Multiplicative Type A/B trap coset

Status: proved structural theorem and active proof direction

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. López 2024 already records the Type B Jacobi-nonresidue fact and explicitly notes the mutual-inverse relationship between the Type A and Type B divisor residues. The subgroup/coset packaging below is being treated as a novelty candidate pending broader literature review.

Read with:

• DIAMOND.md
• QUADRATIC-TRAP-SIGNATURE.md
• CHARACTER-SHIELD-COMPLETENESS.md
• SHADOW-KERNEL.md
• FIBER-SHADOW-KERNEL.md
• PRIOR-ART.md

1. Setup

For k>=1, put

and

Let

Define the divisor-generated subgroup

Equivalently, H_k is generated by the residue classes of the prime divisors of k.

2. Trap-coset theorem

Theorem

For every k>=1,

Moreover,

so H_k and -H_k are distinct disjoint cosets.

Proof

Every divisor e|k is a product of prime divisors of k, hence

Also k itself lies in H_k. Since

we have

and therefore

Thus, for every e|k, both e and 4e belong to H_k, so

Hence

To prove that the two cosets are distinct, use the Jacobi character

For every prime divisor ell|k, the divisor-Jacobi calculation already recorded in QUADRATIC-TRAP-SIGNATURE.md gives

Therefore chi_k is identically +1 on the subgroup generated by those primes:

But m_k=3 mod 4, so

Thus -1 cannot lie in H_k. Consequently H_k and -H_k are distinct cosets. QED.

3. The quadratic theorem is only a projection

The previous quadratic theorem says

or more simply that every trap has Jacobi sign -1.

The trap-coset theorem is strictly more informative:

The Jacobi character is therefore one quotient character of the larger finite quotient

The quadratic signature remembers only one bit of this quotient geometry.

4. Trap-coset index

Define

Because -1 notin H_k but the Jacobi character is trivial on H_k, iota(k) is always even and

Since every Type A/B trap lies in one coset of H_k, at least

of the unit residue classes modulo m_k are automatically Type-A/B-safe before the exact divisor subset inside -H_k is examined.

When iota(k)>2, this is strictly stronger than the quadratic +1 shield, which certifies only one half of the unit group.

This does not say that every element of -H_k is a trap. Usually T_k is far smaller than the complete coset. The coset is a structured outer envelope.

5. Full quotient-character formulation

Let

be the character group of the quotient. Every character in this group may be viewed as a multiplicative character on G_k that is trivial on H_k.

The coset -H_k is characterized by

Therefore a single quotient character satisfying

certifies that x cannot be a Type A/B trap at depth k.

The Jacobi shield is the special order-two case obtained from the character chi_k.

This suggests a multiplicative coset shield stronger than the quadratic shield: exploit the full quotient G_k/H_k, not merely its Jacobi image.

6. Inversion structure and prior-art boundary

For e|k, write d=k/e. Then

and

Thus T_k is closed under inversion. López 2024 already explicitly notes that the Type A and Type B divisor residue sets are mutual inverses, so that fact is not claimed here as new.

The subgroup envelope is compatible with this symmetry because

7. Finite exact signal through k = 1200

An independent exact enumeration through k<=1200 verifies the theorem layer by layer and shows that the quotient can be much richer than one quadratic bit.

Across those 1200 layers:

• iota(k) is always even;
• the median index is 2;
• the mean index is about 7.5183;
• the largest observed index is

where