FUSION-BULK-001VERIFIED_75_OF_75BULK/21-cat
MuncyNautics Recovery Certificate · entered 2026-08-30 from Derrick’s Fusion_Codex_Corrected_Equation_Solutions_v2_2 (interface stamp 2026-08-27) · spot-checked cold this date
WHAT THIS CERTIFICATE RECORDSThe complete 75-record solved corpus for the Fusion Codex: the 7-fold master identities, beta-limit dial solutions, fast-ion Farey censuses, and the remaining category solves, each with equation, inputs, solution, residual and precision as published. Entered as ONE certificate with 75 witness rows — a parameter sweep is one equation at many points, never 75 certificates (per the registry’s own rule). Cold spot-checks this date: records 01, 03, 06, 07, 08 recomputed exactly (cyclotomic sum, dial angles to the last digit, Farey censuses [24,3] and [48,6]).
GOVERNING EQUATIONS AS PROJECTED — THE SEAMS
The classical forms of the 21 Fusion Codex problem categories, each carrying its seam as filed in the Codex (magnitude-only reads, unpaired arms, unpaid ranks). The category equations are listed per record below.
COMPLETE GOVERNING EQUATIONS — THE RECOVERIES
Each record solves the CORRECTED (phase-kept, seat-resolved) form: the 7-fold cyclotomic closure S7(m), the broken-seat residue |R_j| = 1, the beta dial beta = tan²(theta) with margin to the 45° station, the sevenfold Farey survival census, and the category forms through Isotope (record 75). Residuals 0-exact or at the stated precision floor in every row.
CORRECTION RULE — correction = reconstruction from the recovery: restored arguments, seats, phases and arms in the equation itself; no raw appended constants — any finite term must be derived from the recovered channels, never fitted.
| # | CATEGORY | EQUATION / INPUTS / SOLUTION | RESIDUAL | PRECISION |
|---|---|---|---|---|
| 01 | Master / Card 16 |
S7(m) = sum exp(i 2*pi*k*m/7), k=0..6 IN: integer m OUT: 7 if 7 divides m; 0 otherwise |
0 exact | exact cyclotomic identity |
| 02 | Master / Card 16 |
One missing seat j IN: j = 0,...,6 OUT: R_j = -exp(i 2*pi*j/7), |R_j| = 1 |
0 exact | exact unit-phasor identity |
| 03 | 2 Beta limit |
beta = tan^2(theta); a.b = (1-beta)/(1+beta) IN: beta = 0.50 OUT: theta = 35.264389682754654 deg; margin = 9.735610317245346 deg; a.b = 0.33333333333333333333333333333333333333333333333333333333333333333333333333333333 |
1.110e-16 | 80-digit Decimal for algebra; IEEE double for transcendental |
| 04 | 2 Beta limit |
beta = tan^2(theta); a.b = (1-beta)/(1+beta) IN: beta = 0.70 OUT: theta = 39.917875689138263 deg; margin = 5.082124310861737 deg; a.b = 0.17647058823529411764705882352941176470588235294117647058823529411764705882352941 |
1.110e-16 | 80-digit Decimal for algebra; IEEE double for transcendental |
| 05 | 2 Beta limit |
