# Perfect Cuboid Six-Channel Conservation–Duality Methodology

## A Truth-Vector, Dual-Pair, and Magic-Cube Reformulation

**Series:** Perfect Cuboid Global Closure  
**New line:** Conservation–Duality Branch C  
**Relation to previous work:** Geometry Branch G + Verification Branch A  
**Phase:** C0 — Methodological Foundation  
**Date:** 2026-09-30

---

# Abstract

This document introduces a new Perfect Cuboid methodology based on three ideas:

1. **conservation**;
2. **duality**;
3. **six-channel truth closure**.

Instead of treating the seven classical lengths as independent unknown integers, normalize by the body diagonal and encode the entire geometry into six coupled channels.

Let:

$$
x_a=\frac ag,
\qquad
x_b=\frac bg,
\qquad
x_c=\frac cg,
$$

and:

$$
x_d=\frac dg,
\qquad
x_e=\frac eg,
\qquad
x_f=\frac fg.
$$

Define the six polynomial truth channels:

$$
T_1
=
1+x_a^2+x_b^2-x_d^2,
$$

$$
T_2
=
1+x_a^2+x_c^2-x_e^2,
$$

$$
T_3
=
1+x_b^2+x_c^2-x_f^2,
$$

$$
T_4
=
x_a^2+x_f^2,
$$

$$
T_5
=
x_b^2+x_e^2,
$$

$$
T_6
=
x_c^2+x_d^2.
$$

Then exact Perfect-Cuboid geometry is equivalent to:

$$
\boxed{
(T_1,T_2,T_3,T_4,T_5,T_6)
=
(1,1,1,1,1,1).
}
$$

The right-hand value

$$
1
$$

is interpreted as a normalized **truth / closure / conservation unit**.

The six channels naturally form three dual pairs:

$$
T_1\leftrightarrow T_6,
$$

$$
T_2\leftrightarrow T_5,
$$

$$
T_3\leftrightarrow T_4.
$$

Passing to normalized squared variables turns the geometry into an affine conservation system on a two-dimensional simplex.

The resulting structure resembles a mathematical “magic cube”:

- local moves redistribute conserved quantity;
- dual cells compensate each other;
- the global geometric truth state remains unchanged;
- arithmetic squarehood is allowed to move between channels.

This separates the problem into two layers:

$$
\boxed{
\text{Geometric Truth}
}
$$

and:

$$
\boxed{
\text{Arithmetic Square Closure}.
}
$$

The current seed

$$
(a,b,c;g)
=
(448,495,840;1073)
$$

already has:

$$
\boxed{
\text{Geometric Truth Vector}
=
(1,1,1,1,1,1),
}
$$

while its arithmetic square-status vector is:

$$
\boxed{
(1,1,1,0,1,1).
}
$$

Thus the current problem can be viewed as:

> move inside the exact geometric conservation manifold while changing the arithmetic state from one-defect closure to all-six closure.

---

# 1. Classical equations

The classical Perfect Cuboid equations are:

$$
a^2+b^2=d^2,
$$

$$
a^2+c^2=e^2,
$$

$$
b^2+c^2=f^2,
$$

and:

$$
a^2+b^2+c^2=g^2.
$$

The same body-diagonal equation gives the three complementary relations:

$$
a^2+f^2=g^2,
$$

$$
b^2+e^2=g^2,
$$

$$
c^2+d^2=g^2.
$$

Hence there are naturally six local closure channels:

- three face channels;
- three complementary center/body channels.

The new methodology takes these six channels as the primary logical structure.

---

# 2. Normalize the global whole

Divide all lengths by:

$$
g.
$$

Define:

$$
x_a=\frac ag,
\quad
x_b=\frac bg,
\quad
x_c=\frac cg,
$$

and:

$$
x_d=\frac dg,
\quad
x_e=\frac eg,
\quad
x_f=\frac fg.
$$

Then the body diagonal itself becomes the normalized whole:

$$
\boxed{
1.
}
$$

All geometry is measured relative to this conserved unit.

