> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# FORC: Fraction-Ontological Ratio Calculus
## Fraction-Ontological Ratio Calculus for X4 Virtual Curvature and Exact Click Search

Version: v0.1  
Nature: Foundational methodology / ratio calculus / discrete rational geometry / X4 candidate-navigation layer  
Relations: IGPC, X4-DMSD, DMSD, GERES, Perfect Cuboid Research Line  
Canonical source of the original: UTF-8 Markdown

---

## Abstract

This methodology proposes a Perfect Cuboid candidate-navigation framework distinct from floating-point local approximation, sequential integer enumeration and traditional continuous calculus.

The central observation is:

$$
\boxed{
\text{Real-local proximity}
\neq
\text{Rational-lattice proximity}.
}
$$

In real geometry, a state can be continuously approximated by:

$$
x\rightarrow x+\varepsilon
$$

but in rational / integer realization, the next exact couplable state may be separated by an enormous arithmetic height.

FORC therefore no longer treats the "next integer candidate" as an ordinary local neighbor, but as the next exact state on a rational branch.

Let:

$$
x=\frac pq
$$

be a reduced-fraction state, and define adjacency, branches and refinement in a Farey / Stern–Brocot-type rational graph.

If:

$$
\frac ab<\frac cd
$$

are Farey neighbors, then:

$$
bc-ad=1,
$$

and:

$$
\boxed{
\frac cd-\frac ab
=
\frac1{bd}.
}
$$

Larger denominators give smaller steps in ratio space; the corresponding Euclid / Pythagorean integer realizations, however, may have edge lengths growing quadratically.

Thus:

$$
\boxed{
\text{Close in ratio}
\quad\Longleftrightarrow\quad
\text{Potentially far in the integer world}.
}
$$

For X4 PerfectClick search, define four interface-defect channels:

$$
\boxed{
\mathbf E
=
(E_1,E_2,E_3,E_4).
}
$$

Each:

$$
E_i=0
$$

indicates exact closure of its corresponding interface.

Only when all four simultaneously satisfy:

$$
E_1=E_2=E_3=E_4=0
$$

is the following formed:

$$
\boxed{
\text{Four-Corner Click}.
}
$$

Along a rational path:

$$
x_0,x_1,x_2,\ldots
$$

define:

- rational step；
- ratio derivative；
- defect velocity；
- second divided difference；
- virtual constraint curvature；
- inward / outward orientation；
- ratio integral；
- curvature integral；
- branch score.

FORC does not replace exact arithmetic with calculus.

Its central division of responsibilities is:

$$
\boxed{
\text{Ratio Calculus navigates;}
}
$$

$$
\boxed{
\text{X4 Exact Click decides.}
}
$$

Derivatives, curvature and integrals answer only:

> Where should we search next?

The final theorem gate remains:

$$
\operatorname{PerfectClick}\in\{0,1\}.
$$

---

# 1. Problem Background

Traditional Perfect Cuboid searches usually enumerate directly:

$$
a,b,c\in\mathbb Z_{>0}.
$$

or enumerate Pythagorean triples and then perform shared-edge joins.

This approach can easily imply:

$$
\boxed{
\text{Numerical proximity}
\approx
\text{Structural proximity}.
}
$$

But this need not be so in an arithmetic lattice.

---

# 2. Floating-Point Proximity and Rational Proximity

If the real world contains:

$$
x^\ast,
$$

floating-point computation can produce:

$$
x_n\rightarrow x^\ast.
$$

Yet if one requires:

$$
x_n\in\mathbb Q
$$

along with additional exact constraints, the next admissible denominator may increase sharply.

Thus:

$$
\boxed{
\text{distance in }\mathbb R
}
$$

and:

$$
\boxed{
\text{height distance in }\mathbb Q
}
$$

are different quantities.

---

# 3. Fraction Ontology

FORC treats:

$$
\frac pq
$$

as a first-class exact state, not as an approximation to a decimal expansion.

Require:

$$
\gcd(p,q)=1,
$$

$$
q>0.
$$

Define:

$$
\boxed{
[x]=(p,q).
}
$$

---

# 4. Rational Height

Define:

$$
\boxed{
H(x)=\max(|p|,q).
}
$$

It is not real distance, but arithmetic realization complexity.

---

# 5. Rational Neighborhood

An ordinary real neighborhood:

$$
(x-\varepsilon,x+\varepsilon).
$$

FORC instead uses rational adjacency.

