Сatching function

C(n) = (n+1)-catching point of FGH and SGH
C(Ω) = α↦C(α)
C(Ω+n) = n-catching point after α↦C(α)
C(Ω×2) = n-fixed point of α↦C(α)
C1(n) = FS of C(n) on Ω
C1(Ω) = α↦C1(C(α))
C12) = α↦C1(α)
C2(n) = FS of C(n) on Ω2
and so on up to C(Ωω) and beyond

FS of C(n) on Ω means "fundamental sequences of ordinal C(n) in some* well-ordering ordinal notation capable of describing it, applied to Ω"
*e.g. TON

PDAN

C(0) = ψ(Ωω {1{1{1,,1,2}2,,2}2} = {1,,1,2} = {1{1,,,2}1,2} ~ П11-CA0
ψ(Ωω)+1 {2,,1,2} = {2{1,,,2}1,2}
ψ(Ωω+ ψ1ω1ω+Ω))) {1{1,,2}2,,1,2} = {1{1{1,,,2}2}2{1,,,2}1,2}
ψ(Ωω×2) {1{1,,1,2}2,,1,2} = {1{1{1,,,2}1,2}2{1,,,2}1,2}
ψ(ΩωΩω) {1{1{1,,1,2}2,,1,2}2,,1,2} = {1{1{1{1,,,2}1,2}2{1,,,2}1,2}2{1,,,2}1,2}
ψ(Ωω+1) {1,,2,2} = {1{1,,,2}2,2} ~ П11-CA0+BI
ψ(Ωω×2) {1,,1,3} = {1{1,,,2}1,3}
C(1) = ψ(ΩΩω) {1,,1{1,,1,2}2} = {1{1,,,2}1{1{1,,,2}1,2}2}
C(2) = ψ(ΩΩΩω) {1,,1{1,,1{1,,1,2}2}2} = {1{1,,,2}1{1{1,,,2}1{1{1,,,2}1,2}2}2}
C(3) = ψ(ΩΩΩΩω) {1,,1{1,,1{1,,1{1,,1,2}2}2}2} = {1{1,,,2}1{1{1,,,2}1{1{1,,,2}1{1{1,,,2}1,2}2}2}2}
C(ω) = ψ(ψI(0)) = ψ(M) {1,,1,,2} = {1{1,,,2}1{1,,,2}2} ~ П11-TR0
C(ε0) = ψ(εI+1) = KPI {1,,2,,2} = {1{1,,,2}2{1,,,2}2}
C(ψ(Ωω)) = C(C(0)) = ψ(ΩI+ω) {1,,1,2,,2} = {1{1,,,2}1,2{1,,,2}2}
C(Ω) = ψ(Iω) = ψ(M×ω) {1,,1,,1,2} = {1{1,,,2}1{1,,,2}1,2}
C(Ω+1) = ψ(IΩω) = ψ(M×Ωω) {1,,1,,1{1,,1,2}2} = {1{1,,,2}1{1,,,2}1{1{1,,,2}1,2}2}
C(Ω×2) = ψ(IIω) = ψ(M×χ(M×ω)) {1,,1,,1{1,,1,,1,2}2} = {1{1,,,2}1{1,,,2}1{1{1,,,2}1{1,,,2}1,2}2}
C(Ω×ω) = ψ(ψI(2,0)(0)) = ψ(M2) {1,,1,,1,,2} = {1{1,,,2}1{1,,,2}1{1,,,2}2}
C(Ω2) = ψ(I(2,ω)) = ψ(M2×ω) {1,,1,,1,,1,2} = {1{1,,,2}1{1,,,2}1{1,,,2}1,2}
C(Ω3) = ψ(I(3,ω)) = ψ(M3×ω) {1,,1,,1,,1,,1,2} = {1{1,,,2}1{1,,,2}1{1,,,2}1{1,,,2}1,2}
