# What base 3 hides

Base 2 lives on the square lattice, and the constant buried in its arithmetic is pi: two random
integers are coprime with density `6/pi^2 = 1/zeta(2)`. Ask the same question in base 3 and the
answer is not pi. Base 3 brings 3-fold symmetry, and a lattice with 3-fold symmetry is forced to
be hexagonal; its arithmetic is the ring of Eisenstein integers, and the constant that falls out is
built from an L-value in the same family as Catalan's constant — a number with no known closed form
in terms of pi. Base 2 hid something elementary;
base 3 hides something genuinely deeper.

## The trap: visibility is base-blind

There is a wrong way to look for the base-3 constant, and it is worth naming. A lattice point
`(a, b)` is visible from the origin exactly when `gcd(a, b) = 1` — a condition on the coordinates
alone, indifferent to whether those coordinates draw a square grid or a hexagonal one. The density
of visible points is `1/zeta(2) = 6/pi^2 = 0.607927...` on *any* rank-2 lattice. Counting on a
`3000 x 3000` grid gives `0.608042`. So simply re-running a base-2 construction with base-3 tiles
returns pi again: the grid lines have not moved. The base-3 difference has to come from the number
theory that is native to base 3, not from repeating the base-2 experiment.

![Visible points of the hexagonal lattice](bases-fig.png)

The hexagonal lattice with its visible points marked; the faint points are hidden behind a nearer
one. The density of marked points tends to `6/pi^2` here just as on the square grid — the shape of
the lattice does not enter. The figure is drawn by `lab/base3/makefig.py`.

## Fresh structure per level

One thing base 3 does change is exact and easy. Refine the unit interval by a base `b`: level `L`
places nodes at `k/b^L`. The nodes that are new at level `L` — not already present at level
`L - 1` — are those where `b` does not divide `k`, and there are `b^L - b^(L-1)` of them. The
fraction of fresh structure per level is therefore `(b-1)/b`: base 2 refreshes `1/2`, base 3
refreshes `2/3`, base 5 refreshes `4/5`. When `b` is prime this count is exactly Euler's totient
`phi(b^L)` — the same function that makes primes the maximally novel scales. Higher base injects
more new resonance per level, and that is a theorem, not a measurement.

## The base-3 constant

A lattice with 3-fold symmetry is forced to be hexagonal — 6-fold in fact, since every lattice is
centrally symmetric — and its arithmetic is the Eisenstein
integers `Z[omega]`, `omega = e^(2 pi i / 3)`, with norm `N(a + b omega) = a^2 - a b + b^2`. This
ring has unique factorization, so coprimality makes sense in it, and the density of coprime pairs
of Eisenstein integers is `1/zeta_K(2)`, where `zeta_K` is the Dedekind zeta function of the field
`K = Q(sqrt(-3))`. That zeta function splits into familiar pieces:

```
zeta_K(s) = zeta(s) * L(s, chi_-3)
chi_-3(n) = +1 if n = 1 mod 3, -1 if n = 2 mod 3, 0 if 3 divides n
```

Every number below is computed to 40 digits via the Hurwitz zeta (trigamma) form in
`lab/base3/precision.py` and cross-checked by an independent route in `lab/base3/sieves.py`: the
direct Dirichlet series for the L-value, and for `zeta_K(2)` the raw
lattice sum `(1/6) * sum 1/N(z)^2` over all nonzero Eisenstein integers, which reproduces the
product `zeta(2) * L(2, chi_-3)` to ten digits.

| constant | value | role |
|---|---|---|
| `zeta(2) = pi^2/6` | 1.644934066848226 | the base-2 story: elementary, a pi-power |
| `L(2, chi_-3)` | 0.781302412896486 | the base-3 L-value: no known closed form |
| `zeta_K(2)`, `K = Q(sqrt(-3))` | 1.285190955484149 | product of the two rows above |
| `1/zeta_K(2)` | 0.778094489175179 | density of coprime Eisenstein pairs |
| Catalan `G = L(2, chi_-4)` | 0.915965594177219 | the square-lattice sibling |
| `1/(zeta(2) * G)` | 0.663700804613853 | density of coprime Gaussian pairs |

The density claim is not left to analogy. Running the Euclidean algorithm in `Z[omega]` —
nearest-integer division under the hexagonal norm — on 400,000 random pairs gives a coprime
fraction of `0.77809`, against the predicted `0.778094`. The constant is really there in the
lattice, sieved, not asserted.

## Not a power of pi

Is `L(2, chi_-3)` secretly a pi-power, the way `zeta(2) = pi^2/6` is? Everything points the other
way. The natural candidate is a rational multiple of `pi^2/sqrt(3)`, and an integer-relation search
finds none: the continued fraction of `L(2, chi_-3) * sqrt(3) / pi^2` runs on with ordinary
partial quotients, ruling out every rational with denominator below `10^11`. For comparison,
the same test applied to `zeta(2)/pi^2` terminates instantly at `1/6`.

There is a structural reason to expect this. The character `chi_-3` is *odd* (`chi(-1) = -1`), and
for odd characters it is the even-argument L-values that resist closed forms — the same parity
barrier behind `zeta(3)`, which has no known `pi^3` form. The exact sibling of our constant is
Catalan's constant `G = L(2, chi_-4) = 0.915965...`, one of the famous unresolved constants of
analysis: not known to be a pi-power, not even proven irrational. `L(2, chi_-3)` sits in the same
class; its one known relative is Gieseking's constant, via
`L(2, chi_-3) = (4/(3*sqrt(3))) * Cl2(pi/3)` with `Cl2(pi/3) = 1.0149416064...`.

