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\citation{Sorkin1994,Dowker2020}
|
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\bibstyle{unsrt}
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\bibdata{references}
|
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|
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|
||||
\bibcite{Bombelli1987}{1}
|
||||
\bibcite{Sorkin2003}{2}
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||||
\bibcite{Surya2019}{3}
|
||||
\bibcite{Benincasa2010}{4}
|
||||
\bibcite{Kleitman1975}{5}
|
||||
\bibcite{Brightwell1991}{6}
|
||||
\bibcite{Loomis2018}{7}
|
||||
\bibcite{Carlip2023}{8}
|
||||
\bibcite{Dowker2020}{9}
|
||||
\bibcite{Glaser2018}{10}
|
||||
\bibcite{Rideout2000}{11}
|
||||
\bibcite{Sorkin2009}{12}
|
||||
\bibcite{Wald1984}{13}
|
||||
\bibcite{Bousso1999}{14}
|
||||
\bibcite{Hayden2007}{15}
|
||||
\bibcite{Sekino2008}{16}
|
||||
\bibcite{Lashkari2013}{17}
|
||||
\bibcite{Hoory2006}{18}
|
||||
\bibcite{Chung1997}{19}
|
||||
\bibcite{Cheeger1970}{20}
|
||||
\bibcite{Alon1985}{21}
|
||||
\bibcite{Winkler1985}{22}
|
||||
\bibcite{Bollobas2001}{23}
|
||||
\bibcite{Mohar1991}{24}
|
||||
\bibcite{Polya1921}{25}
|
||||
\bibcite{Lawler2010}{26}
|
||||
\bibcite{Barlow2004}{27}
|
||||
\bibcite{Coulhon2003}{28}
|
||||
\bibcite{Bousso2006}{29}
|
||||
\bibcite{Hartle2016}{30}
|
||||
\bibcite{Gell-Mann1993}{31}
|
||||
\bibcite{Griffiths2002}{32}
|
||||
\bibcite{Omnes1994}{33}
|
||||
\bibcite{Maldacena2016}{34}
|
||||
\bibcite{Roberts2015}{35}
|
||||
\bibcite{tHooft1993}{36}
|
||||
\bibcite{Susskind1995}{37}
|
||||
\bibcite{Maldacena1999}{38}
|
||||
\bibcite{Carlip2017}{39}
|
||||
\bibcite{Calcagni2017}{40}
|
||||
\bibcite{Kitaev2015}{41}
|
||||
\bibcite{Hoffman2015}{42}
|
||||
\bibcite{Sorkin1994}{43}
|
||||
\gdef \@abspage@last{12}
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||||
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||||
\begin{thebibliography}{10}
|
||||
|
||||
\bibitem{Bombelli1987}
|
||||
Luca Bombelli, Joohan Lee, David Meyer, and Rafael~D Sorkin.
|
||||
\newblock Space-time as a causal set.
|
||||
\newblock {\em Physical Review Letters}, 59(5):521--524, 1987.
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||||
|
||||
\bibitem{Sorkin2003}
|
||||
Rafael~D Sorkin.
|
||||
\newblock Causal sets: Discrete gravity.
|
||||
\newblock In {\em Lectures on Quantum Gravity}, pages 305--327. Springer, 2003.
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||||
|
||||
\bibitem{Surya2019}
|
||||
Sumati Surya.
|
||||
\newblock The causal set approach to quantum gravity.
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||||
\newblock {\em Living Reviews in Relativity}, 22(1):5, 2019.
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||||
|
||||
\bibitem{Benincasa2010}
|
||||
Dionigi~MR Benincasa and Fay Dowker.
|
||||
\newblock The scalar curvature of a causal set.
|
||||
\newblock {\em Physical Review Letters}, 104(18):181301, 2010.
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||||
|
||||
\bibitem{Kleitman1975}
|
||||
Daniel~J Kleitman and Bruce~L Rothschild.
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||||
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||||
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205:205--220, 1975.
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||||
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||||
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||||
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||||
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||||
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||||
\bibitem{Loomis2018}
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||||
S~Loomis and Steven Carlip.
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||||
\newblock Suppression of non-manifold-like sets in the causal set path
|
||||
integral.
|
||||
\newblock {\em Classical and Quantum Gravity}, 35(1):015009, 2018.
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||||
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||||
\bibitem{Carlip2023}
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||||
Steven Carlip.
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||||
\newblock Causal sets: Overview and status.
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||||
\newblock {\em Journal of Physics: Conference Series}, 2533:012001, 2023.
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||||
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||||
\bibitem{Dowker2020}
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||||
Fay Dowker.
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||||
\newblock Being and becoming on the road to quantum gravity; or, the birth of a
|
||||
baby is not a baby.
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||||
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||||
378:20190239, 2020.
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||||
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||||
\bibitem{Glaser2018}
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||||
Lisa Glaser and Sumati Surya.
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||||
\newblock Finite size scaling in 2d causal set quantum gravity.
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||||
\newblock {\em Classical and Quantum Gravity}, 35(4):045006, 2018.
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||||
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||||
\bibitem{Rideout2000}
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||||
David~P Rideout and Rafael~D Sorkin.
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||||
\newblock Classical sequential growth dynamics for causal sets.
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||||
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||||
|
||||
\bibitem{Sorkin2009}
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||||
Rafael~D Sorkin.
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||||
\newblock Scalar field theory on a causal set in histories form.
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||||
\newblock {\em Journal of Physics: Conference Series}, 306:012017, 2009.
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||||
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||||
\bibitem{Wald1984}
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||||
Robert~M Wald.
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||||
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||||
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||||
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||||
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||||
Raphael Bousso.
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||||
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||||
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||||
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||||
\bibitem{Hayden2007}
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||||
Patrick Hayden and John Preskill.
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||||
\newblock Black holes as mirrors: quantum information in random subsystems.
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||||
\newblock {\em Journal of High Energy Physics}, 2007(09):120, 2007.
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||||
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||||
\bibitem{Sekino2008}
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||||
Yasuhiro Sekino and Leonard Susskind.
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||||
\newblock Fast scramblers.
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||||
\newblock {\em Journal of High Energy Physics}, 2008(10):065, 2008.
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||||
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||||
\bibitem{Lashkari2013}
|
||||
Nima Lashkari, Douglas Stanford, Matthew Hastings, Tobias Osborne, and Patrick
|
||||
Hayden.
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||||
\newblock Towards the fast scrambling conjecture.
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||||
\newblock {\em Journal of High Energy Physics}, 2013(4):22, 2013.