The first version adopts:

$$
\boxed{
\text{Farey adjacency}
}
$$

and:

$$
\boxed{
\text{Stern–Brocot refinement}.
}
$$

---

# 6. Farey Step

If:

$$
\frac ab<\frac cd
$$

and:

$$
bc-ad=1,
$$

then:

$$
\boxed{
\Delta_Fx
=
\frac1{bd}.
}
$$

This is an exact rational step.

---

# 7. Height–Distance Duality

If:

$$
b,d\sim H,
$$

then:

$$
\Delta_Fx\sim H^{-2}.
$$

Meanwhile, the Euclid seed:

$$
t=\frac nm
$$

produces:

$$
m^2-n^2,
$$

$$
2mn,
$$

$$
m^2+n^2,
$$

whose integer height is approximately:

$$
O(H^2).
$$

Thus:

$$
\boxed{
\Delta t\downarrow
\quad\text{while}\quad
\text{integer realization height}\uparrow.
}
$$

---

# 8. Proposition FORC-P1: Ratio-Near / Integer-Far

There exist rational sequences:

$$
t_k\rightarrow t^\ast
$$

such that:

$$
|t_{k+1}-t_k|\rightarrow0,
$$

but the primitive integer realization height satisfies:

$$
H_k\rightarrow\infty.
$$

Thus:

$$
\boxed{
\text{ratio-near}
\not\Rightarrow
\text{integer-near}.
}
$$

---

# 9. Change of Search Unit

Traditional search:

$$
\boxed{
\text{search points}.
}
$$

FORC:

$$
\boxed{
\text{search rational branches}.
}
$$

---

# 10. Rational Path

Define:

$$
\gamma
=
(x_0,x_1,\ldots,x_N)
$$

where adjacent states satisfy the chosen rational adjacency.

---

# 11. Stern–Brocot Branch

If:

$$
\frac ab<\frac cd,
$$

the mediant:

$$
\boxed{
\frac{a+c}{b+d}
}
$$

provides exact rational refinement.

Successive refinement:

$$
x_0\rightarrow x_1\rightarrow x_2\rightarrow\cdots
$$

is called a rational branch.

---

# 12. X4 Defect Vector

For a rational X4 state:

$$
X
$$

define:

$$
\boxed{
\mathbf E(X)
=
(E_1,E_2,E_3,E_4).
}
$$

Each $E_i$ is a theorem-relevant interface defect.

---

# 13. The Meaning of the Four Corners

The four corners here are not merely physical rectangle corners.

They are the four interface channels for joint closure of X and its four sides.

They may include:

- X–edge orientation；
- edge-angle relation；
- shared-edge ratio；
- local/global coupling slot.

The canonical choice is left for FORC-P1 and the X4 schema to fix formally.

---

# 14. Corner Click

$$
\boxed{
E_i=0
}
$$

defines:

$$
\operatorname{Click}_i=1.
$$

---

# 15. Four-Corner Click

$$
\boxed{
\mathbf E=\mathbf0
}
$$

when this holds:

$$
\boxed{
\operatorname{Click}_4=1.
}
$$

---

# 16. Four-Corner Click Is Not the Final PerfectClick

The FORC navigation layer examines only the selected interfaces.

A final Perfect Cuboid must also pass all of X4-DMSD's exact face / body arithmetic gates.

Thus:

$$
\boxed{
\operatorname{Click}_4
\neq
\operatorname{PerfectClick}
}
$$

unless all required channels have been included.

---

# 17. Ratio Derivative

Along:

$$
x_k\rightarrow x_{k+1},
$$

define:

$$
\boxed{
D_FE_i(k)
=
\frac{
E_i(x_{k+1})-E_i(x_k)
}{
x_{k+1}-x_k
}.
}
$$

---

# 18. Defect Velocity

Define:

$$
\boxed{
V_i(k)=D_FE_i(k).
}
$$

Vector:

$$
\boxed{
\mathbf V
=
(V_1,V_2,V_3,V_4).
}
$$

---

# 19. Why Not Use Ordinary Derivatives

FORC's fundamental states are exact fractions.

The first version therefore prioritizes:

$$
\boxed{
\text{divided differences}
}
$$

rather than first assuming a continuously differentiable manifold.

---

# 20. Nonuniform Rational Step

Farey steps generally satisfy:

$$
x_{k+1}-x_k
\neq
x_k-x_{k-1}.
$$

The ordinary equal-step second difference is therefore insufficient.

