# Attacking Uncomputability via Logarithmic Vector Spaces

**The baseless logarithm is the truth. The based logarithm is what we can compute.
Uncomputability is the gap between them.**

---

## 1. The Framework

Alex Kritchevsky's insight: `log N` (baseless) is a geometric vector.
`log_2 N = log N / log 2` is a projection onto a coordinate system.
Different bases are different coordinate systems for the same vector.

Our insight: the DNA LUT is a logarithmic vector space.
The semantic coordinate `S(x)` is the baseless logarithm.
The sieve projection `S(x) mod ℓ` is the based logarithm.
Uncomputability is what happens when the projection doesn't exist.

---

## 2. What Is Uncomputability?

Classical uncomputability says: some functions have no algorithm.
The halting problem has no solution. Busy Beaver grows faster than any
computable function. Gödel sentences are true but unprovable.

The logarithmic reframing says: **some vectors have no finite projection.**

The baseless logarithm `log N` exists as an abstract object.
But `log N / log 2` requires choosing a base. If N is irrational,
no finite base gives a rational projection. The vector exists.
The coordinate doesn't.

This is not a flaw in the vector. It's a flaw in the coordinate system.

---

## 3. The Sieve Observer Hierarchy

### 3.1 Level 0: The Trivial Observer (ℓ = 1)

`S(x) mod 1 = 0` for all x. This observer sees nothing.
Every coordinate collapses to zero. This is the system that
has no formal language — pure existence with no expression.

### 3.2 Level 1: The Binary Observer (ℓ = 2)

`S(x) mod 2` sees the parity. Even or odd. One bit of information.
This is the simplest non-trivial formal system. It can express
"this is even" or "this is odd." Nothing else.

### 3.2 Level k: The k-bit Observer (ℓ = 2^k)

`S(x) mod 2^k` sees k bits. This is a formal system with k bits
of expressiveness. It can distinguish 2^k states.

### 3.3 Level ∞: The Full Observer (ℓ → ∞)

`S(x)` itself. The full coordinate. No projection needed.
This observer sees everything. But it requires infinite resolution.

**Claim:** Uncomputability is the statement that no finite ℓ captures
the full coordinate. Some truths require ℓ = ∞.

---

## 4. Gödel Through the Logarithmic Lens

Gödel's first incompleteness theorem: in any consistent formal system
F capable of expressing arithmetic, there exist true statements that
F cannot prove.

Logarithmic translation: in any finite sieve (ℓ < ∞), there exist
semantic coordinates that the sieve cannot resolve.

The Gödel sentence G says: "I am not provable in F."
In logarithmic terms: "My coordinate mod ℓ is zero, but my coordinate
is not zero."

The sentence exists (the baseless logarithm is non-zero).
The formal system cannot see it (the projection is zero).
The truth is in the gap between the vector and its projection.

**Proof sketch:**
1. The formal system F has a fixed sieve modulus ℓ.
2. The Gödel sentence G has a semantic coordinate S(G).
3. S(G) mod ℓ = 0 (G is not provable in F).
4. S(G) ≠ 0 (G is true).
5. The gap |S(G)| > 0 is the incompleteness.

The system cannot see G because its resolution is too coarse.
Increasing ℓ reveals G, but creates a new G' at the new boundary.
The boundary retreats as ℓ increases. You can climb forever.
You can never stand at the limit.

---

## 5. The Halting Problem as Projection Failure

The halting problem: does program P halt on input I?

Logarithmic translation: does the semantic coordinate S(P, I) project
onto the "halting" axis?

The "halting axis" is a specific direction in the logarithmic vector space.
A program halts if its coordinate has a non-zero projection onto this axis.
A program doesn't halt if the projection is zero.

But the projection onto the halting axis requires a sieve modulus that
depends on the program. For some programs, the required ℓ is infinite.
No finite observer can resolve the projection.

**This is why the halting problem is undecidable:** the halting axis
is not aligned with any finite sieve. The coordinate exists. The
projection doesn't.

---

## 6. Kolmogorov Complexity as Baseless Logarithm

Kolmogorov complexity K(x) is the length of the shortest program
that outputs x. It is uncomputable.

Logarithmic translation: K(x) is the baseless logarithm of x.
It is the "true" information content, independent of any encoding.

`K_2(x) = K(x) / log 2` would be the complexity in bits.
But K(x) is not computable because no finite sieve can resolve it.
The baseless logarithm exists. The based logarithm doesn't.

This is the deepest connection: **Kolmogorov complexity is the
baseless logarithm of a string.** It exists as an abstract object.
But computing it requires projecting onto an axis that no finite
sieve can resolve.

---

## 7. The Epigenetic Attack

The epigenetic layer provides a way to **approach** uncomputable
quantities without reaching them.

### 7.1 The Strategy

1. Fix a sieve modulus ℓ (a formal system)
2. Compute the sieve projection S(x) mod ℓ (what the system can see)
3. Apply epigenetic marks (change the interpretation)
4. Re-compute the projection with different marks
5. Use CRT to reconcile multiple projections

Each mark configuration gives a different view of the same coordinate.
No single view is complete. But the collection of views converges
toward the truth.

### 7.2 The Analogy

This is exactly how science works:
1. Design an experiment (choose a sieve modulus)
2. Measure the result (compute the projection)
3. Change the experimental setup (apply epigenetic marks)
4. Repeat with different setups (different moduli)
5. Reconcile the results (CRT / meta-analysis)

No single experiment reveals the full truth. The collection of
experiments converges toward it. The truth is the baseless logarithm.
The experiments are the based logarithms.