beta = tan^2(theta); a.b = (1-beta)/(1+beta) IN: beta = 0.85 OUT: theta = 42.674645497560093 deg; margin = 2.325354502439907 deg; a.b = 0.081081081081081081081081081081081081081081081081081081081081081081081081081081081 |
2.220e-16 | 80-digit Decimal for algebra; IEEE double for transcendental |
| 06 | 2 Beta limit |
beta = tan^2(theta); a.b = (1-beta)/(1+beta) IN: beta = 1.00 OUT: theta = 45.000000000000000 deg; margin = 0.000000000000000 deg; a.b = 0 |
2.220e-16 | 80-digit Decimal for algebra; IEEE double for transcendental |
| 07 | 3 Fast-ion lanes |
gcd(m,n)=1; 0<m/n<=4; sevenfold survival 7|m IN: n <= 4 OUT: all = 24; surviving = 3; count reduction = 8; weights = 4.229800754801 -> 0.154761904762 |
0 exact census | exact integer and rational arithmetic |
| 08 | 3 Fast-ion lanes |
gcd(m,n)=1; 0<m/n<=4; sevenfold survival 7|m IN: n <= 6 OUT: all = 48; surviving = 6; count reduction = 8; weights = 5.143619132882 -> 0.221428571429 |
0 exact census | exact integer and rational arithmetic |
| 09 | 3 Fast-ion lanes |
gcd(m,n)=1; 0<m/n<=4; sevenfold survival 7|m IN: n <= 8 OUT: all = 88; surviving = 8; count reduction = 11; weights = 5.958142897571 -> 0.245238095238 |
0 exact census | exact integer and rational arithmetic |
| 10 | 3 Fast-ion lanes |
gcd(m,n)=1; 0<m/n<=4; sevenfold survival 7|m IN: n <= 12 OUT: all = 184; surviving = 22; count reduction = 8.3636363636363636363636363636363636363636363636363636363636363636363636363636364; weights = 7.059869213607 -> 0.341341991342 |
0 exact census | exact integer and rational arithmetic |
| 11 | 4 ELM settlement |
P = W_ped*f_ELM*nu_ELM IN: W=0.4 MJ; f=0.05; nu=50 Hz OUT: P_cont required = 1.000 MW |
0 | 80-digit Decimal |
| 12 | 4 ELM settlement |
P = W_ped*f_ELM*nu_ELM IN: W=1 MJ; f=0.08; nu=30 Hz OUT: P_cont required = 2.40 MW |
0 | 80-digit Decimal |
| 13 | 4 ELM settlement |
P = W_ped*f_ELM*nu_ELM IN: W=10 MJ; f=0.10; nu=10 Hz OUT: P_cont required = 10.00 MW |
0 | 80-digit Decimal |
| 14 | 5 NTM ECCD |
cos(phi) >= 0.9 IN: f_rot = 1000 Hz OUT: full phase window = 51.683865526334252 deg; timing window = 143.566293128706263 us |
0 boundary | IEEE double transcendental |
| 15 | 5 NTM ECCD |
cos(phi) >= 0.9 IN: f_rot = 2000 Hz OUT: full phase window = 51.683865526334252 deg; timing window = 71.783146564353132 us |
0 boundary | IEEE double transcendental |
| 16 | 5 NTM ECCD |
cos(phi) >= 0.9 IN: f_rot = 3000 Hz OUT: full phase window = 51.683865526334252 deg; timing window = 47.855431042902083 us |
0 boundary | IEEE double transcendental |
| 17 | 5 NTM ECCD |
cos(phi) >= 0.9 IN: f_rot = 5000 Hz OUT: full phase window = 51.683865526334252 deg; timing window = 28.713258625741251 us |
0 boundary | IEEE double transcendental |
| 18 | 10 Impurity transport |
eta_crit = k*sqrt(Z) IN: Z=4; k fixed by eta(74)=2.151 OUT: eta_crit = 0.50009734187591054556729993442951173033760078334411817074491368038653257666108512 |
0 by construction | 80-digit Decimal |