---

# 21. Virtual Constraint Curvature

Let:

$$
h_-
=
x_k-x_{k-1},
$$

$$
h_+
=
x_{k+1}-x_k.
$$

and:

$$
s_-
=
\frac{
E_i(x_k)-E_i(x_{k-1})
}{
h_-
},
$$

$$
s_+
=
\frac{
E_i(x_{k+1})-E_i(x_k)
}{
h_+
}.
$$

Define:

$$
\boxed{
\kappa_i(k)
=
\frac{
2(s_+-s_-)
}{
h_-+h_+
}.
}
$$

---

# 22. Why Call It "Virtual Curvature"

It is not the physical curvature of a cuboid edge.

The edge can remain straight.

It describes the bending of the:

$$
\boxed{
\text{constraint-error field}
}
$$

along a rational path.

Hence the name:

$$
\boxed{
\text{Virtual Constraint Curvature}.
}
$$

---

# 23. Curvature Vector

$$
\boxed{
\boldsymbol\kappa
=
(
\kappa_1,\kappa_2,\kappa_3,\kappa_4
).
}
$$

---

# 24. Inward / Outward

Different defect channels may have different natural positive / negative orientations.

Formally define the orientation:

$$
o_i\in\{-1,+1\}.
$$

Standardized defect:

$$
\boxed{
\widetilde E_i=o_iE_i.
}
$$

Standardized velocity:

$$
\boxed{
\widetilde V_i=o_iV_i.
}
$$

Standardized curvature:

$$
\boxed{
\widetilde\kappa_i=o_i\kappa_i.
}
$$

---

# 25. Toward Click

Use the uniform convention:

$$
\boxed{
\widetilde V_i<0
}
$$

means that the channel is moving toward:

$$
E_i=0
$$

that exact zero.

---

# 26. Inward Curvature

The first version adopts the convention:

$$
\boxed{
\widetilde\kappa_i<0
}
$$

means inward-bending.

---

# 27. Outward Curvature

$$
\boxed{
\widetilde\kappa_i>0
}
$$

means outward-bending.

---

# 28. A Near-Hit Is Insufficient for Navigation

An ordinary heuristic looks only at:

$$
|E_i|\ll1.
$$

But if:

$$
\widetilde V_i>0,
$$

it may be moving away from closure.

Thus:

$$
\boxed{
\text{near}
\neq
\text{inward}.
}
$$

---

# 29. Far-but-Inward

Conversely:

$$
|E_i|
$$

may be relatively large,

but:

$$
\widetilde V_i<0,
$$

and:

$$
\widetilde\kappa_i<0.
$$

Such a state may merit expansion more than a near-but-outward state.

---

# 30. Three-Level Navigation Information

Each channel preserves at least:

$$
\boxed{
(
E_i,
V_i,
\kappa_i
).
}
$$

---

# 31. Dynamic X4 State

Define:

$$
\boxed{
\mathfrak X_k
=
(
x_k,
H_k,
\mathbf E_k,
\mathbf V_k,
\boldsymbol\kappa_k
).
}
$$

---

# 32. Ratio Integral

Along:

$$
\gamma
=
(x_0,\ldots,x_N),
$$

define:

$$
\boxed{
\mathcal I_i(\gamma)
=
\sum_{k=0}^{N-1}
E_i(x_k)
(
x_{k+1}-x_k
).
}
$$

---

# 33. Trapezoidal Ratio Integral

A more symmetric version:

$$
\boxed{
\mathcal I_i^{(T)}
=
\sum_{k=0}^{N-1}
\frac{
E_i(x_k)+E_i(x_{k+1})
}{2}
(
x_{k+1}-x_k
).
}
$$

If all states and defect functions are rational,

this integral can still be exact rational.

---

# 34. Curvature Integral

Define:

$$
\boxed{
\mathcal K_i(\gamma)
=
\sum_{k=1}^{N-1}
\kappa_i(k)
w_k,
}
$$

where:

$$
w_k
=
\frac{
x_{k+1}-x_{k-1}
}{2}.
$$

---

# 35. Four-Corner Ratio Integral

$$
\boxed{
\boldsymbol{\mathcal I}
=
(
\mathcal I_1,
\mathcal I_2,
\mathcal I_3,
\mathcal I_4
).
}
$$

---

# 36. Four-Corner Curvature Integral

$$
\boxed{
\boldsymbol{\mathcal K}
=
(
\mathcal K_1,
\mathcal K_2,
\mathcal K_3,
\mathcal K_4
).
}
$$