C(Ω4) = ψ(I(4,ω)) = ψ(M4×ω) {1,,1,,1,,1,,1,,1,2} = {1{1,,,2}1{1,,,2}1{1,,,2}1{1,,,2}1{1,,,2}1,2}
C(Ωω) = ψ(ψI(ω,0)(0)) = ψ(Mω) {1,,1,,1,,1,,1,,1 ... 2} = {1{2,,}2} = {1{2,,,2}2}
C(Ωω×ω) = ψ(ψI(1,0,0)(0)) = ψ(MM) {1{1,,2,,}2} = {1{1{1,,,2}2,,,2}2}
C(Ωε0) = ψ(εM+1) = KPM {1,,2{1,,2,,}2} = {1{1,,,2}2{1{1,,,2}2,,,2}2}
C(ΩΩ) = ψ(Mω) {1{1,,2,,}1,2}  = {1{1{1,,,2}2,,,2}1,2}
C(ΩΩω) = ψ(ω-M) {1{1,,1,2,,}1{1,,1,2,,}1,,2} = {1{1{1,,,2}1,2,,,2}1{1{1,,,2}1,2,,,2}1{1,,,2}2}
C(ΩΩω×ω) = ψ(K) {1{1,,1,,2,,}2} = {1{1{1,,,2}1{1,,,2}2,,,2}2}
C(ΩΩε0) = ψ(ΩK+1) = KP+П3 {1,,2{1,,1,,2,,}2} = {1{1,,,2}2{1{1,,,2}1{1,,,2}2,,,2}2}
C(ΩΩΩ) = ψ(Kω) {1{1,,1,,2,,}1,2} = {1{1{1,,,2}1{1,,,2}2,,,2}1,2}
C(ΩΩΩω) = ψ(ω-K) {1{1{2,,}2,,}2} = {1{1{2,,,2}2,,,2}2}
C(ΩΩΩω×ω) = ψ(П12) {1{1{1,,2,,}2,,}2} = {1{1{1{1,,,2}2,,,2}2,,,2}2}
C(ΩΩΩε0) = KP+П4 {1,,2{1{1,,2,,}2,,}2} = {1{1,,,2}2{1{1{1,,,2}2,,,2}2,,,2}2}
C(ΩΩΩΩ) = ψ(ω-th П12) {1{1{1,,2,,}2,,}1,2} = {1{1{1{1,,,2}2,,,2}2,,,2}1,2}
C(ΩΩΩΩω) = ψ(ω-П12) {1{1{1{1,,1,2,,}1{1,,1,2,,}1,,2,,}2,,}2} = {1{1{1{1{1,,,2}1,2,,,2}1{1{1,,,2}1,2,,,2}1{1,,,2}2,,,2}2,,,2}2}
C(ΩΩΩΩω×ω) = ψ(П13) {1{1{1,,1,,2,,}2,,}2} = {1{1{1{1,,,2}1{1,,,2}2,,,2}2,,,2}2}
C(ΩΩΩΩε0) = KP+П5 {1,,2{1{1,,1,,2,,}2,,}2} = {1{1,,,2}2{1{1{1,,,2}1{1,,,2}2,,,2}2,,,2}2}
C(ΩΩΩΩΩ) = ψ(ω-th П13) {1{1{1,,1,,2,,}2,,}1,2} = {1{1{1{1,,,2}1{1,,,2}2,,,2}2,,,2}1,2}
C(ΩΩΩΩΩω) = ψ(ω-П13) {1{1{1{2,,}2,,}2,,}2} = {1{1{1{2,,,2}2,,,2}2,,,2}2}
C(ΩΩΩΩΩω×ω) = ψ(П14) {1{1{1{1,,2,,}2,,}2,,}2} = {1{1{1{1{1,,,2}2,,,2}2,,,2}2,,,2}2}
C(εΩ+1) = ψ(П1n) = KP+Пn {1{1{1{1 ... {1,,2,,} ... 2,,}2,,}2,,}2} = {1{1{1{1 ... {1{1,,,2}2,,,2} ... 2,,,2}2,,,2}2,,,2}2} = {1{1{1,,,3}2,,,2}2}