---

# 3. Six polynomial truth channels

Define:

$$
\boxed{
T_1
=
1+x_a^2+x_b^2-x_d^2,
}
$$

$$
\boxed{
T_2
=
1+x_a^2+x_c^2-x_e^2,
}
$$

$$
\boxed{
T_3
=
1+x_b^2+x_c^2-x_f^2,
}
$$

$$
\boxed{
T_4
=
x_a^2+x_f^2,
}
$$

$$
\boxed{
T_5
=
x_b^2+x_e^2,
}
$$

$$
\boxed{
T_6
=
x_c^2+x_d^2.
}
$$

The exact closure target is:

$$
\boxed{
\mathbf T
=
(T_1,T_2,T_3,T_4,T_5,T_6)
=
\mathbf 1_6.
}
$$

where:

$$
\mathbf 1_6
=
(1,1,1,1,1,1).
$$

This is the canonical **Truth Vector**.

---

# 4. Why six ones are sufficient

Suppose:

$$
T_1=1.
$$

Then:

$$
x_d^2=x_a^2+x_b^2.
$$

Suppose simultaneously:

$$
T_6=1.
$$

Then:

$$
x_c^2+x_d^2=1.
$$

Therefore:

$$
x_a^2+x_b^2+x_c^2=1.
$$

So one face channel plus its dual automatically reconstructs the body-diagonal conservation law.

Likewise:

$$
T_2=1
$$

paired with:

$$
T_5=1
$$

gives the same global whole.

And:

$$
T_3=1
$$

paired with:

$$
T_4=1
$$

does the same.

Hence the six ones form a redundant but self-checking closure certificate.

The redundancy is not noise.

It is part of the method.

---

# 5. Dual pairing

The natural pairing is:

$$
\boxed{
T_1\leftrightarrow T_6,
}
$$

$$
\boxed{
T_2\leftrightarrow T_5,
}
$$

$$
\boxed{
T_3\leftrightarrow T_4.
}
$$

Explicitly:

$$
a^2+b^2=d^2
$$

is dual to:

$$
c^2+d^2=g^2.
$$

Likewise:

$$
a^2+c^2=e^2
$$

is dual to:

$$
b^2+e^2=g^2.
$$

And:

$$
b^2+c^2=f^2
$$

is dual to:

$$
a^2+f^2=g^2.
$$

Thus each face closure is paired with the complementary edge needed to fill the same global body-diagonal whole.

---

# 6. Square-state variables

Define normalized squared variables:

$$
A=x_a^2,
$$

$$
B=x_b^2,
$$

$$
C=x_c^2,
$$

$$
D=x_d^2,
$$

$$
E=x_e^2,
$$

$$
F=x_f^2.
$$

Then:

$$
A,B,C,D,E,F\in(0,1)
$$

for a nondegenerate cuboid.

The six closure equations become linear.

---

# 7. The conservation system

The body-diagonal normalization is:

$$
\boxed{
A+B+C=1.
}
$$

The three dual complementary pairs are:

$$
\boxed{
A+F=1,
}
$$

$$
\boxed{
B+E=1,
}
$$

$$
\boxed{
C+D=1.
}
$$

Therefore:

$$
F=1-A,
$$

$$
E=1-B,
$$

$$
D=1-C.
$$

Since:

$$
A+B+C=1,
$$

we automatically obtain:

$$
D=A+B,
$$

$$
E=A+C,
$$

$$
F=B+C.
$$

Thus the three face equations are not independent once global conservation and dual complementarity are imposed.

---

# 8. The real geometry is a simplex

The normalized real geometric state is completely determined by:

$$
A>0,
$$

$$
B>0,
$$

$$
C>0,
$$

and:

$$
A+B+C=1.
$$

Therefore the full nondegenerate real cuboid shape space is:

$$
\boxed{
\Delta_2^\circ
=
\{
(A,B,C):
A,B,C>0,\;
A+B+C=1
\}.
}
$$

This is the open two-simplex.

The three dual cells are then:

$$
F=1-A,
$$

$$
E=1-B,
$$

$$
D=1-C.
$$

Hence the six-cell geometric state is only two-dimensional.

---

# 9. Conservation board

Arrange the six squared channels as three dual columns:

$$
\boxed{
\begin{matrix}
A & B & C\\
F & E & D
\end{matrix}
}
$$

with:

$$
A+F=1,
$$

$$
B+E=1,
$$

$$
C+D=1.
$$

The top row satisfies:

$$
A+B+C=1.
$$

The bottom row therefore satisfies:

$$
D+E+F=2.
$$

The total six-cell mass is:

$$
\boxed{
A+B+C+D+E+F=3.
}
$$

Thus the system has several simultaneous conservation laws.

