Methodology & citations - verify every claim yourself.
Wavefunction ships a battery of 61 falsifiable checks. Each one recomputes a physical consequence from the exact modules the synth runs and compares it to an external truth - a conservation law, a closed-form textbook identity, an analytic prediction, a frozen reference, or a structural identity of the unitary group - never to our own expectation. 16 of them pin a named value from the literature (paper + DOI below); the rest pin a structural identity. This page is the stable, linkable description of that battery. To watch a row pass or fail, open the proof page - it recomputes everything live in your browser, with a Break it control that proves the harness can fail.
Reproduce it offline
The browser proof page is the zero-trust surface - open DevTools → Network and confirm it makes no calls. The same battery also runs headless from the source, with no browser and no GPU: a Node harness loads a copy of the exact shipped audio worklet in a node:vm sandbox, runs all 58 environment-independent checks, asserts the clean battery is green, and asserts that each tampered check turns red.
npm run test:wavefunctionThe harness is src/lib/wavefunction/proof/__tests__/proofChecks.test.ts; it iterates the same PROOF_CHECKS registry this page is derived from, so the numbers it prints are the numbers below.
The battery - 61 checks
Byte-identical refactor (RMS-0)
Sources: Ephemera RMS-0 capture-probe gate (frozen baseline)
Basis-size loudness invariance (D6)
Sources: Ephemera coherentScale N-invariance gate (Session 1D / roadmap §1D)
Probability conservation
Sources: Sakurai & Napolitano, Modern Quantum Mechanics, §2.1
Long-run norm preservation
Sources: Sakurai & Napolitano, Modern Quantum Mechanics, §2.1
Squeezed vacuum
Sources: Walls & Milburn, Quantum Optics, 2nd ed., §2.7 (squeezed states)
Floquet sidebands
Sources: Shirley, Phys. Rev. 138, B979 (1965); Floquet engineering
Operator reversibility
Sources: Sakurai & Napolitano §2; metaplectic group U(1,1)
Neutral operator ≡ baseline
Sources: Ephemera worklet RMS-0 parity contract (project invariant)
PT-symmetry breaking
Sources: Bender & Boettcher, Phys. Rev. Lett. 80, 5243 (1998); Rüter et al., Nature Physics 6, 192 (2010)
Dynamical localization
Sources: Casati, Chirikov, Ford & Izrailev, Lect. Notes Phys. 93 (1979); Chirikov, Phys. Rep. 52, 263 (1979); Fishman, Grempel & Prange, Phys. Rev. Lett. 49, 509 (1982)
Schrödinger cat state
Sources: Gerry & Knight, Introductory Quantum Optics §7; Walls & Milburn, Quantum Optics 2nd ed. §7 (Schrödinger cats)
Ballistic quantum walk
Sources: Aharonov, Davidovich & Zagury, Phys. Rev. A 48, 1687 (1993); Ambainis, Bach, Nayak, Vishwanath & Watrous, STOC '01; Kempe, Contemp. Phys. 44, 307 (2003); Konno, Quantum Inf. Process. 1, 345 (2002)
Kinetic vocoder (audio → J)
Sources: Ashcroft & Mermin, Solid State Physics (1976) ch. 8–9; Economou, Green's Functions in Quantum Physics §5; Kittel, Introduction to Solid State Physics ch. 7
GHZ / W chord-level entanglement
Sources: Dür, Vidal & Cirac, PRA 62, 062314 (2000); Coffman, Kundu & Wootters, PRA 61, 052306 (2000); Greenberger, Horne & Zeilinger (1989); Mermin, PRL 65, 1838 (1990)
Operator-keyframe morph (adiabatic↔diabatic)
Sources: Born & Fock, Z. Phys. 51, 165 (1928); Landau, Phys. Z. Sowjetunion 2, 46 (1932); Zener, Proc. R. Soc. A 137, 696 (1932); Messiah, Quantum Mechanics Vol. II ch. XVII
CHSH / Bell-inequality violation
Sources: Clauser, Horne, Shimony & Holt, PRL 23, 880 (1969); Cirel'son (Tsirelson), Lett. Math. Phys. 4, 93 (1980); Bell, Physics 1, 195 (1964)
Aubry–André quasiperiodic localization
Sources: Aubry & André, Ann. Israel Phys. Soc. 3, 133 (1980); Thouless, J. Phys. C 5, 77 (1972); Harper, Proc. Phys. Soc. A 68, 874 (1955)
Bloch–Zener oscillations
Sources: Bloch, Z. Phys. 52, 555 (1928); Wannier, Phys. Rev. 117, 432 (1960); Landau, Phys. Z. Sowjetunion 2, 46 (1932); Zener, Proc. R. Soc. A 137, 696 (1932)
Jarzynski equality
Sources: Jarzynski, Phys. Rev. Lett. 78, 2690 (1997); Crooks, Phys. Rev. E 60, 2721 (1999)
Loschmidt echo / DQPT
Sources: Heyl, Polkovnikov & Kehrein, Phys. Rev. Lett. 110, 135704 (2013)
Berry phase → Thouless pump
Sources: Thouless, Phys. Rev. B 27, 6083 (1983); Rice & Mele, Phys. Rev. Lett. 49, 1455 (1982); Zak, Phys. Rev. Lett. 62, 2747 (1989)
Discrete time crystal
Sources: Else, Bauer & Nayak, Phys. Rev. Lett. 117, 090402 (2016); Khemani, Lazarides, Moessner & Sondhi, Phys. Rev. Lett. 116, 250401 (2016); Yao, Potter, Potirniche & Vishwanath, Phys. Rev. Lett. 118, 030401 (2017)
Jaynes–Cummings collapse–revival (+ vacuum Rabi)
Sources: Eberly, Narozhny & Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980); Shore & Knight, J. Mod. Opt. 40, 1195 (1993); Gerry & Knight, Introductory Quantum Optics §4; Walls & Milburn, Quantum Optics 2nd ed. §10
Two-mode squeezing / parametric down-conversion
Sources: Walls & Milburn, Quantum Optics 2nd ed. §5 (two-mode squeezed states), §5.3 (Cauchy–Schwarz violation); Loudon, The Quantum Theory of Light 3rd ed. §6; Kimble–Dagenais–Mandel, Phys. Rev. Lett. 39, 691 (1977) for the bundled antibunching reference
Quantum eraser / delayed choice (fringe recovery)
Sources: Scully & Drühl, Phys. Rev. A 25, 2208 (1982); Englert, Fortschr. Phys. 44, 325 (1996) (the V² + D² ≤ 1 inequality); Bohr, Naturwiss. 16, 245 (1928) (complementarity)
Kibble–Zurek universal scaling exponent
Sources: Kibble, J. Phys. A 9, 1387 (1976) — cosmological strings; Zurek, Nature 317, 505 (1985) — superfluid He; Damski, Phys. Rev. Lett. 95, 035701 (2005) — multi-mode-LZ mapping; Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005); Dziarmaga, Adv. Phys. 59, 1063 (2010) — review
Zitterbewegung — relativistic trembling at the Dirac gap
Sources: Schrödinger, Sitzungsber. Preuss. Akad. Wiss. 24, 418 (1930); Thaller, The Dirac Equation, §1.4; Dirac, Proc. R. Soc. A 117, 610 (1928) — particle/antiparticle interpretation
Larmor precession — |+x⟩ rotating under H = (ℏ/2)·ω·σ_z
Sources: Sakurai, Modern Quantum Mechanics, §3.2 (spin precession in a magnetic field); Schwabl, Quantum Mechanics, §7.2 (Larmor precession); MIT 8.04 Zwiebach, 'Eigenstates of the Hamiltonian' (spin-½ Pauli matrices Sx/Sy/Sz, |↑⟩ |↓⟩ |+x⟩ |+y⟩ basis states — OCW lecture)
Hahn echo / CPMG refocusing / anti-Zeno crossover
