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: Accuracy-controlled fixed-Casimir shooting revisits the rejected cylindrical candidates with analytic derivatives and independent endpoint checks.

Timed investigation

This calculation tests orbit existence and stability through numerical continuation and independent closure checks. The numerical search is not an interval existence proof.

Initial finding

At h=0.0128h=0.0128, the archived state has refined recurrence error 5.3145×10−65.3145\times10^{-6}. A new analytic variational corrector, using the same fixed-Casimir chart, reduced the independently integrated trial residual to 2.0765×10−102.0765\times10^{-10} in one correction of norm 3.702×10−63.702\times10^{-6}. This is evidence of a nearby accurately recurrent solution rather than an obstruction at this homotopy value. Final refinement and further continuation are recorded below as they complete.

The analytic local derivative agrees with the independent complex-step implementation to 4.55×10−134.55\times10^{-13} over random states on both smooth sides of the nuclear-profile boundary. The analytic chart derivative agrees with centered differences to 1.67×10−101.67\times10^{-10}. No claim of a two-sided Jacobian at the nonsmooth profile join is made.

Independent closure and first stability result

Explicitly splitting integration at each finite-profile boundary gives a corrected h=0.0128h=0.0128 period T=20.48965735631915T=20.48965735631915. The original independent state-only DOP853 routine returns a refined closure error 5.28×10−125.28\times10^{-12}; its last endpoint refinement is 9.01×10−119.01\times10^{-11}. A separate full-six-coordinate Radau integration closes to 1.67×10−111.67\times10^{-11}. Source Casimirs remain fixed by the chart.

The full 27-variable state (six positions, six momenta, fifteen source components) has largest Floquet multiplier modulus 1.138471302788. Refining the full monodromy changes its largest matrix entry by only 4.41×10−104.41\times10^{-10} and leaves the growth exponent 0.00632935913910.0063293591391 per model-time unit unchanged at the displayed precision. This is a resolved linear instability, not neutral-mode rounding. The nonlinear perturbation check and mode localization below confirm it.

This result restores numerical existence near the old candidate while rejecting its stability in the full fixed-centre model. It does not establish or exclude another stable branch or a moving-nucleus molecular solution.

The growing mode lies outside the four-coordinate invariant plane: its squared projection into that plane is 1.72×10−251.72\times10^{-25}. Direct nonlinear integrations, with the source perturbed by conjugation so its Casimirs remain fixed, reproduce the predicted three-period amplification 1.4755919041.475591904. Reducing perturbation amplitude from 10−510^{-5} to 10−610^{-6} reduces the scaled linearization error from 4.11×10−44.11\times10^{-4} to 4.10×10−54.10\times10^{-5}. This independently confirms the transverse instability rather than merely reading an eigenvalue from a numerical matrix.

Evidence files:

cylindrical_orbit_stability.py reproduces these checks, including the otherwise omitted spatial perturbations.

Continuation beyond the original endpoint

The corrected branch reaches h=0.100h=0.100, not the purely cylindrical endpoint h=1h=1. Ten candidates pass fresh, boundary-segmented DOP853 and Radau integrations: both closure errors below 10−810^{-8}, endpoint difference below 10−910^{-9}, initial source-Casimir error below 2×10−122\times10^{-12}, and section error below 10−810^{-8}. These tests recompute acceptance rather than trust saved solver labels. A rejected large-step attempt is retained, then corrected further rather than silently counted as successful.

hh refined state-only recurrence largest full-coordinate multiplier
0.0128 5.28×10−125.28\times10^{-12} 1.138471302788
0.016 3.65×10−113.65\times10^{-11} 2.180978231594
0.020 3.56×10−103.56\times10^{-10} not yet tested
0.0256 6.67×10−106.67\times10^{-10} 4.104109176741
0.040 1.42×10−111.42\times10^{-11} 6.371377414549
0.070 1.88×10−111.88\times10^{-11} 9.385879984862
0.100 2.48×10−112.48\times10^{-11} 9.184642958009

Earlier table rows use the original independently refined integrator; the last three use the new segmented DOP853 verification. The complete, uniform results are in cross-method-verification.json. The h=0.100h=0.100 and turning-point h=0.0096h=0.0096 solver reports retain a failed legacy refinement gate: their medium-accuracy unsegmented integrations were not sufficiently converged. The fresh segmented cross-method test supersedes that numerical rejection without weakening its thresholds.