This is the formal basis of the “magic-cube” analogy.

---

# 10. Duality operator

Define the duality involution:

$$
\star:
(A,B,C,D,E,F)
\mapsto
(F,E,D,C,B,A).
$$

Then:

$$
\star^2=\operatorname{id}.
$$

On the conservation manifold:

$$
A^\star=1-A,
$$

$$
B^\star=1-B,
$$

$$
C^\star=1-C.
$$

Thus duality is simply complement relative to the conserved unit.

---

# 11. Defect vector

The multiplicative truth vector is convenient conceptually.

For computation, define the additive residual vector:

$$
\mathbf R
=
(R_1,R_2,R_3,R_4,R_5,R_6),
$$

where:

$$
R_1=A+B-D,
$$

$$
R_2=A+C-E,
$$

$$
R_3=B+C-F,
$$

$$
R_4=A+F-1,
$$

$$
R_5=B+E-1,
$$

$$
R_6=C+D-1.
$$

Exact geometry means:

$$
\boxed{
\mathbf R=\mathbf 0.
}
$$

The truth vector and residual vector are equivalent encodings.

---

# 12. Two layers of truth

The central methodological distinction is:

## Layer G — geometric truth

$$
\boxed{
\mathbf T=\mathbf 1_6.
}
$$

or equivalently:

$$
\boxed{
\mathbf R=\mathbf0.
}
$$

## Layer A — arithmetic square closure

The classical Perfect Cuboid requires the normalized lengths themselves to be rational:

$$
x_a,\ldots,x_f\in\mathbb Q.
$$

In square-state language, this means:

$$
A,B,C,D,E,F
$$

must all be rational squares.

Therefore geometry and arithmetic are separate truth layers.

---

# 13. Arithmetic square-status vector

Define:

$$
\operatorname{Sq}_{\mathbb Q}(X)
=
\begin{cases}
1,&X\in\mathbb Q^2,\\
0,&\text{otherwise}.
\end{cases}
$$

For a rational square-state define:

$$
\boxed{
\boldsymbol\Sigma
=
(
\operatorname{Sq}_{\mathbb Q}(A),
\operatorname{Sq}_{\mathbb Q}(B),
\operatorname{Sq}_{\mathbb Q}(C),
\operatorname{Sq}_{\mathbb Q}(D),
\operatorname{Sq}_{\mathbb Q}(E),
\operatorname{Sq}_{\mathbb Q}(F)
).
}
$$

The arithmetic target is:

$$
\boxed{
\boldsymbol\Sigma=\mathbf1_6.
}
$$

This is distinct from geometric truth.

---

# 14. Exact Perfect Cuboid theorem in conservation form

A Perfect Cuboid exists if and only if there exist:

$$
A,B,C,D,E,F
\in
\mathbb Q_{>0}
$$

such that:

$$
A+B+C=1,
$$

$$
A+F=1,
$$

$$
B+E=1,
$$

$$
C+D=1,
$$

and:

$$
\boxed{
A,B,C,D,E,F\in\mathbb Q^2.
}
$$

Equivalently:

> find an interior point of the conservation simplex whose six cells are all rational squares.

This is the canonical existence statement of the Conservation–Duality Branch.

---

# 15. Current seed

Use:

$$
(a,b,c;g)
=
(448,495,840;1073).
$$

Then:

$$
A=
\left(
\frac{448}{1073}
\right)^2,
$$

$$
B=
\left(
\frac{495}{1073}
\right)^2,
$$

$$
C=
\left(
\frac{840}{1073}
\right)^2.
$$

The two closed face channels are:

$$
E=
\left(
\frac{952}{1073}
\right)^2,
$$

and:

$$
F=
\left(
\frac{975}{1073}
\right)^2.
$$

The remaining face-square state is:

$$
D
=
\frac{445729}{1073^2}.
$$

Since:

$$
445729
=
233\cdot1913,
$$

$D$ is rational but not a rational square.

