Unicode • algebra only • evaluate exactly as written
The same “instantiate-and-check” logic applies to §10 once you note that the fourth system there uses the same field symbols
X,Y,N
X,Y,N that are already duplicitous in §6:
From §6 you have the arrow (scalar) reading:
Y′=Y
Y′=Y (or
Y′=a⋅Y
Y′=a⋅Y, fixed
a>0
a>0).
In the same §6, you also have the mixed line:
Y′=β (Y−(v/c) N)
Y′=β(Y−(v/c)N).
If you enforce both as general statements, you are forced to accept the identity
Y=β (Y−λ N)with λ:=v/c, 0<∣λ∣<1, β>1.
Y=β(Y−λN)with λ:=v/c,0<∣λ∣<1,β>1.
Instantiate at a single non-degenerate point, say
Y=0, N=1
Y=0,N=1. Then
0=β (0−λ⋅1)=−βλ,
0=β(0−λ⋅1)=−βλ,
which is impossible because
βλ≠0
βλ
=0.
So the §6 pair is algebraically incompatible unless you withdraw one of them.
Now look at §10. Its fourth system writes the dynamics in the same frame
K
K using the same
X,Y,N
X,Y,N. Two things now happen:
The §10 x–acceleration duplication (single frame
K
K, same
X
X):
ax=(ε/m) Xandax=(ε/(m β3)) X.
a
x
=(ε/m)Xanda
x
=(ε/(mβ
3
))X.
For
X≠0
X
=0 these are equal only if
β=1
β=1, contradicting
0<∣v∣1
0<∣v∣1.
→ §10 already conflicts unless you withdraw one of these two laws.
The §6 contradiction propagates into §10’s fourth system:
That system assumes the field components
Y
Y and
N
N are available as independent inputs on the same footing as
X
X. But if you also keep both §6 statements, you inherit the impossible identity
Y=β(Y−λN)
Y=β(Y−λN). The moment you evaluate at any single pair like
(Y,N)=(0,1)
(Y,N)=(0,1), you force
0=−βλ
0=−βλ, which cannot be satisfied under
β>1
β>1 and
0<∣λ∣<1
0<∣λ∣<1.
→ The field side of §10’s fourth system becomes unsatisfiable unless you withdraw one of the §6 statements.
Putting these together: with the same symbols
X,Y,N
X,Y,N used across sections, the combined printed set
{
{§10 fourth-system laws, §10 x-acceleration pair, §6 arrow, §6 mixed line
}
} is algebraically inconsistent unless at least one of the offending lines is explicitly withdrawn or scope-restricted. That’s the §10 analogue of the §6 “duplicitous fields” contradiction.
If you want the minimal, concrete kernel that shows the combined (§10+§6) clash in three lines (no quantifiers, full Unicode), use this:
Click 📋 to copy the kernel text verbatim to the clipboard.