but can worsen inference from noisy readings; no automatic refinement policy is justified.

Re-admit the four primitive intervals before constructing the old clock envelope from indices \((0,2,3)\). Let its bounds be \(I_b,I_J,I_G,I_\rho\). Intersect the effective-scale interval with the exact positive prior \([G_-\rho_-,G_+\rho_+]\) before any quotient. Likewise preserve the positive primitive clock/gain endpoints when an outward interval lower bound happens to round to zero.

Enclose \(K(b,a_1)\) over \(b\in I_b\) using (9), obtaining \(I_K\) with positive lower bound. Interval projection of (4), (15) gives

\[ I_Z=\frac{4I_C}{H^2I_J},\qquad I_\rho^{\rm new}=I_\rho\cap \frac{I_Z+[-D_C,D_C]}{I_K},\qquad I_G^{\rm new}=I_G\cap\frac{I_J}{I_\rho^{\rm new}}. \tag{16} \]

Use exact endpoint products and positive quotients before interval materialization, particularly for \(H^2\) and positive clock scales. The result retains the old nominal and original actual-angle bounds; it does not silently reinvert \(K\) or iterate these projections as a fit.

Every full trajectory and held sensor satisfying the four observations belongs to all these necessary constraints. The shared errors have not become independent realizations: if \(w=(w_1,w_2)\) is an old inverse row, its four reading coefficients are

\[ (w_1,0,w_2-w_1,-w_2), \]

whereas the curvature coefficients are \((1,-2,1,0)\). Both act on the same four bounded errors. The separate interval constraints conservatively discard correlations between \(I_J\), \(I_C\), geometry and state. Intersection is sound, but nonempty intersections or their Cartesian products do not prove a common error realization or a realizable trajectory.

Strict exclusion gives incompatible. Missing base inference, unresolved curvature positivity or quotient arithmetic gives unavailable for the refinement, retaining the coarse report for inspection. A surviving bounded_candidate reports necessary outer bounds, not independently calibrated time or gain. Source admission, original-angle meaning and optional whole-window acuteness retain their separate scopes.

An informative finite-error regime without evaluating a response¶

The following separately declared arithmetic example shows that the finite bounds can distinguish clock and gain, rather than only their product. It selects no source realization or recorded values:

\[ \begin{gathered} H=2^{-24},\quad X=Y=2^{-48},\quad (a_1,a_2)=(1/4,3/4),\\ G\in[1,2],\quad\rho\in[1/2,2],\quad \delta=2^{-90},\quad t_i\le\delta. \end{gathered}\tag{17} \]

Keep the full original angle priors. Then \(h_*=2^{-23}\). The shared conditioning argument gives coordinate radius

\[ R_*:=72000E(h_*)/h_*+144000\delta/h_*<1/12500. \]

The auxiliary radial prior is still \([1/4,2]\). For compatible observations, the ideal arithmetic enclosures therefore have widths

\[ w_J\le4R_*<1/3000,\qquad w_b\le\frac{2R_*}{1/4-2R_*}<1/1500. \tag{18} \]

Direct rational substitution into (8), (10)-(12), (15) also yields \(D_C=2\varepsilon_2+2HM_3<10^{-9}\).

The natural positive-coefficient interval (9), even over the whole original angle prior, admits the uniform lower bound \(K_->1/18000=:k_0\). For example, the elementary sine/cosine bounds

\[ \sin(1/8)>3/25,\quad\sin(1/4)>6/25,\quad \cos(1/4)>24/25,\quad\sin b>9/10,\quad\cos b>7/100 \]

imply

\[ P(1/4)\sin b+R(1/4)\cos b>14262/78125, \qquad \gamma>1/3216, \]

whose product exceeds \(k_0\). Alternating Taylor bounds give these inequalities; the degree-six cosine lower bound at \(3/2\), for example, is \(0.0701171875>7/100\). The same coefficient representation has \(P(1/4)<1/8\), \(R(1/4)<5/4\); sine and cosine are one-Lipschitz. Thus its ideal interval width satisfies

\[ w_K\le\frac{11g}{8}w_b. \tag{19} \]

This statement concerns the natural positive interval, not an unjustified replacement of a wider computed interval by a pointwise minimum.