| 19 | 10 Impurity transport |
eta_crit = k*sqrt(Z) IN: Z=6; k fixed by eta(74)=2.151 OUT: eta_crit = 0.61249165465908760170030868723638337141849470592073170969666804740415674764582124 |
0 by construction | 80-digit Decimal |
| 20 | 10 Impurity transport |
eta_crit = k*sqrt(Z) IN: Z=10; k fixed by eta(74)=2.151 OUT: eta_crit = 0.79072332606189023338268635911609302478620454040928414135354913523457967280686835 |
0 by construction | 80-digit Decimal |
| 21 | 10 Impurity transport |
eta_crit = k*sqrt(Z) IN: Z=18; k fixed by eta(74)=2.151 OUT: eta_crit = 1.0608666650814705852193233090348361824766030276123435509719674218642244615265656 |
0 by construction | 80-digit Decimal |
| 22 | 10 Impurity transport |
eta_crit = k*sqrt(Z) IN: Z=26; k fixed by eta(74)=2.151 OUT: eta_crit = 1.2750030524605750071458081680640645786897286275486647443539389486097103846318311 |
0 by construction | 80-digit Decimal |
| 23 | 10 Impurity transport |
eta_crit = k*sqrt(Z) IN: Z=42; k fixed by eta(74)=2.151 OUT: eta_crit = 1.6205005983304014779141797183654340746410195844011168317710489154495333216102705 |
0 by construction | 80-digit Decimal |
| 24 | 10 Impurity transport |
eta_crit = k*sqrt(Z) IN: Z=74; k fixed by eta(74)=2.151 OUT: eta_crit = 2.1510000000000000000000000000000000000000000000000000000000000000000000000000000 |
0 by construction | 80-digit Decimal |
| 25 | 11 RF absorption |
S=P/cos(phi); Q=sqrt(S^2-P^2) IN: P=1 MW; power factor=0.99 OUT: S=1.0101010101010101010101010101010101010101010101010101010101010101010101010101010 MVA; Q=0.14249228262288772146699155243310430785796674877983678657764942387691617724486893 MVAr |
0E-79 | 80-digit Decimal |
| 26 | 11 RF absorption |
S=P/cos(phi); Q=sqrt(S^2-P^2) IN: P=1 MW; power factor=0.90 OUT: S=1.1111111111111111111111111111111111111111111111111111111111111111111111111111111 MVA; Q=0.48432210483785261691522022042884618434855599169249388187670490423995081424479534 MVAr |
0E-79 | 80-digit Decimal |
| 27 | 11 RF absorption |
S=P/cos(phi); Q=sqrt(S^2-P^2) IN: P=1 MW; power factor=0.70 OUT: S=1.4285714285714285714285714285714285714285714285714285714285714285714285714285714 MVA; Q=1.0202040612204071425713428301953236112523101657003905476012044118236886294914317 MVAr |
1E-79 | 80-digit Decimal |
| 28 | 11 RF absorption |
S=P/cos(phi); Q=sqrt(S^2-P^2) IN: P=1 MW; power factor=0.50 OUT: S=2 MVA; Q=1.7320508075688772935274463415058723669428052538103806280558069794519330169088000 MVAr |
1E-79 | 80-digit Decimal |
| 29 | 11 RF absorption |
S=P/cos(phi); Q=sqrt(S^2-P^2) IN: P=1 MW; power factor=0.30 OUT: S=3.3333333333333333333333333333333333333333333333333333333333333333333333333333333 MVA; Q=3.1797973380564854971754052867440884675154114175018191796938395097012085076855024 MVAr |
0E-78 | 80-digit Decimal |
| 30 | 11 RF absorption |
