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9755170 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 | digraph FloquetV2 {
rankdir=LR;
node [shape=box];
"foundation:maxwell" [label="Stated nondispersive canonical Maxwell model and positive material parameters."];
"foundation:projective" [label="Direct polar identities, common positive angular velocity, and projective winding."];
"foundation:ode" [label="Bounded measurable linear ODEs, determinant one, continuity of transfer matrices."];
"foundation:fractional" [label="Exact fractional-linear pointwise subtraction and derivative identities."];
"foundation:segments" [label="Explicit constant-generator transfer matrices and their products."];
"foundation:convex" [label="Compact convex moment fibers, finite-dimensional support, and weak-* compactness."];
"foundation:linear_algebra" [label="Cayley\u2013Hamilton, Vandermonde, Hankel, and singular-value identities."];
"foundation:inverse_function" [label="Inverse/constant-rank theorem and local derivative estimates."];
"foundation:analytic_jets" [label="Analytic transfer family and finite polynomial/exponential expansions."];
"external:real_QE" [label="Effective real-closed-field quantifier elimination; Collins CAD is an external algorithmic input."];
"lem:linearize" [label="3.1 An exact, unrelaxed linearization"];
"lem:boundary" [label="3.2 Boundary moments cannot carry positive gain"];
"lem:normal" [label="3.3 Finite normal duals at every interior moment"];
"thm:resource" [label="4.1 Global resource-constrained extremality and equality"];
"thm:resourceDefect" [label="4.2 Resource defect and unique dual parameters"];
"thm:concavity" [label="5.1 Strict concavity and at most two adjacent winning windings"];
"thm:resourceFree" [label="5.2 Free-period optimum with a fixed long-run resource"];
"lem:margin" [label="6.1 A two-multiplier switching margin"];
"thm:phasestability" [label="6.2 Phase and clock stability from the exact defect"];
"thm:globalstability" [label="6.3 Global resource-constrained square-root rigidity"];
"prop:resourceSharp" [label="6.4 Sharpness while preserving the resource exactly"];
"thm:main" [label="7.1 Preserved exact bounds-only optimum"];
"thm:defect" [label="7.2 Preserved bounds-only exact defect"];
"cor:boundstability" [label="7.3 Preserved bounds-only global stability"];
"prop:sharp" [label="7.4 The exponent $1/2$ cannot generally be improved"];
"thm:free" [label="8.1 Sharp growth per unit time"];
"thm:quantum" [label="9.1 Exact finite-cycle and asymptotic quantum laws"];
"cor:quantumargmax" [label="9.2 Resource-constrained classical--quantum extremality"];
"prop:quantumfinite" [label="9.3 Finite-cycle photon optimization is a different problem"];
"thm:quantumstability" [label="9.4 Finite-observation quantum rigidity"];
"lem:boxselector" [label="10.1 The exact corner selector"];
"lem:boxclock" [label="10.2 A strictly monotone clock"];
"thm:boxfixed" [label="10.3 Complete fixed-period Maxwell corner-purity theorem"];
"cor:boxfree" [label="10.4 Free-period rectangle rate and quantum optimum"];
"prop:boxstability" [label="10.5 Rectangle structural stability"];
"lem:polylinear" [label="11.1 Fractional barycentric linearization"];
"thm:polyresource" [label="11.2 Affine-resource calibration, purity, and finite switching"];
"thm:abstract" [label="11.3 Preserved calibrated rotating-polytope extremality"];
"thm:boxresource" [label="12.1 Explicit two-resource Maxwell extremality"];
"lem:boxnormal" [label="12.2 Boundary moments and normal duals for two Maxwell resources"];
"lem:boxpalindrome" [label="12.3 Time reversal locks the two diagonal visits"];
"prop:boxresourceconcavity" [label="12.4 Resource-ray concavity and rate calibration"];
"cor:boxresourceStability" [label="12.5 Two-resource Maxwell stability and quantum extremality"];
"prop:inversefailure" [label="13.1 Exact two-layer isodiscriminant counterexample"];
"thm:inverse" [label="14.1 Finite derivatives determine every reduced layer"];
"lem:prony" [label="14.2 Finite exponential determination"];
"lem:peel" [label="14.3 Recovering the leading temporal layer"];
"prop:datacount" [label="15.1 Preserved regular-data lower bound"];
"lem:projection" [label="15.2 A direct generic projection bound"];
"thm:jetCompression" [label="15.3 Linear-count compression of exact transfer data"];
"prop:CAD" [label="15.4 Certified projection search for algebraic inputs"];
"prop:localInverse" [label="15.5 Local inverse conditioning at regular points"];
"thm:rigidityCompression" [label="15.6 Saturation is an inverse certificate"];
