Mediator organization and finite nonlinear superposition¶
This owner asks whether changing an acquired mediator's organization changes the nonlinear mixed response of the same complete three-C9 system. The one-probe class contrast and the retained class-two nonlinear result answer different questions. Neither supplies this cross-class contrast.
The calculation below cancels shared analytic contributions before enclosing errors. At the unchanged two-probe design it produces a rigorous but unresolved finite contrast. It also proves a limitation of the inherited independent-class remainder estimate. This is not a symmetry theorem, a record-overlap result or evidence that the actual contrast vanishes. No new nonlinear trajectory, formation assessment or reserved response is evaluated. The execution plan owns the remaining research obligation.
F1-F2: two acquired families and one matched observation¶
Use the full 27-node support of three C9 components, with contacts \((4,13)\) and \((13,22)\), unit capacities and actual central degrees \((3,4,3)\). The two class tuples are \((1,k,1)\), \(k=1,2\). All 54 original form and phase coordinates obey the same supplied law
The clock is \(\tau=et\), \(e=1023/1024\); phase deviations use continuous lifts around \(\Theta_{i,j}=k_i(2\pi/9)(j-4)\). Form and time retain their declared structural units.
The original source and handoff remain conditional premises, separately for both classes. They start from flat phases and class-dependent form ramps
Origins are aligned from source time. The original per-coordinate errors are at most \(10^{-10}\), with separate zero sums in each component and channel. Formation time is 100, additional unprobed dwell is \(10^{13}\), and the inherited guarded decay exponent is 512. Each component's original source storage ceiling is \(10^9\). The matched handoff theorem bounds both form and target-phase Euclidean norms by \(\epsilon=10^{-32}\). These assertions are not rebuilt from a final response or a cached verdict. The different class preparations retain their different costs; no energy-matching assumption is made.
For each class, four continuations start from the same complete actual reached state, with donor central form jumps \(a\) at zero and \(b\) at delay \(s\), present or absent according to history \(00,10,01,11\). All coordinates carry through both events. The two classes may have independent source residuals. Define
where \(R_{k,ij}\) is receiver central form at the common horizon. Keep the existing design, fixed before the analytic channel calculation:
Each of the eight final scalar readings has independent unknown error of magnitude at most \(\delta\); independence here grants separate allowed errors, not a probability law. No laboratory calibration is implied.
F3: the common heat channels cancel¶
Let \(\alpha=2\pi/9\), \(c_k=\cos(k\alpha)\), and \(d_c=c_1-c_2>0\). The normalized heat semigroup \(P(t)=\exp(-At)\) is the same for both classes. For \(t\ge s\), set \(p(t)=a[I-P(t)]e_D\), \(q(t)=b[I-P(t-s)]e_D\). The cubic map inherited from the sine law is
Outer cycle edges have weight \(c_1\), mediator cycle edges have weight \(c_k\), and the two contacts have weight one. Consequently the finite heat coefficient decomposes exactly as
The common outer and contact channels cancel before interval arithmetic. No equality of actual source residuals between classes is used here. Writing \(\Delta p=p_j-p_i\), \(\Delta q=q_j-q_i\), the remaining coefficient is
Every mediator edge and its actual joined degree remain in this expression. It is orientation independent and is an analytic coefficient, not an observed nonlinear response.
Central-port symmetry does not make (7) identically zero. For fixed amplitudes, scale \(s=\lambda s_0\), \(T=\lambda T_0\), and put \(u=T_0-s_0\). The two mediator edges incident on node 13 give
Indeed \(p_{13}(t)=-at/4+O(t^2)\), its mediator neighbors have \(p=O(t^2)\), and \(P(v)_{R,13}/d_{13}=v/12+O(v^2)\). The two edge contributions add; all other mediator edges enter later. Equation (8) concerns the heat channel, not the complete nonlinear time jet at fixed \(\gamma\). It supplies no finite lower bound at (4).
The earlier class-blind fourth-order time term and amplitude-parity restriction remain valid in their stated scope. Reflection fixing the ports can reverse winding orientation under a reflected complete preparation. It does not identify winding magnitudes one and two. Arbitrary admitted residuals need not have nominal reflection symmetry and must retain the following source allowance.
Finite channel, full-law and source errors¶
Keep heat polynomial order 32. Set
The contraction proof for the complete heat polynomial applies unchanged to the restricted mediator channel. At every node the selected mediator incidence count is at most the original degree. Its normalized cubic trilinear bound therefore has the same constant four, without an edge-count multiplier. Three phase columns and one propagated row supply the four factors in (9). Hence \(|\mathcal C_M-\mathcal C_{M,32}|\le E_M\). Multiply this allowance by an upper bound for \(\gamma^4d_c\), while retaining outward cosine and \(\gamma\) factors for the rational polynomial coefficient itself.