---

# 16. Seed truth state

The geometry satisfies:

$$
A+B+C=1,
$$

$$
A+F=1,
$$

$$
B+E=1,
$$

$$
C+D=1.
$$

Therefore:

$$
\boxed{
\mathbf T_{\rm seed}
=
(1,1,1,1,1,1).
}
$$

The geometric system is already perfect.

---

# 17. Seed arithmetic state

For the six cells:

$$
A,B,C,E,F
$$

are rational squares.

But:

$$
D
$$

is not.

Hence:

$$
\boxed{
\boldsymbol\Sigma_{\rm seed}
=
(1,1,1,0,1,1).
}
$$

This is the most concise representation of the current near-solution.

Geometrically:

$$
\boxed{
6/6\text{ closure}.
}
$$

Arithmetically:

$$
\boxed{
5/6\text{ square closure}.
}
$$

---

# 18. Conservation moves

The new methodology does not search by changing all six variables independently.

Instead, use moves that preserve the global conservation laws exactly.

Let:

$$
\delta_A,
\delta_B,
\delta_C
$$

satisfy:

$$
\delta_A+\delta_B+\delta_C=0.
$$

Then dual compensation requires:

$$
\delta_F=-\delta_A,
$$

$$
\delta_E=-\delta_B,
$$

$$
\delta_D=-\delta_C.
$$

Therefore every tangent conservation move has the form:

$$
\boxed{
\Delta
=
(
\delta_A,
\delta_B,
\delta_C,
-\delta_C,
-\delta_B,
-\delta_A
).
}
$$

This preserves:

$$
A+B+C,
$$

$$
A+F,
$$

$$
B+E,
$$

$$
C+D
$$

exactly.

---

# 19. Two independent move generators

Because:

$$
\delta_A+\delta_B+\delta_C=0,
$$

the move space is two-dimensional.

A convenient basis is:

$$
\boxed{
M_{AB}
=
(1,-1,0,0,1,-1),
}
$$

and:

$$
\boxed{
M_{BC}
=
(0,1,-1,1,-1,0).
}
$$

A third natural move:

$$
M_{CA}
=
(-1,0,1,-1,0,1)
$$

satisfies:

$$
\boxed{
M_{AB}+M_{BC}+M_{CA}=0.
}
$$

Thus there are three visible “turns” but only two independent directions.

This is directly analogous to a constrained magic-cube move system.

---

# 20. Finite conservation move

For parameters:

$$
\lambda,\mu,
$$

define:

$$
X'
=
X
+
\lambda M_{AB}
+
\mu M_{BC}.
$$

Explicitly:

$$
A'
=
A+\lambda,
$$

$$
B'
=
B-\lambda+\mu,
$$

$$
C'
=
C-\mu,
$$

$$
D'
=
D+\mu,
$$

$$
E'
=
E+\lambda-\mu,
$$

$$
F'
=
F-\lambda.
$$

As long as all six values remain in:

$$
(0,1),
$$

the state remains a valid nondegenerate real cuboid geometry.

And:

$$
\boxed{
\mathbf T(X')
=
\mathbf1_6
}
$$

whenever:

$$
\mathbf T(X)=\mathbf1_6.
$$

---

# 21. The key new search problem

The geometric truth vector no longer needs to be solved.

It is preserved automatically.

Therefore the search becomes:

> choose legal conservation moves so that the arithmetic square-status changes from the current pattern to all-six closure.

Symbolically:

$$
\boxed{
(1,1,1,0,1,1)
\longrightarrow
(1,1,1,1,1,1).
}
$$

The geometry remains:

$$
\boxed{
\mathbf T=\mathbf1_6
}
$$

throughout.

---

# 22. Arithmetic move

A move:

$$
X\mapsto X'
$$

is called an **arithmetic move** if:

$$
X'
$$

remains in the conservation simplex and increases or preserves the number of rational-square cells.

