diff --git a/CITATION.cff b/CITATION.cff index 7a91226e..52d8f161 100644 --- a/CITATION.cff +++ b/CITATION.cff @@ -1,792 +1,67 @@ -cff-version: 1.2.0 -message: "If you use Research Stack, OTOM, Rainbow Raccoon Compiler, or the associated formal/compression artifacts, please cite this repository." +cff-version: "1.2.0" +message: "If you use this software, please cite it as below." type: software -title: "Research Stack (OTOM)" -abstract: >- - Research Stack is a formal-methods and compression research repository for - integer-routed symbolic systems, Omindirection logogram atoms, the Rainbow - Raccoon Compiler admission gate, Lean-backed semantics, and associated - documentation, infrastructure, and experimental adapters. +title: "SilverSight: Deterministic Equation Search via Fisher Geometry" authors: - - family-names: "Schneider" - given-names: "Brandon" - alias: "allaunthefox" - - name: "Research Stack Contributors" -repository-code: "https://github.com/allaunthefox/Research-Stack" -url: "https://github.com/allaunthefox/Research-Stack" -date-released: 2026-05-08 -license: Apache-2.0 -keywords: - - formal-verification - - lean4 - - compression - - symbolic-systems - - omindirection - - rainbow-raccoon-compiler - - hutter-prize - - fpga - - manifolds - - reproducible-research -preferred-citation: - type: software - title: "Research Stack (OTOM)" - authors: - - family-names: "Schneider" - given-names: "Brandon" - alias: "allaunthefox" - - name: "Research Stack Contributors" - repository-code: "https://github.com/allaunthefox/Research-Stack" - url: "https://github.com/allaunthefox/Research-Stack" - date-released: 2026-05-08 - license: Apache-2.0 - + - family-names: "Allaun" + given-names: "" +repository-code: "https://github.com/allaunthefox/SilverSight" +license: MIT references: - - type: thesis - title: "Pascal's Triangle, Pascal's Pyramid, and the Trinomial Triangle" - authors: - - family-names: "Saucedo" - given-names: "Antonio Jr." - date-published: 2019-06 - institution: - name: "California State University, San Bernardino" - collection-title: "Electronic Theses, Projects, and Dissertations" - url: "https://scholarworks.lib.csusb.edu/etd/855" - notes: "Combinatorics reference for Pascal triangle extensions, Fermat-number patterns, and hockey-stick identities; useful as a domain source, not as a Lean proof authority." - - type: thesis - title: "Addressing Rural Mental Health Crises: An Alternative to Police" - authors: - - family-names: "Weatheral-block" - given-names: "Faith Ann" - date-published: 2024-05 - institution: - name: "California State University, San Bernardino" - collection-title: "Electronic Theses, Projects, and Dissertations" - url: "https://scholarworks.lib.csusb.edu/etd/1957" - notes: "Exploratory qualitative social-work project for rural crisis-response domain fixtures and route-token provenance; not evidence for quantitative safety or formal claims." - - # ── Stability-driven Recurrent Scaling and Single Precision IPMs (2026-06-19 session) ── - - type: article - title: "Stabilizing Recurrent Dynamics for Test-Time Scalable Latent Reasoning in Looped Language Models" authors: - - family-names: "Yang" - given-names: "Xiao-Wen" - - family-names: "Han" - given-names: "Ziyu" - - family-names: "Zhang" - given-names: "Xi-Hua" - - family-names: "Wei" - given-names: "Wen-Da" - - family-names: "Shao" - given-names: "Jie-Jing" - - family-names: "Guo" - given-names: "Lan-Zhe" - - family-names: "Li" - given-names: "Yu-Feng" - date-published: 2026-05 - url: "https://arxiv.org/abs/2605.26733" - notes: "Introduces STARS (STAbility-driven Recurrent Scaling) for looped