For \(g\ge\gamma\), set \(D_T=1-2g^2T^2>0\) and \(\ell_T=1-2gT>0\). The inherited full-law bound is
Its derivation retains phase-feedback and reflection-odd internal corrections. It is uniform for both admitted classes. Tangent linearity cancels the common initial state and the two inputs within each four-history group. The nonlinear remainder does not cancel between classes merely because the probe schedule is common. Similarly each actual-to-ideal history has error at most \(\epsilon/\ell_T\) in the original coordinates, with no division of phase error by \(\gamma\). Thus
If \([H_-,H_+]\) encloses \(\gamma^4d_c\mathcal C_M\), including (9) and transcendental arithmetic, the resulting true-response enclosure is
These are sufficient outer bounds; their Cartesian endpoints need not be jointly realizable responses.
Preparation, event work and identities remain separate¶
The common per-history envelope
bounds form; phase deviation is at most \(\epsilon+2gtQ_A(t)\). For either class, contact work is at most \(8\epsilon^2\), and the joined preprobe excess storage is at most \(22\epsilon^2\). Each donor form jump \(q\) has exact work \(q(Lx^-)_D+3q^2/2\). Consequently
The delayed-only history uses the smaller \(Q_0(s)\), and absent jumps use zero. These bounds consume the actual preevent history. With \(g=1/3000\), both work bounds are below \(2\,10^{-6}\), contact work is below \(10^{-12}\), and
The uniform acute-ball coercivity from the identity owner applies to both classes. All three identities and unforced recovery after the last event retain their conditional source premises. Each present impulse shifts the actual weighted form mean by \(3q/58\); phase mean is unchanged. Neither mean is reset. Failure to resolve (12) does not invalidate these separate identity and work conclusions.
F4: three distinct observation claims¶
The eight supplied scalar error bounds imply
A true contrast enclosure strictly beyond \(8\delta\), or below \(-8\delta\), guarantees a recorded sign. It also suffices to separate the two classes' four-record sets, since each class's mixed statistic has error at most \(4\delta\). These are sufficient tests of the stated observable, not necessary conditions for distinguishing whole records.
A different comparison takes all eight records as one observation and contrasts the nonlinear model with a declared class-blind alternative having noiseless \(D=0\). That alternative's own eight errors permit \([-8\delta,8\delta]\). Separation through this statistic therefore uses a sufficient strict true margin greater than \(16\delta\). The complete tangent law is one such declared postcontact alternative; its mixed statistic vanishes separately in each class. No exclusion of every possible alternative law follows.
Fixed-design outcome and a proved limitation of this estimate¶
One exact order-32 channel calculation at (4) gives \(\mathcal C_{M,32}\) approximately \(-1.26311402922904\,10^{-15}\). It is a rational polynomial integral, not a full nonlinear response. Including the restricted tail (9) and outward factors establishes the exact terminating-rational inequalities
The scaled tail is less than \(1.89\,10^{-40}\). The coefficient is negative in the declared class-one-minus-class-two orientation, but the sign of an analytic coefficient is not a finite full-law conclusion.
Using the shared outward upper bound for \(\gamma\) in (10), exact rational arithmetic gives
The actual interval constructed by (12), rounded outward for display, is
Its exact lower endpoint is negative and its exact upper endpoint is positive. True sign, recorded sign and both record-separation tests are therefore unresolved by this estimate. The preparation and design were not changed after the coefficient calculation. The earlier class-two forward enclosure is unchanged and is not reused as previously unseen cross-class data.
This limitation persists even if source and sensor errors were zero. The positive terms in (10), \(D_T,\ell_T<1\), and \(g>1/3300\) give, using only the history with \(A_*=1/1000\),
This exceeds the entire magnitude of (17). Thus the independent-class remainder construction necessarily crosses zero at this design even without source or sensor error. Improving only heat-polynomial order or arithmetic precision cannot close that gap. In addition, the isolated coefficient's magnitude in (17) is below \(8\delta\); discarding the full-law error would still not justify the recorded-sign test.
Equations (19)-(20) prove a limitation of these sufficient estimates. They do not prove \(D=0\), a symmetry between the classes, an actual noise-overlap witness, or absence of organization dependence. In particular, overlap of outer intervals is not a pair of realizable complete records.
Implementation boundary and the refined comparison¶
The conditional calculator uses the existing shared exact heat polynomial and fixed geometry. Its common channels cancel symbolically; only the mediator channel and its justified tail enter (12). It retains primitive source radii, event budgets and separate work/identity guards, and executes neither source acquisition nor a forward response. The API controls and independent algebra controls check those distinctions, the common-channel cancellation, the mediator tail, noise thresholds and method limitation. An unresolved result does not convert failure of a sufficient inequality into physical nonidentifiability.
The separate complete cubic-amplitude comparison resolves the full-law sign at this unchanged finite design. It retains phase feedback and the quadratic reflection-odd corrections which return to the central observation at cubic order; this heat approximation includes those terms only in (10). The refined coefficient carries both event times and retains original-coordinate actual-source errors and a rigorous higher-amplitude remainder. Nominal parity does not restrict actual residuals.
That tighter bound proves a negative true contrast and, separately, an allowed scalar noise-cancellation witness for each actual source pair. It does not imply overlap of the full eight-record sets. These are results of the linked refined comparison, not of the present weaker estimate, whose unresolved interval and error-floor proof remain valid. The ontology likewise keeps an inherited organization-dependent property distinct from a physical identification or a uniquely selected microscopic law.