At h=0.016h=0.016 and 0.02560.0256, three-turn nonlinear perturbations also reproduce the Floquet growth, while preserving source Casimirs. These tested points are unstable, despite their accurately closed orbits. The h=0.040h=0.040 and 0.0700.070 nonlinear probes also reproduce the predicted growth. The h=0.100h=0.100 result is a Floquet diagnostic only.

At h=0.0064h=0.0064 and 0.00960.0096, the transverse blocks have no resolved exponentially growing multiplier. Full matrices have small splittings near unit neutral multipliers, of order 10−610^{-6} to 10−510^{-5}, which are not a demonstration of physical instability or of nonlinear stability.

Orbit-size and family-selection controls

The optional affine two-row slice (--gauge) chooses a local member of an underdetermined family. Its two residual null directions have not both been proved to be pure coordinate gauge. Also, setting q0=1,q1=0q_0=1,q_1=0 does not itself set the orbit's maximum visible radius.

An alternative --physical-section imposes q˙0=0\dot q_0=0 and the second active unobserved momentum equal to zero, in addition to the original section. At h=0.0096h=0.0096 this gives period 20.55880773524165820.558807735241658, closure errors 4.73×10−134.73\times10^{-13} (segmented DOP853) and 4.71×10−124.71\times10^{-12} (segmented Radau), and sampled maximum visible radius 1.00000000000008331.0000000000000833. The radial second derivative at the section is −0.8029593-0.8029593, confirming a local maximum. Its full-coordinate Floquet calculation likewise finds no resolved transverse exponential mode; near-unit splitting changes with refinement and does not establish nonlinear stability.

The continued paths retain six radial maxima per full period, with raised radial minima: the turning-point h=0.0096h=0.0096 path has minimum visible radius 0.01290640.0129064, while the h=0.100h=0.100 path spans approximately [0.14022,1.02787][0.14022,1.02787]. They are petal-like recurrent paths, not exact visible centre crossings. The affine-sliced samples are not a fixed-maximum-radius stability scan.

Why the reduced stability test misses a mode

Let AA be the active four-coordinate space with two visible and two unobserved coordinates. Linearization about q,p∈Aq,p\in A and C∈so(A)C\in\mathfrak{so}(A) splits the full 27-variable system into blocks of sizes

27=14+6+6+1.27=14+6+6+1.

The 14-block is the original active orbit. Each remaining spatial coordinate contributes its position, momentum, and four mixed source components. The last component rotates the two inactive directions and is constant to linear order. The blocks decouple exactly because the potential depends only on the full and visible squared radii and all inactive base coordinates vanish. Random-state checks give exactly zero off-block entries and active-block agreement to 2.23×10−162.23\times10^{-16}.

At h=0.0128h=0.0128, the growing mode belongs to the third visible direction and its source components, not the third unobserved direction. Its six-block has the reciprocal real pair 1.1384713,0.878370841.1384713,0.87837084, one unit-circle pair, and two unit multipliers. The unobserved-side six-block has only unit-modulus multipliers within numerical accuracy. Omitting the third visible perturbation therefore removes a physically relevant instability from this diagnostic.

Reproduction commands

accurate_cylindrical_continuation.py reads the archived evidence, preserves it, corrects new states, and separately computes recurrence acceptance. Run from the repository root, supplying a fresh output path:

OPENBLAS_NUM_THREADS=1 python3 -B experiments/accurate_cylindrical_continuation.py --h .0128 --output experiments/results/two-path-study/reproduction-0128.json