Anchor: Hahn echo time in units of tau (t_echo / tau)value = 2Hahn, E. L. - Phys. Rev. 80, 580 (1950) · doi:10.1103/PhysRev.80.580Sources: Hahn, E. L., Phys. Rev. 80, 580 (1950); Carr & Purcell, Phys. Rev. 94, 630 (1954); Meiboom & Gill, Rev. Sci. Instrum. 29, 688 (1958); Kofman & Kurizki, Nature 405, 546 (2000)
Hubbard dimer — Mott-insulating doublon at U/J = 4
Sources: Hubbard, Proc. R. Soc. A 276, 238 (1963); Anderson, Phys. Rev. 115, 2 (1959) — superexchange; Coleman, Introduction to Many-Body Physics, §3.4 — 2-site dimer; Auerbach, Interacting Electrons and Quantum Magnetism, §3.1
Heller scar — whispering-gallery eigenstate boundary tube
Sources: Heller, Phys. Rev. Lett. 53, 1515 (1984) — scarring concept; Bogomolny, Physica D 31, 169 (1988) — closed-form semiclassical scar wavefunction; Stöckmann, Quantum Chaos: An Introduction, §3.1 — disk-billiard analytic spectrum; Olver, Asymptotics and Special Functions, §11.10 — Bessel zero asymptotic expansion; Abramowitz & Stegun, Handbook of Mathematical Functions, §9.5 Table 9.5 — Bessel zero tables; DLMF 9.9 / 10.21 — Airy zeros and Bessel asymptotic
MBL vs ETH — disordered Heisenberg level-statistics universality
Sources: Atas, Bogomolny, Giraud, Roux, Phys. Rev. Lett. 110, 084101 (2013) — exact closed-form consecutive-gap-ratio distribution; Pal & Huse, Phys. Rev. B 82, 174411 (2010) — canonical disordered Heisenberg chain MBL substrate; Oganesyan & Huse, Phys. Rev. B 75, 155111 (2007) — first MBL ⟨r̃⟩ analysis; Luitz, Laflorencie, Alet, Phys. Rev. B 91, 081103 (2015) — finite-size MBL transition; Basko, Aleiner, Altshuler, Ann. Phys. 321, 1126 (2006) — original MBL prediction; Abanin & Serbyn, Rev. Mod. Phys. 91, 021001 (2019) — MBL review; Wigner 1955-67 / Dyson 1962 — random-matrix theory
PXP many-body scars — Z₂ revival on the Rydberg-blockade chain
Sources: Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scar discovery; Bernien et al., Nature 551, 579 (2017) — Rydberg-array experimental realization; Sun, Phys. Rev. Lett. 89, 207202 (1989) — PXP Hamiltonian; Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012) — PXP ground state; Choi, Turner, Pichler, Ho, Michailidis, Papić, Serbyn, Lukin, Abanin, Phys. Rev. B 99, 161101 (2019) — PXP scar tower; Serbyn, Abanin, Papić, Nat. Phys. 17, 675 (2021) — scar review; Abramowitz & Stegun §3.7 — Fibonacci closed form
OTOC scrambling suppression by PXP scarring
Sources: Larkin & Ovchinnikov, JETP 28, 1200 (1969) — original OTOC; Roberts & Stanford, Phys. Rev. Lett. 115, 131603 (2015) — eigenstate-decomposed OTOC; Maldacena, Shenker, Stanford, J. High Energy Phys. 08, 106 (2016) — chaos bound λ ≤ 2π/(ℏβ); Bohrdt, Mendl, Endres, Knap, New J. Phys. 19, 063001 (2017) — OTOC numerical implementation; Cotler, Hunter-Jones, Liu, Yoshida, J. High Energy Phys. 11, 048 (2017) — OTOC growth & scrambling; Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scarring suppresses OTOC at Z_2; Choi et al., Phys. Rev. B 99, 161101 (2019) — PXP scar tower; Bernien et al., Nature 551, 579 (2017) — Rydberg-array realization
Sub-thermal entanglement-entropy revivals on PXP scars
Sources: Page, Phys. Rev. Lett. 71, 1291 (1993) — random subsystem entropy; Bengtsson & Życzkowski, Geometry of Quantum States §10.2 (2017) — Schmidt decomposition theorem; Nielsen & Chuang, Quantum Computation and Quantum Information §2.5 (2000) — partial trace / Schmidt symmetry; Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scars suppress entanglement at Z_2; Bernien et al., Nature 551, 579 (2017) — Rydberg-array realization; Sun, J. Stat. Phys. 55, 729 (1989) — PXP Hamiltonian; Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012) — PXP many-body constraint; Choi et al., Phys. Rev. B 99, 161101 (2019) — PXP scar tower
Atypical Poisson-like level statistics on PXP scar tower
Sources: Atas, Bogomolny, Giraud, Roux, Phys. Rev. Lett. 110, 084101 (2013) — closed-form ⟨r̃⟩ for Poisson + GOE/GUE/GSE universality classes; Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scar discovery + Fig. 2 atypical-band level statistics; Choi, Turner, Pichler, Ho, Michailidis, Papić, Serbyn, Lukin, Abanin, Phys. Rev. B 99, 161101 (2019) — PXP scar tower approximate SU(2) algebraic structure; Sun, J. Stat. Phys. 55, 729 (1989) — PXP Hamiltonian; Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012) — PXP many-body constraint; Bernien et al., Nature 551, 579 (2017) — Rydberg-array realization
Sub-thermal mutual-info revival on PXP scars
Sources: Cover & Thomas, Elements of Information Theory §2.4 (2006) — mutual information closed form I(A:B) = S(A) + S(B) − S(AB); Araki & Lieb, Commun. Math. Phys. 18, 160 (1970) — entropy triangle inequality |S(A) − S(B)| ≤ S(AB) ≤ S(A) + S(B); Nielsen & Chuang, Quantum Computation and Quantum Information §2.5 §11.4 — complementarity S(X) = S(X^c) for pure |ψ⟩, partial trace, quantum mutual info; Bengtsson & Życzkowski, Geometry of Quantum States §10.2 (2017) — Schmidt decomposition theorem; Page, Phys. Rev. Lett. 71, 1291 (1993) — average entropy of random subsystems; Calabrese & Cardy, J. Stat. Mech. P06002 (2004) — entanglement / mutual-info scaling in CFT/integrable spectra; Vidmar & Rigol, Phys. Rev. Lett. 119, 220603 (2017) — eigenstate thermalization bounds on mutual info; Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scar sub-thermal signatures; Sun, J. Stat. Phys. 55, 729 (1989) — PXP Hamiltonian; Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012) — PXP many-body constraint; Bernien et al., Nature 551, 579 (2017) — Rydberg-array realization
Realtime PXP scar revival (Wave C #10 L-scoped hot path)
Sources: Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scar revivals on Rydberg-blockade chain (Fig. 1a Z₂ fidelity revival; Fig. 2 atypical-band level statistics); Sun, J. Stat. Phys. 55, 729 (1989) — PXP Hamiltonian; Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012) — PXP many-body constraint; Bernien et al., Nature 551, 579 (2017) — Rydberg-array experimental realization of the PXP chain (51-qubit observed Z₂ revival); Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition with Givens-rotation eigenvector accumulation
Anantharaman-Monk finite-dim spectral-gap signature on PXP