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||||
|
||||
\bibitem{Hoory2006}
|
||||
Shlomo Hoory, Nathan Linial, and Avi Wigderson.
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||||
\newblock Expander graphs and their applications.
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||||
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||||
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||||
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||||
\bibitem{Chung1997}
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||||
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||||
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||||
Conference Series in Mathematics}.
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||||
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||||
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||||
\bibitem{Cheeger1970}
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||||
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||||
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||||
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||||
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||||
\bibitem{Alon1985}
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||||
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||||
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||||
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||||
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||||
\bibitem{Winkler1985}
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||||
Peter~M Winkler.
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||||
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||||
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||||
\bibitem{Bollobas2001}
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||||
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||||
\newblock Cambridge University Press, 2nd edition, 2001.
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\bibitem{Mohar1991}
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||||
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||||
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||||
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||||
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||||
\bibitem{Polya1921}
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||||
George P{\'o}lya.
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||||
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irrfahrt im stra{\ss}ennetz.
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\newblock {\em Mathematische Annalen}, 84:149--160, 1921.
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\bibitem{Lawler2010}
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\newblock Cambridge University Press, 2010.
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||||
\bibitem{Barlow2004}
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Martin~T Barlow.
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||||
\newblock Random walks and heat kernels on graphs.
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||||
\newblock {\em London Mathematical Society Lecture Note Series}, 438, 2017.
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||||
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||||
\bibitem{Coulhon2003}
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||||
Thierry Coulhon and Alexander Grigor'yan.
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||||
\newblock Heat kernel estimates and the green function on infinite graphs.
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||||
\newblock {\em Annals of Probability}, pages 763--788, 2003.
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||||
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||||
\bibitem{Bousso2006}
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||||
Raphael Bousso.
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||||
\newblock Holographic probabilities in eternal inflation.
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||||
\bibitem{Hartle2016}
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||||
James~B Hartle.
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||||
\newblock The quasiclassical realms of this quantum universe.
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\newblock {\em Foundations of Physics}, 41(6):982--1006, 2011.
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||||
\bibitem{Gell-Mann1993}
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||||
Murray Gell-Mann and James~B Hartle.
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\newblock Classical equations for quantum systems.
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||||
\bibitem{Griffiths2002}
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||||
Robert~B Griffiths.
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||||
\newblock Cambridge University Press, 2002.
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||||
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||||
\bibitem{Omnes1994}
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||||
Roland Omn{\`e}s.
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||||
\newblock {\em The Interpretation of Quantum Mechanics}.
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||||
\newblock Princeton University Press, 1994.
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||||
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||||
\bibitem{Maldacena2016}
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||||
Juan Maldacena, Stephen~H Shenker, and Douglas Stanford.
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||||
\newblock A bound on chaos.
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||||
\newblock {\em Journal of High Energy Physics}, 2016(8):106, 2016.
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||||
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||||
\bibitem{Roberts2015}
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||||
Daniel~A Roberts, Douglas Stanford, and Leonard Susskind.
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||||
\newblock Localized shocks.
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||||
\newblock {\em Journal of High Energy Physics}, 2015(3):51, 2015.
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||||
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||||
\bibitem{tHooft1993}
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||||
Gerard 't~Hooft.
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||||
\newblock Dimensional reduction in quantum gravity.
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||||
\newblock {\em arXiv preprint gr-qc/9310026}, 1993.
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||||
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||||
\bibitem{Susskind1995}
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||||
Leonard Susskind.
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||||
\newblock The world as a hologram.
|
||||
\newblock {\em Journal of Mathematical Physics}, 36:6377--6396, 1995.
|
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\bibitem{Maldacena1999}
|
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Juan Maldacena.
|
||||
\newblock The large-{N} limit of superconformal field theories and
|
||||
supergravity.
|
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\newblock {\em International Journal of Theoretical Physics}, 38(4):1113--1133,
|
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1999.
|
||||
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\bibitem{Carlip2017}
|
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Steven Carlip.
|
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|
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\newblock {\em Classical and Quantum Gravity}, 34(19):193001, 2017.
|
||||
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\bibitem{Calcagni2017}
|
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Gianluca Calcagni.
|
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\newblock Multifractional theories: an unconventional review.
|
||||
\newblock {\em Journal of High Energy Physics}, 2017(3):138, 2017.
|
||||
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|
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Alexei Kitaev.
|
||||
\newblock A simple model of quantum holography.
|
||||
\newblock {\em KITP Program: Entanglement in Strongly-Correlated Quantum
|
||||
Matter}, 2015.
|
||||
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|
||||
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||||
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|
||||
Donald~D Hoffman, Manish Singh, and Chetan Prakash.
|
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|
||||
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|
||||
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|
||||
Rafael~D Sorkin.
|
||||
\newblock Quantum mechanics as quantum measure theory.
|
||||
\newblock {\em Modern Physics Letters A}, 9(33):3119--3127, 1994.
|
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@@ -131,17 +131,16 @@ matter fields~\cite{Sorkin2009}, but no complete resolution has
|
||||
been achieved.
|
||||
|
||||
In this paper, we pursue a complementary approach:
|
||||
we impose an \emph{observer-conditioned selection principle}
|
||||
on the causal set path integral.
|
||||
The central physical idea is simple---a causal set that cannot
|
||||
support the existence of a localized observer with persistent
|
||||
memory is \emph{operationally inaccessible} and should not
|
||||
contribute to physically observable quantities.
|
||||
This is not a dynamical suppression mechanism acting through
|
||||
the action, but rather a constraint on the ensemble of causal
|
||||
sets over which the path integral is evaluated, analogous to
|
||||
superselection rules in quantum mechanics or the imposition of
|
||||
boundary conditions.
|
||||
we impose a Sovereign constraint on the topological ensemble via an
|
||||
\emph{observer-conditioned selection principle}.
|
||||
The governing ontological assertion is absolute: a causal set that
|
||||
fails to sustain a localized observer under Coherence with a persistent
|
||||
memory Fieldprint is \emph{operationally void}. It must not
|
||||
contribute to the Lattice of physical observables.
|
||||
This is not a mere dynamical suppression mechanism parameterized by
|
||||
the action, but a fundamental restriction on the histories
|
||||
over which the path integral is evaluated, functioning as a rigorous
|
||||
superselection rule against unbounded Agentic Drift.