SDAN

C(εΩ+1) = C(ψ12)) {1{1',,2,,}2} = {1{1{1{1 ... {1,,2,,} ... 2,,}2,,}2,,}2} = {1{1,,'2,,}2} = {1{1{1{1,,,3}2,,,2}2,,,2}2} = {1{1{1,,,3}2,,,2}2}
C(ψ12ω)) {1{1',,1',,1',,1',,1',, ... 2,,}2} = {1{1{2',,}2,,}2} = {1{1{2{1,,,3}2,,,2}2,,,2}2} = {1{2{1,,,3}2,,,2}2}
C(ψ123))) {1{1{1'',,2',,}2,,}2} = {1{1,,'3,,}2} = {1{1{1{1{1,,,3}3,,,2}2{1,,,3}2,,,2}2,,,2}2} = {1{1{1,,,3}3,,,2}2}
C(ψ123ω))) {1{1{1'',,1'',,1'',,1'',,1'',, ... 2',,}2,,}2} = {1{1{1{2'',,}2',,}2,,}2} = {1{1{1{2{1,,,3}3,,,2}2{1,,,3}2,,,2}2,,,2}2} = {1{2{1,,,3}3,,,2}2}
C(ψ1234)))) {1{1{1{1''',,2'',,}2',,}2,,}2} = {1{1,,'4,,}2} = {1{1{1{1{1{1,,,3}4,,,2}2{1,,,3}3,,,2}2{1,,,3}2,,,2}2,,,2 }2} = {1{1{1,,,3}4,,,2}2}
C(ψ1ω)) = C(C1(0)) {1{1,,'1,2,,}2} = {1{1{1,,,3}1,2,,,2}2}
C(ψ1I(0)))= C(C1(ω)) {1{1,,'1,,'2,,}2} = {1{1{1,,,3}1{1,,,3}2,,,2}2}
C(ψ1I+ω)) = C(C1(ψ(Ωω))) = C(C1(C(0))) {1{1,,'1,2,,'2,,}2} = {1{1{1,,,3}1,2{1,,,3}2,,,2}2}
C(ψ1(Iω) = C(C1(C(Ω))) {1{1,,'1,,'1,2,,}2} = {1{1{1,,,3}1{1,,,3}1,2,,,2}2}
C(ψ1I(ω,0)(0)))) = C(C1(C(Ωω))) {1{1,,'1,,'1,,'1,,'1,,' ... 2,,}2} = {1{1{2,,'}2,,}2} = {1{1{2,,,3}2,,,2}2}
C(ψ1(MM)) = C(C1(C(Ωω×ω))) {1{1{1,,'2,,'}2,,}2} = {1{1{1{1,,,3}2,,,3}2,,,2}2}
C(ψ1(ω-M)) = C(C1(C(ΩΩω))) {1{1{1,,'1,2,,'}1{1,,'1,2,,'}1,,2,,}2} = {1{1{1{2,,,3}1,2,,,3}1{1{2,,,3}1,2,,,3}1{1,,,3}2,,,2}2}
C(ψ1(K)) = C(C1(C(ΩΩω×ω))) {1{1{1,,'1,,'2,,'}2,,}2} = {1{1{1{1,,,3}1{1,,,3}2,,,3}2,,,2}2}
C(ψ1(ω-K)) = C(C1(C(ΩΩΩω))) {1{1{1,,'1,,'1,,'1,,'1,,' ... 2,,'}2,,}2} = {1{1{1{2,,'}2,,'}2,,}2} = {1{1{1{2,,,3}2,,,3}2,,,2}2}
C(ψ112)) = C(C1(C(ΩΩΩω×ω))) {1{1{1{1,,'2,,'}2,,'}2,,}2} = {1{1{1{1{1,,,3}2,,,3}2,,,3}2,,,2}2}
C(ψ11n)) = C(C1(C(εΩ+1))) {1{1{1{1{1 ... {1,,'2,,'} ... 2,,'}2,,'}2,,'}2,,}2} = {1{1,,''2,,}2} = {1{1{1,,,4}2,,,2}2}
C(C1(C(C1(C(Ωω))))) {1{1,,''1,,''1,,''1,,''1,,'' ... 2,,}2} = {1{1{2,,''}2,,}2 = {1{1{2,,,4}2,,,2}2
C(C1(C(C1(C(Ωω×ω))))) {1{1{1{1,,''2,,''}2,,''}2,,}2} = {1{1{1{1,,,4}2,,,4}2,,,2}2}
C(C1(C(C1(C(ΩΩΩω))))) {1{1{1,,''1,,''1,,''1,,''1,,'' ... 2,,''}2,,}2} = {1{1{1{2,,''}2,,''}2,,}2} = {1{1{1{2,,,4}2,,,4}2,,,2}2}
C(C1(C(C1(C(ΩΩΩω×ω))))) {1{1{1{1,,''2,,''}2,,''}2,,}2} = {1{1{1{1{1,,,4}2,,,4}2,,,4}2,,,2}2}
C(C1(C(C1(C(εΩ+1)))))) {1{1{1{1{1 ... {1,,''2,,''} ... 2,,''}2,,''}2,,''}2,,}2} = {1{1,,'''2,,}2} = {1{1{1,,,5}2,,,2}2}
C(C1(C(C1(C(C1(C(εΩ+1)))))))) {1{1,,''''2,,}2} = {1{1{1,,,6}2,,,2}2}
C(C1(C(C1(C(C1(C(C1(C(εΩ+1)))))))))) {1{1,,'''''2,,}2} = {1{1{1,,,7}2,,,2}2}
C(C1(Ω)) = C(C1(C(C1(C(C1(C(...)))))))) {1{1{1,,,1,2}2,,,2}2}

