Closure map · odd Sagnac sector · reduced phase bookkeeping
Recursive closure
A fully executable visual explanation of how signed rest-distance corrections converge, how direction reversal isolates the odd Sagnac sector, and where reduced Poincare–Cartan action enters the phase readout.
qₖ
Interactive laboratory
Watch one return-time correction become a closed result
This is not an animation of a photon physically flying around a ring. It is an animation of the calculation: each glowing point is one correction step, and each step adds the currently remaining light-travel time before shrinking and signing the remainder for the selected direction.
How to read the two plots: The upper trace follows one directed route,
ℓn+1 = σβℓn. Blue points are the signed remainder length ℓn still carried into the next correction; the blue line shows how that remainder contracts or alternates. The lower trace subtracts the two directions and keeps only the odd powers, β + β³ + β⁵ + …. Gold points are the accumulated directional difference, approaching the gold exact limit.β says how large the next correction is compared with the current remainder. Values closer to 1 converge more slowly and make more of the closure visible.
This sets the normalized initial remainder, not β. Every displayed remainder, accumulated time, and exact limit scales by ℓ0/L, while the convergence factor remains |β|. The upper axis adapts to keep both directions visible.
The slider changes how many terms are revealed. It does not change the infinite-limit formula.
Accumulated normalized time tN/T0—Exact directed limit t∞/T0—Accumulated normalized odd difference ΔtN/T0—Exact normalized odd limit Δt∞/T0—
Ready: r₀ is loaded; no correction has been added yet.
The chain
Four layers, kept separate
Signed rest distance
\[T_0=\frac{L}{c},\quad h=\frac{\ell_0}{L},\quad \ell_0^{(\sigma)}=h,\quad t_0^{(\sigma)}=0,\qquad \ell_{n+1}^{(\sigma)}=\sigma\beta\ell_n^{(\sigma)},\quad \frac{t_{n+1}^{(\sigma)}}{T_0}=\frac{t_n^{(\sigma)}}{T_0}+\ell_n^{(\sigma)}\]
Closure map: the normalized signed remainder ℓₙ/L (written as ℓₙ in the recurrence) starts at h=ℓ₀/L and contributes one light-travel increment at each directed step. The sign records direction; |β|<1 is the convergence condition; h only sets scale. It is a projected bookkeeping map, not a full phase-space trajectory.
Sagnac difference
\[q_0=2T_0h\beta,\quad q_{k+1}=\beta^2q_k,\qquad \Delta t_0=0,\quad \Delta t_{k+1}=\Delta t_k+q_k\]
Sagnac difference: this is the direction-odd projection: subtracting the +β and −β geometric sums cancels every even power. qk is the next odd time contribution; the accelerated recurrence is a compact representation, not a second physical route.
Poincare–Cartan bookkeeping
\[\Theta_{\mathrm{PC}}=p_i\,dq^i-H\,dt,\qquad I_{\mathrm{PC}}[\Gamma]=\oint_\Gamma\Theta_{\mathrm{PC}}\]
Poincaré–Cartan bookkeeping: this formula names the extended-phase-space one-form and its closed-contour integral. At fixed energy in a stationary optical/Hamilton–Jacobi reduction, its odd contribution is Δ𝓘odd=EΔt, up to the declared contour orientation.
Detector layer
\[\Delta t_{\rm axle}=\frac{4\mathcal A\Omega}{c^2-\Omega^2R^2},\quad \Delta\tau_{\rm det}=\frac{\Delta t_{\rm axle}}{\gamma},\quad \Delta\phi=\omega_{\rm det}\Delta\tau_{\rm det}\]
Phase readout: this formula converts the axle-frame return-time difference into detector proper time and then phase. Here 𝒜=πR² is ring area; SSZ metric A=D² is a different quantity. Calibration and uncertainty still belong to the measurement protocol.
Continuous origin and convergence
Loop integral, series and exact error
The loop expression and the discrete correction series are two representations of the same constant-speed closure. Each tail formula below names its own index, so a displayed step is not confused with an odd-sector term.