models, regularizing spectral radius of the Jacobian using power iterations with JVPs." - - - type: article - title: "A Differentiable Interior-Point Method in Single Precision" - authors: - - family-names: "Arrizabalaga" - given-names: "Jon" - - family-names: "Tracy" - given-names: "Kevin" - - family-names: "Manchester" - given-names: "Zachary" - date-published: 2026-05 - url: "https://arxiv.org/abs/2605.17913" - notes: "Formulates a differentiable primal-dual interior-point method with implicit complementarity via a softplus retraction map, guaranteeing bounded KKT systems for low-precision or fixed-point arithmetic." - - # ── Hard-sphere packing and polydispersity (2026-06-19 session) ────────── - - - type: article - title: "Close packing density of polydisperse hard spheres" - authors: - - family-names: "Farr" - given-names: "R." - - family-names: "Groot" - given-names: "R. D." - date-published: 2009-12 - doi: 10.1063/1.3276799 - journal: "The Journal of Chemical Physics" - notes: "Anchor paper for polydisperse close-packing theory." - - - type: article - title: "Fractionation effects in phase equilibria of polydisperse hard-sphere colloids" - authors: - - family-names: "Fasolo" - given-names: "M." - - family-names: "Sollich" - given-names: "P." - date-published: 2004-10 - doi: 10.1103/physreve.70.041410 - journal: "Physical Review E" - notes: "Full fractionation phase equilibria for polydisperse hard spheres; terminal polydispersity ~14%." - - - type: article - title: "Brownian dynamics of polydisperse colloidal hard spheres: Equilibrium structures and random close packings" - authors: - - family-names: "Schaertl" - given-names: "W." - - family-names: "Sillescu" - given-names: "H." - date-published: 1994 - doi: 10.1007/bf02183148 - journal: "Journal of Statistical Physics" - notes: "Early polydisperse hard-sphere BD simulation." - - - type: article - title: "Random-close packing limits for monodisperse and polydisperse hard spheres" - authors: - - family-names: "Baranau" - given-names: "V." - - family-names: "Tallarek" - given-names: "U." - date-published: 2014 - doi: 10.1039/c3sm52959b - journal: "Soft Matter" - notes: "Definitive RCP limits for monodisperse (~0.64) and polydisperse spheres." - - - type: article - title: "On the jamming phase diagram for frictionless hard-sphere packings" - authors: - - family-names: "Baranau" - given-names: "V." - - family-names: "Tallarek" - given-names: "U." - date-published: 2014 - doi: 10.1039/c4sm01439a - journal: "Soft Matter" - notes: "Jamming phase diagram for frictionless spheres." - - - type: article - title: "Dense packing of binary and polydisperse hard spheres" - authors: - - family-names: "Santiso" - given-names: "E." - - family-names: "Müller" - given-names: "E. A." - date-published: 2002 - doi: 10.1080/00268970210125313 - journal: "Molecular Physics" - notes: "Packing of binary/tridisperse/distributed sphere mixtures." - - - type: article - title: "Freezing of polydisperse hard spheres" - authors: - - family-names: "Kofke" - given-names: "D." - - family-names: "Bolhuis" - given-names: "P." - date-published: 1999 - doi: 10.1103/physreve.59.618 - journal: "Physical Review E" - notes: "Fractionating phase behavior of polydisperse hard spheres." - - - type: article - title: "Freezing line of polydisperse hard spheres via direct-coexistence simulations" - authors: - - family-names: "Castagnède" + - family-names: "Giani" given-names: "A." - - family-names: "Filion" - given-names: "L." - - family-names: "Smallenburg" - given-names: "F." - date-published: 2025 - doi: 10.1063/5.0281621 - journal: "The Journal of Chemical Physics" - notes: "Recent direct-coexistence simulation of polydisperse freezing line." - - - type: article - title: "Frenkel's entropy-exchange mechanism in monodisperse, nearly hard-sphere colloids: minimal perturbations to access fluid-crystal coexistence" - authors: - - family-names: "Wang" - given-names: "J. G." - - family-names: "Dhumal" - given-names: "U." - - family-names: "Zakhari" - given-names: "M. E. A." - - family-names: "Zia" - given-names: "R." - date-published: 2025 - doi: 10.1017/jfm.2026.11287 - journal: "Journal of Fluid Mechanics" - notes: "Entropy-exchange mechanism to access monodisperse hard-sphere fluid-crystal coexistence." - - - type: article - title: "The elusive fluid-and-crystal coexistence state in simulations of monodisperse, hard-sphere colloids" - authors: - - family-names: "Wang" - given-names: "J. G." - - family-names: "Dhumal" - given-names: "U." - - family-names: "Zakhari" - given-names: "M. E. A." - - family-names: "Zia" - given-names: "R." - date-published: 2024 - doi: 10.1002/aic.70275 - journal: "AIChE Journal" - notes: "Elusive nature of unbiased monodisperse hard-sphere coexistence simulation." - - - type: article - title: "Local composition fluctuations act as precursors for crystal nucleation in polydisperse hard spheres" - authors: - - family-names: "De Jager" - given-names: "M." - - family-names: "Castagnède" - given-names: "A." - - family-names: "Smallenburg" - given-names: "F." - - family-names: "Filion" - given-names: "L." - date-published: 2025 - journal: "arXiv" - notes: "Composition fluctuation precursors in polydisperse nucleation." - - - type: article - title: "Rheology and structure of polydisperse three-dimensional packings of spheres" - authors: - - family-names: "Cantor" - given-names: "D." - - family-names: "Azéma" - given-names: "É." - - family-names: "Sornay" - given-names: "P." - - family-names: "Radjai" - given-names: "F." - date-published: 2018 - doi: 10.1103/physreve.98.052910 - journal: "Physical Review E" - notes: "Shear strength nearly unchanged despite microstructural differences in polydisperse packings." - - - type: article - title: "Impact of polydispersity and confinement on diffusion in hydrodynamically interacting colloidal suspensions" - authors: - - family-names: "Gonzalez" - given-names: "E." - - family-names: "Aponte-Rivera" + - family-names: "Win" + given-names: "S." + - family-names: "Conti" given-names: "C." - - family-names: "Zia" - given-names: "R." - date-published: 2021 - doi: 10.1017/jfm.2021.563 - journal: "Journal of Fluid Mechanics" - notes: "Polydispersity + confinement effect on colloidal diffusion." - + title: "Photon-Varied Gaussian States" + year: 2025 + journal: "arXiv:2505.XXXXX" - type: article - title: "Percus-Yevick structure factors made simple" authors: - - family-names: "Botet" - given-names: "R." - - family-names: "Kwok" + - family-names: "Chabaud" + given-names: "U." + - family-names: "Mehraban" given-names: "S." - - family-names: "Cabane" - given-names: "B." - date-published: 2020 - doi: 10.1107/s1600576720014041 - journal: "Journal of Applied Crystallography" - notes: "Simplified PY structure factors for monodisperse/polydisperse fluids." - - - type: thesis - title: "Microstructure and macroscopic properties of polydisperse systems of hard spheres" - authors: - - family-names: "Ogarko" - given-names: "V." - date-published: 2014 - doi: 10.3990/1.9789036536691 - institution: - name: "University of Twente" - notes: "Polydisperse hard-sphere EOS, moments, glassy regime." - + title: "Holomorphic representation of quantum states" + year: 2022 - type: article - title: "The plane-wall effect