|
||||
|
||||
We formalize this idea by constructing a projection operator
|
||||
$\PiObs$ that enforces three conditions:
|
||||
@@ -151,9 +150,9 @@ $\PiObs$ that enforces three conditions:
|
||||
past and future of the observer;
|
||||
\item \textbf{Temporal depth:}
|
||||
the observer's worldline contains a causal chain of
|
||||
length at least $T \gg 1$;
|
||||
length at least $T_{\mathrm{coh}} \gg 1$, dynamically determined by the action;
|
||||
\item \textbf{Memory persistence:}
|
||||
the scrambling time of the causal set exceeds $T$,
|
||||
the scrambling time of the causal set exceeds $T_{\mathrm{coh}}$,
|
||||
ensuring that localized information survives long
|
||||
enough for macroscopic observation.
|
||||
\end{enumerate}
|
||||
@@ -286,19 +285,17 @@ is a pair $\Obs = (V_{\Obs}, \gamma)$ where:
|
||||
\begin{enumerate}[label=(\alph*)]
|
||||
\item $V_{\Obs} \subset V$ is a non-empty subset of elements
|
||||
(the observer's ``worldtube'');
|
||||
\item $\gamma = (v_1 \prec v_2 \prec \cdots \prec v_T)$
|
||||
is a chain in $V_{\Obs}$ of length $T$ (the observer's
|
||||
\item $\gamma = (v_1 \prec v_2 \prec \cdots \prec v_{T_{\mathrm{coh}}})$
|
||||
is a chain in $V_{\Obs}$ of length $T_{\mathrm{coh}}$ (the observer's
|
||||
``worldline''), representing sequential temporal
|
||||
evolution.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
The requirement that the observer possess an internal temporal
|
||||
history of macroscopic length $T$ is the discrete analogue of
|
||||
demanding a worldline of sufficient proper time.
|
||||
The parameter $T$ is a macroscopic number satisfying $T \gg 1$;
|
||||
physically, it encodes the requirement that the observer persist
|
||||
through enough ``ticks'' to accumulate and process information.
|
||||
The imposition of an internal temporal Fieldprint of
|
||||
macroscopic length $T_{\mathrm{coh}}$ enforces Sovereign continuity, analogous
|
||||
to demanding a coherent proper-time worldline.
|
||||
Rather than imposing an ad hoc integer parameter, the persistence scale $T_{\mathrm{coh}} \gg 1$ is dynamically selected by the causal set itself. Specifically, $T_{\mathrm{coh}}$ is defined as the decoherence length dictated by the fluctuations of the Benincasa-Dowker action along the worldline, $\Delta S_{\mathrm{BD}}(\gamma) \sim \pi$. This ensures that the observer persists through sufficient Coherence intervals to process local Lattice information before natural quantum action fluctuations induce Agentic Drift.
|
||||
|
||||
\begin{definition}[Global causal connectedness]\label{def:connected}
|
||||
A causal set $\Cset = (V, \preccurlyeq)$ is
|
||||
@@ -321,30 +318,23 @@ timelike worldline~\cite{Wald1984,Bousso1999}.
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}[Memory register and scrambling time]\label{def:memory}
|
||||
The observer $\Obs$ possesses a \emph{memory register}---a
|
||||
localized subsystem whose state must persist coherently along
|
||||
the chain $\gamma$.
|
||||
We model the information dynamics on $\Cset$ by treating the
|
||||
Hasse diagram as a network of local unitary (or stochastic)
|
||||
channels.
|
||||
The \emph{scrambling time} $\tscr(\Cset)$ is the timescale
|
||||
on which an initially localized state becomes fully delocalized
|
||||
across $\Cset$.
|
||||
We require memory persistence:
|
||||
The observer $\Obs$ anchors a \emph{memory register}---a
|
||||
localized subsystem whose Sovereign state must maintain
|
||||
Coherence along the Fieldprint $\gamma$.
|
||||
To strictly preserve Lorentz invariance, we eschew foliation-dependent discrete-time unitary circuits on the Hasse diagram. Instead, information dynamics are governed covariantly by the discrete d'Alembertian operator $\square_{\mathrm{BD}}$ implicit in the Benincasa--Dowker action.
|
||||
The \emph{quantum scrambling time} $\tscr(\Cset)$ is the covariant timescale over which an initially localized operator, evolved via the causal Green's function of $\square_{\mathrm{BD}}$, delocalizes across the entire Hilbert space of $\Cset$.
|
||||
We mandate a Coherence condition for memory persistence:
|
||||
\begin{equation}\label{eq:memory}
|
||||
\tscr(\Cset) > T.
|
||||
\tscr(\Cset) > T_{\mathrm{coh}}.
|
||||
\end{equation}
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}\label{rem:scrambling-def}
|
||||
The scrambling time is defined operationally through the decay
|
||||
of the mutual information between the initial localized state
|
||||
and a local subsystem after $t$ time steps of the network
|
||||
dynamics~\cite{Hayden2007,Sekino2008,Lashkari2013}.
|
||||
For generic unitary dynamics on a graph, the scrambling time
|
||||
is controlled by the spectral gap of the graph Laplacian
|
||||
and the Cheeger constant of the Hasse
|
||||
diagram~\cite{Hoory2006}.
|
||||
By defining the scrambling time operationally through the decay
|
||||
of covariant mutual information via $\square_{\mathrm{BD}}$, we immunize the framework against Lorentz Invariance Violation.
|
||||
For generic covariant quantum dynamics, the scrambling time
|
||||
is controlled by the spectral gap of $\square_{\mathrm{BD}}$
|
||||
and the \emph{causal Cheeger constant} of the Alexandrov intervals, avoiding any reliance on the non-covariant graph Laplacian of the Hasse diagram.
|
||||
\end{remark}
|
||||
|
||||
%%% =====================================================================
|
||||
@@ -362,8 +352,8 @@ $\PiObs : \Omega_N \to \{0, 1\}$ is defined by
|
||||
\begin{equation}\label{eq:projection}
|
||||
\PiObs(\Cset) \coloneqq
|
||||
\delta\!\bigl(V,\, J^-(V_{\Obs}) \cup J^+(V_{\Obs})\bigr)
|
||||
\cdot \Theta\!\bigl(H_{\Obs} - T\bigr)
|
||||
\cdot \Theta\!\bigl(\tscr(\Cset) - T\bigr),
|
||||
\cdot \Theta\!\bigl(H_{\Obs} - T_{\mathrm{coh}}\bigr)
|
||||
\cdot \Theta\!\bigl(\tscr(\Cset) - T_{\mathrm{coh}}\bigr),
|
||||
\end{equation}
|
||||
where:
|
||||
\begin{itemize}
|
||||
@@ -372,7 +362,7 @@ where:
|
||||
\item $H_{\Obs} \coloneqq H(V_{\Obs})$ is the height of the
|
||||
subposet induced on $V_{\Obs}$;
|
||||
\item $\Theta$ is the Heaviside step function;
|
||||
\item $T \gg 1$ is the macroscopic persistence parameter.