Continuous loop
\[\Delta t=\oint_\gamma\left(\frac{1}{c-v}-\frac{1}{c+v}\right)d\ell\]
For constant v and fixed ring length, this is the physical h=1 loop expression and equals the difference of the two directed geometric limits. The laboratory height control instead explores the same normalized recurrence at a chosen h=ℓ₀/L.
Directed tail
\[t_\infty^{(\sigma)}-t_N^{(\sigma)}=hT_0\frac{(\sigma\beta)^N}{1-\sigma\beta}\]
N counts directed corrections already added. The counter-rotating sign alternates, while the absolute tail shrinks geometrically for |β|<1.
Odd-sector tail
\[\Delta t_\infty-\Delta t_K^{\rm odd}=\frac{2hT_0\beta^{2K+1}}{1-\beta^2}\]
K counts the odd contributions (\beta,\beta^3,\ldots,\beta^{2K-1}) already included. This compact tail is the accelerated odd-sector series; the loop form is preferable when v or the rotational metric term varies along the path.
Open spectral bridge
From closed action cycles to zeta ordinates — a testable hypothesis
The Recursive-Zeta-Zero-Finding-Suite supplies an independently computed numerical reference sequence γₙ and prime-side diagnostics. This page does not identify those ordinates with the single β-closure or claim a Hilbert–Pólya operator; it states the sharper multi-cycle question that can actually be tested.
Scope lock: a Poincaré–Cartan form is an action one-form for many Hamilton systems. A zeta connection would require a specific self-adjoint operator, a defined domain, closed primitive cycles, a trace formula and exact spectral matching. The numerical prime-trace comparison is a reconstruction of the known explicit formula, not independent evidence for the Riemann hypothesis.
Semiclassical action question
\[\mathcal I(E_n)=2\pi\hbar\left(n+\frac{\mu}{4}\right),\qquad E_n\stackrel{?}{=}\gamma_n\]
A possible Hamiltonian model would quantize closed-orbit actions and produce eigenvalues comparable with the independently computed zeta ordinates. The comparison is a research criterion, not a result already established by this recurrence.
Repeated primitive cycles
\[A_{p,1}=\frac{\log p}{\sqrt p},\quad A_{p,r+1}=\frac{A_{p,r}}{\sqrt p},\qquad \Phi_{p,r+1}=\Phi_{p,r}+E\log p\]
For each prime p, r counts repetitions. The recursion is exactly equivalent to the corresponding truncated prime-power sum; it organizes the explicit formula but does not derive a classical orbit family.
Oscillatory density
\[d_{\rm osc}(E)=-\frac1\pi\sum_p\sum_{r\ge1}\frac{\log p}{p^{r/2}}\cos(E r\log p)\]
Prime periods log p and their repetitions r log p are the essential scales. Shuffled-period and integer-period controls should fail; a high correlation with this formula is expected because it is the same known arithmetic structure being reconstructed.
What would complete the bridge
\[\text{self-adjoint }H+\text{defined domain}+\text{closed cycles}+\text{trace formula}\Longrightarrow\{E_n\}=\{\gamma_n\}?\]
The unresolved work is constructive: specify H and its domain, prove self-adjointness, derive the primitive actions and amplitudes, and establish exact spectral completeness. Until then the result remains a disciplined hypothesis.
Reference implementation: Riemann Zeta Zero-Finding Suite ↗. Its hardened numerical path distinguishes complete mpmath Siegel evaluation from exploratory main-sum scans and labels interval diagnostics as non-rigorous unless fully validated.
Evidence boundary
What this visual establishes — and what it cannot
It establishes
Exact finite-sum identities, convergence for |β|<1, direction reversal, the odd-series limit and the algebraic axle expression within the declared kinematic setup.
It does not establish
A complete rotating SSZ spacetime, symplecticity of the projected matrix map, invariance under arbitrary ring deformation, or a measured detector phase.
Next evidence layer
Use the maintained repository’s seven focused tests, including synchronized odd-step and exact-tail checks, then connect any physical claim to an independent forward model, calibration, uncertainty budget and preregistered decision rule.
Repository disclaimer: The Poincare–Cartan theorem concerns Hamiltonian transport of a closed extended-phase-space contour. It does not make arbitrary spatial contour changes invariant. See the full repository disclaimer ↗.
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How to read Recursive Closure
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