Reciprocity tower shadows — Free Computation Foundation

Geometry

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Status: proved theorem family inside the Type A/B minimal-depth program

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. It gives three exact infinite families of fully shadowed Type A/B layers.

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1. Jacobi-saturated layers

For a Type A/B depth j, put

m=4j-1

and let

N_m^-= \left\{ u\in(\mathbb Z/m\mathbb Z)^\times: \left(\frac{u}{m}\right)=-1 \right\}.

The quadratic trap theorem gives

T_j\subseteq N_m^-.

Call the layer Jacobi-saturated if

\boxed{T_j=N_m^-.}

At such a layer, the single Jacobi bit already describes the exact Type A/B trap set.

2. Classification theorem

Theorem

The Jacobi-saturated Type A/B layers are exactly

\boxed{j\in\{1,2,4\}.}

Equivalently,

j = 1,  m = 3
j = 2,  m = 7
j = 4,  m = 15

are the only layers for which every Jacobi-negative unit is an exact Type A/B trap.

Proof

Suppose T_j=N_m^-.

From the multiplicative trap-coset theorem,

T_j\subseteq-D_j\subseteq N_m^-.

Hence equality at the two ends forces

T_j=-D_j=N_m^-.

The full multiplicative saturation classification in DYADIC-TRAP-LATTICE.md therefore implies

j=2^a

for some a>=0.

The endpoint a=0 gives j=1, and direct inspection gives

T_1=\{2\}=N_3^-.

Now let a>=1 and write

r=a+2, \qquad m=2^r-1.

The dyadic theorem gives

D_j=\langle2\rangle, \qquad

|D_j|=r.</div>

If the layer is Jacobi-saturated, then D_j is the complete Jacobi-positive kernel, whose size is phi(m)/2. Thus every Jacobi-positive unit would have to be one of

1,2,2^2,\ldots,2^{r-1}\pmod m.

For r=3 and r=4, this is true:

j=2,\quad m=7,

and

j=4,\quad m=15.

We show it fails for every r>=5 by explicitly producing a Jacobi-positive unit below m that is not a power of two.

Write

r=2^s u, \qquad u\text{ odd}.

Case 1: r is odd

Take

x=9=3^2.

Since r is odd,

3\nmid2^r-1,

so x is a unit modulo m. It is a square, hence Jacobi-positive. Also

9<m

for r>=5, and 9 is not a power of two. Thus x notin D_j.

Case 2: u>1

Take

t=2^s, \qquad x=(2^t+1)^2.

Here

\frac r{\gcd(r,t)}=u

is odd. The standard gcd identity

\gcd(2^r-1,2^t+1)=1

therefore shows that x is a unit modulo m.

Because u>=3,

r\ge3t,

and for t>=2,

(2^t+1)^2<2^{3t}-1\le2^r-1=m.

The smallest relevant case t=1,r>=6 gives x=9<63<=m directly.

Again x is a square, hence Jacobi-positive, and is visibly not a power of two. So x notin D_j.

Case 3: u=1

Then r is a power of two. Since r>=5, in fact

r\ge8.

Take

x=49=7^2.

The order of 2 mod 7 is 3, which does not divide a power of two. Hence

7\nmid2^r-1,

so x is a unit modulo m.

Also

49<2^8-1\le m,

and 49 is not a power of two. Again this is a Jacobi-positive unit outside D_j.

Thus no r>=5 is Jacobi-saturated.

Therefore the only Jacobi-saturated depths are

\boxed{1,2,4.}

QED.

3. Reciprocity tower lemma

Let j be any Type A/B layer and put

m=4j-1.

For any positive odd integer c, define the square-lift depth

\boxed{ K_c=\frac{mc^2+1}{4}. }

Then

4K_c-1=mc^2.

Lemma

For every divisor e|K_c,

\boxed{ \left(\frac e m\right)=+1. }

Proof

First,

4K_c=mc^2+1

shows

\gcd(K_c,m)=1.

Let ell be an odd prime divisor of K_c. Then

mc^2\equiv-1\pmod\ell.

Also ell cannot divide c, since otherwise the same congruence would give 0=-1 mod ell.

Thus

-m\equiv c^{-2}\pmod\ell,

\left(\frac{-m}{\ell}\right)=+1.

Because

m\equiv3\pmod4,

quadratic reciprocity gives the identity

\left(\frac{-m}{\ell}\right) = \left(\frac{\ell}{m}\right).

Therefore

\left(\frac{\ell}{m}\right)=+1.

If 2|K_c, then c^2=1 mod 8 and

mc^2+1\equiv0\pmod8,

m\equiv7\pmod8.

Hence

\left(\frac2m\right)=+1.

Every prime divisor of K_c therefore has Jacobi symbol +1 mod m. Multiplicativity gives

\left(\frac e m\right)=+1

for every e|K_c. QED.

4. Reciprocity tower shadow theorem

Theorem

If the base layer j is Jacobi-saturated, then every odd square lift K_c is fully shadowed by j:

\boxed{ T_{K_c}\bmod m \subseteq T_j. }

For c>1, this is a genuine earlier-to-later direct shadow relation.

