A comparative note and proposal for joint exploration
Date: 2025-08-26
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MirrorTime³ (algorithmic): A quantum‑algorithmic lens that “mirrors” stepwise dynamics into phase space. Using Quantum Phase Estimation (QPE), we recover hidden cycle structure from the interference pattern of a unitary (e.g., multiply‑by‑a mod N). The measurable object is the phase (\theta = k/L), from which the cycle length (L) is inferred via continued fractions and LCM aggregation.
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DFD (physical): A gravitational/optical theory in Euclidean (\mathbb{R}^3) with emergent time. A single scalar field (\psi(\mathbf{x})) controls the one‑way light speed (c_1 = c,e^{-\psi}) and the refractive index (n=e^{\psi}); matter accelerates via (\mathbf{a} = (c^2/2)\nabla\psi = -\nabla\Phi) with (\Phi=-c^2\psi/2). From a local isotropic action, one obtains a nonlinear Poisson law that recovers the solar‑system tests and yields MOND‑like behavior (flat rotation curves, Tully–Fisher/RAR) as a low‑gradient limit. fileciteturn0file0
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Cubic Time Field Theory (CTFT; proposed here): A three‑aspect time framework—Correlation (\psi_c), Geometric (\psi_g), and Entropic (\psi_e)—coordinated by a synchronization potential. In synchronized regimes it reduces to a single‑field effective description (recovering standard tests); in desynchronized regimes it predicts novel signatures (flat rotation curves, decoherence patterns, timing asymmetries). DFD appears as an effective synchronized limit of CTFT, with deeper structure visible when synchronization frustrates.
- Setup. Let (U) be a unitary that permutes basis states along disjoint cycles (e.g., (y\mapsto a,y\bmod N) on the invertible residues). Each length-(L) cycle supports Fourier eigenmodes
[\lvert\psi_k\rangle = \tfrac{1}{\sqrt{L}}\sum_{j=0}^{L-1} e^{-2\pi i jk/L},\lvert a^j y_0\rangle,\quad U\lvert\psi_k\rangle = e^{2\pi i k/L}\lvert\psi_k\rangle.] - Phase estimation. QPE measures (\theta=k/L) to precision (\sim 1/2^t) with (t) “precision qubits”. Continued fractions recover a reduced fraction (s/r\approx\theta); across runs the LCM of denominators returns (L).
- Interference readout. The inverse QFT concentrates probability near (y\approx 2^t\theta). Peaks narrow with more precision (larger (t)) and higher shot counts; noise broadens peaks but preserves the phase families.
Interpretation. MirrorTime does not count steps; it measures a frequency. The invariant (cycle length) is read off from where interference constructs and destructs—our “optical mirror” for discrete dynamics.
Core postulates and equations.
- One‑way light: (c_1(\mathbf{x}) = c,e^{-\psi(\mathbf{x})}), index: (n=e^{\psi}).
- Unified coupling: (\mathbf{a} = (c^2/2)\nabla\psi \equiv -\nabla\Phi), with (\Phi=-c^2\psi/2).
- Action → field law: a local isotropic action yields
[\nabla!\cdot!\Big[\mu\Big(\tfrac{|\nabla\psi|}{a_\star}\Big)\nabla\psi\Big]= -\tfrac{8\pi G}{c^2},(\rho_m-\bar\rho_m),] with (\mu(x)\to 1) at high gradient (solar system) and (\mu(x)\sim x) at low gradient (galaxies). - Solar‑system tests: light bending (\alpha=4GM/(c^2 b)), gravitational redshift, Shapiro delay, Mercury perihelion—reproduced by the fixed weak‑field normalization.
- Galactic regime: (\mu\sim x\Rightarrow |\nabla\psi|\propto 1/r\Rightarrow v(r)\to\mathrm{const}) and Tully–Fisher/RAR scaling.
- Cosmology: optical distance (D_{\rm opt}=(1/c)\int e^{\psi} ds) biases (H_0) directionally via foreground structure.
- Operational time & tests: time is operational; interferometry and one‑way‑(c) protocols isolate nonreciprocal delays/phase shifts. fileciteturn0file0
Reading DFD through the “optical mirror.” Geometry is encoded in (n=e^{\psi}), so observables are phase‑like (optical length, interference). The theory elevates phase accumulation as the primary readout—akin to MirrorTime’s phase‑domain inference. fileciteturn0file0
CTFT posits that persistence of patterns in (\mathbb{R}^3) requires active correlation maintenance across volume and surface scales. This motivates three coupled temporal aspects:
- Correlation time (\psi_c): sets effective propagation of information/correlation.