Let \(I_A=4I_C/H^2\). Its diameter is at most \(d_A=64\delta/H^2=2^{-36}\). Since the actual response belongs to all the constraints, the conservative lower bounds

\[ (I_A)_-\ge\tfrac12(\tfrac12 k_0-D_C)-d_A>0, \qquad (I_A)_-/4-D_C>0 \]

also certify positivity of the numerator of the clock quotient. After intersecting the original priors, \((I_J)_-\ge1/2\). Set

\[ \beta=w_K/k_0,\qquad t=2w_J+\beta+2w_J\beta. \]

Positive endpoint division in (16) gives

\[ w_\rho\le\frac{2d_A}{k_0} +(2+D_C/k_0)t+\frac{2D_C}{k_0}<\frac1{80}. \tag{20} \]

To see the width estimate, before prior intersection the clock quotient width equals

\[ \frac{d_A}{J_-K_-} +\frac{A_-}{J_+K_+} \left(\frac{J_+K_+}{J_-K_-}-1\right) +D_C\left(\frac1{K_-}+\frac1{K_+}\right) \]

when \(d_A\) is the actual numerator diameter, and is bounded by this expression when \(d_A\) is its allowance. Compatibility gives \(A_-/(J_+K_+)\le2+D_C/K_+\), and the product ratio minus one is at most \(t\). This proves (20). The final strict inequality follows even after replacing \(w_J,w_b,D_C\) by \(1/3000,1/1500,10^{-9}\). Similarly, \(\rho_-\ge1/2\) and compatibility with \(G\le2\) give

\[ w_G\le2w_J+4w_\rho<1/16.\tag{21} \]

The original actual-angle width remains at most \(w_b+Y/4\).

These are conditional exact-arithmetic budgets, not generated observations. An executable report still has to admit its outward coefficient and quotient arithmetic. The noise budget in (17) is newly stated for a second-difference calculation; it neither recalibrates a laboratory sensor nor changes the earlier frozen experiments. A reserved numerical test, if admitted later, must preserve its own first outcome.

Why one extra reading supplies new information on an ideal subfamily¶

The following local theorem justifies the observation design more strongly than counting unknowns. Fix receiver parameter \(c\), common means, \(u=v=0\), zero sensor errors and an interior point of the positive \((b,G,\rho)\) priors. Keep \(a_1<a_2\), both events and the full continuous law. Allow only these three parameters to vary. In particular, the unknown geometry remains part of the admitted source family.

Define

\[ A_j(b)=-k_j\cos(b-a_j/2),\qquad k_j=2\gamma\sin(3a_j/2)>0. \]

The exact full-flow increments \(D_1=r(H)-r(0)\), \(D_2=r(2H)-r(H)\) depend smoothly and analytically near \(H=0\) on this subfamily, including the actual first endpoint and second jump. With \(J=G\rho\), their duration-normalized map extends smoothly as

\[ (D_1/H,D_2/H)=(JA_1(b),JA_2(b))+O(H). \tag{22} \]

The extension includes derivatives with respect to the parameters, by smooth dependence of the full flow and fixed jump maps. Moreover,

\[ W:=A_1'A_2-A_1A_2' =-k_1k_2\sin((a_2-a_1)/2)\ne0. \tag{23} \]

For every sufficiently small fixed positive \(H\), the exact two-increment Jacobian in \((b,G)\), at fixed \(\rho\), is therefore nonsingular. The implicit-function theorem provides a local curve \(b(\rho),G(\rho)\) of exactly equal full-flow increments through the chosen interior point. Since \(q^Tx^-=0\), the same held offset also gives the same original reading. Thus the existing three readings have an actual local collision family, not just equal leading coefficients. No numerical horizon for this theorem is claimed.

Adding the finite contrast gives the smooth limiting map

\[ \left(D_1/H,D_2/H,4C/H^2\right) \longrightarrow (JA_1(b),JA_2(b),J\rho K(b,a_1)). \tag{24} \]

Its Jacobian in \((b,J,\rho)\) has determinant

\[ J^2K(b,a_1)W\ne0.\tag{25} \]

The inverse-function theorem therefore gives local injectivity for each sufficiently small fixed positive \(H\). Positive \(\rho\) makes the coordinate change \(J=G\rho\) regular, so the same statement holds for \((b,G,\rho)\). The three contrasts are invertible linear combinations of the three offset-free readings at \(H/2,H,2H\); no derivative access is assumed. One added scalar reading is thus sufficient and minimal to remove the exhibited local ambiguity while keeping the original protocol and this ideal three-parameter subfamily.

Neither theorem grants global injectivity, identification of the receiver or fine residuals, nor local injectivity after those nuisance coordinates are also free. Positive residual and measurement allowances use the finite necessary bounds (8)-(16). The exact common-law-rate/clock equivalence from the previous owner also survives any extra reading of the identical state history if a free multiplier of every evolution row is admitted. The present law holds that multiplier fixed.