To be precise about the strength of the claim: no search can *prove* a closed form absent, and no
impossibility proof is known. What is true is that none is known, none is expected, and any
candidate relation now has to clear a very high bar.

## Catalan prices every base-2 design

Catalan's constant is not only the square lattice's coprimality companion in the
table above; it prices the lattice energy of every base-2 design at once. Give a
2D design its four parity indicators `a_ee, a_eo, a_oe, a_oo` — which corners of
the parity square are filled — and sum `(i^2 + j^2)^(-s)` over the nonzero
lattice points whose parity class is filled. With `t = 2^(-s)` and `Re(s) > 1`:

```
Z_c(s)  =  4 * zeta(s) * beta(s) * Q_c(t)
Q_c(t)  =  a_ee * t^2  +  a_oo * t * (1 - t)  +  (a_eo + a_oe) * (1 - t)/2
```

The full-lattice factor `4 * zeta(s) * beta(s)` is classical — Jacobi's
two-square theorem summed, a lattice-sum identity going back to Lorenz and
Hardy — and parity-restricted lattice sums evaluating in `zeta` and `beta` are
classical too (Zucker 1974; Borwein et al., *Lattice Sums Then and Now*).
What is local here is the per-design polynomial `Q_c`, pinned by three exact
shell relations: doubling gives `S_ee = t^2 * S`; the 45-degree rotation
`(i, j) -> ((i+j)/2, (i-j)/2)` maps odd-odd pairs bijectively onto mixed-parity
pairs while halving the norm, giving `S_oo = t * S_mix`; and the coordinate
swap gives `S_eo = S_oe`. (Proved; Verified by `lab/visibility/energy.py` as
integer identities on the Dirichlet coefficients to `n = 200000`, and against
truncated lattice sums for all fifteen nonempty designs, worst gap `9.4e-10`
at `s = 2`.)

At `s = 2`, `t = 1/4` and `4 * zeta(2) * beta(2) = (2/3) * pi^2 * G`, so

```
Z_c(2) / (pi^2 * G)  =  (a_ee + 3*a_oo + 6*a_eo + 6*a_oe) / 24
```

a positive rational for every nonempty design — fifteen designs, fifteen small
rationals from `1/24` to `2/3`. The design that draws the Sierpinski triangle
at base 2 — code 7, corners `(0,0)`, `(0,1)`, `(1,0)` — gives
`13 * pi^2 * G / 24 = 4.8967847822`, with `G` the 40-digit value in the table
above. Every nonempty base-2 design's energy at `s = 2` is a rational multiple
of `pi^2` times Catalan. (Proved; Verified.)

The dimensions sort themselves the way this page has come to expect:

- **3D.** The parity classes are still pinned by `n mod 8` (Verified to
  `n = 6000`), but `sum r_3(n) * n^(-s)` has no Euler product and no known
  factorisation into standard L-functions — `r_3` is tied to class numbers —
  so no analogue of `4 * zeta * beta` exists to restrict. That is the absence
  of a known mechanism, not a proved impossibility.
- **4D.** Grade `Z^4` by the count of odd coordinates and Jacobi's four-square
  formula decomposes the energy of every one of the 65536 4D designs. The sum
  converges only for `Re(s) > 2`, so `s = 2` is out of reach; Catalan enters at
  `s = 3` through the factor `beta(s) * beta(s-1)`, with
  `beta(3) * beta(2) = pi^3 * G / 32`. Its coefficient in a design's energy is
  `n_1 - n_3` — filled corners with one odd coordinate minus those with three —
  so exactly the designs with `n_1 = n_3` are *Catalan-blind*: 17920 of 65536,
  which is `C(8,4) * 2^8`. (Proved; Verified to `n = 2400`, the blind census by
  exhaustive enumeration.)

So Catalan sits at `s = 2` in 2D, disappears in 3D for want of a
factorisation, and returns at `s = 3` in 4D — present for most designs and
cancelled exactly for a counted subfamily.

## Base picks the field

The two cases line up into one rule. The base selects a lattice symmetry, the symmetry selects an
imaginary quadratic field, and the field's Dedekind zeta is the hidden constant.

| base | lattice | ring | field | hidden constant |
|---|---|---|---|---|
| 2 | square (4-fold) | `Z[i]` | `Q(i)` | pi via `6/pi^2`; Catalan at `s = 2` |
| 3 | hexagonal (6-fold) | `Z[omega]` | `Q(sqrt(-3))` | `L(2, chi_-3)` — Catalan-class |

The arithmetic inside each row is proved; the pairing of base with lattice is the pattern, and it
comes with a built-in boundary: beyond the 2-fold symmetry every lattice has, only 4-fold and
6-fold rotations are possible — no 5-fold lattice exists, so there is no lattice for base 5 to
live on. Whatever base 5 hides, it is not found this way — which
is its own kind of answer.

> Base 2 hid pi. Base 3 hides a Catalan-class constant, with a `sqrt(3)` riding along as the
> hexagonal signature. Raising the base does not give more of the same; it climbs from the
> elementary world of pi-powers into the genuinely unresolved arithmetic of odd L-values.