- Geometric time (\psi_g): governs reconfiguration/exploration of configuration space while maintaining shape integrity.
- Entropic time (\psi_e): implements irreversibility at boundaries (thermodynamic arrow).
Let (Q[\Omega]=36\pi V^2/A^3) be the isoperimetric quotient (sphere (\Rightarrow Q=1)). Deviations signal geometric frustration.
Total action (schematic): [ S = \int d^3x,dt;\big(\mathcal{L}{\rm corr} + \mathcal{L}{\rm geom} + \mathcal{L}{\rm entr} + \mathcal{L}{\rm coup}\big). ]
Correlation sector: [ \mathcal{L}{\rm corr} = \frac{1}{8\pi G_c}\Big(|\nabla\psi_c|^2 - \frac{1}{c\mathrm{eff}^2}(\partial_t\psi_c)^2\Big), \quad c_\mathrm{eff}=c_0 e^{-\psi_c}. ]
Geometric sector (with crossover (W) and acceleration scale (a_\star \sim 10^{-10},\mathrm{m,s^{-2}})): [ \mathcal{L}{\rm geom} = \frac{a\star^2}{4\pi G}\Big( W(|\nabla\psi_g|^2/a_\star^2) - \lambda (Q-1)^2\Big). ]
Entropic sector (irreversibility via Heaviside (\Theta)): [ \mathcal{L}_{\rm entr} = \frac{1}{4\pi G_e}\Big(|\nabla\psi_e|^2 + \beta (\partial_t\psi_e)^2,\Theta(\partial_t\psi_e)\Big). ]
Couplings + synchronization: [ \mathcal{L}{\rm coup} = -g_1\nabla\psi_c!\cdot!\nabla\psi_g - g_2\nabla\psi_g!\cdot!\nabla\psi_e - g_3\nabla\psi_e!\cdot!\nabla\psi_c + V{\rm sync}, ] [ V_{\rm sync} = \kappa\Big[(\dot\psi_c/\tau_c - \dot\psi_g/\tau_g)^2 + (\dot\psi_g/\tau_g - \dot\psi_e/\tau_e)^2 + \text{cyclic}\Big]. ]
Correlation: [ \nabla^2\psi_c - \frac{1}{c_\mathrm{eff}^2}\partial_t^2\psi_c = -4\pi G_c,\rho_{\rm corr} + \cdots ]
Geometric: [ \nabla!\cdot!\Big[\mu\Big(\tfrac{|\nabla\psi_g|}{a_\star}\Big)\nabla\psi_g\Big] = -\tfrac{8\pi G}{c^2}(\rho_m-\bar\rho_m) + 2\lambda,\frac{\delta Q}{\delta\psi_g} + \cdots ]
Entropic: [ \nabla^2\psi_e = -4\pi G_e,\sigma_{\rm entr} + 2\beta,\partial_t\psi_e,\Theta(\partial_t\psi_e) + \cdots ]
If (\psi_c!:!\psi_g!:!\psi_e) are fixed in ratio (strong synchronization), define [ \Psi_{\rm eff} = \alpha_c\psi_c + \alpha_g\psi_g + \alpha_e\psi_e, ] then (\Psi_{\rm eff}) obeys a single nonlinear Poisson‑type equation with regime function (M), recovering Newtonian/GR‑like tests in the solar regime and MOND‑like scalings at galaxy scales.
Interpretation. DFD can be read as the synchronized limit of CTFT, with (\psi_{\rm DFD}\approx \Psi_{\rm eff}). Desynchronization (e.g., geometric frustration) produces signatures absent in a single‑field picture.