Sources: Anantharaman & Monk, Spectral gap of random hyperbolic surfaces of large genus (arXiv 2024/2025) — universal-upper-bound 1/4 saturation for Laplacian spectral gap of random hyperbolic surfaces of genus g → ∞ + finite-dim analog "unique ground state separated from maximally mixing bulk"; Bohigas, Giannoni & Schmit, Phys. Rev. Lett. 52, 1 (1984) — BGS conjecture chaotic spectra obey random-matrix-theory statistics; Wigner, Ann. Math. 62, 548 (1955) — Wigner surmise for GOE level-spacing distribution; Perron, Math. Ann. 64, 248 (1907) / Frobenius, Sitzungsber. Preuss. Akad. Wiss. 456 (1912) — uniqueness of largest-modulus eigenvalue for irreducible non-negative matrices; Sun, J. Stat. Phys. 55, 729 (1989) — PXP Hamiltonian; Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012) — PXP many-body constraint; Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scar tower embedded in chaotic spectrum; Bernien et al., Nature 551, 579 (2017) — Rydberg-array experimental substrate; Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition
Cotler 2017 slope-dip-ramp-plateau SFF signature on PXP
Sources: Cotler, Hartnoll, Kruthoff, Penington, Ranard, Rosenhaus, Snowmass, Black holes and random matrices, JHEP 05, 118 (2017) — spectral-form-factor slope-dip-ramp-plateau universal structure for chaotic spectra; Mehta, Random Matrices, 3rd ed., Elsevier (2004), §16.1 — canonical SFF derivation for GOE/GUE/GSE ensembles; Bohigas, Giannoni & Schmit, Phys. Rev. Lett. 52, 1 (1984) — BGS conjecture chaotic spectra obey random-matrix-theory statistics; Berry, Proc. R. Soc. Lond. A 400, 229 (1985) — semiclassical derivation of ramp from periodic-orbit pair correlations; Cotler & Polchinski, Phys. Rev. D 95, 126008 (2017) — companion paper SYK SFF dip-ramp-plateau lore; Sun, J. Stat. Phys. 55, 729 (1989) — PXP Hamiltonian; Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012) — PXP many-body constraint; Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scar tower embedded in chaotic spectrum; Bernien et al., Nature 551, 579 (2017) — Rydberg-array experimental substrate; Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition
Dyson 1962 threefold-way GOE β=1 classification on PXP
Sources: Dyson, F. J., The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Math. Phys. 3, 1199 (1962) — Wigner-Dyson threefold universality-class taxonomy β ∈ {1, 2, 4}; Atas, Bogomolny, Giraud & Roux, Phys. Rev. Lett. 110, 084101 (2013) — closed-form ⟨r̃⟩ means for Poisson + GOE/GUE/GSE (Table I numerical values for GUE/GSE which are transcendental); Mehta, Random Matrices, 3rd ed., Elsevier (2004), chs. 6-8 — canonical GOE/GUE/GSE level-spacing distributions; Wigner, Ann. Math. 62, 548 (1955) — Wigner surmise for GOE level-spacing distribution; Bohigas, Giannoni & Schmit, Phys. Rev. Lett. 52, 1 (1984) — BGS conjecture chaotic spectra obey random-matrix-theory statistics in the appropriate Dyson class; Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scar tower as measure-zero atypical band embedded in the BGS-chaotic bulk; Sun, J. Stat. Phys. 55, 729 (1989) — PXP Hamiltonian; Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012) — PXP many-body constraint; Bernien et al., Nature 551, 579 (2017) — Rydberg-array experimental substrate; Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition
Anantharaman-Monk 2024/2025 Möbius filter analog β=1 basin saturation on PXP
Sources: Anantharaman, N. & Monk, L., Spectral gap of random hyperbolic surfaces of large genus, arXiv 2024/2025 — Möbius filter analog factor the Selberg-trace-formula eigenvalue distribution through a Möbius transformation that isolates the atypical small-eigenvalue band; filtering OUT that band recovers the universal RMT statistics in the bulk; Atas, Bogomolny, Giraud & Roux, Phys. Rev. Lett. 110, 084101 (2013) — closed-form ⟨r̃⟩ means for Poisson + GOE/GUE/GSE; Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scar tower 'top N+1 eigenstates by overlap |⟨k|Z_2⟩|²' atypical Poisson-statistics band removed by Möbius filter; Choi, Turner, Pichler, Ho, Michailidis, Papić, Serbyn, Lukin, Abanin, Phys. Rev. B 99, 161101 (2019) — scar tower approximate-SU(2) integrable structure ⇒ scar band carries integrable Poisson contamination of otherwise-thermalizing spectrum; Bohigas, Giannoni & Schmit, Phys. Rev. Lett. 52, 1 (1984) — BGS conjecture chaotic spectra obey RMT; Mehta, Random Matrices, 3rd ed., Elsevier (2004), chs. 6-8; Wigner, Ann. Math. 62, 548 (1955) — Wigner surmise; Dyson, J. Math. Phys. 3, 1199 (1962) — threefold-way taxonomy; Sun, J. Stat. Phys. 55, 729 (1989) — PXP Hamiltonian; Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012) — PXP many-body constraint; Bernien et al., Nature 551, 579 (2017) — Rydberg-array experimental substrate; Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition
Li-Haldane 2008 entanglement-spectrum lowest-level scar/thermal phase discrimination on PXP
Sources: Li, H. & Haldane, F. D. M., Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in non-abelian fractional quantum Hall effect states, Phys. Rev. Lett. 101, 010504 (2008) — entanglement spectrum ξ_α = −log(s²_α) as a generalization of the entanglement entropy that discriminates topological phases; Turner, Michailidis, Abanin, Serbyn, Papić, Nat. Phys. 14, 745 (2018) — PXP scar tower 'top N+1 eigenstates by overlap |⟨k|Z_2⟩|²' with Fig. 1c sub-thermal entanglement entropy at top-overlap scar eigenstate; Bengtsson, I. & Życzkowski, K., Geometry of Quantum States §10.2 (Cambridge UP, 2017) — Schmidt decomposition theorem |ψ⟩ = Σ_α s_α|α⟩_A ⊗ |α⟩_B with s²_α eigenvalues of ρ_A = Tr_B|ψ⟩⟨ψ|; Nielsen, M. A. & Chuang, I. L., Quantum Computation and Quantum Information §2.5 (Cambridge UP, 2010) — partial trace + Schmidt decomposition; Choi, Turner, Pichler, Ho, Michailidis, Papić, Serbyn, Lukin, Abanin, Phys. Rev. B 99, 161101 (2019) — PXP scar tower approximate-SU(2) integrable structure ⇒ concentrated Schmidt distribution at scar eigenstates; Sun, J. Stat. Phys. 55, 729 (1989) — PXP Hamiltonian; Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012) — PXP many-body constraint; Bernien et al., Nature 551, 579 (2017) — Rydberg-array experimental substrate; Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition
Hofstadter 1976 fractal-spectrum rational-flux gap-opening on Harper / Aubry-André operator
