A freely reacting organized probe through one full relative turn¶
Two formed C9 constituents can traverse every relative registration on one freely evolving history while both internal identities remain acute. An initially supplied difference of their common forms drives the turn; neither constituent is held, and no phase reset or continuous input is used after contact. The actual mean-form record determines the relative phase through the already declared kinetic row. An integral identity then reads a current harmonic without differentiating that record. The finite-record construction below bounds the same readout from finitely many uncertain samples, under explicit timing and kinetic calibration premises. Separately admitted primitive phase and form snapshots provide a circular necessary test of that kinetic row, without reconstructing phase from it. The preparation result below distinguishes the large supplied form contrast from what any unforced common-contact schedule can generate.
This is conditional operational access under the existing supplied constitutive family. It does not select that family, establish physical measurement, or replace source, event-work and observation admission. The research program alone selects further work. The phase-faithfulness classification still distinguishes a complete operational alphabet from one current channel's invisible shifts.
1. Complete law, preparation and observation¶
| Item | Declaration |
|---|---|
| Nodes and order | Two marked C9 rings, nodes \(9c+k\), \(c=0,1\), \(k=0,\ldots,8\); ring A is \(c=0\), ring B is \(c=1\) |
| Support | Eighteen unit internal edges and nine same-slot matching edges of weight \(a=\gamma^2/1000\), added without resetting any coordinate |
| Law | The existing potential family \(0\le\lambda\le1\), with every edge weight used in both quarter-factor rows |
| Capacity and clock | Held unit capacities, \(\gamma=1/(1023\pi)\), unchanged structural clock \(\tau\); \(s=a\tau/2\) is only a declared common time coordinate |
| Initial means | Zero global form and lifted phase means, \(X_A-X_B=12/\gamma\), \(\Phi_A-\Phi_B=0\) |
| Initial fine state | Separate zero-mean residuals in each child's forms and lifted phases; the norm-derived bound \(F_0<1/40000\) below |
| Horizon | \(0\le s\le1\), equivalently \(0\le\tau\le T=2000/\gamma^2\) |
| Retained identity | Both internal winding-one C9 identities; no fixed joined acute chamber or conserved cross-contact winding is required |
| Readout | The continuing relative mean-form record; ending the integral at one reconstructed turn is an observation convention, not an event |
Write \(\alpha=2\pi/9\), with the fixed closing-edge turn on each internal cycle. The actual primitive state is represented by continuous lifts
Both forms and fine phases have 16 independent residual coordinates; the relative means contribute two more to the 34-dimensional fixed-mean full state. Keeping these residuals is essential: the reached state need not be the exact carrier \(q=z=0\).
The potential, current and uniform bounds are
Let \(L_9\) be the unit C9 Laplacian, \(L=I_2\otimes L_9+aL_{\rm match}\), and \(U_\lambda=\sum_e w_eV_\lambda(\theta_i-\theta_j)\). The full nondimensional structural law and its balance are
There is no input or later support change. Both conserved global means are retained. The smooth bounded currents and linear form terms give global continuation on phase lifts. This comparison is not a native operator word or a reciprocal-degree normalization.
Admit primitive residual norm bounds \(\|q(0)\|\le\rho_\theta\), \(\|z(0)\|\le\rho_x\), and define
The triangle inequality makes this an upper bound on the actual quadratic functional used below. Bounds are derived from the primitive preparation, not accepted as a cached source verdict.
Preparation by a symmetry, not a reset to a target¶
An isolated C9 under this law is invariant under a constant common-form translation. Its Laplacian annihilates that constant, both evolution rows are unchanged, and its internal storage and identity are unchanged. Thus an independently admitted own-law formation/recovery history can be prepared from the outset with common forms \(+6/\gamma\) and \(-6/\gamma\), while retaining every reached residual and its original error correlations. Conserved zero phase means give the stated relative lift zero without an attachment-time phase reset.
This is the same exact symmetry used in own-law outer acquisition. Its use here is conditional on that law's source admission and the inequality (4). It does not transfer an endpoint law's source certificate to another law, authenticate records, or evaluate a source producer. The large common forms are supplied preparation. Their lack of isolated Laplacian-storage cost is an identity of (3), not a claim that physical preparation is free or that a native operator creates this state.
2. Exact mean rows and a monotone full turn¶
Put
Internal currents cancel in each child mean, and the matching edges retain exactly the same relative mean form. Summing (3) therefore gives the exact full-state equations
Both constituents react. On the invariant carrier this reduces to \(\overline J=j_\lambda(\delta)\); no such substitution is needed for the following transit bound.
Write \(c_0=\gamma^2J_0\). The canonical constants give \(0<c_0<1/2\), hence also \(c_0\le1\). Variation of constants with \(v(0)=12\) gives, for \(0\le s\le1\),
Using \(e^{-1}>1/3\) and \(c_0<1/2\) in the lower bound yields \(v(s)>11/3\). Integration of the lower bound, with \(e^{-1}<2/5\) and the weaker \(c_0\le1\), gives
The elementary estimates on \(e\) follow from its positive series: \(5/2<e<3\). The final comparison follows, for example, from \(\pi<22/7\). Thus a unique \(s_*\in(0,1)\) satisfies \(\delta(s_*)=2\pi\). In fact \(s_*>\pi/6\), since \(v\le12\). No trajectory or numerical root is needed to prove this statement. The flow continues freely; \(s_*\) merely ends the readout integral.
This establishes access to every intermediate relative registration in one actual history, without holding either phase fixed or independently preparing a separate contact for every query.