Implementation and evidence boundary¶

The curvature inverse reuses the existing clock and two-input owners through freshly admitted primitives. It adds the fourth reading, the full derivative bounds and one necessary refinement; it accepts no cached report as evidence. The contract owns the input ordering and availability behavior, and the tests exercise full graph derivatives, finite contrasts, representation boundaries and fresh complete-flow controls. Numerical implementation controls are not a reserved response or an independent physical observation.

This result concerns inference from a supplied source, calibrated input amplitudes, held affine observation law and fixed normalized-sine dynamics. It derives neither the source nor its support, event funding, future maintenance or a unique fundamental law. Any finite response campaign must first freeze its complete preparation, clock/sensor model, all four reading times, error budgets and stopping rules in the sole execution plan.

Prospective four-reading software protocol¶

This section admits a new reserved software evaluation after the conditional proof above. Its preparation, response generator, public inverse packets and stopping rules are fixed before any selected response is evaluated. It does not reinterpret an earlier known-clock record as a new observation. The execution plan owns this single active gate. The frozen machine-readable protocol and source archive, under the stem curvature-inference-v1, retain the exact constants and the prospective committed source revision.

F1-F2: hidden source and declared observation¶

Use the complete support, full thirty-six-coordinate law and weighted source chart of (1)-(3). The public prior and supplied input are fixed as

\[ \begin{gathered} b\in[11/8,3/2],\quad c\in[2/3,1],\quad G\in[1,2],\quad\rho\in[1/2,2],\\ X=Y=2^{-48},\quad H=2^{-24},\quad (a_1,a_2)=(1/4,3/4),\quad\delta=2^{-90}. \end{gathered}\tag{26} \]

Angles are in radians, \(H\) uses the declared observation clock and \(\tau=\rho s\) applies to both continuous rows. Gain converts the signed form readout to recorded units; \(\delta\) and numerical reading radii use those recorded units. There is no independent laboratory calibration.

The generator receives these three exact cases, indexed by \(k=0,1,2\). The final column selects the full source residual recipe; it is not a new coordinate of the model.

Case \(k\)	\(b\)	\(c\)	\(G\)	\(\rho\)	Recipe \(r\)
0	\(45/32\)	\(3/4\)	\(3/2\)	\(1\)	0
1	\(45/32\)	\(3/4\)	\(6/5\)	\(5/4\)	0
2	\(47/32\)	\(11/12\)	\(7/4\)	\(7/8\)	2

For node indices \(i=0,\ldots,17\), set

\[ \widetilde u_i=\frac{(i+2)(r+2)}{2^{62}},\qquad \widetilde v_i=\frac{(5i+3r)\bmod23+1}{2^{62}}, \quad u=\widetilde u-\langle\widetilde u\rangle_M\mathbf1, \quad v=\widetilde v-\langle\widetilde v\rangle_M\mathbf1. \tag{27} \]

Set \(m_x=(r+2)/11\) and \(m_\theta=-(r+2)/13\), with the nominal phases centered by the same full weighted mean. Retain every coordinate and the realized exact arrays. Weighted centering preserves the residual norm bound, and \(40\cdot76^2/2^{124}<2^{-96}\) admits both squared norms. The recipe has nonzero residuals at every node. Its original actual long-arc mean is

\[ B_{\rm initial}=b-5/2^{65},\tag{28} \]

because \(v_1-v_0=5/2^{62}\). The retained initial interval source also supplies an outer interval for that same initial statistic; it is not an endpoint mean or an exact recovery of every residual.

Cases 0 and 1 have the same complete source and the same \(J=G\rho=3/2\). Their gains and clocks differ. Equality of this leading response scale does not assert equal exact endpoint responses, and these nonzero-residual cases are not the ideal local-collision family in (22)-(25).

The sensor has one held offset \(O_k=(2k+3)/11\) per case. Its four realized errors are fixed before response generation:

\[ (\eta_0,\eta_{1/2},\eta_1,\eta_2) =2^{-91}(k-1,\ 1-k,\ (-1)^k,\ k-1). \tag{29} \]

Each has magnitude at most \(\delta\). There is exactly one recorded reading at each of \(0,H/2,H,2H\), under the same gain, offset and clock. Reuse those four primitive intervals in all differences and controls. The numerical half-width of each retained reading must be at most \(\delta\), separately from the per-reading error allowance.