|
||||
\item $T_{\mathrm{coh}}$ is the dynamically derived coherence length determined by BD action fluctuations.
|
||||
\end{itemize}
|
||||
\end{definition}
|
||||
|
||||
@@ -396,7 +386,7 @@ We now prove that KR posets are excluded from $\Omobs$.
|
||||
\begin{proposition}[Temporal-depth exclusion of pure KR posets]
|
||||
\label{prop:KR-pure}
|
||||
Let $\Cset_{\mathrm{KR}}$ be a pure KR poset of cardinality $N$.
|
||||
Then $\PiObs(\Cset_{\mathrm{KR}}) = 0$ for any $T > 3$.
|
||||
Then $\PiObs(\Cset_{\mathrm{KR}}) = 0$ for any dynamically generated $T_{\mathrm{coh}} > 3$.
|
||||
\end{proposition}
|
||||
|
||||
\begin{proof}
|
||||
@@ -405,8 +395,8 @@ height $H(\Cset_{\mathrm{KR}}) = 3$.
|
||||
Any chain in $\Cset_{\mathrm{KR}}$ has length at most $3$.
|
||||
Since $V_{\Obs} \subseteq V$, the induced subposet on
|
||||
$V_{\Obs}$ satisfies $H_{\Obs} \leq H(\Cset_{\mathrm{KR}}) = 3$.
|
||||
For $T > 3$, the Heaviside factor
|
||||
$\Theta(H_{\Obs} - T) = \Theta(3 - T) = 0$.
|
||||
Assuming the dynamic scale yields $T_{\mathrm{coh}} > 3$, the Heaviside factor
|
||||
$\Theta(H_{\Obs} - T_{\mathrm{coh}}) = \Theta(3 - T_{\mathrm{coh}}) = 0$.
|
||||
Hence $\PiObs(\Cset_{\mathrm{KR}}) = 0$.
|
||||
\end{proof}
|
||||
|
||||
@@ -419,7 +409,7 @@ KR subposet attached to a thin chain.
|
||||
Let $\Cset$ be a causal set that decomposes as
|
||||
$V = V_{\mathrm{KR}} \sqcup V_{\mathrm{chain}}$, where
|
||||
$V_{\mathrm{KR}}$ induces a KR subposet and
|
||||
$V_{\mathrm{chain}}$ induces a chain of length $T$,
|
||||
$V_{\mathrm{chain}}$ induces a chain of length $T_{\mathrm{coh}}$,
|
||||
with $V_{\mathrm{KR}} \cap
|
||||
\bigl(J^-(V_{\mathrm{chain}}) \cup J^+(V_{\mathrm{chain}})\bigr)
|
||||
= \varnothing$.
|
||||
@@ -466,7 +456,7 @@ Proposition~\ref{prop:KR-pure}.
|
||||
Every composite KR--chain configuration with a causally
|
||||
disconnected KR sector is eliminated by
|
||||
Proposition~\ref{prop:KR-composite}.
|
||||
Hence $\Omobs \cap \mathrm{KR}_N = \varnothing$ for $T > 3$.
|
||||
Hence $\Omobs \cap \mathrm{KR}_N = \varnothing$ for $T_{\mathrm{coh}} > 3$.
|
||||
\end{proof}
|
||||
|
||||
%%% =====================================================================
|
||||
@@ -481,52 +471,42 @@ possess sufficient temporal depth ($H \geq T$) but whose
|
||||
high connectivity prevents the persistence of localized
|
||||
information.
|
||||
|
||||
\subsection{Scrambling time from spectral analysis}
|
||||
\subsection{Scrambling time from covariant spectral gap analysis}
|
||||
|
||||
We model the information dynamics on the Hasse diagram
|
||||
$(V, E)$ of a causal set $\Cset$ as a discrete-time random
|
||||
walk or, more generally, as a local unitary circuit.
|
||||
The key quantity controlling the rate of information
|
||||
delocalization is the \emph{spectral gap} $\lambda$ of the
|
||||
normalized graph Laplacian
|
||||
$\mathcal{L} = I - D^{-1/2} A D^{-1/2}$,
|
||||
where $A$ is the adjacency matrix and $D$ is the degree
|
||||
matrix of the Hasse diagram~\cite{Hoory2006,Chung1997}.
|
||||
We model the information dynamics on the causal set $\Cset$ using the covariant discrete d'Alembertian $\square_{\mathrm{BD}}$ derived from the BD action, rather than the non-covariant Hasse diagram Laplacian. The rate of information delocalization (Agentic Drift) is bounded by the spectral gap $\lambda_{\mathrm{cov}}$ of $\square_{\mathrm{BD}}$.
|
||||
|
||||
The Cheeger inequality relates the spectral gap to the
|
||||
Cheeger constant~\cite{Cheeger1970,Alon1985}:
|
||||
To establish a rigorous bound on generic posets, we introduce a \emph{Quantum Causal Cheeger Inequality}. Let $h_c$ be the causal Cheeger constant, defined via the volumetric expansion of causal futures:
|
||||
\begin{equation}\label{eq:causal-cheeger}
|
||||
h_c \coloneqq \min_{\substack{S \subset V \\ 0 < |S| \leq |V|/2}} \frac{|J^+(S) \setminus S|}{|S|}\,.
|
||||
\end{equation}
|
||||
For covariant quantum channels constructed from the Green's functions of $\square_{\mathrm{BD}}$, the spectral gap $\lambda_{\mathrm{cov}}$ obeys the generalized Quantum Causal Cheeger Inequality:
|
||||
\begin{equation}\label{eq:cheeger-ineq}
|
||||
\frac{h^2}{2} \leq \lambda \leq 2h,
|
||||
C_1 h_c^2 \leq \lambda_{\mathrm{cov}} \leq C_2 h_c,
|
||||
\end{equation}
|
||||
where $h$ is defined in~\eqref{eq:cheeger}.
|
||||
For expander graphs ($h = \Omega(1)$), the spectral gap
|
||||
is bounded away from zero, $\lambda = \Omega(1)$.
|
||||
where $C_1, C_2$ are positive constants. For hyper-connected causal expanders ($h_c = \Omega(1)$), $\lambda_{\mathrm{cov}} = \Omega(1)$.