on monodisperse and polydisperse sphere packings" authors: - - family-names: "Zhou" - given-names: "X." - - family-names: "Huang" - given-names: "Z." - - family-names: "Li" - given-names: "S." - date-published: 2025 - doi: 10.1016/j.powtec.2025.120669 - journal: "Powder Technology" - notes: "Wall effects in monodisperse vs polydisperse packings." - - - type: article - title: "Random packing fraction of binary similar particles: Onsager's model revisited" - authors: - - family-names: "Brouwers" - given-names: "H." - date-published: 2022 - doi: 10.3367/ufne.2023.11.039606 - journal: "Uspekhi Fizicheskih Nauk" - notes: "Binary packing density theory." - - - type: article - title: "Mechanical response of particle packings at jamming onset" - authors: - - family-names: "Huang" - given-names: "Z." - - family-names: "Zhou" - given-names: "X." - - family-names: "Li" - given-names: "S." - date-published: 2025 - doi: 10.1039/d5sm00762c - journal: "Soft Matter" - notes: "Bulk modulus reduction from rattlers at jamming onset." - - - type: article - title: "Dynamical coexistence in moderately polydisperse hard-sphere glasses" - authors: - - family-names: "Campo" - given-names: "M." - - family-names: "Speck" - given-names: "T." - date-published: 2019 - doi: 10.1063/1.5134842 - journal: "The Journal of Chemical Physics" - notes: "Dynamical heterogeneity in polydisperse glasses." - - - type: article - title: "Reentrant melting in polydispersed hard spheres" - authors: - - family-names: "Bartlett" - given-names: "P." - - family-names: "Warren" - given-names: "P. B." - date-published: 1999 - doi: 10.1103/physrevlett.82.1979 - journal: "Physical Review Letters" - notes: "Predicted reentrant melting at high polydispersity; later challenged by full fractionation calculations." - - - type: article - title: "Molecular dynamics simulations of crystallization of hard spheres" - authors: - - family-names: "Volkov" - given-names: "I." - - family-names: "Cieplak" - given-names: "M." - - family-names: "Koplik" - given-names: "J." - - family-names: "Banavar" - given-names: "J." - date-published: 2002 - doi: 10.1103/physreve.66.061401 - journal: "Physical Review E" - notes: "MD comparison of monodisperse vs polydisperse crystallization rates." - - - type: article - title: "Effects of Polydispersity on Structuring and Rheology in Flowing Suspensions" - authors: - - family-names: "Rosenbaum" - given-names: "E." - - family-names: "Massoudi" - given-names: "M." - - family-names: "Dayal" - given-names: "K." - date-published: 2019 - doi: 10.1115/1.4043094 - journal: "Journal of Applied Mechanics" - notes: "Shear-induced ordering suppressed by small polydispersity." - - - type: article - title: "Sedimentation of monodisperse and bidisperse hard-sphere colloidal suspensions" - authors: - - family-names: "Al-Naafa" - given-names: "M." - - family-names: "Selim" - given-names: "M." - date-published: 1992 - doi: 10.1002/aic.690381012 - journal: "AIChE Journal" - notes: "Early monodisperse/bidisperse sedimentation theory." - - - type: article - title: "Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: results for the permeability and drag force" - authors: - - family-names: "van der Hoef" - given-names: "M. A." - - family-names: "Beetstra" - given-names: "R." - - family-names: "Kuipers" - given-names: "J. A. M." - date-published: 2005 - doi: 10.1017/s0022112004003295 - journal: "Journal of Fluid Mechanics" - notes: "Drag force changes up to 5× in bidisperse arrays." - - # ── Vortex-glass and granular superconductor literature (2026-06-19 session) ── - - - type: article - title: "Orbital glass in HTSC: a new state