Petal response residual

Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: The prescribed demo lift changes a lock invariant conserved by every orientation-only kernel, while finite-source inverse tests locate a dynamical mismatch despite instantaneous tensor compatibility.

Target actually substituted

Use the ideal, zero-imbalance construction in the visualization, with its existing nuclear-relative C2 tensor C2 identification. For L=1,2,3,4 set r=(0.6/0.8)^(1/L), beta=1/sqrt(L), N=2r²+beta²+1 and

$$z(\theta)=\frac{(r e^{-i\theta},r e^{i\theta}, \beta e^{i\theta},1)^T}{\sqrt N},\qquad G=\operatorname{diag}(1,-1,-1,0),\qquad\theta=\omega t.$$

Applying the existing selected degree-L map, even/odd amplitude basis, and relative-phase clock reproduces w/R=|cos(L theta)| exp(i theta). Thus the readout and its radial modulation already follow mathematically from this prescribed lift. The missing derivation is not another arbitrary radial formula: it is source dynamics that supplies this lift, or a physically justified different lift giving the same spatial orbit. The L=0 ring requires its separate clock completion and is not part of this test.

Exact obstruction for every orientation-only V(s)

The moment-map matrix J(z) has nuclear-plane block A=[a]_cross and s=-tr(A²)/2=|a|². Direct substitution gives

$$s(\theta)=\frac{(2r^2-\beta^2-1)^2+ 4r^2(\beta^2+1+2\beta\cos(3\theta))}{N^2}, \qquad \frac{ds}{dt}=-\frac{24\omega r^2\beta}{N^2}\sin(3\theta).$$

The cosine coefficient is positive for each preparation. In contrast, every orientation-only Hamiltonian V(s) generates dot J=(2V'(s)/I_e)[PJP,J], for which dot A=0 and dot s=0. Consequently no choice of V(s), sign, or strength generates this prescribed nonstationary demo lift in this fixed nuclear split. This includes the static induced-response kernel; it does not include all dynamical sources.

These are changes of the frame-lock invariant, not changes in electron depth. The result is conditional on the stated relative embedding; arbitrary reorientation of the generator relative to the nucleus has not been excluded.

Instantaneous coupling is not the obstruction

The target generator decomposes exactly as

$$G=-\tfrac14 I+\tfrac14\sigma_z\otimes I+ \tfrac14 I\otimes\sigma_z+\tfrac34\sigma_z\otimes\sigma_z.$$

The response operator hX=g sum Xij sigma_i tensor sigma_j contains only mixed generators, not the two displayed within-plane terms. This is an operator-level distinction, not proof of an instantaneous trajectory obstruction: different operators can have the same action on one ray. Indeed real least squares solves

$$(I-zz^\dagger)h_Xz=\hbar\omega(I-zz^\dagger)Gz$$

to numerical roundoff at the sampled phases of all four preparations when X is unrestricted. That calculation does not enforce a physical source's amplitude bound, evolution, or energy cost.

Test the full dynamical source, not an assigned response history

For Delta=4, g=sqrt(25/32), hbar=2 and omega=0.2 or 1, propagate the full ten-state source Hamiltonian hN(Q(theta)) along the prescribed lift over one clock turn. Its propagator W determines every possible initial source density matrix by rhoN(theta)=W rhoN(0) W†. This includes every pure source state and also allows mixtures. The contact model's product-state closure is retained; a fully entangled electron-source theory is not being tested.

Expand rhoN(0) in an orthonormal real basis of 100 Hermitian matrices. At 81 sampled phases the required projective electron equation is a real linear system Ax=b. No fitting of source gap, coupling, orientation, or clock rate is done. An unconstrained least-squares residual provides a unit test vector v. Every physical density matrix has ||x||=||rhoN||F<=1, so Cauchy-Schwarz yields the useful witness

$$\|b-Ax\|\geq v^T b-\|A^T v\|.$$

This bound does not depend on interpreting a nonpositive least-squares matrix as a physical source. It applies to all physical initial sources of this finite model, at these fixed parameters and samples.

Relative means divided by the sampled target's Euclidean norm, not a pointwise maximum or a measured spatial error. Integrations at relative tolerances 2e-10 and 2e-12 are compared; each witness must exceed 100 times the Frobenius change in the design matrix, plus 1e-7. This is numerical evidence with a refinement check, not an interval-certified theorem or an all-parameter exclusion. A zero witness would be inconclusive, not success.

What the missing term means

For a supplied response history define the horizontal mismatch

$$r_z=(I-zz^\dagger)(\hbar\omega G-h_X)z.$$

An on-path Hermitian correction of minimum Frobenius norm is delta h=|r_z><z|+|z><r_z|, with ||delta h||F²=2||r_z||². It restores the required ray velocity by construction. This is an inverse specification of the missing drive, not a derived physical interaction; its source, reciprocal backreaction, and off-path extension remain to be provided. Adding it as a prescribed drive would not complete the theory.

The concrete requirement is a response mechanism that sustains the needed tensor history and, for this lift, the three-clock-frequency lock modulation. Independent within-plane fields, additional source modes, spatial motion, or a changed physical lift are candidate extensions, not established answers. Simply retuning the static V(s) kernel cannot work for the stated lift.

The earlier two-electron result remains a valid shared-source precession solution. It does not inherit the petals from the display. Closing this single-electron lift/source test precedes claiming two interacting petal trajectories; collective occupied-frame gauge freedom must also be respected.

Reproduction and related notes

The full response extension now tests the proposed addition of six within-plane channels. With all channels sharing the original gap and coupling, the relaxed response is stationary and the physical-source inverse tests exclude the sampled prescribed lifts for one and two electrons. Thus the additional operator components alone do not complete the dynamics. The baseline precession result is unchanged.

The final normal and optimized Python runs each completed 219 evaluations across 16 conditions. The inverse test accepts the independently established one-electron rotating solution with maximum residual 1.08e-14. Analytic lock and derivative residuals were at most 4.44e-16; instantaneous tensor-fit residuals were at most 3.61e-15. Source design-matrix refinement changes ranged from 9.05e-10 to 1.13e-8. Deliberately reversing the lock derivative, changing the witness subtraction to addition, or reversing the on-path correction each caused an explicit runtime failure. A feasible positive-control system checks that the witness does not falsely exclude a known solution.

All 48 finite diagnostic scripts and source checks passed in the regression run; the final added rotating-solution positive control also passed in the separate normal and optimized reruns.

• petal_response_residual_checks.py — explicit substitution, readout, source inverse tests, and refinement witnesses.
• Geometric response operator — moment-map and response conventions.
• Gapped contact source — full source Hamiltonian being tested.
• Two-electron shared response — the established precession extension, not a petal closure.
• Research status — current proof boundaries.