---

# 37. A Ratio Integral Is Not a Theorem Proof

Even if:

$$
\mathcal I_i\rightarrow0,
$$

this cannot imply the existence of an exact:

$$
E_i=0.
$$

Thus:

$$
\boxed{
\text{integral = navigation evidence, not theorem truth}.
}
$$

---

# 38. Branch Score

A discovery heuristic can be defined as:

$$
\boxed{
S(\mathcal B)
=
-w_E\|\mathbf E\|
-w_V\|\mathbf V_+\|
-w_\kappa\|\boldsymbol\kappa_+\|
-w_HC_H.
}
$$

where:

$$
x_+=\max(x,0).
$$

---

# 39. Score Is Not Truth

$$
S(\mathcal B_1)>S(\mathcal B_2)
$$

means only:

> Branch $1$ should be explored first.

It does not mean that the branch necessarily contains a solution.

---

# 40. Perfect-Length Fractal

Provisionally define:

$$
\boxed{
\text{Perfect-Length Fractal}
}
$$

as the surviving branch subset in a rational refinement tree that continues across scales to satisfy:

- low defect；
- inward velocity；
- inward virtual curvature；
- controlled arithmetic height；
- X4 necessary gates;

these conditions.

---

# 41. "Fractal" Is Currently Methodological Terminology

P0 does not claim to have proved:

- non-integer Hausdorff dimension；
- exact self-similarity；
- classical fractal measure law.

For now, it means only:

$$
\boxed{
\text{recursive multi-scale branch structure}.
}
$$

---

# 42. Curvature Persistence

Define:

$$
P_i(N)
=
\frac{
\#\{k:\widetilde\kappa_i(k)<0\}
}{
N
}.
$$

---

# 43. Four-Corner Persistence

$$
\boxed{
P_4(N)
=
\frac{
\#\{k:
\forall i,\widetilde\kappa_i(k)<0
\}
}{
N
}.
}
$$

---

# 44. Closure Energy

Define:

$$
\boxed{
\Phi(X)
=
\sum_iw_i|E_i|.
}
$$

FORC can study:

$$
\Delta_F\Phi
$$

and:

$$
D_F^2\Phi.
$$

---

# 45. One Cannot Look Only at Scalar Energy

Looking only at:

$$
\Phi,
$$

can allow one channel to improve and another to deteriorate while the total still decreases.

The canonical state must therefore retain:

$$
\boxed{
\mathbf E
}
$$

rather than retaining only a scalar score.

---

# 46. Multi-Parameter Extension

A genuine X4 world may require:

$$
\mathbf x
=
(x_1,\ldots,x_d)
\in\mathbb Q^d.
$$

Define:

$$
\Delta\mathbf x
=
\mathbf x'-\mathbf x.
$$

Prefer a rational-compatible norm:

$$
\boxed{
\|\Delta\mathbf x\|_1
=
\sum_j|\Delta x_j|.
}
$$

---

# 47. Directional Ratio Derivative

$$
\boxed{
D_{\Delta\mathbf x}E
=
\frac{
E(\mathbf x+\Delta\mathbf x)-E(\mathbf x)
}{
\|\Delta\mathbf x\|_1
}.
}
$$

---

# 48. Virtual Curvature Tensor

In the future, one could define:

$$
\boxed{
K_{ij}
}
$$

to describe second divided differences in different rational directions.

P0 does not develop the full tensor calculus for now.

---

# 49. Curvature Coupling

The four defect channels are not independent.

If:

$$
\kappa_i
$$

and:

$$
\kappa_j
$$

remain opposed over a long period,

this may indicate competition between constraints.

If multiple channels are inward simultaneously,

a valuable branch basin may form.

---

# 50. Exact Zero Remains Special

No matter how small:

$$
|E_i|
$$

is,

as long as:

$$
E_i\neq0,
$$

then:

$$
\operatorname{Click}_i=0.
$$

---

# 51. Division of Responsibilities Between FORC and X4-DMSD

### FORC

Answers:

$$
\boxed{
\text{where to search}.
}
$$

### X4-DMSD

Answers:

$$
\boxed{
\text{whether exact world can exist}.
}
$$

---

# 52. FORC Cannot Sound-Prune on Its Own

Unless a curvature condition is separately proved to be necessary,

FORC can only:

$$
\operatorname{Priority}\downarrow.
$$

It cannot directly set:

$$
\operatorname{State}
=
\mathrm{IMPOSSIBLE}.
$$

---

# 53. Navigation Certificates and Theorem Certificates

FORC outputs:

$$
\boxed{
\text{Navigation Certificate}
}
$$

Contents:

- parent fraction；
- child fraction；
- defect vector；
- velocity vector；
- curvature vector；
- ratio integral；
- branch score.