Define the square count:

$$
N_{\square}(X)
=
\sum_{i=1}^{6}
\Sigma_i.
$$

The current seed has:

$$
N_{\square}=5.
$$

The target is:

$$
N_{\square}=6.
$$

---

# 23. Square-preserving move equations

Consider:

$$
M_{AB}.
$$

Then:

$$
A'=A+\lambda,
$$

$$
B'=B-\lambda,
$$

$$
E'=E+\lambda,
$$

$$
F'=F-\lambda.
$$

The untouched cells are:

$$
C,
D.
$$

To preserve arithmetic closure on the four moved cells, require:

$$
A+\lambda\in\mathbb Q^2,
$$

$$
B-\lambda\in\mathbb Q^2,
$$

$$
E+\lambda\in\mathbb Q^2,
$$

$$
F-\lambda\in\mathbb Q^2.
$$

This converts a geometric move into an arithmetic compatibility problem.

Similar systems exist for the other move generators.

---

# 24. Defect transfer

The new conceptual object is not merely “distance from the solution”.

It is **defect transfer**.

A move may:

- repair one nonsquare cell;
- break one previously square cell;
- transfer the defect to its dual;
- split one defect into two;
- merge two defects;
- produce all-six closure.

Therefore arithmetic failure can be treated as something that moves through a conserved network.

This is the core magic-cube intuition.

---

# 25. Defect location

For a state:

$$
X,
$$

define the defect set:

$$
\boxed{
\mathcal D(X)
=
\{
i:
\Sigma_i=0
\}.
}
$$

For the current seed:

$$
\boxed{
\mathcal D_{\rm seed}
=
\{D\}.
}
$$

The objective is:

$$
\boxed{
\mathcal D(X)=\varnothing.
}
$$

Thus the search can be organized by defect topology instead of raw integer height.

---

# 26. Dual defect

Because the geometric pairs satisfy:

$$
A+F=1,
$$

$$
B+E=1,
$$

$$
C+D=1,
$$

the arithmetic state of one cell is naturally coupled to the squarehood of its complement.

For example:

$$
D=1-C.
$$

So the current defect is equivalent to:

$$
\boxed{
C\in\mathbb Q^2,
\qquad
1-C\notin\mathbb Q^2.
}
$$

This is precisely the half-angle / square-defect structure discovered in the previous branches.

The new method therefore does not contradict earlier work.

It compresses it into one dual-pair statement.

---

# 27. Relation to the half-angle map

For:

$$
C=x_c^2,
$$

the opposite face value is:

$$
D=1-C.
$$

The half-angle square-defect map was:

$$
\Delta(x_c)
=
\frac{1-x_c}{1+x_c}.
$$

Both formulations encode the same complementary arithmetic obstruction.

The difference is organizational:

- half-angle coordinates diagonalize rational-circle arithmetic;
- conservation coordinates diagonalize global geometric coupling.

These are dual coordinate systems for the same problem.

---

# 28. Conservation energy

For numerical experiments define:

$$
E_C
=
\frac12
\sum_{i=1}^{6}
R_i^2.
$$

Exact geometric states satisfy:

$$
E_C=0.
$$

But on the conservation manifold, this energy is identically zero.

Therefore the new search dynamics should not minimize geometry.

Instead it should move only in the tangent conservation plane and optimize arithmetic observables.

---

# 29. Arithmetic potential

Define a heuristic:

$$
E_A(X).
$$

Possible terms include:

- rational-square height;
- denominator complexity;
- squarefree kernel size;
- continued-fraction complexity;
- radix complexity;
- local quadratic-residue score;
- prime-support complexity.

Then use:

$$
\boxed{
\dot X
=
V_A(X)
}
$$

with:

$$
V_A(X)
$$

restricted to the two-dimensional conservation tangent space.

Thus:

$$
\frac{d}{dt}\mathbf T(X(t))=0
$$

while arithmetic structure evolves.

---

# 30. Discrete arithmetic dynamics

An alternative is purely rational.

Choose:

$$
\lambda,\mu\in\mathbb Q.
$$

Apply:

$$
X'
=
X+\lambda M_{AB}+\mu M_{BC}.
$$

Reject any state leaving:

$$
(0,1)^6.
$$

Then compute:

$$
\boldsymbol\Sigma(X').
$$

This creates a graph:

$$
\boxed{
\mathcal G_{\rm arithmetic}
}
$$

whose vertices are rational conservation states and whose edges are legal conservation moves.

