# Supplementary scripts for
#   P. Lopatto, "Supersymmetric Analysis of a Random Band Matrix at Criticality".
#
# The calculations used in the proofs are written out explicitly in the text.
# Supplementary scripts check selected identities numerically or symbolically.
# See Section 1.3, "Computations confirmed by computer". Each script is named in the paper
# where it is used; paths below are relative to this folder (anc/).
#
# These are selected diagnostic checks, not verification of the proofs. Some scripts report
# residuals without failing on discrepancies; inspect their output as well as their exit status.
#
# Requirements: Python 3 with numpy, scipy, sympy and mpmath.
# How to run: from the folder that contains the script, `python3 <script>.py`. The scripts in
#   grassmann/ and polar/ import sibling modules from their own folder.
# Statement numbers refer to the version of the paper distributed with these files; the label
# in brackets is the LaTeX label in the source.

requirements: [python3, numpy, scipy, sympy, mpmath]

files:
  - path: grassmann/grassmann.py
    kind: module
    description: Minimal Grassmann algebra with sympy coefficients, used by the other scripts in grassmann/.
  - path: grassmann/gue_check.py
    checks: Proposition 3.13 [p:grass-gue], identity (3.12) [e:grass-gue], the worked GUE example
    method: numerical, against the eigenvalue density for N <= 3 and Monte Carlo for N = 4
  - path: grassmann/berezin_change_check.py
    checks: the arrangement of the Jacobian supermatrix in Theorem 3.15 [t:grass-change]
    method: symbolic
  - path: grassmann/supergauss_check.py
    checks: Proposition 3.12 [p:grass-supergauss] for a (1|1) supermatrix
    method: symbolic
  - path: grassmann/supercharge_check.py
    checks: Definition 3.16 and Lemma 3.17 [d:grass-supercharge, l:grass-supercharge] for n = 1
    method: symbolic
  - path: rep_cycle_check.py
    checks: the identities for the Schafer--Wegner cycle, Definition 4.6 [d:rep-cycle] and Appendix B
    method: numerical
  - path: rep_homotopy_check.py
    checks: the real parts along the deformation, (4.14) [e:rep-homotopy], Proposition 4.8 [p:rep-contour] and Appendix B
    method: numerical
  - path: rep_dictionary_check.py
    checks: Lemma 4.14 [l:rep-dictionary], matrix and chain sides separately
    method: numerical (about a minute)
  - path: polar/grass.py
    kind: module
    description: Grassmann algebra and supermatrices with sympy coefficients, used by the other scripts in polar/.
  - path: polar/chart.py
    kind: module
    description: The polar chart of the rank-one target space.
  - path: polar/casimir.py
    kind: module
    description: The generator fields (5.6) [e:sigma-action] in polar coordinates and the Casimir operator; writes basis.pkl as a cache.
  - path: polar/check_regular_action.py
    checks: the parameterized regular-coordinate construction in Lemma 5.4 [l:sigma-action], its factorization and action on Q, changes of section, and odd leading-degree bounds
    method: exact rank-one algebra for all sixteen generators at two body representatives; rejects the former parameter-free construction
  - path: polar/check_polar_density.py
    checks: the divergence identity (6.20) [e:sigma-polar-divergence] behind the density d^-2 of Lemmas 6.2 and 6.3 [l:sigma-polar-density, l:sigma-radial-integral]; confirms d^-2 and rejects d^-1 and d^-3
    method: exact computer algebra
    output: polar/check_polar_density.log
  - path: polar/test_singlet.py
    checks: the values of the Casimir operator on radial superfunctions, Lemmas 6.7 and 6.8 [l:sigma-singlet, l:sigma-radial-part]; see Remark 6.9 [r:sigma-scripts]
    method: exact computer algebra at two rational points; asserts all 16 coefficients after the -1/2 normalization
  - path: polar/verify_tangency_exact.py
    checks: that the fields of casimir.py reproduce [A, Q] on all sixteen entries of Q, for all sixteen elementary supermatrices A; see Remark 10.10 [r:res-polar-check]
    method: exact computer algebra
  - path: polar/verify_op_exact.py
    checks: formula (10.22) of Lemma 10.7 [e:res-polar, l:res-polar]; see Remark 10.10 [r:res-polar-check]
    method: exact computer algebra (several minutes)
    output: polar/verify_op_exact.log
  - path: polar/verify_op.py
    kind: auxiliary
    description: A randomized version of verify_op_exact.py, imported by verify_sym.py.
  - path: polar/verify_sym.py
    kind: auxiliary
    description: A randomized symmetry check of the operator of verify_op.py.
  - path: coarse_constants.py
    checks: numerical constants of Sections 10 and 12, including the second moment of Lemma 12.12 [l:coarse-second-moment] used in Remark 12.13 [r:coarse-mean-physics]
    method: numerical quadrature (mpmath)
  - path: metallic_coefficients.py
    checks: the coefficients I^[0], ..., I^[3] of Lemma 13.8 [l:met-coefficients]; see Remark 13.9 [r:met-script]
    method: exact arithmetic (sympy)
  - path: crossover_ratio.py
    checks: computes the curve V(l)/l of Theorem 1.19 [t:crossover] drawn in Figure 1.2 [f:intro-crossover]
    method: numerical quadrature (scipy)
    output: crossover_ratio.tsv
  - path: tube_ordinary_check.py
    checks: the ordinary (non-super) case of the harmonicity of the tubular half-density; see Remark 16.4 [r:tube-harmonic-checks]
    method: numerical
  - path: tube_super_check.py
    checks: the super case of the same statement, with an exact Grassmann algebra; see Remark 16.4 [r:tube-harmonic-checks]
    method: numerical complex-coefficient checks in a finite Grassmann jet algebra at selected base points
  - path: tube_cross_identity.py
    checks: the cross identities behind the same statement; see Remark 16.4 [r:tube-harmonic-checks]; imports tube_super_check.py
    method: numerical complex-coefficient checks in a finite Grassmann jet algebra at random base points
  - path: tube_odd_wick.py
    checks: the odd Wick identity (16.42) [e:tube-odd-wick] for one pair of odd normal coordinates
    method: exact computer algebra
  - path: fock_dissipativity.py
    checks: finite numerical models of the gauged heat generator and odd charges discussed in Remark 16.11 [r:stab-fock]
    method: numerical