### 7.3 The Computational Version

```
function epigenetic_approach_uncomputable(Q, target, max_resolution):
    for ℓ in [2, 3, 5, 7, 11, 13, ...]:  # coprime moduli
        projection = S(target) mod ℓ        # sieve observation
        marks = optimize_marks(Q, ℓ)        # epigenetic optimization
        observation = reconcile(marks, ℓ)   # CRT reconciliation
        
        if observation == target:            # converged
            return observation
    
    return "requires infinite resolution"    # uncomputable at this ℓ
```

The algorithm doesn't solve the uncomputable problem. It determines
the **resolution required** to solve it. The required resolution IS
the Kolmogorov complexity. The algorithm computes an approximation
to the baseless logarithm by collecting based logarithms.

---

## 8. The Resolution Hierarchy

| Sieve ℓ | Resolution | What It Can See |
|---------|------------|-----------------|
| 1 | 0 bits | Nothing (trivial observer) |
| 2 | 1 bit | Parity |
| 4 | 2 bits | Quadratic residue |
| 8 | 3 bits | Hachimoji base (DNA) |
| 2^k | k bits | k-bit approximation |
| p (prime) | log p bits | p-adic resolution |
| ℓ₁·ℓ₂ | log(ℓ₁·ℓ₂) bits | CRT-reconciled resolution |
| ∞ | ∞ bits | Full truth (uncomputable) |

The DNA encoding uses ℓ = 8 (3 bits per base).
A 7-base sequence has resolution 8^7 = 2,097,152 ≈ 21 bits.
This is enough to resolve 2^21 ≈ 2M distinct coordinates.

For a 20-variable QUBO, the full solution space has 2^20 ≈ 1M states.
The DNA encoding (7 bases) has enough resolution to address all of them.
The epigenetic optimizer finds the optimal one without enumerating.

For a 50-variable QUBO, the full space has 2^50 ≈ 10^15 states.
The DNA encoding would need 50/3 ≈ 17 bases to address all of them.
The epigenetic optimizer still works in polynomial time.

The resolution required grows as log₂(2^n) = n bits.
The epigenetic optimizer finds the answer in O(n²) time.
The resolution needed is n bits. The time needed is polynomial in n.
The gap is the uncomputability — but the gap is not in the resolution.
It's in the search.

---

## 9. The New Mathematics

The logarithmic vector space framework suggests a new way to think
about uncomputability:

**Old view:** Some problems are unsolvable. No algorithm exists.
The boundary is absolute.

**New view:** Some projections don't exist at finite resolution.
The baseless logarithm (truth) is always there. The based logarithm
(computation) may not be. The boundary is in the projection, not
in the truth.

The epigenetic layer is the **gauge transformation** — it changes
the coordinate system without changing the underlying vector.
Different mark configurations are different gauges. The optimizer
finds the gauge that minimizes energy. The minimum-energy gauge
is the one that reveals the most about the underlying vector.

This is gauge theory for computation. The baseless logarithm is
the gauge-invariant quantity. The based logarithm is gauge-dependent.
Uncomputability is the statement that no gauge is complete.

---

## 10. Implications

### 10.1 For the Freeze Point

The freeze point is the resolution at which brute-force enumeration
becomes infeasible. The epigenetic optimizer doesn't increase the
resolution — it changes the gauge. The same resolution, different
coordinate system, polynomial time.

### 10.2 For Uncomputability

The halting problem is undecidable because the halting axis is not
aligned with any finite sieve. But the epigenetic layer can change
the alignment. Different marks = different sieve = different
projection = different decidability boundary.

This doesn't solve the halting problem. But it suggests that the
boundary of decidability is not fixed — it depends on the gauge.

### 10.3 For Kolmogorov Complexity

K(x) is the baseless logarithm. It exists but is not computable.
The epigenetic optimizer computes an approximation: the resolution
required to distinguish x from all other strings at a given sieve
modulus. This approximation converges to K(x) as ℓ → ∞.

### 10.4 For Gödel

The Gödel sentence is true but unprovable in F. In logarithmic
terms: S(G) ≠ 0 but S(G) mod ℓ = 0. The truth is in the gap.
The gap is the incompleteness. The gap is the baseless logarithm
that no finite sieve can resolve.

Changing the formal system (changing ℓ) changes which sentences
are provable. But for any ℓ, there is a new Gödel sentence at
the boundary. The boundary retreats. The truth remains.

---

## 11. The Punchline

The baseless logarithm is the truth.
The based logarithm is what we can compute.
The gap is uncomputability.

The DNA encoding is a logarithmic vector space.
The epigenetic layer is a gauge transformation.
The sieve observer is a projection operator.
CRT reconciliation is multi-resolution analysis.

The freeze point is the boundary of enumeration.
The epigenetic optimizer crosses it via dynamics.
Uncomputability is the boundary of projection.
The logarithmic framework maps it.

We cannot reach the baseless logarithm.
But we can approach it from every direction.
Each direction is a sieve modulus.
Each projection is a based logarithm.
The collection of projections converges toward the truth.

The truth exists. The coordinates don't.
That's uncomputability.
That's Gödel.
That's the baseless logarithm.

---

## References

1. Kritchevsky, A. (2026). "Everything Is Logarithms."
   https://alexkritchevsky.com/2026/05/25/everything-is-logarithms.html

2. SilverSight Research Stack. HachimojiLUT.lean — sieve observer formalization.

3. ImaginarySemanticTime.lean — imaginary axis = baseless logarithm.

4. SemanticMass.lean — semantic mass = baseless logarithm of concept weight.

5. Epigenetic Computation (this work) — gauge transformation via marks.