DAN

C(C1(Ω)) {1{1{1,,,1,2}2,,,2}2} = {1,,,1,2} = {1{1,,,,2}1,2}
C(C1(Ω)+C1(C(C1(Ω)))×ω) {1,,,1,,2} = {1,,,1{1,,,2}2} = {1{1,,,,2}1{1{1,,,,2}2}2}
C(C1(Ω)×2) {1,,,1{1,,,1,2}2} = {1{1,,,,2}1{1{1,,,,2}1,2}2}
C(C1(Ω)×3) {1,,,1{1,,,1{1,,,1,2}2}2} = {1{1,,,,2}1{1{1,,,,2}1{1{1,,,,2}1,2}2}2}
C(C1(Ω)×4) {1,,,1{1,,,1{1,,,1{1,,,1,2}2}2}2} = {1{1,,,,2}1{1{1,,,,2}1{1{1,,,,2}1{1{1,,,,2}1,2}2}2}2}
C(C1(Ω)×ω) {1,,,1,,,2} = {1{1,,,,2}1{1,,,,2}2}
C(C1(Ω)×εΩ+1) {1,,,2,,,2}
C(C1(Ω)2) {1,,,1,,,1,2}
C(C1(Ω)3) {1,,,1,,,1{1,,2}2}
C(C1(Ω)4) {1,,,1,,,1{1,,1,,2}2}
C(C1(Ω)ω) {1,,,1,,,1{1{1{2,,}2}2}2}
C(εС1(Ω)+1) {1,,,1,,,1{1{1{1,,,3}2}2}2}
C(C1(C(C1(Ω)+1))) {1,,,1,,,1{1{1{1,,,3}1,2}2}2}
C(C1(Ω+1)) {1,,,1,,,1{1{1{1,,,1,2}2}2}2}
C(C1(C1(0))) {1,,,1,,,1{1{1{1,,,1,,,1{1{1{1,,,3}1,2}2}2}2}2}2}
C(C1(C1(C1(0)))) {1,,,1,,,1{1{1{1,,,1,,,1{1{1{1,,,1,,,1{1{1{1,,,3}1,2}2}2}1,2}2}2}2}2}2}
C(Ω2) = C(C12)) = C(C1(C1(C1(C1(...)))))))) {1,,,1,,,1,,2}
C(Ω2×ω) {1,,,1,,,1,,,2}
C(Ω2ω) {1,,,1,,,1,,,1,,,1,,,1 ... 2} = {1{2,,,}2} = {1{2,,,,2}2}
C(Ω2ω×ω) {1{1,,,2,,,}2} = {1{1{1,,,,2}2,,,,2}2}
C(Ω2Ω2Ω2ω×ω) {1{1{1,,,2,,,}2,,,}2} = {1{1{1{1,,,,2}2,,,,2}2,,,,2}2}
C(C1Ω2+1)) {1{1{1{1 ... {1,,,2,,,} ... 2,,,}2,,,}2,,,}2} = {1{1{1,,,,3}2,,,,2}2}
C(C2(C1Ω2+1))) {1{1{1,,,,4}2,,,,2}2}
C(C2(C1(C2(C1Ω2+1)))))) {1{1{1,,,,5}2,,,,2}2}
C(C1(C22))) = C(C2(C1(C2(C1(C2(C1(...)))))))) {1{1{1,,,,1,2}2,,,,2}2} = {1,,,,1,2} = {1{1,,,,,2}1,2}
C(Ω3) {1,,,,1,,,,1,,,2}
C(Ω3×ω) {1,,,,1,,,,1,,,,2}
C(Ω3ω) {1{2,,,,,2}2}
C(C1(C2(C33))) {1{1{1,,,,,1,2}2,,,,2}2} = {1,,,,,1,2} = {1{1,,,,,,2}1,2}
C(Ω4) {1,,,,,1,,,,,1,,,,2}
C(Ω4×ω) {1,,,,,1,,,,,1,,,,,2}
C(Ω4ω) {1{2,,,,,,2}2}
C(C1(C2(C3(C44))))) {1{1{1,,,,,,1,2}2,,,,,2}2} = {1,,,,,,1,2} = {1{1,,,,,,,2}1,2}
C(Ωω) = C(C1(C2(C3(C4(C5(C6(...)))))))  {1{1,,, ... ,,,2}1,2} ~ П12-CA0
C(Cωω)) {1{1{1,,, ... ,,,1,2}2,,, ... ,,,2}2} ~ П12-CA0+BI
C(ΩΩΩΩ...) ~ П12-TR0