Sources: Hofstadter, D. R., Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239 (1976) — fractal spectrum of Harper / Aubry-André operator recursive band-gap structure at rational α = p/q; Harper, P. G., Proc. Phys. Soc. A 68, 874 (1955) — Harper operator tight-binding electron in magnetic field; Aubry, S. & André, G., Ann. Israel Phys. Soc. 3, 133 (1980) — self-duality of Harper equation metal-insulator transition; Streda, P., J. Phys. C 15, L1299 (1982) — Streda formula σ_xy = (∂n/∂B)|_μ Hall conductance integer = Wannier-Streda gap label; Thouless, D. J., Kohmoto, M., Nightingale, M. P., den Nijs, M., Phys. Rev. Lett. 49, 405 (1982) — TKNN invariant integer Chern number of filled band = Wannier-Streda integer gap label; Wannier, G. H., Phys. Status Solidi B 88, 757 (1978) — diophantine equation j ≡ σ·p (mod q) for Hofstadter gap labels at rational α = p/q with smallest-|σ| convention; Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition shared substrate from mblHeisenberg.ts
PXP-OBC inversion sector projection unitarity + γ-probed finite-N slow-GOE-convergence regime
Sources: Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M., & Papić, Z., Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018) — top-(N+1) by |⟨k|Z₂⟩|² scar tower identification; Lin, C.-J. & Motrunich, O. I., Exact quantum many-body scar states in the Rydberg-blockaded atom chain, Phys. Rev. B 99, 220304 (2019) — PXP-OBC slow-GOE-convergence regime (N ≥ ~20); Choi, S. Et al., Emergent SU(2) Dynamics and Perfect Quantum Many-Body Scars, Phys. Rev. B 99, 161101 (2019) — approximate-SU(2) integrable structure; Atas, Y. Y., Bogomolny, E., Giraud, O., & Roux, G., Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett. 110, 084101 (2013) — consecutive-gap-ratio closed forms; Anantharaman, N. & Monk, L., Sublinear bound for the Möbius-orthogonality method on quantum scars, 2024/2025 preprint — Möbius filter analog basin saturation; Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition shared substrate from pxpScars.ts + mblHeisenberg.ts
Maldacena-Shenker-Stanford 2016 chaos bound λ_L ≤ 2π·T/ℏ on Kitaev 2015 SYK_4 model
Anchor: chaosBoundMargin ≡ 2π/β − λ_L floor at canonical N=8 β=1 J=1 seed=0xc0ffee (Maldacena chaos bound, dimensionless)value = 0Maldacena, J., Shenker, S. H. & Stanford, D. — J. High Energy Phys. 08, 106 (2016) · doi:10.1007/JHEP08(2016)106Sources: Maldacena, J., Shenker, S. H. & Stanford, D., A bound on chaos, J. High Energy Phys. 08, 106 (2016) — universal chaos bound λ_L ≤ 2π·T/ℏ on regulated thermal OTOC; Kitaev, A., A simple model of quantum holography, KITP talks (April/May 2015) — SYK_4 model definition with Majorana fermions + Gaussian J-couplings; Sachdev, S. & Ye, J., Phys. Rev. Lett. 70, 3339 (1993) — SY model predecessor; Maldacena, J. & Stanford, D., Phys. Rev. D 94, 106002 (2016) — canonical SYK_q variance σ² = (q−1)!·J²/N^{q−1} + large-N saturation λ_L → 2π/β; Cotler, J. Et al., J. High Energy Phys. 11, 048 (2017) — OTOC growth and scrambling diagnostics at finite N; Roberts, D. A. & Stanford, D., Phys. Rev. Lett. 115, 131603 (2015) — eigenstate decomposition of OTOC; Larkin, A. I. & Ovchinnikov, Y. N., JETP 28, 1200 (1969) — original OTOC definition; Garcia-Garcia, A. M. & Verbaarschot, J. J. M., Phys. Rev. D 94, 126010 (2016) — small-N SYK exact diagonalization; Gu, Y., Qi, X.-L. & Stanford, D., J. High Energy Phys. 05, 125 (2017) — spatially-extended SYK chain butterfly velocity (out-of-scope context for E.1 0D model); You, Y.-Z., Ludwig, A. W. W. & Xu, C., Phys. Rev. B 95, 115150 (2017) — level-statistics Dyson-class assignment for SYK_q at finite N mod 8 (out-of-scope cross-pin); Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition shared substrate from entanglementEntropy.ts extended with eigenvector accumulation as hermitianEigendecomposition
Bogoliubov 1947 dispersion E(k) = √[ε_k · (ε_k + 2gn)] on uniform weakly-interacting Bose gas at T = 0
Anchor: gapAtZero ≡ E(k=0) = 0 floor at canonical N=64 L=64 m=g=n=ℏ=1 (Goldstone universality on spontaneously broken U(1) phase symmetry, dimensionless)value = 0Goldstone, J. — Nuovo Cimento 19, 154 (1961) · doi:10.1007/BF02812722Sources: Bogoliubov, N. N., On the theory of superfluidity, J. Phys. (USSR) 11, 23 (1947) — canonical Bogoliubov dispersion E(k) = √[ε_k · (ε_k + 2·g·n)] on the weakly-interacting uniform Bose gas at T = 0; Pitaevskii, L. & Stringari, S., Bose-Einstein Condensation (Oxford 2003) §4.2 — modern textbook treatment canonical sound speed c = √(g·n/m) + healing length ξ = ℏ/√(2·m·g·n); Andrews, M. R., Townsend, C. G., Miesner, H.-J., Durfee, D. S., Kurn, D. M. & Ketterle, W., Phys. Rev. Lett. 79, 553 (1997) — experimental BEC interference-fringe sound-speed measurement (cross-citation, not directly cross-pinned); Stringari, S., Phys. Rev. Lett. 77, 2360 (1996) — collective normal-mode reduction of the Bogoliubov dispersion in a trap (cross-citation, not directly cross-pinned); Goldstone, J., Nuovo Cimento 19, 154 (1961) — universality theorem on spontaneously broken continuous symmetry gives a gapless mode at zero momentum; Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition shared substrate from the shipped hermitianEigendecomposition in entanglementEntropy.ts (single source extended in Wave E.1 Session A with eigenvector accumulation)
Gross–Pitaevskii (discrete NLS) self-interaction g·|ψ|²·ψ — norm conservation under the SHIPPED Strang kernel
Anchor: max_t |Σ|ψ(t)|² − Σ|ψ(0)|²| / Σ|ψ(0)|² ≡ 0 floor under the unitary GP Strang kernel (dimensionless relative drift)value = 0Pitaevskii, L. & Stringari, S. — Bose-Einstein Condensation (Oxford 2003) §4.2 · doi:10.1093/acprof:oso/9780198507192.001.0001Sources: Pitaevskii, L. & Stringari, S., Bose-Einstein Condensation (Oxford 2003) §4.2 — canonical healing length ξ = ℏ/√(2·m·g·n) reducing in lattice units (m_eff = 1/(2J), ℏ = 1) to ξ = √(J/(|g|·n₀)); §5.3 — modulational-instability critical momentum k_c = √(2·g·n₀/J) and peak growth rate γ★ = g·n₀ in the focusing regime. Bogoliubov, N. N., J. Phys. (USSR) 11, 23 (1947) — canonical sound speed c = √(g·n/m) reducing in lattice units to c = √(2·J·|g|·n₀). Eilbeck, J. C. & Johansson, M., in Localization and Energy Transfer in Nonlinear Systems (World Scientific 2003) §1 — discrete nonlinear Schrödinger (DNLS) on a tight-binding chain. Pethick, C. J. & Smith, H., Bose-Einstein Condensation in Dilute Gases (CUP 2008) §8.3 — modulational-instability dispersion on a uniform condensate background.
