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Multiplicative trap quotient

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Status: exact theorem note and active proof direction

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove Direct-Shadow Completeness, universal López Type A/B coverage, or the Erdős-Straus conjecture. López 2024 already notes the mutual-inverse relationship between the Type A and Type B divisor residues. The quotient construction below is being treated as a potentially novel organization of that structure pending broader prior-art review.

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1. Setup

For a Type A/B depth k, put

m_k=4k-1

and

G_k=(\mathbb Z/m_k\mathbb Z)^\times.

Let the distinct prime divisors of k be

\ell_1,\ldots,\ell_r.

Define the multiplicative subgroup

\boxed{ D_k=\langle \ell_1,\ldots,\ell_r\rangle\le G_k. }

The Type A/B trap set is

T_k=\{-e,-4e\pmod{m_k}:e\mid k\}.

2. Multiplicative trap-coset theorem

Theorem

For every k>=1,

\boxed{T_k\subseteq -D_k.}

Proof

Every divisor e|k is a product of powers of the prime divisors of k, so

e\in D_k.

Also

4k\equiv1\pmod{4k-1}.

Since k in D_k, its inverse belongs to D_k, and therefore

4\equiv k^{-1}\in D_k.

Hence

4e\in D_k

for every e|k.

Multiplication by -1 now gives

-e,-4e\in-D_k.

Thus

\boxed{T_k\subseteq-D_k.}

QED.

3. The quotient

Define the finite abelian quotient

\boxed{ \Gamma_k=G_k/D_k. }

Every Type A/B trap has the same image in this quotient:

\boxed{ [t]=[-1]\in\Gamma_k \qquad(t\in T_k). }

Therefore:

Multiplicative quotient shield

If a unit residue x mod m_k satisfies

\boxed{[x]\ne[-1]\text{ in }\Gamma_k,}

then

x\notin T_k.

This is an exact sufficient non-hit certificate.

4. Multiplicative trap index

Define

\boxed{ \iota(k)=[G_k:D_k]. }

This measures how many multiplicative cosets remain after quotienting by the subgroup generated by the prime divisors of k.

The trap set occupies exactly one distinguished quotient class.

A large value of iota(k) therefore gives a much finer coarse partition than the single Jacobi bit.

5. Relation to the quadratic-signature quotient

Let

\chi_k:G_k\to V_k

be the vector of local Legendre symbols at the distinct prime factors of m_k, encoded over F_2.

As recorded in QUADRATIC-SIGNATURE-QUOTIENT.md,

H_k=\chi_k(D_k)

and the quadratic quotient is

Q_k=V_k/H_k.

Because D_k lies in the kernel of the composite map

G_k\xrightarrow{\chi_k}V_k\to Q_k,

the map factors through Gamma_k:

\boxed{ \Gamma_k\twoheadrightarrow Q_k. }

Hence

\boxed{|Q_k|\mid\iota(k).}

The ordinary Jacobi character is a further one-bit quotient.

Thus the avoidance hierarchy is

\boxed{ T_k \subseteq -D_k \subseteq \chi_k^{-1}(\chi_k(-1)+H_k) \subseteq \{u:(u/m_k)=-1\}. }

Each step moving left is a finer approximation to the exact Type A/B trap set.

6. Why this is stronger than quadratic signs

The quadratic-signature quotient is necessarily a 2-group quotient. The multiplicative quotient need not be.

For example, finite computations show indices such as

6,10,12,18,30,\ldots

and much larger values.

Those odd factors encode multiplicative information invisible to every quadratic character.

Therefore a candidate trapped on the correct Jacobi side, and even in the correct complete local quadratic-signature coset, may still lie outside -D_k and be certified safe by Gamma_k.

7. Finite structure through k <= 3000

Exact subgroup enumeration through k<=3000 gives:

index iota(k) = 2: 1763 layers
index iota(k) > 2: 1237 layers
number of distinct observed indices: 54

The most common larger indices begin:

4:  368 layers
6:  186
8:  156
12: 115
16: 111
24:  46
48:  36
10:  30
32:  23

The largest observed value in this range is

\boxed{\iota(2819)=800,}

with

4(2819)-1=11275, \qquad \varphi(11275)=8000, \qquad

|D_{2819}|=10.</div>

These are finite structural data, not an asymptotic theorem.