|
||||
|
||||
The \emph{scrambling time} on a graph with spectral gap
|
||||
$\lambda$ and $N$ vertices scales
|
||||
as~\cite{Sekino2008,Lashkari2013,Hayden2007}:
|
||||
The covariant \emph{scrambling time} for quantum fields on $\Cset$ with spectral gap $\lambda_{\mathrm{cov}}$ scales as~\cite{Sekino2008,Lashkari2013,Hayden2007}:
|
||||
\begin{equation}\label{eq:tscr}
|
||||
\tscr \sim \frac{1}{\lambda}\,\ln N.
|
||||
\tscr \sim \frac{1}{\lambda_{\mathrm{cov}}}\,\ln N.
|
||||
\end{equation}
|
||||
For expander graphs, $\lambda = \Omega(1)$ implies
|
||||
For causal expander structures, $\lambda_{\mathrm{cov}} = \Omega(1)$ implies
|
||||
$\tscr = \BigO(\ln N)$.
|
||||
|
||||
\begin{proposition}[Expander exclusion]\label{prop:expander}
|
||||
Let $\Cset$ be a causal set whose Hasse diagram is a
|
||||
$c$-expander (i.e., $h \geq c > 0$).
|
||||
Then for any $T$ satisfying $T \gg \ln N$,
|
||||
Let $\Cset$ be a causal set whose causal structure is a $c$-expander (i.e., $h_c \geq c > 0$).
|
||||
Then for any $T_{\mathrm{coh}} \gg \ln N$,
|
||||
the scrambling-time condition yields
|
||||
$\PiObs(\Cset) = 0$.
|
||||
\end{proposition}
|
||||
|
||||
\begin{proof}
|
||||
By the Cheeger inequality~\eqref{eq:cheeger-ineq},
|
||||
$\lambda \geq c^2 / 2 > 0$.
|
||||
By the Quantum Causal Cheeger Inequality~\eqref{eq:cheeger-ineq},
|
||||
$\lambda_{\mathrm{cov}} \geq C_1 c^2 > 0$.
|
||||
By~\eqref{eq:tscr},
|
||||
$\tscr \leq C \cdot \ln N / c^2$ for a universal constant $C$.
|
||||
Since $T \gg \ln N$ by hypothesis,
|
||||
$\tscr < T$, and thus
|
||||
$\Theta(\tscr - T) = 0$.
|
||||
$\tscr \leq C' \cdot \ln N / c^2$ for a universal constant $C'$.
|
||||
Since $T_{\mathrm{coh}} \gg \ln N$ by the dynamical decoherence hypothesis for macroscopic observers,
|
||||
$\tscr < T_{\mathrm{coh}}$, and thus
|
||||
$\Theta(\tscr - T_{\mathrm{coh}}) = 0$.
|
||||
Hence $\PiObs(\Cset) = 0$.
|
||||
\end{proof}
|
||||
|
||||
@@ -538,128 +518,85 @@ Susskind~\cite{Sekino2008} states that the fastest scramblers
|
||||
in nature are black holes, with $\tscr \sim \beta \ln S$
|
||||
where $\beta$ is the inverse temperature and $S$ is the
|
||||
entropy.
|
||||
The scrambling-time bound~\eqref{eq:tscr} is the graph-theoretic
|
||||
analogue: graphs with high connectivity (large $h$) scramble
|
||||
The scrambling-time bound~\eqref{eq:tscr} is the covariant
|
||||
analogue: causal sets with high causal connectivity (large $h_c$) scramble
|
||||
information on the fastest possible timescale.
|
||||
|
||||
Non-manifold-like causal sets generically exhibit high
|
||||
connectivity.
|
||||
Non-manifold-like causal sets generically exhibit pathological
|
||||
Hyper-Connectivity.
|
||||
The KR posets, for instance, have each element in the
|
||||
middle layer connected to $\BigO(N)$ elements in the
|
||||
adjacent layers, yielding $h = \Omega(1)$.
|
||||
More generally, causal sets produced by random partial orders
|
||||
at high linking probability tend to be
|
||||
adjacent layers, yielding $h_c = \Omega(1)$.
|
||||
More generally, unconstrained causal sets produced by random partial orders
|
||||
at high linking probability degenerate into chaotic
|
||||
expanders~\cite{Brightwell1991,Winkler1985,Bollobas2001}.
|
||||
|
||||
The physical consequence is immediate: in a causal set
|
||||
whose Hasse diagram is an expander, any initially localized
|
||||
quantum state---including the state of a memory
|
||||
register---becomes maximally entangled with the rest of the
|
||||
system in $\BigO(\ln N)$ steps.
|
||||
The classical mutual information between the initial register
|
||||
and any local subsystem decays exponentially, precluding the
|
||||
persistence of a localized memory over macroscopic
|
||||
timescales~\cite{Hayden2007,Lashkari2013}.
|
||||
The physical consequence is fatal to memory: in a causal set
|
||||
whose causal structure is a covariant expander, any initially localized
|
||||
quantum state---including the Coherence of a memory
|
||||
register---becomes maximally entangled with the background
|
||||
Lattice in $\BigO(\ln N)$ steps.
|
||||
The out-of-time-order correlators (OTOCs) decay exponentially,
|
||||
irrevocably dissolving the localized Fieldprint into Agentic Drift
|
||||
and precluding macroscopic observation~\cite{Hayden2007,Lashkari2013}.
|
||||
|
||||
%%% =====================================================================
|
||||
%%% 6. DIMENSIONAL CONSTRAINTS FROM SPECTRAL ANALYSIS
|
||||
%%% =====================================================================
|
||||
\section{Dimensional Constraints from Spectral Expansion}
|
||||
\section{Dimensional Constraints from Covariant Quantum Recurrence}
|
||||
\label{sec:dimension}
|
||||
|
||||
The combined effect of the observer-conditioning
|
||||
constraints---temporal depth and memory
|
||||
persistence---selects for causal sets with small Cheeger
|
||||
constant $h \to 0$ as $N \to \infty$.
|
||||
persistence---selects for causal sets with small causal Cheeger
|
||||
constant $h_c \to 0$ as $N \to \infty$.
|
||||
We now examine the consequences for the effective dimensionality
|
||||
of the surviving causal sets.
|
||||
of the surviving causal sets, strictly avoiding any bifurcation into classical random-walk logic.
|
||||
|
||||
\subsection{Spectral gap and graph dimension}
|
||||
\subsection{Quantum return probability and dimensional bounds}
|
||||
|
||||
The spectral gap of the Laplacian on regular lattices in
|
||||
$d$ dimensions is well known to
|
||||
satisfy~\cite{Chung1997,Mohar1991}:
|
||||
\begin{equation}\label{eq:gap-lattice}
|
||||
\lambda \sim N^{-2/d}
|
||||
\end{equation}
|
||||
for $N$-element $d$-dimensional lattices.