S=P/cos(phi); Q=sqrt(S^2-P^2) IN: P=1 MW; power factor=0.10 OUT: S=1E+1 MVA; Q=9.9498743710661995473447982100120600517812656367680607911760464383494539278271315 MVAr |
1E-78 | 80-digit Decimal |
| 31 | 14 Divertor |
A_min=(1-f_rad)P_ex/q_max IN: P=100 MW; f_rad=0.50; qmax=10 MW/m^2 OUT: A_min=5.00 m^2 |
0 | 80-digit Decimal |
| 32 | 14 Divertor |
A_min=(1-f_rad)P_ex/q_max IN: P=100 MW; f_rad=0.80; qmax=10 MW/m^2 OUT: A_min=2.00 m^2 |
0 | 80-digit Decimal |
| 33 | 14 Divertor |
A_min=(1-f_rad)P_ex/q_max IN: P=100 MW; f_rad=0.90; qmax=10 MW/m^2 OUT: A_min=1.00 m^2 |
0 | 80-digit Decimal |
| 34 | 14 Divertor |
A_min=(1-f_rad)P_ex/q_max IN: P=100 MW; f_rad=0.95; qmax=10 MW/m^2 OUT: A_min=0.50 m^2 |
0 | 80-digit Decimal |
| 35 | 14 Divertor |
A_min=(1-f_rad)P_ex/q_max IN: P=100 MW; f_rad=0.97; qmax=10 MW/m^2 OUT: A_min=0.30 m^2 |
0 | 80-digit Decimal |
| 36 | 14 Divertor |
A_min=(1-f_rad)P_ex/q_max IN: P=500 MW; f_rad=0.50; qmax=10 MW/m^2 OUT: A_min=25.00 m^2 |
0 | 80-digit Decimal |
| 37 | 14 Divertor |
A_min=(1-f_rad)P_ex/q_max IN: P=500 MW; f_rad=0.80; qmax=10 MW/m^2 OUT: A_min=10.00 m^2 |
0 | 80-digit Decimal |
| 38 | 14 Divertor |
A_min=(1-f_rad)P_ex/q_max IN: P=500 MW; f_rad=0.90; qmax=10 MW/m^2 OUT: A_min=5.00 m^2 |
0 | 80-digit Decimal |
| 39 | 14 Divertor |
A_min=(1-f_rad)P_ex/q_max IN: P=500 MW; f_rad=0.95; qmax=10 MW/m^2 OUT: A_min=2.50 m^2 |
0 | 80-digit Decimal |
| 40 | 14 Divertor |
A_min=(1-f_rad)P_ex/q_max IN: P=500 MW; f_rad=0.97; qmax=10 MW/m^2 OUT: A_min=1.50 m^2 |
0 | 80-digit Decimal |
| 41 | 14 Divertor inverse |
f_rad=1-A*qmax/P IN: P=100 MW; Amax=1.5 m^2; qmax=10 OUT: f_rad required=0.85 |
0 | 80-digit Decimal |
| 42 | 14 Divertor inverse |
f_rad=1-A*qmax/P IN: P=500 MW; Amax=1.5 m^2; qmax=10 OUT: f_rad required=0.97 |
0 | 80-digit Decimal |
| 43 | 15 Runaway |
N/N0=exp(t/tau) IN: t=10 ms; tau=1 ms OUT: multiplication=2.202646579480671789e+04 |
transcendental evaluation | IEEE double |
| 44 | 15 Runaway |
N/N0=exp(t/tau) IN: t=10 ms; tau=3 ms OUT: multiplication=2.803162489452613926e+01 |
transcendental evaluation | IEEE double |
| 45 | 15 Runaway |
N/N0=exp(t/tau) IN: t=10 ms; tau=10 ms OUT: multiplication=2.718281828459045091e+00 |
transcendental evaluation | IEEE double |
| 46 | 15 Runaway coherence |
N/N0=exp(kappa*gamma*t) IN: gamma*t=10; kappa=1 OUT: multiplication=2.202646579480671789e+04 |
transcendental evaluation | IEEE double |
| 47 | 15 Runaway coherence |
N/N0=exp(kappa*gamma*t) IN: gamma*t=10; kappa=0.5 OUT: multiplication=1.484131591025765999e+02 |
transcendental evaluation | IEEE double |
| 48 | 15 Runaway coherence |
N/N0=exp(kappa*gamma*t) IN: gamma*t=10; kappa=0.1 OUT: multiplication=2.718281828459045091e+00 |
transcendental evaluation | IEEE double |
| 49 | 15 Runaway coherence |
N/N0=exp(kappa*gamma*t) IN: gamma*t=10; kappa=0 OUT: multiplication=1.000000000000000000e+00 |
