The Principle of Generative Order: A Variational Unification of Sampling Theory, Holography, and Dream Construction
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2026
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| _version_ | 1866902030978646016 |
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| author | jiazheng liu |
| author_facet | jiazheng liu |
| contents | <p>We demonstrate that three independent lines of inquiry—Whittaker's cardinal function (1915), the holographic principle ('t Hooft/Susskind, 1994), and entanglement renormalization (Swingle, 2012)—are manifestations of a single variational principle: the minimization of a previously unrecognized entropy functional, which we term boundary entropy \mathcal{S}_{\mathrm{bdy}} .</p> <p> </p> <p>For any analytic function f and sampling resolution w > 0 , the boundary entropy is defined as</p> <p> </p> <p>\mathcal{S}_{\mathrm{bdy}}[f;w] = \limsup_{\lambda \to \infty}\frac{1}{\lambda}\ln |\hat{f} (\lambda)| + \ln w,</p> <p> </p> <p>where \hat{f} is the Fourier transform. We prove:</p> <p> </p> <p>1. Uniqueness of the minimizer: Among all functions cotabular with a given discrete sample \{f(a + rw)\}_{r\in \mathbb{Z}} , the cardinal function C_{w}(x) is the unique global minimizer of \mathcal{S}_{\mathrm{bdy}} , attaining \mathcal{S}_{\mathrm{bdy}}[C_{w};w] = -\infty . This is Whittaker's construction restated as an entropy principle.</p> <p>2. Holographic realization: On a black hole horizon with Planck-scale pixels (w = l_{\mathrm{p}}) , the requirement that each pixel carries one bit of information imposes a finite entropy density. The physical state of the horizon minimizes boundary entropy under the constraint of finite energy, yielding \mathcal{S}_{\mathrm{bdy}} = \ln 2 per pixel. This reproduces the Bekenstein-Hawking entropy as</p> <p> </p> <p>S_{\mathrm{BH}} = \frac{A}{4l_{\mathrm{p}}^{2}} k_{B} = \frac{A}{l_{\mathrm{p}}^{2}} k_{B}\cdot \frac{S_{\mathrm{bdy}}}{\ln 2}.</p> <p> </p> <p>1. Emergent geometry: The tensor network generated by entanglement renormalization (Swingle) is shown to be a discretization of the unique interpolation kernel that minimizes \mathcal{S}_{\mathrm{bdy}} . The radial coordinate of anti-de Sitter space is identified with the logarithm of the sampling scale w . This provides a derivation, not an analogy, of AdS from entanglement.</p> <p>2. Cognitive implementation: Recent experiments on dream reactivation (Morris et al., 2025) demonstrate that the human brain, given a sparse auditory cue (\sim 10 bits), reconstructs a full immersive experience (\sim 10^{6} bits). We prove that any such reconstruction must, under physiological bandwidth constraints, approximate the cardinal function of the cue. The ratio of transmitted to generated information tends to zero as cue complexity increases—a signature of generative order.</p> <p>3. Universal bound: For any finite sample \{y_{r}\}_{r = 0}^{N - 1} with Shannon entropy H , the boundary entropy of its cardinal interpolant satisfies</p> <p> </p> <p>-S_{\mathrm{bdy}}\geq H - \log_{2}N + O(1).</p> <p> </p> <p>Equality holds iff the sample is uniform. This establishes a direct inequality between information-theoretic uncertainty and frequency-domain parsimony.</p> <p> </p> <p>We conclude that boundary entropy minimization is a universal principle governing the transition from discrete data to continuous representation. This principle operates identically in mathematics (Whittaker), quantum gravity (holography), quantum information (AdS/CFT), and neuroscience (dreaming). We propose the term Generative Order for this phenomenon.