8. Direct residual refinement on the frozen k <= 1200 bundle

The exact k<=1200 candidate bundle contains 41,470 directly novel candidates.

At the successive coarse resolutions:

Jacobi character residual:             11,056
full quadratic-signature residual:     10,684
direct multiplicative-coset residual:  10,258

Thus the full local quadratic vector locally rescues

11056-10684=372

candidates beyond the Jacobi shield, while the still-finer multiplicative quotient locally rescues

11056-10258=798

beyond the Jacobi direct residual.

Equivalently, it locally rescues another

\boxed{426}

candidates beyond the full quadratic-signature resolution.

The word locally is essential here. These counts ask only whether one earlier layer by itself makes the quotient class unavoidable. They do not yet prove that all multiplicative-coset constraints can be avoided simultaneously.

9. The exact residual inside -D_k

The subgroup coset remains a coarse superset of the true trap set, often a very large one.

Normalize by multiplication by -1. Then

-T_k=\{e,4e:e\mid k\}\subseteq D_k.

Write

k=\prod_{i=1}^r\ell_i^{a_i}

and let

\phi_k:\mathbb Z^r\to D_k, \qquad (b_1,\ldots,b_r)\mapsto\prod_i\ell_i^{b_i}\pmod{m_k}.

If

\mathbf a=(a_1,\ldots,a_r)

and

\mathcal B_{\mathbf a}=\{\mathbf b:0\le b_i\le a_i\},

then the divisor family is phi_k(B_a).

Since

4\equiv k^{-1}=\phi_k(-\mathbf a),

the second Type A/B family is the translated box

\phi_k(\mathcal B_{\mathbf a}-\mathbf a).

Therefore

\boxed{ -T_k = \phi_k(\mathcal B_{\mathbf a}) \cup \phi_k(\mathcal B_{\mathbf a}-\mathbf a). }

This is an exact two-box representation of the Type A/B trap set inside the subgroup D_k.

It identifies the next refinement after the multiplicative quotient: once a candidate is forced into -D_k, the remaining exact obstruction is not an arbitrary subset of the subgroup but the image of two tightly constrained exponent boxes.

10. Connection to López prior art

López 2024 explicitly observes that the Type -n and Type -4d divisor sets are mutual inverses modulo 4dn-1. That inversion relation is prior art and must not be presented as an FCF discovery.

The present research uses that same arithmetic in a different structural organization:

  1. collect all divisors at fixed product depth k=dn;
  2. pass to the subgroup generated by the prime divisors of k;
  3. quotient the full unit group by that subgroup;
  4. identify all Type A/B traps with one quotient class;
  5. retain the exact trap set inside the class as two exponent boxes.

Targeted searching on 2026-08-14 did not locate this exact quotient-plus-two-box formulation inside the minimal-depth/shadow framework, but that negative search does not establish priority.

11. New theorem program

The hierarchy now reads

\boxed{ \begin{array}{c} \text{Jacobi character}\\ \downarrow\\ \text{full quadratic signature}\\ \downarrow\\ \text{multiplicative quotient }\Gamma_k\\ \downarrow\\ \text{two-box geometry inside }D_k\\ \downarrow\\ \text{exact Type A/B traps} \end{array} }

Each level gives a more precise description of what a Type A/B trap can look like.

The immediate questions are:

  1. Does the multiplicative quotient shield have a direct-shadow completeness theorem analogous to the quadratic-signature finite collapse?
  2. What are the invariant factors of Gamma_k, and which of them create the large observed indices?
  3. Can the two-box representation explain the exact direct-shadow relation between layers?
  4. Can modulus ancestry induce homomorphisms between the quotients Gamma_j and Gamma_k that force trap-coset shadowing?
  5. Inside a direct multiplicative residual, can the two exponent boxes always be avoided unless an exact direct shadow already exists?

The fifth question is now a particularly direct bridge back toward DSC-P.