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29887 bytes, as of the version from 2026-10-27 01:07 (current). Recorded here, not run.
{
"tid": "T187",
"kind": "besselj",
"rationale": "Integer orders $\\nu=m$ label cylindrical Fourier modes $e^{im\\varphi}$. Positive half-integer orders $\\nu=\\ell+1/2$ correspond to spherical harmonic radial orders through the standard relation with spherical Bessel functions. These are structural order families, not a rational order mesh. The prefixes $m,\\ell=0,\\ldots,5$ and six positive critical arguments per order are explicit finite lookup depths, not canonical cutoffs. Each argument $c_k$ is the unique zero of the derivative of the tabulated cylindrical Bessel function in its stated rational interval. The entries are extremal heights, not zero locations. The labels enumerate the isolating intervals; a global root-count proof is not claimed. At order zero the zero of $J_0^{\\prime}$ at the origin is excluded. For half-integer order these are extrema of $J_\\nu$ or $Y_\\nu$, not of the differently scaled spherical radial functions.",
"unresolved": "No application-specific demand is asserted for every individual height. Both order families and the stationary arguments have intrinsic justification; the finite lookup depths are stated choices. This certificate proves isolation within each defining interval, not a global ordinal count.",
"sources": [
"https://dlmf.nist.gov/10.73",
"https://dlmf.nist.gov/10.47",
"https://dlmf.nist.gov/10.21",
"https://dlmf.nist.gov/10.6",
"https://dlmf.nist.gov/10.51"
],
"rows": [
{
"params": {
"nu": "0",
"x": "critical-1"
},
"argument": "critical",
"bracket": [
"3923/1024",
"981/256"
],
"label": "$c_{1}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
{
"params": {
"nu": "0",
"x": "critical-2"
},
"argument": "critical",
"bracket": [
"7183/1024",
"449/64"
],
"label": "$c_{2}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
{
"params": {
"nu": "0",
"x": "critical-3"
},
"argument": "critical",
"bracket": [
"10417/1024",
"5209/512"
],
"label": "$c_{3}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
{
"params": {
"nu": "0",
"x": "critical-4"
},
"argument": "critical",
"bracket": [
"13643/1024",
"3411/256"
],
"label": "$c_{4}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
{
"params": {
"nu": "0",
"x": "critical-5"
},
"argument": "critical",
"bracket": [
"16865/1024",
"8433/512"
],
"label": "$c_{5}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
{
"params": {
"nu": "0",
"x": "critical-6"
},
"argument": "critical",
"bracket": [
"10043/512",
"20087/1024"
],
"label": "$c_{6}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
{
"params": {
"nu": "1/2",
"x": "critical-1"
},
"argument": "critical",
"bracket": [
"1193/1024",
"597/512"
],
"label": "$c_{1}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
{
"params": {
"nu": "1/2",
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"label": "$c_{2}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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{
"params": {
"nu": "1/2",
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"label": "$c_{3}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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{
"params": {
"nu": "1/2",
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"label": "$c_{4}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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{
"params": {
"nu": "1/2",
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"label": "$c_{5}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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{
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"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{1}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{2}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{3}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{4}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"params": {
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"label": "$c_{5}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{1}$",
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"label": "$c_{3}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{4}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{5}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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{
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"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{1}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{3}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{1}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{2}$",
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"label": "$c_{3}$",
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"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{4}$",
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"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{3}$",
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"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{3}$",
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"label": "$c_{4}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{3}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{1}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{2}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{3}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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{
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"label": "$c_{4}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"params": {
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"label": "$c_{5}$",
"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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{
"params": {
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"reason": "Integer order labels a cylindrical Fourier mode; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
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"label": "$c_{1}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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"label": "$c_{2}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
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{
"params": {
"nu": "11/2",
"x": "critical-3"
},
"argument": "critical",
"bracket": [
"14981/1024",
"7491/512"
],
"label": "$c_{3}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
{
"params": {
"nu": "11/2",
"x": "critical-4"
},
"argument": "critical",
"bracket": [
"18409/1024",
"9205/512"
],
"label": "$c_{4}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
{
"params": {
"nu": "11/2",
"x": "critical-5"
},
"argument": "critical",
"bracket": [
"10883/512",
"21767/1024"
],
"label": "$c_{5}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
},
{
"params": {
"nu": "11/2",
"x": "critical-6"
},
"argument": "critical",
"bracket": [
"6271/256",
"25085/1024"
],
"label": "$c_{6}$",
"reason": "Half-integer order corresponds to a spherical harmonic radial order; the positive argument is an isolated derivative zero, so the stored value is an intrinsic extremal height."
}
]
}