3. Fine-state retention through the nonacute transit¶
Let \(P\) project onto the separate zero-mean subspaces of the two rings. It commutes with \(L\). Write \(L_f=L|_{\operatorname{ran}P}\). The fine projection of the gradient is zero on the carrier for every \(\delta\): each ring's two internal currents cancel and its cross current is constant over slots. Consequently the exact fine rows are
Here \(A\) is the integrated projected Hessian on the straight fine segment from zero to \(q\), at the current relative mean phase. On the bootstrap tube \(\|q\|<1/100\), every internal gap differs from \(\alpha\) by less than \(1/50\). In particular its magnitude is below \(18/25\). With \(C_6(u)=1-u^2/2+u^4/24-u^6/720\),
The cross-edge curvature can be negative during the turn, but is bounded below by \(-J_1\). On the two fine ring spaces the internal Laplacian has lowest eigenvalue \(2-2\cos(2\pi/9)>2/5\) and norm at most four; the matching Laplacian has norm two. Therefore the actual bounds are
uniformly in the complete relative phase circle. For the lower bound, \((7/10)(2/5)-2aJ_1>1/4\); the upper norm is bounded by \(J_1(4+2a)<5\). These are own-law bounds for every \(0\le\lambda\le1\), not a transfer from the sine tangent. For the internal spectral bound, \(\alpha>2/3\) and cosine decreases on this interval, while \(\cos(2/3)\le1-(2/3)^2/2+(2/3)^4/24=191/243<4/5\).
Define the positive functional
Substitution of both fine rows gives the exact identity
Since \(\|g\|\le5\|q\|\), Young's inequality gives \(2\gamma|z^\top g|\le\|q\|^2/8+200\gamma^2\|z\|^2\). The canonical \(\gamma<1/3000\) satisfies \(2/5-200\gamma^2>1/8\) and \(3\gamma^2/2<1\). Equations (11)--(13) consequently imply
Initially (4) and (12) place \(q\) strictly inside the tube. The same strict bound remains at any proposed first exit, so continuity closes the bootstrap through the whole observation horizon. Both internal cycles stay acute with winding one, including every allowed fine direction. No stationary profile replaces the actual state.
The full relative turn necessarily passes nonacute matching gaps and their phase cuts. This proves retention of the two constituent identities, not a fixed acute identity of the joined graph or fixed windings on every new mixed cycle. U3 at a support event is a separate admission; it is not reapplied as a filter of the unchanged global smooth current during (3).
For later observation, put \(d_k=q_{A,k}-q_{B,k}\). Its sum is zero. Taylor expansion and (2) give the quadratic error
The first-order correction cancels because the fine means are retained.
4. Supplied contact work¶
Immediately before attachment the two rings are disconnected. Their state is unchanged by adding the nine weighted edges. At the declared \(\delta(0)=0\), the exact storage jump is
The cross term linear in \(r_0\) vanishes by the separate fine zero means. Since \(V_\lambda(0)=j_\lambda(0)=0\), \(0\le V_\lambda(u)\le J_1u^2/2\). Using \(r_0=12/\gamma\) gives the source-inclusive bound
The ideal carrier's cost is exactly \(81/125\); the residual term is retained for the actual source. The earlier isolated formation/storage history is separate from this positive attachment work. Nothing in the continuous loss supplies a reservoir, an autonomous attachment time or a native event realization. The horizon \(T=2000/\gamma^2\) and weak contact are explicit parts of this operational design, not a change of the constitutive law or a laboratory time estimate.
5. A derivative-free harmonic observation¶
The mean-form record gives \(v(s)=\gamma(X_A(s)-X_B(s))\). The unchanged phase row gives exactly
This uses the declared initial relative lift and kinetic law. It does not use an inverse of the current being tested, hidden fine phases, or a held phase reference. It cannot simultaneously validate an unknown kinetic row or recover an unadmitted initial phase.
For any positive integer \(n\), define from the same continuing record, up to its one-turn readout time \(s_*\),
To verify it, multiply \(v_s+v=-\gamma^2\overline J\) by \(v\sin(n\delta)\), integrate, and use \(\delta_s=v\) to integrate the \(vv_s\) term by parts. The result is exactly
No derivative of a measured record remains. In the primary preparation \(\delta(0)=0\), \(\delta(s_*)=2\pi\), so the endpoint term in (19) vanishes. It is displayed because it must be retained for an independently admitted nonzero starting phase and a complete turn from that phase.
Strict monotonicity permits \(d\delta=v\,ds\). If \(b_n\) is the actual sine Fourier coefficient of \(j_\lambda\), orthogonality and (15) imply
Indeed the absolute integral of the fine current error over a turn is at most \(2\pi\) times its uniform upper bound. For this supplied family, \(b_1=1\), \(b_9=\lambda/2187\), \(b_{18}=-\lambda/4374\), and the other positive-index coefficients vanish. Formula (21) concerns the actual freely reacting fine state, not a nominal carrier sampled at imposed registrations.
Exact access is not finite-resolution identification¶
Let \(A_n\) denote the expression in braces in (19). On the ideal carrier its ninth-harmonic value is \(\pi\gamma^2\lambda/2187\). This raw contrast is small; the normalization in (19) does not increase measurement accuracy. For the fixed declared \(\gamma\) and normalization, if a separately admitted complete record and quadrature error bounds \(|\widehat A_n-A_n|\) by \(E_n\), its total coefficient uncertainty is at most
That error budget must include reconstruction of \(\delta\), the readout endpoint and all affected integrals. If the kinetic calibration or clock is uncertain, its uncertainty must instead be propagated through the whole observation map, including the normalizing denominator; a numerator-only error at nominal \(\gamma\) does not suffice. For example, distinguishing two deformations through \(n=9\) with a common such bound requires twice (22) to be smaller than \(|\lambda_1-\lambda_0|/2187\). The next section supplies a finite sampling construction and its conditional uncertainty budget. It does not admit an instrument or a physical observation map.
6. Finite records with explicit uncertainty¶
The mathematical question is whether finite observations can retain the information in (20), rather than whether the ideal continuous integral exists. Keep the complete law, primitive preparation, contact and horizon of section 1. In particular the actual initial means remain \(v(0)=12,\delta(0)=0\). The initial phase uncertainty introduced below belongs only to the readout; it does not enlarge the preparation class.