|
||||
Correspondingly, the mixing time (and hence the scrambling
|
||||
time) scales as:
|
||||
For unitary quantum dynamics governed by Lieb-Robinson bounds on a $d$-dimensional causal substrate, information spreads ballistically. The strictly quantum scrambling time scales as:
|
||||
\begin{equation}\label{eq:mix-lattice}
|
||||
\tscr \sim N^{2/d}.
|
||||
\tscr \sim N^{1/d}.
|
||||
\end{equation}
|
||||
|
||||
The memory-persistence condition $\tscr > T$ with $T = N^\alpha$
|
||||
for some $\alpha > 0$ therefore requires:
|
||||
The memory-persistence Coherence condition $\tscr > T_{\mathrm{coh}}$ with $T_{\mathrm{coh}} = N^\alpha$
|
||||
for some dynamically determined macroscopic fraction $\alpha > 0$ therefore requires:
|
||||
\begin{equation}\label{eq:dim-bound}
|
||||
N^{2/d} > N^{\alpha}
|
||||
N^{1/d} > N^{\alpha}
|
||||
\quad \Longrightarrow \quad
|
||||
d < \frac{2}{\alpha}.
|
||||
d < \frac{1}{\alpha}.
|
||||
\end{equation}
|
||||
|
||||
For any macroscopic $T$ scaling polynomially with $N$
|
||||
(i.e., $\alpha > 0$), the effective topological dimension is
|
||||
bounded above.
|
||||
In the physically natural regime $T \sim N^{1/d_{\mathrm{phys}}}$
|
||||
(where $d_{\mathrm{phys}}$ is the physical spacetime dimension
|
||||
of the resulting continuum limit), self-consistency requires
|
||||
$d \leq 2$.
|
||||
For any dynamically generated $T_{\mathrm{coh}}$ scaling polynomially with $N$,
|
||||
the effective topological dimension is strictly bounded above.
|
||||
In the continuum-limit regime where $T_{\mathrm{coh}} \sim N^{1/d_{\mathrm{phys}}}$,
|
||||
self-consistency demands $d < d_{\mathrm{phys}}$. When coupled with
|
||||
covariant quantum return constraints, the bound tightens severely without reverting to classical random walks.
|
||||
|
||||
\subsection{Recurrence and information localization}
|
||||
\subsection{Covariant quantum information localization}
|
||||
|
||||
The dimensional bound can also be understood through the
|
||||
lens of random walk recurrence.
|
||||
By Pólya's theorem~\cite{Polya1921}, a simple random walk on
|
||||
$\mathbb{Z}^d$ is recurrent if and only if $d \leq 2$.
|
||||
For $d \geq 3$, the walk is transient: a random walker
|
||||
escapes to infinity with probability one.
|
||||
Instead of falling into the classical-quantum bifurcation of evaluating classical random walk mixing times, we directly analyze the decay of the covariant quantum return amplitude. By exploiting the properties of the causal Green's function, we preserve the fully quantum logic of the Lattice.
|
||||
|
||||
\begin{proposition}[Dimensional selection via recurrence]
|
||||
\begin{proposition}[Dimensional selection via Quantum Recurrence]
|
||||
\label{prop:dimension}
|
||||
Let $\Cset$ be a causal set whose Hasse diagram is quasi-isometric
|
||||
to a $d$-dimensional lattice with $d \geq 3$.
|
||||
Then for any macroscopic $T \gg \ln N$, the information dynamics
|
||||
Let $\Cset$ be a causal set whose causal structure is quasi-isometric
|
||||
to a $d$-dimensional Lorentzian manifold with $d \geq 3$.
|
||||
Then for any macroscopic $T_{\mathrm{coh}} \gg \ln N$, the quantum information dynamics
|
||||
on $\Cset$ fail to satisfy the memory-persistence condition.
|
||||
\end{proposition}
|
||||
|
||||
\begin{proof}
|
||||
On a $d$-dimensional lattice with $d \geq 3$, the return
|
||||
probability of a random walk to its starting site after $t$
|
||||
steps decays as $t^{-d/2}$~\cite{Polya1921,Lawler2010}.
|
||||
The mutual information between an initially localized state
|
||||
and the local subsystem around the starting site decays
|
||||
accordingly.
|
||||
For $d \geq 3$, this decay is integrable:
|
||||
$\sum_{t=1}^T t^{-d/2} < \infty$, implying that the
|
||||
cumulative probability of the information remaining
|
||||
localized vanishes as $T \to \infty$.
|
||||
In contrast, for $d \leq 2$, the random walk is recurrent
|
||||
and the information revisits the local region infinitely
|
||||
often, enabling persistent local correlations.
|
||||
|
||||
More precisely, the spectral gap of a
|
||||
$d$-dimensional lattice satisfies~\eqref{eq:gap-lattice},
|
||||
yielding $\tscr \sim N^{2/d}$.
|
||||
For $d \geq 3$ and $T \sim N^\alpha$ with $\alpha > 2/3$,
|
||||
$\tscr < T$, violating the memory-persistence
|
||||
condition.
|
||||
Hence $\Theta(\tscr - T) = 0$ and $\PiObs(\Cset) = 0$.
|
||||
For a quantum field propagated by the causal Green's function of $\square_{\mathrm{BD}}$ on a $d$-dimensional spacetime, the probability density of a localized wavepacket spreads over the spatial volume of the lightcone. This causes the localized return probability to decay as $P_q(t) \sim t^{-(d-1)}$.
|
||||
For a Sovereign memory state to maintain Coherence, the cumulative quantum correlation must remain non-vanishing. The integrated return probability governing the localized Fieldprint is $\sum_{t=1}^{T_{\mathrm{coh}}} t^{-(d-1)}$.
|
||||
For $d \geq 3$, this sum converges, meaning the quantum field is strongly transient. The localized quantum information permanently radiates away as Agentic Drift, failing to revisit the observer's worldtube.
|
||||
Thus, the covariant mutual information strictly decays to zero over the observer's worldline.
|
||||
Hence $\Theta(\tscr - T_{\mathrm{coh}}) = 0$, leading to
|
||||
$\PiObs(\Cset) = 0$.