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18615302 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Principle of Generative Order: A Variational Unification of Sampling Theory, Holography, and Dream Construction jiazheng liu Generative Order, Boundary Entropy, Cardinal Function, Variational Principle, Sampling Theory, Holographic Principle, Entanglement Renormalization, MERA Tensor Network, AdS/CFT Duality, Bekenstein-Hawking Entropy, Dream Reconstruction, Shannon Entropy, Bandlimited Interpolation, Paley–Wiener Theorem, Information Density, Planck Scale, Cotabularity Class, Fourier Transform, Analytic Function, Renormalization Group (RG) <p>We demonstrate that three independent lines of inquiry—Whittaker's cardinal function (1915), the holographic principle ('t Hooft/Susskind, 1994), and entanglement renormalization (Swingle, 2012)—are manifestations of a single variational principle: the minimization of a previously unrecognized entropy functional, which we term boundary entropy \mathcal{S}_{\mathrm{bdy}} .</p> <p> </p> <p>For any analytic function f and sampling resolution w > 0 , the boundary entropy is defined as</p> <p> </p> <p>\mathcal{S}_{\mathrm{bdy}}[f;w] = \limsup_{\lambda \to \infty}\frac{1}{\lambda}\ln |\hat{f} (\lambda)| + \ln w,</p> <p> </p> <p>where \hat{f} is the Fourier transform. We prove:</p> <p> </p> <p>1. Uniqueness of the minimizer: Among all functions cotabular with a given discrete sample \{f(a + rw)\}_{r\in \mathbb{Z}} , the cardinal function C_{w}(x) is the unique global minimizer of \mathcal{S}_{\mathrm{bdy}} , attaining \mathcal{S}_{\mathrm{bdy}}[C_{w};w] = -\infty . This is Whittaker's construction restated as an entropy principle.</p> <p>2. Holographic realization: On a black hole horizon with Planck-scale pixels (w = l_{\mathrm{p}}) , the requirement that each pixel carries one bit of information imposes a finite entropy density. The physical state of the horizon minimizes boundary entropy under the constraint of finite energy, yielding \mathcal{S}_{\mathrm{bdy}} = \ln 2 per pixel. This reproduces the Bekenstein-Hawking entropy as</p> <p> </p> <p>S_{\mathrm{BH}} = \frac{A}{4l_{\mathrm{p}}^{2}} k_{B} = \frac{A}{l_{\mathrm{p}}^{2}} k_{B}\cdot \frac{S_{\mathrm{bdy}}}{\ln 2}.</p> <p> </p> <p>1. Emergent geometry: The tensor network generated by entanglement renormalization (Swingle) is shown to be a discretization of the unique interpolation kernel that minimizes \mathcal{S}_{\mathrm{bdy}} . The radial coordinate of anti-de Sitter space is identified with the logarithm of the sampling scale w . This provides a derivation, not an analogy, of AdS from entanglement.</p> <p>2. Cognitive implementation: Recent experiments on dream reactivation (Morris et al., 2025) demonstrate that the human brain, given a sparse auditory cue (\sim 10 bits), reconstructs a full immersive experience (\sim 10^{6} bits). We prove that any such reconstruction must, under physiological bandwidth constraints, approximate the cardinal function of the cue. The ratio of transmitted to generated information tends to zero as cue complexity increases—a signature of generative order.</p> <p>3. Universal bound: For any finite sample \{y_{r}\}_{r = 0}^{N - 1} with Shannon entropy H , the boundary entropy of its cardinal interpolant satisfies</p> <p> </p> <p>-S_{\mathrm{bdy}}\geq H - \log_{2}N + O(1).</p> <p> </p> <p>Equality holds iff the sample is uniform. This establishes a direct inequality between information-theoretic uncertainty and frequency-domain parsimony.</p> <p> </p> <p>We conclude that boundary entropy minimization is a universal principle governing the transition from discrete data to continuous representation. This principle operates identically in mathematics (Whittaker), quantum gravity (holography), quantum information (AdS/CFT), and neuroscience (dreaming). We propose the term Generative Order for this phenomenon.</p> |
| title | The Principle of Generative Order: A Variational Unification of Sampling Theory, Holography, and Dream Construction |
| topic | Generative Order, Boundary Entropy, Cardinal Function, Variational Principle, Sampling Theory, Holographic Principle, Entanglement Renormalization, MERA Tensor Network, AdS/CFT Duality, Bekenstein-Hawking Entropy, Dream Reconstruction, Shannon Entropy, Bandlimited Interpolation, Paley–Wiener Theorem, Information Density, Planck Scale, Cotabularity Class, Fourier Transform, Analytic Function, Renormalization Group (RG) |
| url | https://doi.org/10.5281/zenodo.18615302 |