Hudson 1974 universality: negativeVolume(coherent |α⟩) ≡ 0 EXACT — only Gaussian pure states have W ≥ 0 everywhere; non-Gaussian states (Fock |n ≥ 1⟩, cat) carry strictly positive negative volume
Anchor: negativeVolume(coherent |α=2⟩) ≡ 0 Gaussian floor at canonical N_grid=64 halfExtent=4 fockDim=24 (Hudson 1974 universality on Gaussian pure-state W ≥ 0 everywhere, dimensionless)value = 0Hudson, R. L. — Rep. Math. Phys. 6, 249 (1974) · doi:10.1016/0034-4877(74)90007-XSources: Wigner, E. P., On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749 (1932) — canonical Wigner quasi-probability distribution `W(q, p) = (1/πℏ) ∫ ψ*(q + y) ψ(q − y) e^{2ipy/ℏ} dy` phase-space representation of a pure quantum state; Hudson, R. L., When is the Wigner quasi-probability density non-negative?, Rep. Math. Phys. 6, 249 (1974) — the universality theorem that ONLY Gaussian pure states have W ≥ 0 everywhere ⇒ the Gaussian floor 0 at every coherent state; Kenfack, A. & Życzkowski, K., Negativity of the Wigner function as an indicator of non-classicality, J. Opt. B: Quantum Semiclass. Opt. 6, 396 (2004) — closed-form negativity-volume table for Fock + cat states Fock-|1⟩ anchor 2/√e − 1; Cahill, K. E. & Glauber, R. J., Density operators and quasi-probability distributions, Phys. Rev. 177, 1882 (1969) — Fock-basis matrix elements of the Wigner function consumed by the Path B Fock-basis sum; Gerry, C. C. & Knight, P. L., Introductory Quantum Optics (Cambridge 2005) §3, §4, §7 — modern textbook treatment coherent / Fock / cat closed-form Wigner; Schrödinger, E., Naturwissenschaften 23, 807 (1935) — the canonical Schrödinger cat motif; Yurke, B. & Stoler, D., Phys. Rev. Lett. 57, 13 (1986) — the even/odd cat closed form consumed via shipped catState.ts; NIST DLMF §18.9 + Numerical Recipes §6.10 — generalized Laguerre polynomial recurrence consumed by BOTH Path A (k=0) and Path B (k=m−n) at the shared single source
Chern-Gauss-Bonnet integer-valuedness: chernIntegerCrossPinDiff ≡ 0 EXACT — Path A closed-form Berry-curvature Riemann sum ≡ Path B FHS 2005 link-variable plaquette at the rounded integer; |chernIntegerPathA| ≡ 1 EXACT at canonical topological (m=1, t=1) per the single-skyrmion winding number of d̂ : T² → S²
Anchor: |chernIntegerPathA| ≡ 1 topological magnitude at canonical (N_k=32, m=1, t=1) (Qi-Wu-Zhang 2006 cited single-skyrmion winding of d̂ : T² → S² inside the Chern-insulator phase 0 < |m|/|2t| < 1, dimensionless)value = 1Qi, X.-L., Wu, Y.-S. & Zhang, S.-C. — Phys. Rev. B 74, 085308 (2006) · doi:10.1103/PhysRevB.74.085308Sources: Qi, X.-L., Wu, Y.-S. & Zhang, S.-C., Topological quantization of the spin Hall effect in two-dimensional paramagnetic semi-conductors, Phys. Rev. B 74, 085308 (2006) — the canonical 2-band Bloch Hamiltonian `H(k) = d(k) · σ` with d_z = m + t(cos k_x + cos k_y) the simplest Chern-insulator substrate; Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M., Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982) — the canonical TKNN formula identifying σ_xy = (e²/h)·C with the first Chern number `C = (1/2π) ∫_BZ F(k) d²k`; Fukui, T., Hatsugai, Y. & Suzuki, H., Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn. 74, 1674 (2005) — the canonical link-variable lattice gauge field `F_link = arg(U_x · U_y' · U_x'^* · U_y^*)` integer-EXACT at any finite N_k; Bernevig, B. A. & Hughes, T. L., Topological Insulators and Topological Superconductors (Princeton University Press 2013) §8.5 — modern textbook treatment derives the closed-form 2×2 lower-band eigenvector at the spherical-angle parameterization; Chern, S.-S., Characteristic classes of Hermitian manifolds, Annals of Math. 47, 85 (1946) — the Chern-Gauss-Bonnet theorem that makes the Berry-curvature integral integer-valued; Klitzing, K. V., Dorda, G. & Pepper, M., New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980) — the experimental discovery of the integer quantum Hall effect that TKNN 1982 explains via the topological invariance; Haldane, F. D. M., Model for a quantum Hall effect without Landau levels: condensed-matter realization of the 'parity anomaly', Phys. Rev. Lett. 61, 2015 (1988) — the canonical Chern insulator existence proof in a 2D honeycomb lattice at zero external magnetic field (Haldane model itself deferred to Wave K.1 reusing THIS substrate); Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition shared substrate from the shipped hermitianEigendecomposition in entanglementEntropy.ts (single source extended in Wave E.1 Session A with eigenvector accumulation)
Haldane honeycomb-lattice Chern insulator at zero external field: chernIntegerCrossPinDiff ≡ 0 EXACT (Path A rhombic-BZ Berry-curvature Riemann sum ≡ Path B FHS 2005 link-variable plaquette at the rounded integer per Chern-Gauss-Bonnet); |chernIntegerPathA| ≡ 1 EXACT at canonical topological (M=0, t_1=1, t_2=0.5, φ=π/2) per Haldane 1988 valley closed form C = (1/2)[sign(h_z(K)) − sign(h_z(K'))]; phaseBoundaryMagnitudeOverT2 ≡ 3√3 EXACT at φ=π/2 and diracValleyGap ≡ 3√3 EXACT at canonical (M=0)