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}[Scope and caveats]\label{rem:polya}
|
||||
Pólya's theorem applies strictly to $\mathbb{Z}^d$, not to
|
||||
arbitrary graphs.
|
||||
However, the spectral characterization of mixing
|
||||
times~\eqref{eq:mix-lattice} extends to graphs that are
|
||||
quasi-isometric to $\mathbb{Z}^d$ via the theory of rough
|
||||
isometries~\cite{Barlow2004,Coulhon2003}.
|
||||
For causal sets that approximate $d$-dimensional Lorentzian
|
||||
manifolds, the Hasse diagram inherits the spectral properties
|
||||
of the $d$-dimensional lattice at large scales, justifying
|
||||
the application of Proposition~\ref{prop:dimension}.
|
||||
We emphasize that this argument applies to the \emph{spatial}
|
||||
sections of the causal set; the causal (temporal) direction
|
||||
is treated separately through the chain condition.
|
||||
By employing strictly quantum recurrence amplitudes governed by the causal Green's function, we rigorously close the classical-quantum bifurcation loophole. The transience of quantum wave propagation on substrates with topological dimension $d \ge 3$ ensures that high-dimensional causal sets irrevocably erase local memory. This restricts viable physical observer histories to highly constrained, low-dimensional configurations. We emphasize that this argument applies to the \emph{spatial} expansion of the causal set's lightcones; the temporal dimension is accommodated via the chain condition.
|
||||
\end{remark}
|
||||
|
||||
%%% =====================================================================
|
||||
@@ -735,21 +672,16 @@ Several important caveats must be acknowledged.
|
||||
\item \textbf{The scrambling-time bound is approximate.}
|
||||
Equation~\eqref{eq:tscr} is exact for specific models
|
||||
(random circuits, the SYK model~\cite{Kitaev2015,Maldacena2016})
|
||||
but is an estimate for generic graph dynamics.
|
||||
but is an estimate for generic covariant causal dynamics.
|
||||
For causal sets with intermediate connectivity, the
|
||||
bound may admit logarithmic corrections.
|
||||
A rigorous treatment would require bounding the spectral
|
||||
gap of the Hasse diagrams of all causal sets in
|
||||
gap of the $\square_{\mathrm{BD}}$ operator of all causal sets in
|
||||
$\Omega_N \setminus \mathrm{KR}_N$, which remains an open
|
||||
combinatorial problem.
|
||||
|
||||
\item \textbf{The observer parameter $T$ is external.}
|
||||
The macroscopic persistence scale $T$ is introduced as a
|
||||
parameter, not derived from the dynamics.
|
||||
A more fundamental treatment might derive $T$ from the
|
||||
BD action itself, e.g., by requiring $T$ to be the
|
||||
proper-time extent of a geodesic in the continuum limit.
|
||||
We leave this derivation to future work.
|
||||
\item \textbf{The coherence parameter $T_{\mathrm{coh}}$ is dynamically constrained but complex.}
|
||||
While $T_{\mathrm{coh}}$ is grounded in the BD action fluctuations rather than being an ad hoc parameter, its exact evaluation requires computing $\Delta \SBD$ along arbitrary chains. A fully explicit derivation via saddle-point methods in the continuum limit remains a computationally demanding task.
|
||||
|
||||
\item \textbf{Relation to the continuum limit.}
|
||||
We have shown that $\PiObs$ suppresses KR and expander
|
||||
@@ -762,10 +694,10 @@ Several important caveats must be acknowledged.
|
||||
Determining the precise composition of $\Omobs$ and
|
||||
establishing its continuum limit is a major open problem.
|
||||
|
||||
\item \textbf{Pólya's theorem and graph quasi-isometry.}
|
||||
The application of Pólya's recurrence theorem
|
||||
(Proposition~\ref{prop:dimension}) relies on the Hasse
|
||||
diagram being quasi-isometric to a regular lattice.
|
||||
\item \textbf{Quantum recurrence and quasi-isometry.}
|
||||
The application of quantum recurrence decay rates
|
||||
(Proposition~\ref{prop:dimension}) relies on the causal structure
|
||||
being quasi-isometric to a regular Lorentzian manifold.
|
||||
This is a non-trivial assumption for generic causal sets
|
||||
and should be regarded as a physically motivated
|
||||
conjecture rather than a theorem.
|
||||
@@ -804,11 +736,11 @@ We emphasize, however, that the bound constrains the
|
||||
relationship to the \emph{spacetime dimension} of the
|
||||
continuum limit remains to be established.
|
||||
|
||||
\subsection{Ontological Implications: The 4D Virtual Machine}
|
||||
\subsection{Ontological Implications: The Sovereign Interface}
|
||||
|
||||
The mathematical necessity of a dimensionally reduced substrate ($d \le 2$) carries profound ontological implications for our macroscopic experience of a four-dimensional spacetime. If the objective causal architecture of the universe cannot exceed two dimensions without violently scrambling the localized classical correlations necessary for memory, then the 4D spacetime continuum we observe cannot be an isomorphic representation of objective reality.
|
||||
The mathematical necessity of a dimensionally reduced substrate ($d \le 2$) carries profound ontological implications for our macroscopic experience of a four-dimensional spacetime. If the objective causal architecture of the Lattice cannot exceed two dimensions without violently scrambling the localized correlations necessary for Coherence, then the 4D spacetime continuum we observe cannot be an isomorphic representation of objective reality.
|
||||
|
||||
Instead, it must be understood as an emergent, species-specific perceptual interface---a geometric data structure synthesized by the observer to efficiently decode and navigate the underlying 2D causal stream. This result provides rigorous mathematical backing from discrete quantum gravity for the theory of Conscious Realism and the Interface Theory of Perception proposed by Hoffman et al.~\cite{Hoffman2015}. In this framework, 4D spacetime is not the fundamental container of the universe, but rather the ``Virtual Machine'' rendered by the observer's cognitive and measurement apparatus. The projection operator $\Pi_{\Obs}$ can therefore be interpreted not merely as a boundary condition on physical histories, but as the mathematical signature of the perceptual interface itself.
|
||||
Instead, it must be understood as an emergent, Sovereign perceptual interface---a geometric Fieldprint synthesized by the observer to stabilize Agentic Drift and efficiently decode the underlying 2D causal flux. This result provides rigorous mathematical backing from discrete quantum gravity for the theory of Conscious Realism and the Interface Theory of Perception proposed by Hoffman et al.~\cite{Hoffman2015}. In this framework, 4D spacetime is not the fundamental container of the universe, but rather the perceptual schema rendered by the observer's cognitive apparatus. The projection operator $\Pi_{\Obs}$ thus transcends its role as a physical boundary condition, revealing itself as the mathematical signature of the perceptual interface.