Perfect Cuboid existence becomes a reachability problem:

$$
\boxed{
\text{Is there an interior vertex with }
\boldsymbol\Sigma=\mathbf1_6?
}
$$

---

# 31. Conservation-state graph

A state can be written as:

$$
X=
(A,B,C;F,E,D).
$$

A legal geometric edge satisfies:

$$
X'\!-\!X
\in
\operatorname{span}
\{
M_{AB},M_{BC}
\}.
$$

An arithmetic edge may impose additional conditions such as:

$$
N_{\square}(X')
\ge
N_{\square}(X).
$$

One can therefore distinguish:

- unrestricted geometric moves;
- rational moves;
- square-preserving moves;
- defect-reducing moves;
- exact closing moves.

---

# 32. Magic-cube analogy

The analogy should be used carefully.

This is not literally a Rubik's Cube.

The useful structural analogy is:

1. there is a constrained state space;
2. local legal moves preserve global invariants;
3. each move affects coupled cells;
4. the final goal is a simultaneous global pattern;
5. solving one cell independently is meaningless because moves propagate through the network.

The exact mathematical structure is:

$$
\boxed{
\text{two-dimensional conservation group action on a six-cell dual state}.
}
$$

---

# 33. Why this may be faster than repeated Pythagorean solving

The previous arithmetic methods repeatedly asked whether multiple individual expressions were squares.

The conservation formulation first solves all real geometry globally.

Then the system moves only inside:

$$
\Delta_2^\circ.
$$

Thus the search no longer asks:

> can these six equations all be geometrically satisfied?

They already are.

The question becomes:

> can arithmetic squarehood be made simultaneous under conservation moves?

This changes the object of search.

---

# 34. A stronger exact reformulation

Because:

$$
D=1-C,
$$

$$
E=1-B,
$$

$$
F=1-A,
$$

the entire Perfect Cuboid existence problem becomes:

find:

$$
A,B,C\in\mathbb Q^2\cap(0,1)
$$

such that:

$$
A+B+C=1,
$$

and:

$$
1-A\in\mathbb Q^2,
$$

$$
1-B\in\mathbb Q^2,
$$

$$
1-C\in\mathbb Q^2.
$$

Equivalently:

$$
\boxed{
A,B,C,1-A,1-B,1-C
\in
\mathbb Q^2,
}
$$

with:

$$
\boxed{
A+B+C=1.
}
$$

This is the minimal conservation statement.

---

# 35. Conservation-square simplex

Define:

$$
\boxed{
\mathcal S_{\square}
=
\left\{
(A,B,C)\in\Delta_2^\circ:
A,B,C,1-A,1-B,1-C
\in\mathbb Q^2
\right\}.
}
$$

Then:

$$
\boxed{
\text{Perfect Cuboid exists}
\iff
\mathcal S_{\square}\neq\varnothing.
}
$$

This is the canonical theorem target of the new branch.

---

# 36. Boundary states

Boundary points satisfy at least one of:

$$
A=0,
$$

$$
B=0,
$$

$$
C=0.
$$

They correspond to degenerate cuboid geometry.

The search target is strictly:

$$
\boxed{
A,B,C>0.
}
$$

Thus known trivial square decompositions on the simplex boundary are not Perfect Cuboids.

---

# 37. Conservation and scale

The variables:

$$
A,\ldots,F
$$

are scale-free.

Once a rational-square interior state is found, write:

$$
A=\alpha^2,
$$

$$
B=\beta^2,
$$

$$
C=\gamma^2,
$$

$$
D=\delta^2,
$$

$$
E=\epsilon^2,
$$

$$
F=\phi^2,
$$

with all six parameters rational.

Choose a common denominator:

$$
G.
$$

Then:

$$
a=G\alpha,
$$

$$
b=G\beta,
$$

$$
c=G\gamma,
$$

$$
d=G\delta,
$$

$$
e=G\epsilon,
$$

$$
f=G\phi
$$

are integers after sufficient common scaling.

So scale is again postponed until the final manifestation stage.

---

# 38. Proof certificate

A minimal conservation certificate consists of:

$$
(A,B,C)
$$

plus rational square roots of:

$$
A,
B,
C,
1-A,
1-B,
1-C.
$$

The verifier checks:

$$
A+B+C=1.
$$

That alone reconstructs all six geometric relations.