"prop:nearNoGo" [label="15.7 No finite exact inverse for the entire near-extremal class"];
"foundation:projective" -> "lem:linearize";
"foundation:fractional" -> "lem:linearize";
"lem:linearize" -> "lem:boundary";
"foundation:convex" -> "lem:boundary";
"lem:linearize" -> "lem:normal";
"lem:boundary" -> "lem:normal";
"foundation:convex" -> "lem:normal";
"lem:normal" -> "thm:resource";
"foundation:fractional" -> "thm:resource";
"foundation:segments" -> "thm:resource";
"thm:resource" -> "thm:resourceDefect";
"lem:normal" -> "thm:resourceDefect";
"foundation:fractional" -> "thm:resourceDefect";
"lem:normal" -> "thm:concavity";
"thm:resource" -> "thm:concavity";
"foundation:convex" -> "thm:concavity";
"thm:concavity" -> "thm:resourceFree";
"thm:resourceDefect" -> "thm:resourceFree";
"foundation:fractional" -> "lem:margin";
"lem:margin" -> "thm:phasestability";
"thm:resourceDefect" -> "thm:phasestability";
"foundation:projective" -> "thm:phasestability";
"thm:phasestability" -> "thm:globalstability";
"thm:resource" -> "thm:globalstability";
"foundation:segments" -> "prop:resourceSharp";
"thm:resource" -> "prop:resourceSharp";
"foundation:projective" -> "thm:main";
"foundation:fractional" -> "thm:main";
"foundation:segments" -> "thm:main";
"thm:main" -> "thm:defect";
"foundation:fractional" -> "thm:defect";
"thm:main" -> "cor:boundstability";
"thm:defect" -> "cor:boundstability";
"thm:phasestability" -> "cor:boundstability";
"foundation:segments" -> "prop:sharp";
"thm:main" -> "prop:sharp";
"thm:main" -> "thm:free";
"thm:defect" -> "thm:free";
"foundation:maxwell" -> "thm:quantum";
"foundation:linear_algebra" -> "thm:quantum";
"thm:quantum" -> "cor:quantumargmax";
"thm:resource" -> "cor:quantumargmax";
"thm:globalstability" -> "cor:quantumargmax";
"foundation:segments" -> "prop:quantumfinite";
"thm:quantum" -> "prop:quantumfinite";
"thm:resource" -> "prop:quantumfinite";
"thm:quantum" -> "thm:quantumstability";
"thm:globalstability" -> "thm:quantumstability";
"foundation:ode" -> "thm:quantumstability";
"foundation:maxwell" -> "lem:boxselector";
"foundation:fractional" -> "lem:boxselector";
"lem:boxselector" -> "lem:boxclock";
"lem:boxclock" -> "thm:boxfixed";
"lem:boxselector" -> "thm:boxfixed";
"foundation:projective" -> "thm:boxfixed";
"thm:boxfixed" -> "cor:boxfree";
"thm:quantum" -> "cor:boxfree";
"thm:boxfixed" -> "prop:boxstability";
"thm:phasestability" -> "prop:boxstability";
"thm:quantumstability" -> "prop:boxstability";
"foundation:projective" -> "lem:polylinear";
"foundation:fractional" -> "lem:polylinear";
"lem:polylinear" -> "thm:polyresource";
"foundation:convex" -> "thm:polyresource";
"thm:polyresource" -> "thm:abstract";
"lem:polylinear" -> "lem:boxnormal";
"foundation:convex" -> "lem:boxnormal";
"lem:boxselector" -> "lem:boxnormal";
"lem:boxnormal" -> "lem:boxpalindrome";
"foundation:segments" -> "lem:boxpalindrome";
"foundation:projective" -> "lem:boxpalindrome";
"lem:boxnormal" -> "prop:boxresourceconcavity";
"lem:boxpalindrome" -> "prop:boxresourceconcavity";
"foundation:convex" -> "prop:boxresourceconcavity";
"lem:boxnormal" -> "thm:boxresource";
"lem:boxpalindrome" -> "thm:boxresource";
"prop:boxresourceconcavity" -> "thm:boxresource";
"foundation:projective" -> "thm:boxresource";
"thm:boxresource" -> "cor:boxresourceStability";
"prop:boxstability" -> "cor:boxresourceStability";
"thm:quantum" -> "cor:boxresourceStability";
"thm:quantumstability" -> "cor:boxresourceStability";
"foundation:segments" -> "prop:inversefailure";
"foundation:linear_algebra" -> "lem:prony";
"foundation:analytic_jets" -> "lem:peel";
"foundation:segments" -> "lem:peel";
"lem:prony" -> "thm:inverse";
"lem:peel" -> "thm:inverse";
"foundation:analytic_jets" -> "thm:inverse";
"thm:inverse" -> "prop:datacount";
"foundation:inverse_function" -> "prop:datacount";
"foundation:inverse_function" -> "lem:projection";
"thm:inverse" -> "thm:jetCompression";
"lem:projection" -> "thm:jetCompression";
"prop:datacount" -> "thm:jetCompression";
"thm:jetCompression" -> "prop:CAD";
"foundation:analytic_jets" -> "prop:CAD";
"external:real_QE" -> "prop:CAD";
"prop:datacount" -> "prop:localInverse";
"thm:inverse" -> "prop:localInverse";
"thm:jetCompression" -> "prop:localInverse";
"foundation:inverse_function" -> "prop:localInverse";
"thm:resource" -> "thm:rigidityCompression";
"thm:concavity" -> "thm:rigidityCompression";
"thm:quantum" -> "thm:rigidityCompression";
"thm:boxresource" -> "thm:rigidityCompression";
"thm:resource" -> "prop:nearNoGo";
"prop:datacount" -> "prop:nearNoGo";
"foundation:ode" -> "prop:nearNoGo";
}
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