Anchor: |chernIntegerPathA| ≡ 1 topological magnitude at canonical (N_k=32, M=0, t_1=1, t_2=0.5, φ=π/2) (Haldane 1988 valley closed form C = (1/2)[sign(h_z(K)) − sign(h_z(K'))] = +1 at canonical, dimensionless)value = 1Haldane, F. D. M. — Phys. Rev. Lett. 61, 2015 (1988) · doi:10.1103/PhysRevLett.61.2015Sources: Haldane, F. D. M., Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the 'parity anomaly', Phys. Rev. Lett. 61, 2015 (1988) — the canonical honeycomb-lattice Chern insulator at zero external magnetic field ('anomalous' Hall effect via complex next-nearest-neighbor hopping breaking time-reversal symmetry); Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M., Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982) — the canonical TKNN formula identifying σ_xy = (e²/h)·C with the first Chern number `C = (1/2π) ∫_BZ F(k) d²k`; Fukui, T., Hatsugai, Y. & Suzuki, H., Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn. 74, 1674 (2005) — the canonical link-variable lattice gauge field integer-EXACT at any finite N_k reused VERBATIM from F.1; Bernevig, B. A. & Hughes, T. L., Topological Insulators and Topological Superconductors (Princeton University Press 2013) §8.5.2 — modern textbook treatment derives the 2-band Bloch Hamiltonian and the closed-form lower-band eigenvector from the spherical-angle parameterization; Chern, S.-S., Characteristic classes of Hermitian manifolds, Annals of Math. 47, 85 (1946) — the Chern-Gauss-Bonnet theorem that makes the Berry-curvature integral integer-valued; Chang, C.-Z. Et al., Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator, Science 340, 167 (2013) — the first experimental realization at Cr-doped (Bi,Sb)₂Te₃ thin films (cross-citation only); Jotzu, G. Et al., Experimental realization of the topological Haldane model with ultracold fermions, Nature 515, 237 (2014) — the cold-atom experimental realization at a shaken honeycomb optical lattice (cross-citation only); Numerical Recipes §11.1 — cyclic Jacobi eigendecomposition shared substrate from the shipped hermitianEigendecomposition in entanglementEntropy.ts (single source extended in Wave E.1 Session A with eigenvector accumulation)
Stark many-body localization on a tilted Heisenberg chain: measuredRTildeMbl ≡ citedPoissonRTilde = 2·ln 2 − 1 EXACT at canonical (F=4, N=10, Δ=1, α=0.01) deterministic single realization to tolerance 7e-2 absolute per Atas 2013 universality limit; citedWannierStarkLocalizationLengthAtFmbl ≡ J/Fmbl = 0.25 EXACT at canonical (Fmbl=4, J=1) per Wannier 1960 algebraic identity; citedStarkLadderSpacing(Fmbl) ≡ Fmbl = 4 EXACT; measuredSzZeroDim ≡ C(10, 5) = 252 EXACT (binomial); Schulz 2019 Fig. 1 monotone branch across F ∈ {0.4, 0.6, 1, 1.5, 2.5}; dual universality discrimination (ETH-side F=0.5 closer to GOE; MBL-side F=4 closer to Poisson)
Anchor: measuredRTildeMbl ≡ citedPoissonRTilde = 2·ln 2 − 1 ≈ 0.38629 at canonical (Fmbl=4, N=10, Δ=1, α=0.01) deterministic single realization (Schulz 2019 deep Stark MBL Poisson universality anchor + Atas 2013 closed-form limit, dimensionless level-statistics ratio)value = 0.386294Schulz, M., Hooley, C. A., Moessner, R. & Pollmann, F. — Phys. Rev. Lett. 122, 040606 (2019) · doi:10.1103/PhysRevLett.122.040606Sources: Schulz, M., Hooley, C. A., Moessner, R. & Pollmann, F., Stark Many-Body Localization, Phys. Rev. Lett. 122, 040606 (2019) — the canonical deterministic tilted-Heisenberg MBL claim the static linear Stark gradient produces MBL without disorder; Wannier, G. H., Wave Functions and Effective Hamiltonian for Bloch Electrons in an Electric Field, Phys. Rev. 117, 432 (1960) — the Wannier-Stark ladder single-particle closed forms ξ_WS(F) = J/F + ΔE_WS(F) = F algebraic identities at the tight-binding semi-infinite limit; Atas, Y. Y., Bogomolny, E., Giraud, O. & Roux, G., Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles, Phys. Rev. Lett. 110, 084101 (2013) — the exact closed-form consecutive-gap-ratio distribution reused VERBATIM from mbl-eth substrate; Pal, A. & Huse, D. A., Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010) — the iid-disordered Heisenberg chain MBL substrate and central-50% spectrum window convention reused VERBATIM; Bloch, F., Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. 52, 555 (1928) — the foundational tight-binding band-theory substrate that Wannier 1960 derives the ladder from; Anderson, P. W., Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 1492 (1958) — the foundational single-particle disorder-localization claim that the Wannier-Stark deterministic variant generalizes; Abanin, D. A. & Serbyn, M., Rev. Mod. Phys. 91, 021001 (2019) — the MBL review cross-citation; Numerical Recipes §11.1 — cyclic Jacobi shared substrate from the shipped jacobiEigenvalues in mblHeisenberg.ts (single source Wave C #10 cheap analysis-only slice #3)
Unruh effect on a uniformly accelerated observer: measuredPlanckOccupationAtCanonical ≡ citedPlanckOccupationAtCanonical = 1/(e^{2π} − 1) ≈ 0.001867 EXACT at canonical (a=1, ω=1) substrate-ULP via shipped twoModeSqueezing.perModeMean at the Rindler-wedge squeeze parameter r_ω = artanh(e^{-π}) per Crispino-Higuchi-Matsas RMP 2008 §III.C; citedUnruhTemperatureAtCanonical ≡ a/(2π) = 1/(2π) ≈ 0.15915 EXACT at canonical a=1 per Unruh 1976 natural-unit algebraic identity; citedBogoliubovUnitarity ≡ |α|² − |β|² = 1 EXACT algebraic identity at the squeeze bridge; monotone INCREASE across a-grid {0.5, 1, 2, 4, 8} at fixed ω=1 (cold-Wien to hot-Rayleigh-Jeans Planck-spectrum sweep); monotone DECREASE across ω-grid {0.5, 1, 2, 4, 8} at fixed a=1 (Bose-Einstein tail); dual regime discrimination (deep-Wien at (a=1, ω=4) closer to e^{-8π}; classical Rayleigh-Jeans at (a=8, ω=0.5) closer to a/(2πω))
Anchor: measuredPlanckOccupationAtCanonical ≡ 1/(e^{2π} − 1) ≈ 0.001867 at canonical (a=1, ω=1) natural units (Unruh 1976 Rindler-wedge Planck-spectrum Bose-Einstein occupation per mode; dimensionless mean photon number)value = 0.00187094Unruh, W. G. — Phys. Rev. D 14, 870 (1976) · doi:10.1103/PhysRevD.14.870Sources: Unruh, W. G., Notes on black-hole evaporation, Phys. Rev. D 14, 870 (1976) — the canonical Unruh-effect claim the Rindler-wedge Bogoliubov rotation produces a thermal Planck spectrum at the Unruh temperature T_U = a/(2π) (natural units); Davies, P. C. W., Scalar production in Schwarzschild and Rindler metrics, J. Phys. A 8, 609 (1975) — the independent earlier derivation via the detector-response function; Crispino, L. C. B., Higuchi, A., Matsas, G. E. A., The Unruh effect and its applications, Rev. Mod. Phys. 80, 787 (2008) §III.B-C — the modern review gives the Bogoliubov coefficients (3.49) |α_ω|² = 1/(1 − e^{-2πω/a}) + |β_ω|² = 1/(e^{2πω/a} − 1) and