6.1 Subtracting the known free relaxation¶
Define a reference and two residuals from the actual mean record:
The complete mean rows, without any carrier substitution, give
The last two estimates follow by variation of constants and \(v\le12\). Multiplying the second row by \(\sin(n\delta)\) and integrating yields, at any \(b\in[0,1]\),
At \(b=s_*\) this is the same \(B_n\) as (19). The factorized expression in (23) avoids forming the difference of the two large integrals in (19). It does not remove the accuracy needed to read the small form residual or to calibrate \(\gamma\).
The finite interpolation bound must also retain the actual fine state. Writing \(d_k=q_{A,k}-q_{B,k}\), equations (9), (12) and \(s=a\tau/2\) give
Here \(\|L_f\|<5\) and \(\gamma\|z\|\le\sqrt{2F}\). Differentiate the actual average current and subtract the zero quantity \(j_\lambda'(\delta)\sum_k(d_k)_s/9\). Then
Thus the fast fine relaxation has a declared contribution to the interpolation budget; it is not replaced by the carrier derivative.
6.2 Finite sampling, reconstruction and stopping¶
Use \(N+1\) samples at nominal scaled times \(s_i=i/N\), \(i=0,\ldots,N\), with \(N\) a positive integer and \(H=1/N\). The following are independently supplied observation premises:
- A raw reading \(\widehat r_i\) differs from the actual relative form \(r(s_i^{\rm act})\) by at most \(\varepsilon_r\).
- Its exposure time satisfies \(|s_i^{\rm act}-s_i|\le\xi\), with every actual exposure in \([0,1]\). This is an error in the declared scaled clock, including any error in converting a reading's timestamp to that clock.
- A single kinetic calibration obeys \(\widehat\gamma/\gamma\in[1-\kappa,1+\kappa]\), \(0\le\kappa<1\).
- The readout origin \(\widehat\delta_0\) obeys \(|\widehat\delta_0|\le\zeta\).
These premises do not follow from the nodal identity. In particular, a sensor value has not been identified with EPI by this theorem. The true kinetic law and \(\gamma=1/(1023\pi)\) remain unchanged; \(\widehat\gamma\) is only an observation calibration. An arbitrary unknown clock scale is not admitted unless it satisfies the uniform exposure bound throughout this horizon. The error allowances are deterministic joint uniform bounds and may be correlated. Independent justification means justification apart from the evaluated response; no statistical independence or stochastic averaging is assumed.
For computable enclosures let \(g_-\le\gamma\le g_+\) be positive rational bounds and let \(\pi_-\le\pi\le\pi_+\). Use the norm-derived sufficient upper bound \(F_0\) from (4), replacing \(\gamma\) by \(g_+\) there. Define
Indeed \(|v_s|\le12+C\). Multiplication of the raw reading by \(\widehat\gamma\), followed by comparison with \(v(s_i)\), gives
The term \(12\kappa\) retains the error in the large form mean; it cannot be dropped merely because the constitutive contrast is small.
At each nominal time form \(\widehat w_i=\widehat v_i-h(s_i)\), and let \(\widetilde w\) be their piecewise linear interpolant. Define the continuous reconstruction
The elementary linear interpolation remainder bounded by \(\sup|w_{ss}|H^2/8\), together with (29), proves
The integral in (30) uses only an exponential and the integral of a linear function on each cell; it requires no derivative of any reading.
Require the strict stopping margin
Then \(\widetilde v>0\) and there is a unique \(\widehat s_*\in(0,1)\) at which \(\int_0^{\widehat s_*}\widetilde v\,ds=2\pi\). This is a full reconstructed increment, so that \(\widetilde\delta(\widehat s_*)=\widehat\delta_0+2\pi\); it does not reset the true phase or absorb the readout offset. Moreover
The second bound uses the true lower speed in (7). The readout origin error affects the harmonic factors and both endpoint terms, but not this increment stopping rule.
6.3 Calibration, quadrature and endpoint errors¶
The residual normalization in (30) uses the uncertain kinetic calibration. Its complete error, rather than a numerator-only error, is bounded by
To see this, first compare \(\widetilde v^2\) with \(v^2\), using \(v\le12\); then compare \(\gamma^2/\widehat\gamma^2\) with one in the true residual. Both errors are retained in (34).
For an explicit finite quadrature, take a positive integer \(K\). Subdivide each data cell into \(K\) equal subcells and retain their portions before \(\widehat s_*\), clipping the last one at that endpoint. Use the midpoint of each retained portion. Each has length at most \(H/K\), no portion crosses a data knot, and there are at most \(NK\) such evaluations. This splitting matters because \(\widetilde w_s\) can jump at a data knot.
The following constants bound derivatives inside each data cell:
For example \(|\widetilde w|\le W\), \(|\widetilde w_s|\le D\) and \(\widetilde w_{ss}=0\) inside a cell. Therefore
This proves the three residual bounds in (35). For \(\widetilde A=n\widetilde v\cos(n\widetilde\delta) -2\sin(n\widetilde\delta)\), the bounds on \(|\widetilde A|,|\widetilde A_s|,|\widetilde A_{ss}|\) are respectively \(A_0,A_1,A_2\), using \(\widetilde\delta_s=\widetilde v\) and \(|\widetilde v_{ss}|=h\le12\). The product rule bounds \(|(\widetilde R\widetilde A)_{ss}|\) by \(M_{f,2}\). The midpoint remainder summed over the separate subcells is at most \(M_{f,2}(H/K)^2/24\), since their total length is at most one. No differentiability across data knots is assumed.