of condensed matter" - authors: - - family-names: "Kusmartsev" - given-names: "F." - date-published: 1992 - doi: 10.1007/bf00620505 - journal: "Journal of Superconductivity" - notes: "Orbital-glass state from frustrated Josephson loops in granular HTSC; Meissner disappearance at low fields." - - - type: article - title: "Destruction of the Meissner effect in granular high-temperature superconductors" - authors: - - family-names: "Johnston" - given-names: "K." - alias: "K." - date-published: 1992 - doi: 10.1103/physrevlett.69.2268 - journal: "Physical Review Letters" - notes: "Experimental/computational destruction of Meissner in granular HTSC." - - - type: article - title: "Thermal fluctuations, quenched disorder, phase transitions, and transport in type-II superconductors" - authors: - - family-names: "Fisher" - given-names: "D." - - family-names: "Fisher" - given-names: "M." - - family-names: "Huse" - given-names: "D." - date-published: 1991 - doi: 10.1103/physrevb.43.130 - journal: "Physical Review B" - notes: "Foundational vortex-glass theory; continuous transitions from scaling." - - - type: article - title: "Phase transitions in a disordered granular superconductor near percolation" - authors: - - family-names: "J." - given-names: "J." - - family-names: "L." - given-names: "L." - date-published: 1986 - doi: 10.1103/physrevb.34.4815 - journal: "Physical Review B" - notes: "Granular Josephson-network glass phases near percolation threshold." - - - type: article - title: "Superconducting transition in disordered granular superconductors in magnetic fields" - authors: - - family-names: "Ikeda" - given-names: "R." - date-published: 2005 - doi: 10.1103/physrevb.74.054510 - journal: "Physical Review B" - notes: "Field-driven glass transitions in granular superconductors." - - - type: article - title: "Vortex-glass superconductivity: A possible new phase in bulk high-Tc oxides" - authors: - - family-names: "Fisher" - given-names: "M. P. A." - date-published: 1989 - doi: 10.1103/physrevlett.62.1415 - journal: "Physical Review Letters" - notes: "Original vortex-glass superconductivity proposal (zero linear resistivity)." - - - type: article - title: "Paramagnetic Meissner effect and related dynamical phenomena" - authors: - - family-names: "Li" - given-names: "M." - date-published: 2003 - doi: 10.1016/s0370-1573(02)00635-x - journal: "Physics Reports" - notes: "Review of paramagnetic Meissner effect in granular Bi-2212." - - - type: article - title: "Vortex Glass—Vortex Liquid Transition in BaFe2(As1-xPx)2 and CaKFe4As4 Superconductors from Multi-Harmonic AC Magnetic Susceptibility Studies" - authors: - - family-names: "Ivan" - given-names: "I." - - family-names: "Ionescu" - given-names: "A." - - family-names: "Crisan" - given-names: "D." - - family-names: "Crisan" - given-names: "A." - date-published: 2023 - doi: 10.3390/ijms24097896 - journal: "International Journal of Molecular Sciences" - notes: "Multi-harmonic AC susceptibility identification of VG-VL transition." - - - type: article - title: "Evidence of the Vortex-Glass Transition in Homogeneously Disordered Thick Films of a-MoxSi1-x" - authors: - - family-names: "Okuma" - given-names: "S." - - family-names: "Arai" - given-names: "M." - date-published: 2000 - doi: 10.1143/jpsj.69.2747 - journal: "Journal of the Physical Society of Japan" - notes: "Continuous VG transition from scaling in homogeneous disordered films." - - - type: article - title: "Universal scaling behaviour near vortex-solid/glass to vortex-fluid transition in type-II superconductors in two and three dimensions" - authors: - - family-names: "Kundu" - given-names: "H. K." - - family-names: "Jesudasan" - given-names: "J." - - family-names: "Raychaudhuri" - given-names: "P." - - family-names: "Mukerjee" - given-names: "S." - - family-names: "Bid" - given-names: "A." - date-published: 2019 - doi: 10.1209/0295-5075/128/27001 - journal: "Europhysics Letters" - notes: "Universal scaling of VG-VL transitions in 2D and 3D." - - - type: article - title: "Peak effect, vortex-lattice melting line, and order-disorder transition in conventional and high-Tc superconductors" - authors: - - family-names: "Mikitik" - given-names: "G." - - family-names: "Brandt" - given-names: "E." - date-published: 2001 - doi: 10.1103/physrevb.64.184514 - journal: "Physical Review B" - notes: "Order-disorder transitions and characteristic fields in vortex matter." - - - type: article - title: "Fragile-to-strong glass transition in two-dimensional vortex liquids" - authors: - - family-names: "Maccari" - given-names: "I." - - family-names: "Benfatto" - given-names: "L." - - family-names: "Castellani" + - family-names: "Pizzimenti" given-names: "C." - - family-names: "Lorenzana" - given-names: "J." - - family-names: "De Michele" - given-names: "C." - date-published: 2024 - doi: 10.1103/physrevresearch.7.013160 - journal: "Physical Review Research" - notes: "Fragile-to-strong glass transition in 2D vortex liquids." - + - family-names: "et al." + title: "Wigner negativity of superpositions" + year: 2024 - type: article - title: "Unveiling of Bragg glass to vortex glass transition by an ac driving force in a single crystal of Yb3Rh4Sn13" authors: - - family-names: "Kumar" - given-names: "S." - - family-names: "Singh" - given-names: "R." - - family-names: "Thamizhavel" - given-names: "A." - - family-names: "Tomy" - given-names: "C." - - family-names: "Grover" - given-names: "A." - date-published: 2015 - doi: 10.1088/0953-2048/28/8/085013 - journal: "Superconductor Science and Technology" - notes: "Material-specific H* ~4 kOe for BG-VG transition in Yb3Rh4Sn13." - - - type: article - title: "Paramagnetic Meissner effect in YBa2Cu3O7/La0.7Ca0.3MnO3 superlattices" - authors: - - family-names: "Torre" - given-names: "M. A. L. L." - - family-names: "Peña" - given-names: "V." - - family-names: "Sefrioui" - given-names: "Z." - date-published: 2006 - doi: 10.1103/physrevb.73.052503 - journal: "Physical Review B" - notes: "PME in YBCO/LCMO superlattices with granular manganite layers." - - - type: article - title: "Observation of predicted superconductivity in Gd1.4Ce0.6Sr2Cu2TiOx with x ≈ 10" - authors: - - family-names: "Blackstead" - given-names: "H. A." - - family-names: "Dow" - given-names: "J." - - family-names: "Goldschmidt" - given-names: "D." - - family-names: "Pulling" - given-names: "D. B." - date-published: 1998 - doi: 10.1016/s0375-9601(98)00348-x - journal: "Physics Letters A" - notes: "Granular superconductivity with mesoscopic Meissner and vortex dissipation." - - - type: article - title: "Observation of the Granular Josephson Mechanism and the Vortex-Glass Transition in the Polycrystalline GdBa2Cu3O7-δ Superconductor" - authors: - - family-names: "Vargas-Pineda" - given-names: "E. M." - - family-names: "Rivera-Contreras" - given-names: "L. 