Proof

Take any target trap residue at the lifted layer. It is of the form

-e

-4e

with e|K_c.

By the reciprocity tower lemma,

\left(\frac e m\right)=+1.

Since m=3 mod 4,

\left(\frac{-1}{m}\right)=-1,

and since 4 is a square,

\left(\frac4m\right)=+1.

Therefore

\left(\frac{-e}{m}\right) = \left(\frac{-4e}{m}\right) =-1.

The base is Jacobi-saturated, so every Jacobi-negative unit modulo m lies in T_j. Hence both reduced trap residues lie in T_j.

Thus

T_{K_c}\bmod m\subseteq T_j.

QED.

5. The three universal shadow towers

By the classification theorem, the only possible bases are

j=1,2,4.

Therefore every positive odd c gives the exact shadow families

\boxed{ K_c^{(1)}=\frac{3c^2+1}{4}, }

\boxed{ K_c^{(2)}=\frac{7c^2+1}{4}, }

and

\boxed{ K_c^{(4)}=\frac{15c^2+1}{4}. }

For every odd c>1, respectively,

\boxed{ T_{K_c^{(1)}}\bmod3\subseteq T_1, }

\boxed{ T_{K_c^{(2)}}\bmod7\subseteq T_2, }

and

\boxed{ T_{K_c^{(4)}}\bmod15\subseteq T_4. }

These are three explicit infinite families of later Type A/B layers that contribute no new exact trap information beyond one fixed earlier layer.

6. Polynomial form of the depth families

Writing

c=2n+1, \qquad n\ge0,

the three depth families become

\boxed{ K_n^{(1)}=3n^2+3n+1, }

\boxed{ K_n^{(2)}=7n^2+7n+2, }

and

\boxed{ K_n^{(4)}=15n^2+15n+4. }

For n>=1, each is fully shadowed by the corresponding base layer 1, 2, or 4.

The first few values are:

base j=1:  7, 19, 37, 61, 91, ...
base j=2: 16, 44, 86, 142, 212, ...
base j=4: 34, 94, 184, 304, 454, ...

7. Relation to square-lift towers

The modulus relation is exactly

4K_c-1=(4j-1)c^2.

So these are square-lift towers in the sense of SQUARE-LIFT-TOWERS.md.

The new theorem identifies the precise reason that these three towers are universally shadowed:

every divisor of the lifted depth is forced by quadratic reciprocity into the Jacobi-positive class of the base modulus, and at the three saturated bases the negative Jacobi class is already the exact trap set.

This is stronger than merely observing modulus divisibility.

8. Relation to the dyadic lattice

The base layers 1, 2, and 4 are also exactly the first three multiplicatively saturated dyadic layers.

But the reciprocity tower theorem is not the same as the dyadic Mersenne shadow lattice.

For example,

j=2,\quad c=3

gives

K_c=16,

which lies in both structures.

Meanwhile

j=4,\quad c=3

gives

K_c=34,

which is not dyadic at all, yet it is still completely shadowed by layer 4.

Thus quadratic-reciprocity square lifts generate a substantially broader infinite shadow mechanism.

9. Immediate exact-depth consequence

No prime can have its first Type A/B hit at a layer belonging to one of these towers once the corresponding earlier layer is among the imposed history.

In the unrestricted Type A/B system, for every odd c>1,

\boxed{ C_{AB}(p)\ne K_c^{(1)}, \quad C_{AB}(p)\ne K_c^{(2)}, \quad C_{AB}(p)\ne K_c^{(4)} }

for every prime p for which the congruence comparison is prime-compatible.

More precisely, any prime landing in the lifted trap layer necessarily already lands in the corresponding earlier trap layer, so the lifted layer can never be a new first hit.

10. Why this matters

The shadow graph now contains at least two proved infinite algebraic mechanisms:

Mersenne/dyadic cyclic ancestry, controlled by exponent divisibility;
quadratic-reciprocity square-lift towers, controlled by Jacobi saturation.

So the large finite shadow map is no longer merely a collection of computational coincidences. Distinct infinite theorem families are beginning to emerge from different algebraic causes.

That is exactly the kind of decomposition needed for a structural classification of the irredundant Type A/B core.

11. Next theorem targets

classify square-lift shadows from non-saturated bases by replacing the Jacobi bit with the full local quadratic-signature quotient;
classify them again using the stronger multiplicative quotient Gamma_j;
determine whether every observed square-lift direct shadow is explained by one of these quotient mechanisms;
prove a general quotient-tower shadow criterion;
fold the resulting infinite families into an algebraic description of the exact-depth structural gaps.

12. Novelty boundary

Quadratic reciprocity, Jacobi symbols, and the algebraic identity 4K-1=(4j-1)c^2 are classical. López's Type A/B congruences are prior art.

The candidate contribution is the Jacobi-saturation classification and resulting three infinite reciprocity-driven Type-A/B direct-shadow towers inside the minimal-depth/shadow framework.

A targeted arXiv search on 2026-08-14 did not locate this exact formulation. That negative search does not establish publication priority.