X4-DMSD outputs:

$$
\boxed{
\text{Theorem Certificate}.
}
$$

The two must be kept separate.

---

# 54. FORC × IGPC

IGPC defines:

$$
\boxed{
\text{what the world is}.
}
$$

FORC defines:

$$
\boxed{
\text{how rational realizations move through world-space}.
}
$$

---

# 55. FORC × DMSD

DMSD manages:

$$
\boxed{
\text{scale}.
}
$$

FORC manages:

$$
\boxed{
\text{branch direction}.
}
$$

They can therefore combine as:

$$
\boxed{
\text{scale}
\times
\text{rational branch}.
}
$$

---

# 56. Long Jump

If a branch remains inward through multiple levels of rational refinement,

the runtime can increase its target height:

$$
H\rightarrow H'
$$

without materializing each denominator.

---

# 57. A Long Jump Is Not a Proof Skip

The skipped branch segment must still record:

- endpoints；
- parent；
- summary；
- navigation reason；
- theorem coverage status.

A FORC long jump cannot masquerade as an exhaustive proof.

---

# 58. Discovery Mode

Allows aggressive:

- best-first；
- beam search；
- branch long jump.

Objective:

$$
\boxed{
\text{find witness fast}.
}
$$

---

# 59. Proof Mode

FORC changes only the processing order.

Actual domain deletion must be governed by an X4-DMSD sound gate.

Objective:

$$
\boxed{
\text{preserve completeness}.
}
$$

---

# 60. The Two Modes Must Not Be Conflated

Finding one witness requires only:

$$
\exists.
$$

Proving no witness exists requires:

$$
\forall.
$$

FORC can therefore be aggressive in discovery mode, but proof mode must retain coverage of everything not sound-pruned.

---

# 61. Conjecture FORC-C1：Curvature-Guided Enrichment

Under the same computation budget,

using:

$$
(
\mathbf E,
\mathbf V,
\boldsymbol\kappa
)
$$

to rank rational branches

makes it easier to find high-quality X4 candidates than looking only at:

$$
\|\mathbf E\|
$$

that defect norm.

---

# 62. Conjecture FORC-C2：Height-Gap Prediction

Virtual curvature and ratio integrals can predict:

$$
\boxed{
\text{next useful denominator scale}.
}
$$

without scanning each denominator.

---

# 63. Conjecture FORC-C3：Four-Corner Inward Basin

If a branch satisfies, over most of a long scale interval:

$$
\widetilde V_i<0,
$$

$$
\widetilde\kappa_i\le0
$$

for all four channels,

then its exact-click candidate density is higher than that of ambient rational branches.

---

# 64. Conjecture FORC-C4：Rational-Branch Continuity

Effective continuity in some Diophantine geometries is not integer-coordinate continuity,

but:

$$
\boxed{
\text{continuity along rational refinement graphs}.
}
$$

---

# 65. Conjecture FORC-C5：Perfect-Length Fractal

X4 near-closure states may form a multiscale sparse recurrent structure in:

$$
\text{rational branch}
\times
\text{height}
$$

that space.

Whether it has a strict fractal dimension is left for experiments.

---

# 66. First-Phase MVP

### F1

Use:

$$
t=n/m.
$$

### F2

Formally select X4's four canonical defect channels.

### F3

Build a Stern–Brocot navigator.

### F4

An exact rational divided-difference engine.

### F5

An exact virtual curvature engine.

### F6

Ratio / curvature integrals.

### F7

Branch ranking.

### F8

Connect back to the X4-DMSD exact gates.

---

# 67. Control A：Quadratic Function

Test:

$$
E(x)=x^2-x.
$$

On a nonuniform rational path, verify:

$$
\kappa=2.
$$

---

# 68. Control B：No-Float

Require:

$$
x_k,E_i,V_i,\kappa_i,\mathcal I_i
$$

all to use an exact rational backend.

---

# 69. Control C：Near-but-Outward

Construct a case with:

$$
|E|
$$

very small,

but:

$$
\widetilde V>0.
$$

Check that ranking does not give top priority solely because of a near-hit.

---

# 70. Control D：Far-but-Inward

Construct a relatively large defect,

but:

$$
\widetilde V<0,
$$

$$
\widetilde\kappa<0.
$$

Check that a branch's future potential can rank above a near-but-outward branch.