F3: complete uninterrupted response and information separation¶

Use the full-state readout producer and shared direct source-box Taylor kernel. For each case, the three consecutive structural durations and phase jumps are

\[ (\rho H/2,\rho H/2,\rho H),\qquad(1/4,0,1/2)q. \tag{30} \]

The zero entry means continuous continuation with no event at the passive half-time observation. The first event occurs at observed time zero and the final event at \(H\). Both are phase-only jumps. Carry each complete thirty-six-coordinate endpoint interval into the next segment before applying its declared jump; do not reset, recenter, narrow or reprepare the intermediate state. Source common means remain part of the same trajectory.

Each segment uses order four, the fixed sixteen-attempt strict Picard inclusion budget, shared outward dyadic-128 interval arithmetic and the shared elementary-function enclosures. Retain its initial box, complete tube, all source Taylor coefficients, fifth-order remainder, full endpoint, strict inclusion margin and achieved horizon. Local segment time may start at zero; the record separately retains its global observation and structural time association. No smaller-step search, precision escalation, endpoint selection or budget retry is admitted. A positive smooth-law domain flag establishes neither acute geometry nor maintenance after either pulse.

Construct the sensor readings from the original baseline and the three complete endpoints. The identical middle readings enter every applicable increment and the contrast \(C=r_H-2r_{H/2}+r_0\). Offset cancellation and error correlation must be retained algebraically before interval projection. The original source boxes, hidden parameters, sensor error realizations and response certificates stay on the generator/evidence side.

The inverse receives only these ten public keywords: bulk_angle_bounds, receiver_short_angle_bounds, form_radius, phase_radius, phase_increments, probe_duration, recorded_reading_bounds, readout_error_bound, readout_gain_bounds and clock_rate_bounds. In particular, its duration is observed \(H\), not a hidden structural segment duration. No true \(b,c,G,\rho\), common mean, offset, realized error, state, response certificate or cached verdict may enter the inverse. Retain the exact allowlisted packets and their hashes and invoke the inverse through the separated worker. This is auditable software information exclusion, not cryptographic secrecy or provenance authentication.

F4: fixed stopping criteria and controls¶

The reserved evaluation passes only if every declared case satisfies all source, clock, sensor, full-horizon and information-separation checks and these inference criteria:

The primary four-reading inverse is available as bounded_candidate, with its coarse child, finite-curvature bound and positive divisions admitted. Its necessary marginals cover the exact nominal \(b\), original actual \(B_{\rm initial}\), effective gain \(J\), gain \(G\) and clock \(\rho\). The actual-angle marginal also contains the retained initial-box enclosure of \(B_{\rm initial}\).
Strict widths are below 1/1024 for the actual original angle, 1/2048 for \(J\), 1/16 for \(G\) and 1/80 for \(\rho\). These are marginal resolution checks, not point recovery or joint realizability of every retained tuple.
Cases 0 and 1 have strictly disjoint retained recorded-curvature intervals. The comparison uses their actual four-reading records, not separately supplied derivative values or fitted curvature.
Reusing the same readings with false clock prior [25/32,13/16] or false gain prior [31/16,2] leaves the coarse child available but makes the curvature refinement incompatible. This checks the contribution of the additional reading; coarse outer compatibility is not a proof of an alternative full trajectory.
The false angle prior [11/8,353/256] returns incompatible. The equal cumulative amplitudes (1/4,1/4) return unavailable through the leading-rank limitation. The latter is an altered-declaration method control, not an evaluated equal-pulse physical alternative.
The complete phase-blind alternative \(x_\tau=-Ax,\ \theta_\tau=\gamma Ax\), with the same source, phase events and held sensor/clock priors, is excluded by each retained full-window recorded increment. Its necessary band is \([-B_{\rm heat},B_{\rm heat}]\), where \(B_{\rm heat}=2G_+\rho_+HX+2\delta\). Compare complete interval differences with this band; do not silently drop numerical widths.

All three segment margins must be strictly positive and all three declared horizons complete. The optional whole-window acute flag is retained without requiring it to pass or interpreting its failure as a proved sector exit. Neither work allowance nor future identity/recovery is a stopping criterion for this inference protocol.

Freeze the prospective proof, machine-readable protocol, committed runtime source, evaluator and isolated inverse worker before the first response. Retain their archive inventory and hashes together with the complete source, observations, reports and every stopping result. Preserve the first outcome, including solver unavailability, failed criteria or export failure. Any necessary correction requires a separately identified record and may not rewrite the original prediction or silently replace its first outcome. Do not rerun an earlier frozen producer or reinterpret an old sensor budget.