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A." - - family-names: "Maldonado" - given-names: "R. C." - - family-names: "Bolecek" - given-names: "N. R. C." - date-published: 2019 - doi: 10.1038/s42005-019-0243-4 - journal: "Communications Physics" - notes: "First-order BG-VG transition in Bi-2212 from surface and volume probes." - - - type: article - title: "Critical currents at the Bragg glass to vortex glass transition" - authors: - - family-names: "Hernández" - given-names: "A. D." - - family-names: "Domínguez" - given-names: "D." - date-published: 2003 - doi: 10.1103/physrevlett.92.117002 - journal: "Physical Review Letters" - notes: "Simulated first-order BG-VG transition with critical current signature." - - - type: article - title: "Study of vortex glass-liquid transition in superconducting Fe(Te, Se) thin films on LaAlO3 substrates" - authors: - - family-names: "Kumar" - given-names: "R." - - family-names: "Mitra" - given-names: "A." - - family-names: "Varma" - given-names: "G. 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A." - - family-names: "Mironov" + - family-names: "TROSE9025" + title: "From Spherical Gradients to Dirac Kets: The Hidden Linear Algebra of the Laplacian" + year: 2026 + url: "https://www.reddit.com/r/LinearAlgebra/comments/1ucglpd/" + notes: "Reddit post connecting spherical Laplacian to angular momentum operators and Dirac kets. Key insight for SilverSight: Fisher simplex maps to S^7 under sqrt(p) transform, giving spherical harmonics Y_l^m as the natural spectral basis for program state embedding." + - type: book + authors: + - family-names: "Amari" given-names: "S." - - family-names: "Buzdin" - given-names: "A." - date-published: 2019 - doi: 10.1103/physrevlett.122.117002 - journal: "Physical Review Letters" - notes: "First-order transitions in ferromagnetic superconductor domain phases." + title: "Information Geometry and Its Applications" + year: 2016 + publisher: "Springer" + - type: book + authors: + - family-names: "Vilenkin" + given-names: "N.J." + - family-names: "Klimyk" + given-names: "A.U." + title: "Representation of Lie Groups and Special Functions" + year: 1991 + publisher: "Kluwer" + notes: "Volume 3: Classical and Quantum Groups and Special Functions. Spherical harmonics on S^n." diff --git a/docs/S7_SPECTRAL_BASIS.md b/docs/S7_SPECTRAL_BASIS.md new file mode 100644 index 00000000..9230dee5 --- /dev/null +++ b/docs/S7_SPECTRAL_BASIS.md @@ -0,0 +1,162 @@ +# S⁷ Spectral Basis — Fisher Sphere → Spherical Harmonics + +## Discovery Source + +Reddit post: "From Spherical Gradients to Dirac Kets: The Hidden Linear +Algebra of the Laplacian" — r/LinearAlgebra, user TROSE9025. +URL: https://www.reddit.com/r/LinearAlgebra/comments/1ucglpd/ + +The post shows how the spherical Laplacian decomposes into angular momentum +operators, giving the discrete eigenbasis |l,m⟩. This is the coordinate +system for SilverSight's state space. + +## The Map: Δ₇ → S⁷ + +The Fisher-Rao metric on the 7-simplex maps EXACTLY to the round metric on +the 7-sphere via the square-root transform: + +``` + Δ₇ (probability simplex) --√p--> S⁷ (unit sphere in ℝ⁸) + p_i ≥ 0, Σp_i = 1 x_i = √p_i, Σx_i² = 1 + + g^Fisher_ij = δ_ij/p_i + 1/p_8 → g^round_μν = δ_μν +``` + +This is not an analogy. The Fisher metric IS the round metric in √p +coordinates. Chentsov's theorem (proven in ChentsovFinite.lean) guarantees +this metric is unique — there is no other choice. + +## The Laplacian on S⁷ + +The Laplace-Beltrami operator on S⁷ has eigenfunctions Y_l^m (spherical +harmonics) with eigenvalues l(l+6) for l = 0, 1, 2, ... + +| Mode | Eigenvalue | Physical Meaning | SilverSight Interpretation | +|------|-----------|------------------|---------------------------| +| l=0 | 0 | Constant (average) | Background state, no structure | +| l=1 | 7 | Dipole | Φ vs Σ imbalance, stack bias | +| l=2 | 16 | Quadrupole | Basin structure, FAMM curvature | +| l=3 | 27 | Octupole | Fine structure, scar details | +| l≥4 | l(l+6) | Higher multipoles | Memory patterns, complexity | + +## Program State as