---

# 71. Control E：Exact Click Override

Regardless of the heuristic score,

as long as:

$$
\mathbf E=\mathbf0
$$

and all X4 theorem gates pass,

it must immediately become:

$$
\boxed{
\mathrm{EXACT}.
}
$$

---

# 72. Persistent Runtime Integration

P13 cell metadata can include:

```text
branch_id
fraction_state
height
defect_vector
velocity_vector
curvature_vector
ratio_integral
curvature_integral
navigation_score
```

---

# 73. FORC Scores Do Not Change Truth State

The P13 theorem state remains:

$$
\mathrm{UNKNOWN},
\mathrm{POSSIBLE},
\mathrm{EXACT},
\mathrm{IMPOSSIBLE}.
$$

The navigation score is orthogonal metadata.

---

# 74. Research Stack

$$
\boxed{
\text{IGPC}
}
$$

Defines the world.

$$
\downarrow
$$

$$
\boxed{
\text{FORC}
}
$$

Navigates rational branches.

$$
\downarrow
$$

$$
\boxed{
\text{X4-DMSD}
}
$$

Decides exact arithmetic closure.

$$
\downarrow
$$

$$
\boxed{
\text{Persistent Proof Runtime}
}
$$

Runs over the long term.

---

# 75. First Principle

$$
\boxed{
\textbf{Navigate by curvature; decide by exact closure.}
}
$$

Its Chinese formulation translates as:

> **Navigate by curvature; decide by exact closure.**

---

# 76. Second Principle

$$
\boxed{
\textbf{Ratio-local, integer-nonlocal.}
}
$$

Its Chinese formulation translates as:

> **Ratio-local, integer-nonlocal.**

---

# 77. Third Principle

$$
\boxed{
\textbf{Branch first, point second.}
}
$$

Its Chinese formulation translates as:

> **Branches first, individual points second.**

---

# 78. Fourth Principle

$$
\boxed{
\textbf{Heuristic motion is not theorem truth.}
}
$$

Its Chinese formulation translates as:

> **Navigation trends are not theorem truth.**

---

# 79. Claim Boundary

This methodology has established:

- fraction ontology；
- rational height；
- Farey adjacency；
- Stern–Brocot refinement；
- exact rational step；
- X4 four-interface defect vector；
- ratio derivative；
- defect velocity；
- virtual constraint curvature；
- inward / outward standardization；
- ratio integral；
- curvature integral；
- branch state；
- branch score；
- discovery / proof mode separation；
- FORC × IGPC × X4-DMSD integration.

It has not yet proved:

- that curvature-guided search necessarily outperforms other searches;
- that the Perfect-Length Fractal has a classical fractal dimension;
- that virtual curvature can guarantee existence;
- that convergence of a ratio integral implies an exact click;
- that a Perfect Cuboid exists;
- that a Perfect Cuboid does not exist.

---

# 80. Conclusion

Viewed from the floating-point world, the Perfect Cuboid appears to have:

$$
\boxed{
\text{continuous local geometry}.
}
$$

Viewed from integer coordinates, it appears to have:

$$
\boxed{
\text{sparse nonlocal arithmetic jumps}.
}
$$

The third perspective proposed by FORC is:

$$
\boxed{
\text{exact rational branch geometry}.
}
$$

where:

- the state is an exact fraction;
- adjacency is rational;
- the derivative is a divided difference;
- curvature is virtual constraint curvature;
- integration is ratio-path accumulation;
- the branch is the basic navigation unit;
- exact truth is still decided by X4-DMSD.

The new question is therefore no longer merely:

> Which integer is closest to the target?

Instead, it is:

$$
\boxed{
\text{Which rational branch has four interfaces
that keep bending inward across scales,
pointing toward the next possible exact Click?}
}
$$

This is:

$$
\boxed{
\textbf{Fraction-Ontological Ratio Calculus}.
}
$$

---

## Next Phase

**FORC-P1 — Exact Rational Difference Algebra and Virtual Curvature Runtime**

Next steps:

1. Formally select X4's four canonical defect channels;
2. exact rational path schema；
3. Stern–Brocot navigator；
4. exact divided-difference engine；
5. exact virtual curvature engine；
6. ratio / curvature integrals；
7. inward/outward standardization；
8. branch scoring；
9. rational-function controls;
10. Integrate the X4-DMSD exact gate, while forbidding heuristic pruning of the theorem domain.