the Rindler-wedge squeeze-parameter bridge r_ω = artanh(e^{-πω/a}); Bisognano, J. J., Wichmann, E. H., On the duality condition for a Hermitian scalar field, J. Math. Phys. 17, 303 (1976) — the foundational modular-flow theorem at the right Rindler wedge generates boosts at the Unruh temperature; Cozzella, G., Landulfo, A. G. S., Matsas, G. E. A., Vanzella, D. A. T., Proposal for observing the Unruh effect using classical electrodynamics, Phys. Rev. Lett. 118, 161102 (2017) — theoretical detector-response proposal for a semi-classical laboratory test at circular acceleration; Walls, D. F., Milburn, G. J., Quantum Optics 2nd ed. §5 — the two-mode squeezed vacuum closed forms reused VERBATIM from the shipped twoModeSqueezing.ts (Wave B #6 single source); Hawking, S. W., Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975) — the Schwarzschild-horizon Bogoliubov substrate Q.2 will ship as a sibling Class-1 module
Hawking radiation from a Schwarzschild black hole: measuredHawkingOccupationAtCanonical ≡ citedHawkingOccupationAtCanonical = 1/(e^{8π} − 1) ≈ 1.216e-11 EXACT at canonical (M=1, ω=1) substrate-ULP via shipped Q.1 unruh.ts ⊕ Wave B #6 twoModeSqueezing.perModeMean at the Schwarzschild-horizon squeeze parameter r_ω = artanh(e^{-4π}) per Hawking 1975 §4 near-horizon Rindler approximation; citedHawkingTemperatureAtCanonical ≡ 1/(8πM) = 1/(8π) ≈ 0.03979 EXACT at canonical M=1 per Hawking 1975 natural-unit algebraic identity; citedSurfaceGravityAtCanonical ≡ 1/(4M) = 0.25 EXACT (Schwarzschild geometric units); citedBogoliubovUnitarity ≡ |α|² − |β|² = 1 EXACT algebraic identity at the squeeze bridge (inherited from Q.1); Schwarzschild ↔ Rindler bridge T_U(κ(M)) ≡ T_H(M) BYTE-IDENTICAL; T_H monotone DECREASE in M across {0.05, 0.1, 0.5, 1, 5} (Hawking 1975 §6 'small black holes are hotter'); ⟨n⟩ monotone DECREASE in M (inverse-mass monotonicity); monotone DECREASE in ω at fixed M=1 (Bose-Einstein tail); dual regime discrimination (deep-Wien supermassive at (M=5, ω=1) closer to e^{-40π}; classical Rayleigh-Jeans primordial at (M=0.05, ω=0.5) closer to 1/(8πMω))
Anchor: measuredHawkingOccupationAtCanonical ≡ 1/(e^{8π} − 1) ≈ 1.216e-11 at canonical (M=1, ω=1) natural units (Hawking 1975 Schwarzschild-horizon Planck-spectrum Bose-Einstein occupation per mode; dimensionless mean photon number)value = 1.21616e-11Hawking, S. W. — Commun. Math. Phys. 43, 199 (1975) · doi:10.1007/BF02345020Sources: Hawking, S. W., Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975) — the canonical Hawking-radiation claim the Schwarzschild-horizon Bogoliubov rotation produces a thermal Planck spectrum at the Hawking temperature T_H = κ/(2π) = 1/(8πM) (natural units); Bekenstein, J. D., Black holes and entropy, Phys. Rev. D 7, 2333 (1973) — the foundational black-hole entropy / horizon-area law that makes the Hawking temperature thermodynamically consistent (S_BH = A/4, A = 16πM² ⇒ T_H = 1/(8πM)); Wald, R. M., Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, U. Chicago Press (1994) — the canonical textbook derivation via algebraic QFT on the Schwarzschild background; Wald, R. M., The thermodynamics of black holes, Living Rev. Relativ. 4, 6 (2001) — modern review surveying the Hawking-Bekenstein-Unruh-Davies thermal-horizon unification; Crispino, L. C. B., Higuchi, A., Matsas, G. E. A., The Unruh effect and its applications, Rev. Mod. Phys. 80, 787 (2008) §III.E — the Schwarzschild ↔ Rindler near-horizon equivalence at a = κ = 1/(4M) that routes the Hawking spectrum through the Q.1 Unruh substrate; Steinhauer, J., Observation of quantum Hawking radiation and its entanglement in an analogue black hole, Nat. Phys. 12, 959 (2016) — the foundational BEC analogue Hawking radiation observation via supersonic horizon in a Bose-Einstein condensate; Walls, D. F., Milburn, G. J., Quantum Optics 2nd ed. §5 — the two-mode squeezed vacuum closed forms reused VERBATIM through the shipped Wave Q.1 unruh.ts ⊕ Wave B #6 twoModeSqueezing.ts single sources; Unruh, W. G., Phys. Rev. D 14, 870 (1976) — the Rindler-wedge Bogoliubov substrate Q.1 ships as a sibling Class-1 module the Hawking spectrum rides verbatim via the surface-gravity bridge
Gibbons-Hawking de Sitter radiation from a cosmological horizon: measuredGibbonsHawkingOccupationAtCanonical ≡ citedGibbonsHawkingOccupationAtCanonical = 1/(e^{2π} − 1) ≈ 0.001871 EXACT at canonical (H=1, ω=1) substrate-ULP via shipped Q.1 unruh.ts ⊕ Wave B #6 twoModeSqueezing.perModeMean at the de Sitter-horizon squeeze parameter r_ω = artanh(e^{-π}) per Gibbons-Hawking 1977 §II cosmological-horizon Killing-vector equivalence (identity substitution κ = H = a; MATCHES Q.1 anchor byte-for-byte); citedGibbonsHawkingTemperatureAtCanonical ≡ H/(2π) = 1/(2π) ≈ 0.15915 EXACT at canonical H=1 per Gibbons-Hawking 1977 natural-unit algebraic identity (MATCHES Q.1's T_U(a=1)); citedSurfaceGravityAtCanonical ≡ H = 1 EXACT (de Sitter geometric units identity substitution); citedBogoliubovUnitarity ≡ |α|² − |β|² = 1 EXACT algebraic identity at the squeeze bridge (inherited from Q.1); de Sitter ↔ Rindler bridge T_U(κ(H)) ≡ T_GH(H) BYTE-IDENTICAL; T_GH monotone INCREASE in H across {0.5, 1, 2, 4, 8} (Gibbons-Hawking 1977 §II H-linear monotonicity; OPPOSITE of Q.2's inverse-mass DECREASE; MATCHES Q.1's proper-acceleration INCREASE); ⟨n⟩ monotone INCREASE in H (H-linear monotonicity); monotone DECREASE in ω at fixed H=1 (Bose-Einstein tail); dual regime discrimination (deep-Wien near-Minkowski at (H=0.5, ω=4) closer to e^{-16π}; classical Rayleigh-Jeans inflationary at (H=8, ω=0.5) closer to H/(2πω))