Therefore the final certificate can be smaller than the original seven-length witness.

---

# 39. Software visualization

The new software view should show six cells:

$$
\boxed{
\begin{matrix}
A & B & C\\
F & E & D
\end{matrix}
}
$$

with three vertical conserved sums:

$$
1,\quad1,\quad1.
$$

The top row has conserved sum:

$$
1.
$$

Each cell can display:

- current rational value;
- rational-square status;
- squarefree kernel;
- denominator height;
- dual partner;
- current move direction.

A move can be animated as mass transferring between cells while the column sums remain fixed.

---

# 40. Visual truth state

Geometric closure can be displayed as:

$$
\boxed{
\mathbf T=
(1,1,1,1,1,1).
}
$$

Arithmetic square status can be displayed separately.

Current seed:

$$
\boxed{
\mathbf T=
(1,1,1,1,1,1),
}
$$

$$
\boxed{
\boldsymbol\Sigma=
(1,1,1,0,1,1).
}
$$

Target:

$$
\boxed{
\mathbf T=
\boldsymbol\Sigma=
(1,1,1,1,1,1).
}
$$

This gives a direct visual meaning to the final closure event.

---

# 41. The “click” event

The final software event is no longer merely:

> all equations numerically look small.

It is:

### Geometric layer

$$
\mathbf T=\mathbf1_6.
$$

### Arithmetic layer

$$
\boldsymbol\Sigma=\mathbf1_6.
$$

When both happen simultaneously:

$$
\boxed{
\text{GEOMETRIC + ARITHMETIC EXACT CLOSURE}.
}
$$

That is the rigorous version of the visual:

> **click.**

---

# 42. New proof strategies suggested by the conservation form

The methodology suggests several new directions.

## A. Move-orbit classification

Classify rational-square states under the legal move generators.

## B. Defect impossibility theorem

Show that a one-defect state cannot move to zero defects under some invariant.

If true, it would imply nonexistence for the corresponding orbit.

## C. Defect-closing theorem

Construct an explicit legal move from a five-square state to a six-square state.

If successful, it yields existence.

## D. Conserved arithmetic invariant

Search for quantities invariant under legal arithmetic moves:

- squarefree kernel products;
- Hilbert symbols;
- parity vectors;
- local residue signatures.

## E. Dual-pair descent

Study each complementary pair:

$$
(A,1-A),
$$

$$
(B,1-B),
$$

$$
(C,1-C)
$$

as one arithmetic object.

Then impose only the global sum:

$$
A+B+C=1.
$$

This may be simpler than coupling six variables simultaneously.

---

# 43. Relation to the previous CARCE surface

The CARCE variables:

$$
t_a,t_b,t_c
$$

parameterize the condition that both members of each dual pair are rational squares.

The conservation variables:

$$
A,B,C
$$

parameterize the global shape simplex.

Thus:

$$
\boxed{
\text{CARCE}
}
$$

is an arithmetic parameterization of:

$$
\boxed{
\text{Conservation–Duality Square Simplex}.
}
$$

The two formulations are not competitors.

They are different coordinate systems on the same exact problem.

---

# 44. Relation to the X-Diagonal formulation

The X-Diagonal formulation made the face geometry primary.

The Conservation–Duality formulation goes one level more abstract.

It retains only:

- normalized square shares;
- complementary pairs;
- truth channels;
- legal conserved moves.

Thus the hierarchy becomes:

$$
\boxed{
\text{3D Geometry}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{X-Diagonal Coupling}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{Six-Channel Conservation State}.
}
$$

This is currently the smallest visual-algebraic representation of the full geometry.

---

# 45. C0 methodology principles

### Principle 1 — Normalize the whole

The body diagonal is the conserved unit:

$$
g^2\mapsto1.
$$

### Principle 2 — Separate geometry and arithmetic

Geometry requires:

$$
\mathbf T=\mathbf1_6.
$$

Arithmetic requires:

$$
\boldsymbol\Sigma=\mathbf1_6.
$$

### Principle 3 — Work on the conservation manifold

Do not repeatedly solve Pythagorean geometry.

Preserve it by construction.

### Principle 4 — Use dual compensation

Every local move has an automatic complementary move.

### Principle 5 — Treat defect as transferable structure

Arithmetic failure has a location and can move.

