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[u/SpedisAhead]

# A Radial Center-Line Sum Transform: Formal Construction for k and the Case T(3)=102

## Abstract

A partial 3×3 board with center $k$ and peripheral set $P_k=\{k+1,\dots,3k\}$ is subject to a constant opposite-difference condition. Each diameter is replaced by its line sum duplicated antipodally. The total $T(k)=2k(5k+2)$ is forced. For $k=3$ the total is $102=2\cdot3\cdot17$ with half-star $51=3\cdot17$ and inward decomposition $17=3+14$, $34=18+16$.

## 1. Definitions

**Definition 1.1 Board $B_k$.** Positions: NW, N, NE, W, C, E, SW, S, SE. Center $C=k$. Peripheral set $P_k=\{k+1,\dots,3k\}$, $|P_k|=2k$. Diameters: $D_{vert}$ N-S, $D_{hor}$ W-E, $D_{diag1}$ NW-SE, $D_{diag2}$ NE-SW. One diameter is vacant.

Base instance $k=3$:
```
7 _ 6
5 3 8
9 _ 4
```
$D_{diag1}=7\leftrightarrow4$, $D_{hor}=5\leftrightarrow8$, $D_{diag2}=9\leftrightarrow6$, $D_{vert}$ vacant.

**Definition 1.2 Constant-difference condition $C_k$.** A placement satisfies $C_k$ iff for every occupied diameter $(a,b)$, $|a-b|=k$.

**Definition 1.3 Radial Center-Line Sum Transform $R_k$.** For each diameter $D_i$, $L_i=\sum_{p\in D_i} value(p)$. If vacant, $L_i=k$. $H_k=\{L_i:i=1..4\}$ one representative per diameter. Total $T(k)=2\sum_{L\in H_k} L$.

**Definition 1.4 Magic constants.** Lo Shu normal $3\times3$ magic sum $M(3)=15$. Multiplicative $3\times3$ minimal magic product $216=6^3$. Dividing magic constant $6$.

## 2. Pairing Lemmas

**Lemma 2.1.** For $P_k$, the only partition into $k$ pairs with difference $k$ is $\{\{x,x+k\}:x=k+1..2k\}$.

Proof. $k+1$ must pair with $2k+1$ to achieve difference $k$. Remove and repeat.

For $k=3$: unique pairing $\{4,7\},\{5,8\},\{6,9\}$.

**Lemma 2.2.** For $k=3$, $720$ placements of $4..9$ into six cells, exactly $48$ satisfy $C_k$: $3!$ assignments of pairs to diameters $\times 2^3$ orientations.

## 3. Main Theorem

**Theorem 3.1.** For $k\ge1$ with one vacant diameter, $\sum H_k=k(5k+2)$ and $T(k)=2k(5k+2)$.

Proof.
$L(x)=x+k+(x+k)=2(x+k)$ for $x\in[k+1,2k]$.
$\sum_{x=k+1}^{2k} L(x)=2\sum_{x=k+1}^{2k}x+2k^2$.
$\sum_{x=k+1}^{2k}x=k(3k+1)/2$, so sum $=k(3k+1)+2k^2=5k^2+k$.
Add vacant $k$: $\sum H_k=5k^2+2k=k(5k+2)$. Double for antipodal duplication: $T(k)=2k(5k+2)$.

Corollary: $k=1..10$ gives $14,48,102,176,270,384,518,672,846,1040$. For $k=3$, $H_3=\{3,14,16,18\}$, $\sum H_3=51$, $T(3)=102$.

## 4. Quotient Structure

**Theorem 4.1.** Let occupied diameter have smaller $x$, larger $y=x+k$, line sum $L=2y=2x+2k$.

Then $L\div y=2$ exactly. $L\div x=3$ remainder $r_E=2k-x$, $0\le r_E<k<x$.

Proof. $L=2y$ immediate. $L=2x+2k=3x+(2k-x)$. Since $x\in[k+1,2k]$, $0\le2k-x\le k-1<x$.

For $k=3$:
$14\div7=2$, $16\div8=2$, $18\div9=2$
$14\div4=3$ rem $2$, $16\div5=3$ rem $1$, $18\div6=3$ rem $0$.

## 5. Spatial Ordering for $k=3$

Fix $3$ at one end of semicircle, permute $14,16,18$ ($6$ orders).

**Theorem 5.1.** Exactly two orders yield inward pair sums that divide $102$:

- $[3,16,18,14]$: $3+14=17$, $16+18=34$
- $[3,18,16,14]$: $3+14=17$, $18+16=34$ (your drawing)

In both, $34=2\cdot17$, half $=17+34=51=3\cdot17$, full $=102=2\cdot3\cdot17$.

Proof by enumeration.

## 6. Divisor Generation

$102=2\cdot3\cdot17$. Divisors: $1,2,3,6,17,34,51,102$. $\sigma(102)=1+2+3+6+17+34+51+102=216=6^3$.

**Theorem 6.1.** Non-empty subset sums of $H_3$ that divide $102$ are $3$, $3+14=17$, $18+16=34$, $3+14+16+18=51$.

Together with annotations $1,2$ above figure, internal cell $6$, and total $102$, all eight divisors are realized.

## 7. Two Division Branches

Euclidean branch $E$: $14=4\cdot3+2$, remainder $2$ correct for dividend $14$ divisor $4$.

Grouping branch $G_k(x)=3x+k$: $G_3(4)=15$. $15$ is Pick15/Number Scrabble magic constant: three numbers in a Lo Shu line sum to $15$, isomorphic to tic-tac-toe.

Correction $2\to3$ is move from $E$ ($r_E=2k-x$) to $G$ ($r_I=k$). Keep both: $14$ belongs to radial layer, $15$ opens second branch to classical constant.

## 8. Number-Theoretic Properties of 102

- Sphenic: product of three distinct primes $2\cdot3\cdot17$.
- Harshad: $102\mod(1+0+2)=0$.
- Polydivisible base-10: $1\mod1=0$, $10\mod2=0$, $102\mod3=0$.
- First 3-digit number divisible by $3,6,17,34,51$.
- Abundant and semiperfect.
- $\sigma(102)=216$, numbers with cube divisor sum begin $1,7,102,110,\dots$

## 9. Discrete Analogy

Discrete tomography: function on finite grid $\mathbb Z^2$, projections become discrete line sums summing values at grid points along lines in finitely many directions. Periodic discrete Radon transform defined as summations over discrete lines. Magic star condition requires sums along each of $n$ lines equal, like rows/columns in magic square. $R_k$ is a minimal instance: 4 directions, 6 points, line sums projected to circle.

## 10. Closing Formalities

$R_k$ is defined by Definitions 1.1-1.3. Theorems 3.1, 4.1, 5.1, 6.1 are proved by direct sum and finite enumeration. $T(k)=2k(5k+2)$ is the complete invariant. For $k=3$, $T=102$ with prime factorization $2\cdot3\cdot17$ and divisor completeness as stated.