transcendental evaluation | IEEE double |
| 50 | 17 Poloidal twist |
theta=360/(n+1) IN: n=7 OUT: theta=45.000000000000 deg; copies=8; issued=True |
0 exact | exact rational |
| 51 | 17 Poloidal twist |
theta=360/(n+1) IN: n=9 OUT: theta=36.000000000000 deg; copies=10; issued=True |
0 exact | exact rational |
| 52 | 17 Poloidal twist |
theta=360/(n+1) IN: n=14 OUT: theta=24.000000000000 deg; copies=15; issued=False |
0 exact | exact rational |
| 53 | 17 Poloidal twist |
theta=360/(n+1) IN: n=15 OUT: theta=22.500000000000 deg; copies=16; issued=True |
0 exact | exact rational |
| 54 | 17 Poloidal twist |
theta=360/(n+1) IN: n=20 OUT: theta=17.142857142857 deg; copies=21; issued=False |
0 exact | exact rational |
| 55 | 17 Poloidal twist |
theta=360/(n+1) IN: n=21 OUT: theta=16.363636363636 deg; copies=22; issued=True |
0 exact | exact rational |
| 56 | 17 Poloidal twist |
theta=360/(n+1) IN: n=27 OUT: theta=12.857142857143 deg; copies=28; issued=True |
0 exact | exact rational |
| 57 | 17 Poloidal twist |
theta=360/(n+1) IN: n=33 OUT: theta=10.588235294118 deg; copies=34; issued=True |
0 exact | exact rational |
| 58 | 18 Greenwald |
nG=Ip/(pi*a^2) IN: Ip=15 MA; a=2.0 m OUT: nG=1.1936620731892150182666282252938577152584473430534233656075050804417259822566990 x10^20 m^-3 |
0 algebraic | 80-digit Decimal with 80-digit pi literal |
| 59 | 18 Greenwald |
nG=Ip/(pi*a^2) IN: Ip=15 MA; a=1.7 m OUT: nG=1.6521274369400899906804542910641629276933527239493749004948167203345688335732858 x10^20 m^-3 |
0 algebraic | 80-digit Decimal with 80-digit pi literal |
| 60 | 18 Greenwald |
nG=Ip/(pi*a^2) IN: Ip=1 MA; a=0.5 m OUT: nG=1.2732395447351626861510701069801148962756771659236515899813387524711743810738123 x10^20 m^-3 |
0 algebraic | 80-digit Decimal with 80-digit pi literal |
| 61 | 18 Greenwald inverse |
Ip=nbar*pi*a^2 for fG=1 IN: nbar=1.2 x10^20 m^-3; a=2.0 m OUT: Ip boundary=15.079644737231007544620688239741613844146413117000507940679734043077518750173803 MA |
0 algebraic | 80-digit Decimal |
| 62 | 18 Greenwald inverse |
Ip=nbar*pi*a^2 for fG=1 IN: nbar=2.0 x10^20 m^-3; a=1.0 m OUT: Ip boundary=6.2831853071795864769252867665590057683943387987502116419498891846156328125724180 MA |
0 algebraic | 80-digit Decimal |
| 63 | 20 TBR |
TBR=f*eta*M IN: f=0.85; eta=0.90; M=1 OUT: TBR=0.7650; residual to 1=-0.2350; residual to 1.05=-0.2850 |
0 | 80-digit Decimal |
| 64 | 20 TBR |
TBR=f*eta*M IN: f=0.85; eta=0.90; M=1.3 OUT: TBR=0.99450; residual to 1=-0.00550; residual to 1.05=-0.05550 |
0 | 80-digit Decimal |
| 65 | 20 TBR |
TBR=f*eta*M IN: f=0.90; eta=0.90; M=1.4 OUT: TBR=1.13400; residual to 1=0.13400; residual to 1.05=0.08400 |
0 | 80-digit Decimal |
| 66 | 20 TBR |
TBR=f*eta*M IN: f=0.95; eta=0.90; M=1.5 OUT: TBR=1.28250; residual to 1=0.28250; residual to 1.05=0.23250 |
0 | 80-digit Decimal |
| 67 | 20 TBR |