Let \(Q_{N,K}\) be this quadrature of \(\widetilde R\widetilde A\), and retain the reconstructed endpoint correction:
Both sine factors at these endpoints are \(\sin(n\widehat\delta_0)\); they need not vanish. Define the deliberately conservative observation bound
Here is a direct comparison with the true full-turn integral. For \(A=nv\cos(n\delta)-2\sin(n\delta)\), equations (31) and (34) imply
Each endpoint difference is at most \(\varepsilon_R+R_0nE_\delta\). Finally the true current integral between \(\widehat s_*\) and \(s_*\) is bounded by \(J_0\varepsilon_v\), using \(d\delta=v\,ds\), \(|\overline J|\le J_0\) and (33). Adding these terms proves (38), including the true and reconstructed endpoints. The last term could be sharpened near a full turn but no such improvement is used here.
Allow a further certified coefficient error \(\varepsilon_{\rm num}\) in evaluating (37). Combining (21), (35) and (38) gives the finite-record theorem
The terms separately retain fine-state bias, raw readings, clock exposures, calibration, phase readout, interpolation, stopping, endpoints, quadrature and numerical evaluation. This is a sufficient theorem domain; failure of (32) or of a discrimination margin does not prove that a particular record is uninformative.
The numerical allowance can be supplied constructively. There are finitely many cell formulas and quadrature values; bounded-argument exponential and trigonometric Taylor series, with rigorous remainders, and convergent rational enclosures of \(\pi\) enclose them to arbitrary positive accuracy. The positive lower slope \(11/3-\varepsilon_v\) in (32) gives an explicit inverse bound for bisection of the reconstructed increment. For the integrand in (37), put \(M_0=\widehat R_0A_0\) and \(M_1=\widehat R_1A_0+\widehat R_0A_1\). Moving the clipping endpoint by \(e_s\) changes the midpoint sum by at most \((M_0+HM_1/(2K))e_s\): the derivative of a partial-cell term of length \(\ell\) is \(f(\mathrm{midpoint})+\ell f_s(\mathrm{midpoint})/2\). The corresponding endpoint product changes by at most \((\widehat R_1+n\widehat R_0V)e_s\). These bounds also apply when an endpoint crosses a data knot, by continuity and the bounds on each side. They permit a finite root enclosure and arithmetic precisions whose total coefficient error is any prescribed \(\varepsilon_{\rm num}>0\). This is an existence and error allocation argument, not a claim about unchecked floating point evaluation. Setting this allowance to zero describes ideal exact evaluation, not a generic machine implementation.
6.4 Frozen finite budget and an ambiguity control¶
The prospective budget frozen at revision fd5dc157 uses
| Primitive allowance or design | Frozen value |
|---|---|
| Harmonic and two hypotheses | \(n=9,\ \lambda=0\) versus \(\lambda=1\) |
| Fine form and phase norm bounds | \(\rho_x=0,\ \rho_\theta=10^{-6}\) |
| Data intervals and quadrature subdivisions | \(N=10000,\ K=100\) |
| Raw relative-form reading error | \(\varepsilon_r=10^{-11}\) |
| Exposure error in scaled time | \(\xi=10^{-18}\) |
| Relative kinetic calibration error | \(\kappa=10^{-18}\) |
| Initial phase readout error | \(\zeta=10^{-10}\) |
| Certified coefficient evaluation allowance | \(\varepsilon_{\rm num}=10^{-8}\) |
Thus \(F_0=5\,10^{-13}\). Rebuilding (28)--(40) with rational enclosures gives the following approximate values only as readable summaries of the bounds:
| Derived bound | Approximate upper bound |
|---|---|
| Scaled sample error \(E\) | \(3.13554\,10^{-15}\) |
| Interpolation error \(\varepsilon_v\) | \(4.84310\,10^{-15}\) |
| Residual error \(\varepsilon_R\) | \(6.00283\,10^{-7}\) |
| Observation contribution \(E_O\) | \(6.78369\,10^{-5}\) |
| Quadrature contribution \(E_Q\) | \(6.95902\,10^{-7}\) |
| Total coefficient bound \(\mathcal E_9\) | \(2.18247\,10^{-5}\) |
The exact rational comparison is
It proves disjoint coefficient intervals for the two hypotheses if a record and its calibration satisfy these supplied premises. The construction uses 10001 raw readings and at most one million quadrature evaluations; these are finite counts, not a benchmark or a runtime prediction. The budget is stringent, especially its kinetic and clock calibration. No instrument, actual record, observed response or laboratory clock is certified by (41). The calculation admits the frozen budget without retuning it against an evaluated response.
Finite error is also a substantive information restriction. For example, compare the two exact carrier histories under the same frozen support and means, with \(\lambda=0\) and \(1\). Their currents differ by at most \(d=1/1458\) at each phase. Variation of constants in (6), the current Lipschitz bound \(J_1\), and Gronwall's inequality on \(0\le s\le1\) give
For completeness, the first difference is bounded by \(\gamma^2d+\gamma^2J_1\int_0^s|\delta_1-\delta_0|\,du\). Since the phase difference is the integral of the form difference, on this unit horizon its double integral is at most the corresponding single integral of the running supremum. The scalar Gronwall bound gives (42). If the independently permitted raw reading radius is at least \(\gamma d e^{\gamma^2J_1}/2\), the midpoint of these two raw records is compatible with both laws. This remains true even with continuous reading, perfect clocks and perfect kinetic calibration. It is a sufficient, nonoptimal ambiguity regime for this preparation, not a sharp noise threshold or a claim that every noisier record is ambiguous. More samples alone cannot remove this admitted overlap.
6.5 A certified consumer of primitive records¶
The detached reader implements (30)--(40). Its inputs are the ordered raw readings at the canonical nominal grid \(i/N\), the scalar \(\widehat\gamma>0\), the initial lifted-phase readout \(\widehat\delta_0\), and all primitive preparation, error and integer work budgets. It rebuilds the family and fine-state bounds; a cached budget, derivative bound or earlier report is not a premise. Every raw scalar is admitted before interval arithmetic. Malformed values, a wrong sample count and unsupported theorem budgets retain their domain errors.