Spectral Decomposition + +Instead of raw 25-dim coordinates, expand in the |l,m⟩ basis: + +``` +|ProgramState⟩ = Σ_{l=0}^∞ Σ_m c_{l,m} |l,m⟩ + +c_{l,m} = ⟨l,m|ProgramState⟩ = ∫_{S⁷} Y_l^m*(x) · state(x) dΩ +``` + +For the default quine.py state (stack=['Φ', 'Σ'], 2 FAMM cells, 1 scar): + +``` +|stack⟩ = (1/√2)(|Φ⟩ + |Σ⟩) + = (1/√2)|l=0,m=0⟩ + (1/√2)|l=1,m=Φ-Σ⟩ + 0|l≥2⟩ + +Spectrum: c_00 = 1/√2 ≈ 0.707 (average) + c_1,Φ-Σ = 1/√2 ≈ 0.707 (dipole) + all other c_{l,m} = 0 + +Laplacian expectation: ⟨L²⟩ = Σ l(l+6)|c_{l,m}|² = 7 × 0.5 = 3.5 +``` + +## Why This Basis Is Natural + +| Property | Raw Coords (25-dim) | Spectral (|l,m⟩) | +|----------|-------------------|------------------| +| Orthonormal | No | Yes (Y_l^m are orthonormal on S⁷) | +| Physical meaning | None | l = curvature scale | +| Geodesic path | Curved in Δ₇ | Rotation in |l,m⟩ space | +| State comparison | Euclidean distance | Spectral overlap | +| Scars | Point masses | High-l coefficients | +| FAMM frustration | Metric distortion | Laplacian eigenvalue shift | + +## FAMM Frustration = Laplacian Eigenvalue Shift + +When FAMM cells compete for delay lines, the metric stretches. This is +a **conformal deformation** of S⁷: + +``` +g' = e^{2σ(x)} · g_round + +where σ(x) = Σ_{scars} pressure_k · G(x, x_k) + +G(x, x_k) = Green's function on S⁷ (log-distance kernel) +``` + +The Laplacian eigenvalues shift: + +``` +λ'_l = λ_l + ⟨Y_l|σ|Y_l⟩ + O(σ²) +``` + +High-l modes shift MORE (they probe finer structure). So the scar field +creates a **spectral fingerprint**: the eigenvalue shifts encode the +manifold's wound pattern. + +## Receipt Format (Spectral) + +```json +{ + "receiptID": "sha256(...)", + "spectralDecomposition": { + "basis": "spherical_harmonics_S7", + "manifold": "Fisher_sphere_sqrt_p", + "lmax": 3, + "coefficients": { + "c_00": 0.707, + "c_1_mPhiSigma": 0.707, + "c_2m": [0.0, 0.0, ...], + "c_3m": [0.0, 0.0, ...] + }, + "laplacianSpectrum": { + "eigenvalues": [0, 7, 16, 27], + "expectation": 3.5, + "frustrationShift": 0.0 + }, + "scarImprint": { + "count": 1, + "totalPressure": 0.1, + "spectralFingerprint": "low-l dominant" + } + } +} +``` + +## Connection to Prior Work + +| This Result | Prior SilverSight Work | +|------------|----------------------| +| S⁷ from Δ₇ via √p | ChentsovFinite.lean (metric uniqueness) | +| Spherical harmonics Y_l^m | UniversalMathEncoding.lean (8-state basis) | +| Laplacian eigenvalues | FAMM_BAKER_ANALOGUE.md (curvature bounds) | +| Spectral decomposition | STATE_SPACE_EMBEDDING.md (25-dim coords) | +| Frustration = eigenvalue shift | FAMM.lean (delay-line competition) | +| Scar spectral fingerprint | COEVOLUTION_MODEL.md (scar accumulation) | + +## References + +1. Chentsov, N.N. (1972). Statistical Decision Rules and Optimal Inference. + (Proven in ChentsovFinite.lean for n=8.) + +2. TROSE9025 (2026). "From Spherical Gradients to Dirac Kets: The Hidden + Linear Algebra of the Laplacian." r/LinearAlgebra, Reddit. + https://www.reddit.com/r/LinearAlgebra/comments/1ucglpd/ + +3. Amari, S. (2016). Information Geometry and Its Applications. + (Fisher-Rao metric on probability simplex.) + +4. Vilenkin, N.J. & Klimyk, A.U. (1991). Representation of Lie Groups + and Special Functions. (Spherical harmonics on Sⁿ.) + +## One-Line Summary + +The Fisher information simplex IS the 7-sphere in √p coordinates. The +spherical Laplacian gives the natural |l,m⟩ spectral basis. Program +states are superpositions of spherical harmonics. FAMM frustration is a +conformal deformation that shifts the Laplacian spectrum. Scars leave a +spectral fingerprint. + +This is the coordinate system you were looking for.