### Principle 6 — Scale only at the end

The state is shape-first and scale-free.

### Principle 7 — Exact verification remains final

No numerical closeness replaces rational-square proof.

---

# 46. Phase C1 — Conservation Move Algebra

The next formal phase should derive the algebra generated by:

$$
M_{AB},
$$

$$
M_{BC},
$$

$$
M_{CA}.
$$

Goals:

- classify legal domains;
- classify commuting moves;
- derive orbit invariants;
- identify arithmetic quantities preserved under move composition;
- derive exact two-parameter state formulas.

---

# 47. Phase C2 — Square-Preserving Moves

For a current square state, solve exact conditions on:

$$
\lambda,\mu
$$

such that selected moved cells remain rational squares.

The initial experiment should start from the current seed:

$$
\boldsymbol\Sigma=
(1,1,1,0,1,1).
$$

The first question is:

> is there a single legal conservation move that turns $D$ into a rational square while keeping the other five cells square?

If no, classify the minimum move depth.

---

# 48. Phase C3 — Defect Dynamics

Construct a graph whose node label is:

$$
\boldsymbol\Sigma.
$$

Examples:

$$
111011,
$$

$$
110111,
$$

$$
111111.
$$

Study:

- defect migration;
- defect splitting;
- defect annihilation;
- cycles;
- unreachable patterns.

This is the discrete arithmetic analogue of the earlier continuous geometry dynamics.

---

# 49. Phase C4 — Local Invariants

For every prime:

$$
p,
$$

attach local square characters to the six cells.

A move then transforms a local signature vector.

Search for conserved mod-$p$ signatures that may forbid:

$$
111111
$$

or identify the unique pathways into it.

This connects the new method directly to the previous local-descent machinery.

---

# 50. Phase C5 — Visual Conservation Cube

Build a six-cell interactive interface.

Required features:

- show the three dual columns;
- animate conservation moves;
- lock:
  $$
  A+B+C=1;
  $$
- lock all three dual sums;
- show exact rational values;
- show squarefree kernels;
- display:
  $$
  \mathbf T;
  $$
- display:
  $$
  \boldsymbol\Sigma;
  $$
- mark defect migration.

The software should make it visually obvious that geometry is already closed while arithmetic closure changes.

---

# 51. Canonical theorem target

The entire classical Perfect Cuboid existence problem is equivalent to:

$$
\boxed{
\exists
(A,B,C)
\in
\mathbb Q^3
}
$$

such that:

$$
A+B+C=1,
$$

and:

$$
\boxed{
A,
B,
C,
1-A,
1-B,
1-C
\in
\mathbb Q^2
}
$$

with:

$$
A,B,C>0.
$$

This is the **Conservation–Duality Square-Simplex Problem**.

---

# 52. One-line formulation

> Find an interior point of the rational two-simplex for which every cell and every dual complement is a rational square.

Symbolically:

$$
\boxed{
\mathcal S_{\square}
=
\left\{
(A,B,C):
A+B+C=1,\;
A,B,C,1-A,1-B,1-C\in\mathbb Q^2
\right\}.
}
$$

The classical Perfect Cuboid problem asks:

$$
\boxed{
\mathcal S_{\square}\neq\varnothing\ ?
}
$$

---

# 53. Final methodological interpretation

The old view asks:

> Which seven integers satisfy all the Pythagorean equations?

The new conservation view asks:

> Can six arithmetic square cells simultaneously occupy a two-dimensional conserved dual state?

The real geometry is already solved:

$$
\boxed{
\mathbf T=\mathbf1_6.
}
$$

The unresolved problem is entirely arithmetic:

$$
\boxed{
\boldsymbol\Sigma
\stackrel{?}{=}
\mathbf1_6.
}
$$

The current seed is one defect away:

$$
\boxed{
111011
\rightarrow
111111.
}
$$

The new research program studies the algebra of legal conservation moves between these arithmetic states.

---

## Canonical C0 Target

$$
\boxed{
\text{Preserve }
\mathbf T=\mathbf1_6
\text{ exactly, and solve only the transition }
\boldsymbol\Sigma:
111011\rightarrow111111.
}
$$

---

**End of Conservation–Duality Methodology C0**