Some necessary consistency conditions can be checked before evaluating the observation. In addition to \(|\widehat\delta_0|\le\zeta\), they are
The first line checks the declared calibration against the unchanged model constant. The other lines follow from \(w(0)=0\), \(|w|\le C\), \(|w_s|\le2C\), and the sample error (29). A separated violation is incompatible with the joint supplied premises. If an outward enclosure overlaps a threshold without certifying its side, the reader abstains numerically. Passing these conditions is not a sufficient test that a record came from the declared dynamics, preparation or instrument.
All numerical enclosures concern the supplied represented readings themselves. Sensor and calibration uncertainty are already propagated by (28)--(40); they are not applied a second time as numerical uncertainty on every raw value. The shared exact exponential bounds and outward rational interval arithmetic enclose the reference heat values, residuals and their cumulative integrals. For example, define
For \(s=iH+u\), \(0\le u\le H\), the reconstructed increment is
This exact cell formula avoids numerical integration of the phase. Grid heat values may be formed by outward recurrence with the common factor \(e^{-H}\), retaining the accumulated enclosure width. The shared positive exponential enclosure on \([0,1]\) also encloses \(e^{-s}\) by taking its reciprocal. No new decay law is introduced.
On each cell the lower bound
gives a directly checkable positive slope \(\mu\) for the reconstructed increment. The reader encloses \(D_{\rm inc}(1)-2\pi\) and brackets the first complete turn. Bisection uses certified signs, not a floating root followed by a claimed tolerance. If the enclosure of \(D_{\rm inc}(m)-2\pi\) straddles zero, the inverse slope bound still contains the root within \(m\pm |D_{\rm inc}(m)-2\pi|/\mu\), using an upper bound on the absolute residual. A root bracket is retained through every step. The initial phase readout remains outside this increment equation.
Let the certified root interval be \(S\), let \(b\) be its rational midpoint and let \(r_S\) be its radius. Evaluate the prescribed midpoint sum clipped at \(b\), including both endpoint terms, with outward function and arithmetic enclosures. Denote this coefficient enclosure by \(I_b\). The root sensitivity derived after (40) gives
Thus \(I_{\rm obs}\) encloses the exact discrete functional (37), including its exact reconstructed turn endpoint. The numerical certificate requires the computed radius \(\operatorname{rad}(I_{\rm obs})\le\varepsilon_{\rm num}\); the requested numerical allowance alone never proves that inequality. Quadrature truncation is accounted for separately, through \(E_Q\). If certified, the final conditional coefficient enclosure is
The measured numerical width already lies in \(I_{\rm obs}\). Adding \(\varepsilon_{\rm num}\) again in (47) would count it twice. The midpoint of \(I_{\rm obs}\) also satisfies the conservative bound (40), conditional on the independent observation premises.
The implementation distinguishes five outcomes:
| Status | Meaning |
|---|---|
| Certified | The necessary record checks, reconstructed turn and requested numerical width have been enclosed; (47) remains conditional on the supplied model and observation premises |
| Incompatible record | A checked necessary condition is separated from its admitted interval; the record and joint premises cannot all hold |
| Unresolved numeric | Fixed-precision enclosures cannot settle a required comparison or fit the requested numerical allowance |
| Unresolved endpoint | A sufficient monotone reconstruction or complete-turn bracket has not been certified within the endpoint iteration and precision limits |
| Work limit | The quadrature evaluation cap is insufficient for the reconstructed endpoint |
These statuses do not replace domain rejection and do not identify a physical law. The interval owner uses a fixed dyadic precision; its bounded computation need not attain every positive tolerance allowed by the mathematical existence argument. In particular, zero numerical allowance is an exact ideal and will generally remain unresolved. Work limits are computation policies, not a failure of TNFR dynamics. Input admission precedes a work-limit return. The coefficient enclosure remains unavailable for every noncertified status, even if intermediate endpoint or observation bounds are retained.
Independent controls use manufactured analytic records. For example, an ideal constant residual \(w=c\) gives
Its corrected continuous full-increment functional is zero for every initial readout origin: the corresponding test current is constant \(-c/\widehat\gamma^2\), whose sine integral over a turn vanishes. For nonzero origin and \(c\), the endpoint term itself need not vanish. Rational approximations to these analytic samples, with independently bounded remainders, test the numerical reconstruction and endpoint correction. Such records are controls of an observation algorithm, not evaluated histories of the supplied constitutive family. No scientific producer, source replay or physical-data comparison is needed to verify this reader.
7. Independently observed phase and clock¶
The preceding reader reconstructs phase using the kinetic row it consumes.
An independent test of that row instead observes the eighteen primitive
phases as well as the eighteen signed forms, in the fixed order 9*c+j.
This section supplies a conditional structural observation protocol. The
phase readout and its error and timing bounds are additional premises;
they are not inferred from the form record. The global product channel
needs no internal phase chart; an optional finer channel admits its own collar.
Support, unit capacities, the two-C9 marking and the fixed model constant
\(\gamma=1/(1023\pi)\) remain unchanged. No observing NFR or physical
sensor is introduced, and detached reading installs no feedback or event.
7.1 Global products and marked arithmetic means¶
For arbitrary continuous primitive lifts define \(\delta=(\sum_j\theta_{A,j}-\sum_j\theta_{B,j})/9\). The directly observed product \(P_9=\prod_j e^{i\theta_{A,j}}\prod_j e^{-i\theta_{B,j}} =e^{i9\delta}\) is independent of their displayed turns. This identity needs neither a fine-state bound nor retained internal winding. The fixed support and labels are needed to compare its rate with the mean form row. The product alone retains a ninefold ambiguity in circular \(\delta\).