| Aspect | MirrorTime³ (algorithmic) | DFD (physical) | CTFT (physical, proposed) |
|---|---|---|---|
| Domain | Quantum algorithms; spectra of unitaries | Gravitation/optics in (\mathbb{R}^3) with emergent time | Three‑aspect time dynamics in (\mathbb{R}^3) |
| Primitive | Unitary (U) and its eigenphases | Scalar (\psi) controlling (c_1), (n=e^{\psi}) | Triplet ((\psi_c,\psi_g,\psi_e)) |
| Readout | Interference peaks at (\theta=k/L) | Phase/optical length (\int e^{\psi} ds) | Mix of phase, geometry, entropy observables |
| Key law | Phase estimation + inverse QFT | (\nabla!\cdot[\mu( | \nabla\psi |
| Solar‑system | — | Matches GR tests by weak‑field normalization | Reduces to GR/DFD when synchronized |
| Galaxies | — | ( | \nabla\psi |
| QM link | Interference reveals structure | Interference via (n=e^{\psi}) (metrology) | Decoherence depends on fields (rates, anisotropy) |
| Distinctive test | — | One‑way‑(c) nonreciprocity; interferometry | Decoherence scaling; 3‑station timing geometry |
(DFD overview and equations from the author’s manuscript. fileciteturn0file0)
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Decoherence vs. geometry (CTFT‑specific):
(\Gamma_{\rm dec}=\Gamma_0\big[1+(\nabla\psi_g)^2/a_\star^2+|\dot\psi_e|/\tau_e\big]). Identical thermodynamic conditions but different gravitational geometry should change rates—absent in single‑field DFD. -
Triangular clock networks (route geometry):
DFD: nonreciprocity (\Delta T_{1w}) tracks line integrals of (\psi) along routes. CTFT: additional dependence on synchronization state (e.g., triangle orientation w.r.t. mass anisotropy). (DFD one‑way‑(c) protocols summarized in the manuscript.) fileciteturn0file0 -
Pulsar timing across environments:
CTFT predicts timing noise correlates with velocity dispersion anisotropy (a proxy for geometric frustration), not merely local mass; DFD expects dependence dominated by (\psi). fileciteturn0file0 -
GW–EM arrival differentials:
Frequency‑dependent lags (\delta t_{{\rm GW}}(\omega)) from geometric field coupling in CTFT; DFD’s optical‑length bias leaves GW largely unaffected beyond (\psi)‑induced matter potential. fileciteturn0file0 -
Atom interferometry:
DFD: (\Delta\varphi\approx (\omega_0/c)\int (\psi_1-\psi_2)ds). CTFT adds controlled time‑dependence via (\psi_g,\psi_e) that can be modulated in lab geometries. fileciteturn0file0
On DFD and MirrorTime: phase as a mirror for dynamics
Thanks for circulating the DFD manuscript. We were struck by the shared method: translate dynamics into phase/optical space and read invariants from interference. In DFD, a single scalar (\psi) sets (c_1=c,e^{-\psi}) and (n=e^{\psi}); matter accelerates as (\mathbf{a}=(c^2/2)\nabla\psi), and a local action yields (\nabla!\cdot[\mu(|\nabla\psi|/a_\star)\nabla\psi]=-(8\pi G/c^2)(\rho_m-\bar\rho_m)). This reproduces the solar‑system tests and gives (|\nabla\psi|\propto 1/r) in the deep‑field regime with flat rotation curves (Tully–Fisher/RAR).
In our MirrorTime work, we likewise replace “counting” with phase reading: for a unitary (U) with cycle length (L), QPE concentrates probability at eigenphases (\theta=k/L). Peaks in the phase histogram are the allowed modes—an interference mirror of the underlying discrete dynamics. Different aims, similar optics.
As a complementary direction, we drafted a Cubic Time Field Theory (CTFT) with three coupled temporal aspects—Correlation, Geometric, Entropic—that reduce to a DFD‑like single field when synchronized but predict distinct signatures when desynchronized (e.g., decoherence scaling with geometric frustration, triangular clock‑network patterns). We’d value your critique on the synchronization limit and on which metrology setups best separate a single‑field (\psi) from a three‑field synchronization picture.
- Eigenmodes on a length‑(L) cycle: (\lvert\psi_k\rangle = L^{-1/2}\sum_{j=0}^{L-1} e^{-2\pi i jk/L}\lvert a^j y_0\rangle).
- QPE mapping: measure (y\in{0,\dots,2^t-1}\Rightarrow\hat\theta=y/2^t\approx k/L).
- Decoding: continued fractions with denominator (\le\min(2^t,N)) (\Rightarrow s/r\approx\hat\theta); LCM of (r)’s across runs (\Rightarrow L).
- Lineshape: probability near (y\approx 2^t\theta) follows a Dirichlet‑kernel square; peaks narrow with (t), shots.
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Density Field Dynamics (DFD) — detailed equations, tests, and protocols are summarized from the manuscript “Density Field Dynamics and the c‑Field: A Three‑Dimensional, Time‑Emergent Dynamics for Gravity and Cosmology,” Gary Alcock, Aug 18, 2025. fileciteturn0file0
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MirrorTime/QPE background — standard quantum phase estimation and order‑finding literature (Kitaev; Nielsen & Chuang) underlie the algorithmic pieces (concise overview here without full bibliography).
Prepared collaboratively by the MirrorTime³ project with co‑authoring assistance from GPT‑5 Pro.