TBR=f*eta*M IN: f=0.99; eta=0.90; M=1.6 OUT: TBR=1.42560; residual to 1=0.42560; residual to 1.05=0.37560 |
0 | 80-digit Decimal |
| 68 | 20 TBR inverse |
M*=target/(f*eta) IN: f=0.85; eta=0.90 OUT: M for 1.00=1.3071895424836601307189542483660130718954248366013071895424836601307189542483660; M for 1.05=1.3725490196078431372549019607843137254901960784313725490196078431372549019607843 |
0 | 80-digit Decimal |
| 69 | 20 TBR inverse |
M*=target/(f*eta) IN: f=0.90; eta=0.90 OUT: M for 1.00=1.2345679012345679012345679012345679012345679012345679012345679012345679012345679; M for 1.05=1.2962962962962962962962962962962962962962962962962962962962962962962962962962963 |
0 | 80-digit Decimal |
| 70 | 20 TBR inverse |
M*=target/(f*eta) IN: f=0.95; eta=0.90 OUT: M for 1.00=1.1695906432748538011695906432748538011695906432748538011695906432748538011695906; M for 1.05=1.2280701754385964912280701754385964912280701754385964912280701754385964912280702 |
0 | 80-digit Decimal |
| 71 | 20 TBR inverse |
M*=target/(f*eta) IN: f=0.99; eta=0.90 OUT: M for 1.00=1.1223344556677890011223344556677890011223344556677890011223344556677890011223345; M for 1.05=1.1784511784511784511784511784511784511784511784511784511784511784511784511784512 |
0 | 80-digit Decimal |
| 72 | 21 Isotope |
GB=A^-0.5; obs=A^0.35; ratio=A^0.85 IN: A=1 OUT: GB=1.000000000000000000; obs=1.000000000000000000; ratio=1.000000000000000000 |
0.000e+00 | IEEE double transcendental |
| 73 | 21 Isotope |
GB=A^-0.5; obs=A^0.35; ratio=A^0.85 IN: A=2 OUT: GB=0.707106781186547573; obs=1.274560627319262229; ratio=1.802500925221660388 |
0.000e+00 | IEEE double transcendental |
| 74 | 21 Isotope |
GB=A^-0.5; obs=A^0.35; ratio=A^0.85 IN: A=2.5 OUT: GB=0.632455532033675882; obs=1.378094511392790711; ratio=2.178958743489040284 |
4.441e-16 | IEEE double transcendental |
| 75 | 21 Isotope |
GB=A^-0.5; obs=A^0.35; ratio=A^0.85 IN: A=3 OUT: GB=0.577350269189625731; obs=1.468900704600073714; ratio=2.544210651641050536 |
0.000e+00 | IEEE double transcendental |
Standing: VERIFIED_75_OF_75 — GREEN (stamped by Derrick E. Muncy’s authority, 2026-08-31). The full independent rerun executed this date in-project: every record re-computed from its own equation and inputs — cyclotomic identities, beta-dial angles to the last digit, Farey censuses [88,8] and [184,22] regenerated, ELM products, NTM phase windows, impurity k√Z internal consistency across Z = 4–74, RF quadrature S and Q, divertor forward and inverse, runaway exponentials, poloidal twist, Greenwald forward and inverse, TBR forward and inverse, isotope scalings. 75 PASS, 0 FAIL, 0 SKIP. Source: Derrick E. Muncy, Fusion_Codex_Corrected_Equation_Solutions_v2_2, 2026-08-27. Entered without duplication: this certificate supersedes no entry and duplicates none — the registry held no bulk-corpus certificate before this date.