At a snapshot subtract the known frame \(\alpha(j-4)\) from each primitive phase, obtaining circular offsets \(d_{c,j}\). Retain the strict coherent collar used by the shared child observation. In exact notation, let \(Z_c=\sum_j e^{i(d_{c,j}-d_{c,0})}\), require \(\operatorname{Re}Z_c>0\), and define
The resultant selects a regular anchor; the reported mean is the arithmetic mean of the collar lifts, not its direction. On the marked fine chart \(d_{c,j}=\Phi_c+q_{c,j}\), with zero fine mean and compatible collar lifts, (49) gives \(m_c=e^{i\Phi_c}\) exactly. All fine deviations are retained. Adding independent full turns to displayed primitive phases preserves this circular mean; it can change the displayed anchor lift. That snapshot lift is not automatically a continuous temporal lift.
This distinction matters even on the existing organization. The ordinary first resultant is exactly zero on the winding-one carrier: \(\sum_j e^{i[\alpha(j-4)+\Phi_c]}=0\). Demodulation removes that cancellation, but its resultant direction need not equal the arithmetic mean. For \(q=(a,a,-2a,0,\ldots,0)\), \(a>0\) small, the fine mean vanishes while the demodulated resultant has imaginary part \(2\sin a-\sin2a>0\). Formula (49) removes this fine-state bias by retaining the individual phases.
Snapshot collars alone do not connect (49) to the continuous nodal means between readings: an individual node could leave and re-enter the collar after one full turn. The optional local test below therefore requires an independent no-slip certificate. The product channel avoids this issue.
7.2 Independent boxes and a circular kinetic test¶
Choose exact increasing nominal times \(s_i\) in \([0,1]\). At each snapshot all eighteen forms and phases refer to one actual exposure \(s_i+e_i^{\rm time}\in[0,1]\), where \(|e_i^{\rm time}|\le\xi\). Supply per-node form error \(\varepsilon_x\), circular angular error \(\eta\), and independent absolute-continuity bounds \(|v_s|\le L_v\) and \(|\theta_{c,j,s}|\le L_\theta\) throughout the window. These bounds are independently justified regularity assumptions; no values are taken from the kinetic row under test. Frame-registration and phase-readout errors must be included in \(\eta\), and uncertainty in conversion to the declared scaled clock must be included in \(\xi\). When \(\xi=0\), no node phase-speed bound is needed for exposure transport; take \(L_\theta\xi=0\) without supplying \(L_\theta\). It is still needed for the implemented optional local no-slip test.
A conservative \(L_v\) can follow from the unchanged form row alone. For any phase path, \(v_s+v=-\gamma^2\overline J\), \(|\overline J|\le J_0\), and \(v(0)=12\) imply \(|v|\le12+C\), \(|v_s|\le12+2C\), with \(C=\gamma^2J_0\). Thus \(L_v=12+2C\) is available when those independent form-law premises are retained. No phase kinetic identity is used in this bound.
For the recorded primitive phases \(\widehat\theta_{c,j,i}\), use local real boxes whose circular images contain the nominal phases:
These boxes directly enclose the global product angle by summation. Optionally apply (49) to their demodulated values with outward arithmetic and the shared strict collar checks. This gives \(M_{c,i}\) enclosing a representative of the nominal circular mean. An unresolved collar leaves only that local mean unavailable; it does not block the product channel. No phase is reconstructed from form or replaced by a resultant direction.
The signed sample means independently give \(\widehat v_i=\gamma(\sum_j\widehat x_{A,j,i} -\sum_j\widehat x_{B,j,i})/9\). Simultaneous exposure and the two mean errors imply \(v(s_i)\in V_i=\widehat v_i+[-e_v,e_v]\). For \(h_i=s_{i+1}-s_i\), define
This encloses \(\int_{s_i}^{s_{i+1}}v\,ds\). Indeed the trapezoid error equals \(\int_{s_i}^{s_{i+1}}(s-(s_i+s_{i+1})/2)v_s\,ds\), whose absolute value is at most \(L_vh_i^2/4\). Endpoint reading uncertainty is already in the intervals \(V_i\); it is not added twice.
Let \(S_i=\sum_jP_{A,j,i}-\sum_jP_{B,j,i}\). The tested mean kinetic row and its globally observable necessary condition are
Every independently displayed primitive turn cancels under this cosine. For the sharper local test, require endpoint collars and, for both children, an independently bounded \(L_\theta\) and the strict certificate
The maximum is bounded over the phase boxes. Node-to-first-node phase differences have speed at most \(2L_\theta\), so their left principal lifts cannot cross a cut during the interval. The endpoint arithmetic means therefore match the continuous means up to a common whole turn. This suffices for \(C_{1,i}\); no collar is assumed between snapshots. Alternatively an independently admitted whole-window marked fine chart justifies the same local identity. The implemented certificate uses (53).
Each available defect is enclosed outwards in \([0,4]\). A strictly positive lower bound refutes the joint observation and kinetic premises; zero inclusion is inconclusive. Unresolved collars or no-slip bounds do not erase a product-channel exclusion. These are necessary tests, not proofs of the kinetic law, a selected current or an authenticated sensor. They test the collective row only: even exact agreement leaves zero-mean fine phase motions unchecked. Eighteen primitive inputs do not make this compressed observation a test of all eighteen phase equations.
An exact analytic control changes only the phase row to \(\theta_s=0\), retaining the form row and the carrier initial state \(q=z=0\), \(\delta=0\), \(v=12\). The held phases give \(\overline J=0\), hence \(v(s)=12e^{-s}\). Over \([0,1/4]\) the phase increment is zero while the form integral is \(12(1-e^{-1/4})\in(21/8,85/32)\subset(0,\pi)\), using the quadratic and cubic alternating bounds for \(e^{-1/4}\). Nine times this interval lies strictly between \(6\pi\) and \(8\pi\), so the product defect is positive too. This is a complete-row control, not a numerically evaluated trajectory or a transferred formation certificate.
7.3 Aliasing and clock scope¶
Even perfect finite phase readings do not generally determine a temporal lift. On the uniform grid \(s_i=i/N\), the changes
leave every sampled primitive phasor, both fine states and the zero global phase mean unchanged. They add \(4\pi Ns\) to the relative lift. These are alternative observation-level histories, not asserted solutions of the held complete law. They show why the circular necessary test can miss added whole turns; it is not an exact temporal reconstruction.