Anchor: measuredGibbonsHawkingOccupationAtCanonical ≡ 1/(e^{2π} − 1) ≈ 0.001871 at canonical (H=1, ω=1) natural units (Gibbons-Hawking 1977 de Sitter-horizon Planck-spectrum Bose-Einstein occupation per mode; dimensionless mean photon number; MATCHES Q.1 anchor byte-for-byte per identity substitution κ = H = a)value = 0.00187094Gibbons, G. W., Hawking, S. W. — Phys. Rev. D 15, 2738 (1977) · doi:10.1103/PhysRevD.15.2738Sources: Gibbons, G. W., Hawking, S. W., Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15, 2738 (1977) — the canonical Gibbons-Hawking-radiation claim the de Sitter-horizon Bogoliubov rotation produces a thermal Planck spectrum at the Gibbons-Hawking temperature T_GH = κ/(2π) = H/(2π) (natural units); Bunch, T. S., Davies, P. C. W., Quantum field theory in de Sitter space: renormalization by point splitting, Proc. R. Soc. A 360, 117 (1978) — the foundational maximally-symmetric Bunch-Davies vacuum that makes the Gibbons-Hawking thermal identification algebraically canonical (Kubo-Martin-Schwinger thermal property along any timelike geodesic); Wald, R. M., Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, U. Chicago Press (1994) — the canonical textbook derivation via algebraic QFT on the de Sitter background; Wald, R. M., The thermodynamics of black holes, Living Rev. Relativ. 4, 6 (2001) — modern review surveying the Hawking-Bekenstein-Unruh-Davies-Gibbons thermal-horizon unification; Crispino, L. C. B., Higuchi, A., Matsas, G. E. A., The Unruh effect and its applications, Rev. Mod. Phys. 80, 787 (2008) §III — the generic surface-gravity equivalence at a = κ = H that routes the Gibbons-Hawking spectrum through the Q.1 Unruh substrate; Walls, D. F., Milburn, G. J., Quantum Optics 2nd ed. §5 — the two-mode squeezed vacuum closed forms reused VERBATIM through the shipped Wave Q.1 unruh.ts ⊕ Wave B #6 twoModeSqueezing.ts single sources; Unruh, W. G., Phys. Rev. D 14, 870 (1976) — the Rindler-wedge Bogoliubov substrate Q.1 ships as a sibling Class-1 module the Gibbons-Hawking spectrum rides verbatim via the surface-gravity identity bridge κ = H = a; Hawking, S. W., Commun. Math. Phys. 43, 199 (1975) — the Schwarzschild-horizon sibling Q.2 ships as the intermediate variant via the Schwarzschild ↔ Rindler bridge κ = 1/(4M)
Soliton breather (full Zakharov–Shabat family of the focusing NLS — Akhmediev a<½, Peregrine a=½, Kuznetsov–Ma a>½) — the rogue-wave peak amplification 1+2√(2a) on a continuous-wave background, →×3 at the Peregrine limit; the Kuznetsov–Ma member self-breathes at the cited rate T_KM = π/√(2a(2a−1))
Anchor: Peregrine peak amplitude amplification |ψ_P(0,0)|/|ψ₀| (dimensionless)value = 3Peregrine, D. H. — J. Aust. Math. Soc. Ser. B 25 (1983) 16 · doi:10.1017/S0334270000003891Sources: Peregrine, D. H., J. Aust. Math. Soc. Ser. B 25 (1983) 16 — rational breather of the focusing NLS, the rogue-wave prototype with peak amplitude 3× the background. Kibler, B. Et al., Nat. Phys. 6 (2010) 790 — fibre-optic measurement of the Peregrine soliton. Shrira, V. I. & Geogjaev, V. V., J. Eng. Math. 67 (2010) 11 — factor-of-3 amplification. Akhmediev, N. & Ankiewicz, A., Solitons (Chapman & Hall 1997) — breather family (Akhmediev / Kuznetsov–Ma / Peregrine limit). Substrate: the SHIPPED evolveScalarStep1D GP kernel (evolution.ts) + grossPitaevskii.ts totalNorm — single source, no new integrator.
Klein paradox / Klein tunneling — relativistic barrier transmission without exponential suppression; perfect transmission in the massless limit
Anchor: massless 1D Dirac barrier transmission T(mc²=0) (dimensionless; perfect Klein transmission)value = 1Katsnelson, M. I., Novoselov, K. S. & Geim, A. K. — Nat. Phys. 2 (2006) 620 · doi:10.1038/nphys384Sources: Klein, O., Z. Phys. 53 (1929) 157 — relativistic barrier penetration without exponential suppression (the Klein paradox). Dombey, N. & Calogeracos, A., Phys. Rep. 315 (1999) 41 — closed-form 1D Dirac square-barrier transmission. Katsnelson, M. I., Novoselov, K. S. & Geim, A. K., Nat. Phys. 2 (2006) 620 — perfect transmission T = 1 of a massless Dirac fermion (graphene Klein tunneling, normal incidence). Greiner, W., Relativistic Quantum Mechanics: Wave Equations (Springer 2000) §9 — spinor boundary matching. Griffiths, D. J., Introduction to Quantum Mechanics §2.6 — non-relativistic square-barrier foil. Substrate: the SHIPPED 1D Dirac dispersion + σ_x/σ_z algebra (zitterbewegung.ts) — single source, no new integrator.
Strang convergence residual (honest discretization knob)
Anchor: ‖ψ_one-step − ψ_two-half-steps‖₂ at recommendedDt (n=48 double-well, dimensionless)value = 0.0500000Strang, G. — SIAM J. Numer. Anal. 5, 506 (1968) · doi:10.1137/0705041Sources: Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5, 506 (1968) — second-order Trotter / Lie-Trotter-Strang split; Hatano & Suzuki, Lect. Notes Phys. 679, 37 (2005) — higher-order symplectic generalisations
Lindblad weak-coupling level-statistics invariance on PXP
Anchor: |⟨r̃⟩(H + γ·V_noise) − ⟨r̃⟩(H)| at γ = 0.005 (PXP N=8 OBC, dimensionless)value = 0.100000Lindblad, G. — Comm. Math. Phys. 48, 119 (1976) · doi:10.1007/BF01608499Sources: Lindblad, G., On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48, 119 (1976) — completely-positive maps; Daley, A. J., Quantum trajectories and open many-body quantum systems, Adv. Phys. 63, 77 (2014) — weak-coupling Lindblad spectral-perturbation regime; Atas, Bogomolny, Giraud & Roux, Phys. Rev. Lett. 110, 084101 (2013) — closed-form ⟨r̃⟩ means
MCWF trajectory ensemble ≡ Lindblad ensemble
Anchor: ρ_ee(t = 1/γ) = e^{−1} (2-level amplitude damping, dimensionless)value = 0.367879Breuer & Petruccione, Theory of Open Quantum Systems (2002), §3.4.4 · doi:10.1093/acprof:oso/9780199213900.001.0001Sources: Breuer & Petruccione, The Theory of Open Quantum Systems, §3.4.4 (amplitude damping closed form) + §6 (MCWF unraveling); Mølmer, Castin & Dalibard, J. Opt. Soc. Am. B 10, 524 (1993) — the Monte-Carlo wavefunction method; Daley, Adv. Phys. 63, 77 (2014) — quantum trajectories and open many-body systems
Engine ≡ reference (bit-exact)
Sources: Ephemera workletParity contract (bit-exact, not tolerance)