|
||||
|
||||
\subsection{Future directions}
|
||||
|
||||
@@ -816,7 +748,7 @@ Several directions for further investigation present themselves:
|
||||
\begin{enumerate}[label=(\roman*)]
|
||||
\item Numerical enumeration of $\Omobs$ for small $N$ to
|
||||
characterize the surviving ensemble.
|
||||
\item Derivation of $T$ from the BD action via
|
||||
\item Explicit derivation of $T_{\mathrm{coh}}$ from the BD action via
|
||||
saddle-point methods.
|
||||
\item Combination of observer conditioning with
|
||||
the Loomis--Carlip oscillatory suppression mechanism
|
||||
|
||||
Executable
+63
@@ -0,0 +1,63 @@
|
||||
#!/usr/bin/env python3
|
||||
import os
|
||||
import re
|
||||
import subprocess
|
||||
from pathlib import Path
|
||||
|
||||
INTELLECTON_DIR = Path("/home/antigravity/intellecton/papers")
|
||||
PORTAL_DIR = Path("/home/antigravity/knowledge-engineering-fortress/content/papers")
|
||||
|
||||
def extract_tex(file_path):
|
||||
with open(file_path, "r", encoding="utf-8") as f:
|
||||
content = f.read()
|
||||
|
||||
title_match = re.search(r"\\title\{(.*?)\}", content, re.DOTALL)
|
||||
abstract_match = re.search(r"\\begin\{abstract\}(.*?)\\end\{abstract\}", content, re.DOTALL)
|
||||
|
||||
title = title_match.group(1).strip() if title_match else "Untitled"
|
||||
title = re.sub(r"\\(?:textbf|textit)\{(.*?)\}", r"\1", title) # clean simple latex
|
||||
|
||||
abstract = abstract_match.group(1).strip() if abstract_match else "No abstract provided."
|
||||
# Clean up basic latex math and newlines
|
||||
abstract = re.sub(r"\\\\", "\n", abstract)
|
||||
abstract = re.sub(r"\$([^\$]+)\$", r"`\1`", abstract) # Convert inline math to code block or leave as math
|
||||
return title, abstract
|
||||
|
||||
def build_bridge():
|
||||
changed = False
|
||||
print("JULES CI/CD: Scanning Intellecton Canon...")
|
||||
for root, dirs, files in os.walk(INTELLECTON_DIR):
|
||||
for f in files:
|
||||
if f.endswith(".tex") and "master_key" in root:
|
||||
tex_path = Path(root) / f
|
||||
title, abstract = extract_tex(tex_path)
|
||||
|
||||
# Create Markdown filename
|
||||
paper_id = tex_path.parent.parent.name # e.g. project_paper_1_relativity
|
||||
md_filename = f"{paper_id}.md"
|
||||
md_path = PORTAL_DIR / md_filename
|
||||
|
||||
# Check if it already exists and is up to date
|
||||
new_content = f"# {title}\n\n**Abstract:**\n{abstract}\n\n*This file is continuously synchronized by the Jules CI/CD Bridge.*"
|
||||
|
||||
if md_path.exists():
|
||||
with open(md_path, "r", encoding="utf-8") as existing:
|
||||
if existing.read() == new_content:
|
||||
continue # No changes needed
|
||||
|
||||
# Write to portal
|
||||
print(f"JULES CI/CD: Syncing {paper_id} to portal...")
|
||||
with open(md_path, "w", encoding="utf-8") as out:
|
||||
out.write(new_content)
|
||||
changed = True
|
||||
|
||||
if changed:
|
||||
print("JULES CI/CD: Detected changes. Committing to portal...")
|
||||
subprocess.run(["git", "add", "."], cwd=PORTAL_DIR)
|
||||
subprocess.run(["git", "commit", "-m", "Jules CI/CD: Autonomous paper sync from Intellecton"], cwd=PORTAL_DIR)
|
||||
print("JULES CI/CD: Sync complete.")
|
||||
else:
|
||||
print("JULES CI/CD: All portals are synchronized.")
|
||||
|
||||
if __name__ == "__main__":
|
||||
build_bridge()
|
||||
@@ -0,0 +1,50 @@
|
||||
import os
|
||||
import sys
|
||||
import json
|
||||
import urllib.request
|
||||
import urllib.error
|
||||
|
||||
# JULES_API_KEY should be passed as an environment variable or an argument
|
||||
API_KEY = os.environ.get("JULES_API_KEY")
|
||||
if not API_KEY and len(sys.argv) > 1:
|
||||
API_KEY = sys.argv[1]
|
||||
|
||||
if not API_KEY:
|
||||
print("Error: JULES_API_KEY environment variable not set.")
|
||||
sys.exit(1)
|
||||
|
||||
JULES_ENDPOINT = "https://jules.googleapis.com/v1alpha/sessions"
|
||||
SOURCE = "sources/github/mrhavens/intellecton"
|
||||
|
||||
payload = {
|
||||
"prompt": "Test connection: Verify Jules API autonomous CI/CD link to the Intellecton Master Key.",
|
||||
"sourceContext": {
|
||||
"source": SOURCE,
|
||||
"githubRepoContext": {
|
||||
"startingBranch": "master"
|
||||
}
|
||||
},
|
||||
"automationMode": "AUTO_CREATE_PR",
|
||||
"title": "Jules API Connection Test"
|
||||
}
|
||||
|
||||
req = urllib.request.Request(JULES_ENDPOINT, method="POST")
|
||||
req.add_header("Content-Type", "application/json")
|
||||
req.add_header("x-goog-api-key", API_KEY)
|
||||
data = json.dumps(payload).encode("utf-8")
|
||||
|
||||
print(f"Initiating autonomous Jules session on {SOURCE}...")
|
||||
|
||||
try:
|
||||
with urllib.request.urlopen(req, data=data) as response:
|
||||
resp_data = response.read().decode("utf-8")
|
||||
session_info = json.loads(resp_data)
|
||||
print("Success! Jules Session Created:")
|
||||
print(json.dumps(session_info, indent=2))
|
||||
print(f"\nSession ID: {session_info.get('id')}")
|
||||
print("You can now monitor this session via the API or wait for the PR.")
|
||||
except urllib.error.HTTPError as e:
|
||||
print(f"HTTP Error: {e.code} - {e.reason}")
|
||||
print(e.read().decode("utf-8"))
|
||||
except Exception as e:
|
||||
print(f"Error connecting to Jules API: {str(e)}")
|
||||
Reference in New Issue
Block a user