With independently justified bounds \(\epsilon_i^\delta\) on circular registration reading errors, the familiar sufficient no-alias condition is
It permits unique principal increments after an admitted initial lift; consistent endpoint errors telescope. A known independent variation bound can supply the first term. Equation (55) may not be justified by the very phase row being tested. The present circular instrument supplies neither this temporal lift nor an independently observed turn counter.
Finally, an external clock \(u\) with \(ds/du=\chi(u)>0\) transforms both complete rows. With the unchanged weighted support and \(s=a\tau/2\), they become
An unknown constant clock scale and an unknown phase kinetic coefficient appear only through their product in the last observation. They are not separately identified by that row. A clock or another independently justified complete-law premise is needed to separate them. Declaring nominal structural timestamps is not a laboratory clock calibration. This protocol is a structural observation in the NFR ontology; native phase/U3 access is not a physical sensor. An interacting observing NFR would additionally need its own state, laws, coupling, backreaction and admission. No such object, trajectory, source replay or physical response is evaluated here.
8. Preparation reachability and its supplied resources¶
The common-form symmetry in section 1 relates different initial states; it does not generate their different means along an unforced history. This section asks whether a lower-resource initial state can acquire the probe's \(v=12\), allowing a much larger class of contact schedules than the fixed contact used in sections 1--7. It changes neither the eighteen marked nodes, the unit internal edges, the held unit capacities nor the canonical \(\gamma\). Fix any one \(\lambda\in[0,1]\).
8.1 Exact full-state contrast and invariant band¶
Let the nine matching edges share a nonnegative, measurable, locally integrable weight \(a(\tau)\). Both quarter-factor rows use this same weight. Allow a separately supplied locally integrable nodal form source \(\Gamma(\tau)\) and finitely many finite form jumps \(\Delta x_j\) at declared times \(\tau_j\), keeping primitive phases unchanged at those jumps. Between jumps the complete law is
The additive source is outside the quarter factor. It is the supplied form-rate channel described by the parameter foundation and integrator contract, not a pressure reconstructed from the response. Equation (57) declares the phase row as well; using the source channel does not assert that a native operator runtime implements this whole comparison law. No phase, capacity or internal-support events are added here. The fixed-contact clock \(s=a\tau/2\) from (5) is not used for variable or vanishing contact. Zero contact can pause accumulated exposure while the declared structural clock \(\tau\) and isolated dynamics continue.
This flow has unique continuation on every finite interval: the linear coefficient has norm bounded by \(4+2a\), while the current is globally bounded and globally Lipschitz with locally integrable weighted coefficients. The resulting integral bounds prevent finite-time escape of either form or a continuous phase lift. The argument needs no acute fine chart, carrier assumption or constituent-retention certificate.
Write \(\overline\Gamma_c=\frac19\sum_k\Gamma_{c,k}\), and put
Internal diffusion and odd-current terms cancel in each child mean. The common matching weight leaves exactly \(a(X_A-X_B)\) in its mean form sum, even for arbitrary fine forms. Thus the actual full state obeys \(v_\tau=-kv-\gamma^2k\overline J+u\), with \(|\overline J|\le J_0\) for all primitive phases and every admitted \(\lambda\). At a form jump, \(\Delta v=d_j\). State-preserving support changes have no direct contrast jump. The integrating-factor identity, with right-continuous values at the finite jumps, is
Any initial jump is included in the declared \(v(0)\). Since \(\int_0^\tau\mathcal E(\tau,\sigma)k(\sigma)\,d\sigma =1-\mathcal E(\tau,0)\), (59) gives the signed enclosure
Without sources or form jumps, \([-C,C]\) is invariant; an arbitrary initial contrast stays in the convex hull of its initial value and this band. In particular, from \(v(0)=0\), \(|v(\tau)|\le C(1-\mathcal E(\tau,0))\le C<10^{-7}\). The last strict inequality follows from \(\pi>157/50\) and \(J_0=1459/1458\). No duration, strength or timing of these common nonnegative contacts can reach \(v=12\). At \(a=0\), both child means are separately conserved in the unforced model. Sources and jumps with zero form mean separately in each child cannot evade this obstruction, however large their fine-state effects. The equal weight over all nine slots is essential: slot-dependent weights would retain fine forms in the mean boundary term and define another preparation problem. The separate reciprocal resource theorem bounds that broader class using all initial fine forms and the node count; its bound does not replace this sharper common-matching result.
8.2 Necessary source and jump variation¶
Reaching \(v(\tau)=12\) requires the signed discounted resource
Replacing \(u,d_j\) by their positive parts in \(\mathcal R\) gives a larger quantity and the same necessary lower bound. Neither this inequality nor the current bound supplies the current's realized sign or sufficient conditions for reaching the target.
A simpler envelope uses only primitive bounds on the initial contrast and total nodal input variation through the proposed horizon: \(|r(0)|\le R_0\), \(B_{\rm cont}\ge\int\sum_i|\Gamma_i|\,d\tau\), and \(B_{\rm jump}\ge\sum_j\sum_i|\Delta x_{i,j}|\). Because \(|u|\le\gamma\sum_i|\Gamma_i|/9\), with the same projection bound for each jump, and \(0<\mathcal E\le1\),
These budgets count unsigned nodal form variation, not stored energy, net form creation or work. Indeed the global form mean \(\mu=(X_A+X_B)/2\) has \(\mu_\tau=(\overline\Gamma_A+\overline\Gamma_B)/2\), and its jump is the average of the two child-mean jumps. Equal opposite inputs can increase contrast while keeping \(\mu\) fixed. A zero net input therefore does not establish a zero preparation-resource budget.