Success would establish finite software clock/gain resolution under the supplied normalized-sine law and held observation model. It would not select that law, establish a laboratory clock or sensor, eliminate the exact common all-row-rate/clock equivalence, or identify physical constituents. Any next research gate requires separate admission in the execution plan.

Retained first evaluation¶

The prospective statements above retain their pre-evaluation wording. The first reserved evaluation completed with certified_reserved_curvature_inference: all 109 fixed conditions passed (35 for each of the three cases, plus four paired-source conditions). The attempt record, protocol, source archive, response and manifest retain that outcome. No retry, budget change, state reset or earlier frozen producer replay was used.

All nine full-state segments completed their prescribed horizons with strict Picard inclusion. The passive half-time sample, zero-event continuation, both phase-only events and complete endpoint carry were retained. Each inverse received only its ten public primitives. The initial source, four associated readings, sensor errors and held clock passed their declared admission and information-separation checks.

The following interval endpoints are rounded outwards to nine decimal places from the stored exact rational endpoints. Cases 0 and 1 share the same original complete source and effective gain, but their separate held gains and clocks lie inside these disjoint necessary intervals.

Protocol case	Gain outer interval	Clock-rate outer interval
0	`[1.499571124, 1.500428784]`	`[0.999785284, 1.000214740]`
1	`[1.199658759, 1.200341114]`	`[1.249733572, 1.250266462]`
2	`[1.749472953, 1.750526932]`	`[0.874796884, 0.875203188]`

Each exact nominal angle, original actual mean \(B_{\rm initial}=b-5/2^{65}\), effective gain, gain and clock is covered. The actual-angle marginal also contains its retained initial-source-box interval. Widths below are conservative decimal upper bounds, rounded up to twelve decimal places; the tests against the frozen thresholds use exact rational widths.

Protocol case	Actual-angle width	Effective-gain width	Gain width	Clock-rate width
0	`0.000064767591`	`0.000213475638`	`0.000857658487`	`0.000429455253`
1	`0.000064767593`	`0.000213475638`	`0.000682354095`	`0.000532889190`
2	`0.000055473855`	`0.000211200277`	`0.001053977818`	`0.000406303072`
Frozen strict threshold	`1/1024`	`1/2048`	`1/16`	`1/80`

The paired recorded-curvature intervals are separated. Outward decimal summaries, in recorded units, are

\[ \begin{aligned} C_0&\in[1.3209747830116573,1.3209747830116574]\,10^{-19},\\ C_1&\in[1.6512184641602141,1.6512184641602142]\,10^{-19}. \end{aligned}\tag{31} \]

This is a finite contrast of the four retained readings, not access to an instantaneous derivative. Equal \(G\rho\) did not erase the observed finite-curvature distinction under the supplied law.

For every case, the false clock and false gain priors left the coarse child available as bounded_candidate, while the curvature refinement returned incompatible. The false angle prior was also excluded. Equal amplitudes returned the declared rank-based unavailable; both full-window increments excluded the specified complete phase-blind alternative. These controls retain their prospective meanings: outer compatibility is not existence, and rank abstention is not an equal-response theorem. All three optional whole-window acute flags were false. This is neither proof of a sector exit nor a maintenance certificate.

The outcome establishes finite software resolution of separate clock and sensor-gain marginals under one supplied law, preparation and held affine observation model. Receiver geometry and fine residuals remain nuisance state. No joint point identification, source formation, physical calibration, unique law, event funding or future recovery follows. In particular, the very small stipulated observation errors are a protocol premise, not a laboratory feasibility result. A free multiplier of every evolution row would still have the exact rate/clock equivalence proved earlier.

The archive records source revision 8eb47d261ce971909c343901cc5ed7a7427a8428, with no runtime overlays; its SHA-256 is c6d5992168d7e98d9a2107dab6912455033ee56e3795d16070c86b06b1643ac4. Its prospective proof has 27,353 bytes and SHA-256 3410ebfbe4e1a3e62dae6d9b722f2c2fae55cac0eec8a2ffcae470fb4a689add; that content is preserved above. Archived bytes define the receipt despite routine checkout newline conversions. Hashes associate retained records; they do not independently authenticate execution chronology or acquisition. The read-only evidence audit checks retained primitives and certificate arithmetic without rerunning the producer or inverse. The execution plan records the closed gate and the separate inactive boundary.