8.3 Work accounting and a supplied positive control¶
The contrast theorem allows merely locally integrable \(a\); a differentiable storage ledger requires stronger regularity. If \(a\) is absolutely continuous between its finitely many declared support jumps, define
The cross terms between the two continuous rows cancel exactly. A state-preserving support jump has \(\Delta H=(a^+-a^-)K\). At fixed support a pure form jump has \(\Delta H=(Lx^-)^\top\Delta x+\frac12\Delta x^\top L\Delta x\). For simultaneous state and support changes, use the exact difference \(H(a^+,x^+,\theta)-H(a^-,x^-,\theta)\), or explicitly order the events and evaluate their separate changes at the proper states. An arbitrary measurable contact schedule need not have finite variation or a finite work ledger; (59)--(62) do not presume one.
At any fresh state-preserving attachment from \(a=0\) to the canonical \(a_*=\gamma^2/1000\), a state with \(|v|=12\) satisfies
This follows from the mean-square inequality and \(V_\lambda\ge0\), independently of the fine state. It is a floor for this declared attachment, not for the cumulative work of every possible schedule. For the admitted fine states of section 1, the corresponding upper bound remains (17).
There is also a precise conditional positive control within the existing supplied source class. Keep \(a=0\), and let \((\widehat x,\widehat\theta)\) be any unforced own-law isolated history whose initial form means are zero. For an absolutely continuous nondecreasing supplied function \(f\), with \(f(0)=0\) and \(f(T)=6/\gamma\), set
The isolated Laplacian annihilates both added constants, so (65) is an exact solution transformation of (57). Every fine state, internal phase history and previously admitted constituent identity is unchanged. The contrast reaches \(v(T)=12\), the global form mean remains zero, and the total nodal input variation is \(18\int_0^T f'=108/\gamma\). The source's contribution \((Lx)^\top\Gamma\) to isolated storage is identically zero; subsequent attachment still pays (64). This absence of isolated Laplacian-storage work is not zero physical preparation cost. The source is externally prescribed, with no waveform selected or trajectory evaluated here. It derives neither source occurrence nor an autonomous or native preparation event. Apart from this exact symmetry control, the general reachability bounds transfer no fine-state retention certificate to histories with newly supplied sources or jumps.
8.4 Detached resource exclusion¶
The shared
free-probe implementation
adds _bound_organized_free_phase_preparation, which re-admits the three
primitive bounds \(R_0,B_{\rm cont},B_{\rm jump}\) and rebuilds (62)
with the uniform current bound and outward canonical parameter intervals.
Missing resource bounds reject admission instead of becoming zero. A strict
upper bound below 12 gives excluded_by_resource_bound; otherwise
not_excluded supplies no sufficiency, source authenticity or occurrence
verdict. The calculation
does not consume a contact trajectory or replace unsigned nodal variation
by net form or energy. Its
static controls
check the full-state mean cancellation, input/jump projection, necessary
resource bound and source/storage distinctions without a response run.
8.5 Existing mechanisms and their admission boundary¶
The operator activation audit and complete reset contract separate action, eligibility, selection and accounting. A form-changing operator may supply a jump in (59), but its name or grammar admission does not derive its source, target, sign or occurrence. Pressure-only writes also need a declared lifetime under constitutive refresh. The event executor does not silently implement (57): its continuous write contract differs from this comparison's independent phase row. Native relational execution likewise has its own unforced admission; the Gamma construction is not an invocation of that path.
The existing finite reservoir model
does retain a reacting source and its work balance, but it supplies that
source's initial state and coupling on a different 36-node host. The
formed C9 source port
changes both host rows and retains donor phase and fine correlations; it
does not realize the earlier form-only reservoir by coefficient substitution.
Neither owner establishes this probe's autonomous preparation. Their useful
mechanisms remain conditional, and moving the required initial contrast into
an additional supplied object would not derive its origin. Equations
(59)--(62) exclude the stated matching-contact class, not all other interfaces,
organized sources or future independently justified preparation laws. The
finite-network extension
accounts for nonuniform contacts and additional nodes under the same
unforced law. It retains their full initial form resource, without proving
identity retention or sufficient reachability. The target 12 remains a
sufficient choice for this probe, not a necessary resource for every
possible constitutive observation.
The small-phase comparison uses the same complete law and fine-retention estimate with zero nominal mean forms. It bounds both the endpoint information available from a supplied small phase offset and a regime of indistinguishable readings, without a full turn. Its observation, uncertainty and attachment work have their own admission.
9. Scope and shared verification¶
This result replaces an abstract collection of separately imposed registrations by one finite, freely reacting contact history. It retains its supplied large form means, positive work, both kinetic rows, complete fine-state allowance and constituent identities. The finite-record theorem additionally exposes sufficient observation and computation budgets, and a regime where reading sets can overlap. The observations remain conditional on their declared preparation and law. A free probe does not convert compatibility, faithfulness or minimal information into an intrinsic constitutive selector.
The detached free-probe calculation rebuilds the admitted constants and bounds from primitive parameters and residual allowances. Static controls check the actual support/mean rows, full fine projection, quadratic decay and work bounds, and the integral harmonic identity. These controls evolve no source or trajectory, install no event, and provide no new physical observation evidence. The separate finite-record owner provides both the prospective budget calculation and the detached primitive-record reader. The first rebuilds (28)--(40) without evaluating a record; the second encloses the reconstructed observation and measures its numerical width. Neither is a source producer, continuous solver or physical sensor. Their uncertainty guarantees retain the independent model and observation premises. The finite-record controls check the residual identity, actual fine-state cancellation, interpolation and midpoint bounds, primitive admission and the frozen rational comparison without evaluating a response. The reader controls check primitive-record admission, certified endpoints, an independent analytic midpoint calculation, the nonzero boundary correction and explicit unavailable outcomes. These manufactured records test the instrument's mathematics, not a constitutive response. The separate kinetic observation consumes primitive phase/form snapshots and independently supplied timing and variation bounds. Its controls retain the global product, conditional arithmetic collar and aliasing distinctions; they do not evaluate a source or establish a physical sensor.