Type A/B trap quotient factorization — Free Computation Foundation

Ancestry

Read with:

Status: proved structural theorem and active proof direction

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. It reorganizes the exact multiplicative and quadratic envelopes of a Type A/B trap layer. Literature priority for this packaging remains under review.

Read with:

1. Setup

Fix k>=1 and write

m=4k-1=\prod_{i=1}^r p_i^{a_i}, \qquad G=(\mathbb Z/m\mathbb Z)^\times.

Let

H=\langle\ell\bmod m:\ell\mid k,\ \ell\text{ prime}\rangle.

The multiplicative trap-coset theorem gives

T_k\subseteq-H.

Let

\lambda:G\to\mathbb F_2^r

be the vector of local Legendre signs at the distinct prime factors of m.

Let

V=\lambda(H).

By the quadratic-signature theorem, this is exactly the span generated by the signatures of the prime divisors of k, and

\lambda(T_k)=\eta+V, \qquad \eta=\lambda(-1).

Define

\kappa(k)=r-\dim V.

2. Surjectivity of the local sign map

Lemma

The map

\lambda:G\to\mathbb F_2^r

is surjective.

Proof

For each prime-power factor p_i^{a_i}, choose a unit residue having either desired Legendre sign modulo p_i. Such a unit lifts to p_i^{a_i} without changing its Legendre sign. CRT then combines the independently chosen local residues into one unit modulo m.

Thus every vector in F_2^r occurs. QED.

Let

K=\ker\lambda.

Then

G/K\cong\mathbb F_2^r.

3. Exact quotient factorization theorem

Define the full multiplicative trap-coset index

\iota(k)=[G:H].

Theorem

\boxed{ \iota(k) =2^{\kappa(k)}\Theta(k), }

where

\boxed{ \Theta(k)=[K:H\cap K] }

is a positive integer.

Proof

Insert the intermediate subgroup HK:

[G:H]=[G:HK][HK:H].

Under the quotient map G -> G/K, the subgroup HK/K is exactly the image lambda(H)=V. Hence

[G:HK] =[G/K:HK/K] =[\mathbb F_2^r:V] =2^{r-\dim V} =2^{\kappa(k)}.

By the second isomorphism theorem,

HK/H\cong K/(H\cap K),

[HK:H]=[K:H\cap K]=\Theta(k).

Multiplying gives the result. QED.

4. Interpretation of Theta

The factor

2^{\kappa(k)}

is exactly the quotient information visible to the vector of local quadratic characters.

The residual factor

\boxed{\Theta(k)}

is invisible to all of those Legendre-sign bits.

It measures the remaining multiplicative distinction between H and the complete preimage of its quadratic-signature space.

We therefore call Theta(k) the deep multiplicative index.

The cases split cleanly:

Theta(k)=1

Then

K\subseteq H,

H=\lambda^{-1}(V).

Therefore the multiplicative coset envelope is exactly the full preimage of the quadratic-signature affine space:

\boxed{ -H=\lambda^{-1}(\eta+V). }

At such a layer, quadratic signatures capture the entire multiplicative coset information.

Theta(k)>1

Then multiple distinct cosets of H share the same quadratic signature.

The full multiplicative quotient therefore contains genuinely higher-order information that no collection of Legendre signs can detect.

This is the next character layer beyond quadratic signatures.

5. Exact three-scale trap-density factorization

Because

T_k\subseteq-H,

we can factor the exact unit-group density of the Type A/B trap set as

\frac{|T_k|}{\varphi(m)} = \frac{|T_k|}{|H|} \frac{|H|}{\varphi(m)}.

Using

\frac{|H|}{\varphi(m)} =\frac1{[G:H]} =\frac1{2^{\kappa(k)}\Theta(k)},

we obtain

\boxed{ \frac{|T_k|}{\varphi(4k-1)} = \underbrace{2^{-\kappa(k)}}_{\text{quadratic-signature filter}} \cdot \underbrace{\Theta(k)^{-1}}_{\text{deep multiplicative filter}} \cdot \underbrace{\frac{|T_k|}{|H_k|}}_{\text{exact divisor sparsity}}. }

This is an exact identity.

It separates three distinct mechanisms that were previously mixed together in one raw density:

quadratic-signature geometry;
higher-order multiplicative quotient geometry;
the sparse divisor-generated trap subset inside its multiplicative coset.

6. Relation to earlier hazard failures

The original raw trap density

\rho(k)=\frac{|T_k|}{4k-1}

was too coarse for survival analysis because it mixed primality conditioning, earlier-layer survival, shadowing and local group structure.

Even after restricting to units, the trap density

\frac{|T_k|}{\varphi(4k-1)}

still hides three algebraically different filters.

The quotient factorization shows why a single scalar density can be misleading: two layers with similar trap cardinalities can have radically different quadratic codimension, deep multiplicative index, and internal divisor sparsity.

These quantities should therefore be tracked separately in future survivor and cryptology experiments.

7. Finite exact signal through k <= 1200

Exact enumeration gives the following diagnostics:

layers checked: 1200
Theta(k)=1:       884
median Theta:       1
mean Theta:         about 2.26417
maximum Theta:    105

Thus in roughly 73.7% of the first 1200 layers, the full multiplicative coset carries no information beyond the complete local quadratic-signature vector.

But the remaining layers can carry a substantial deeper quotient.

The largest observed deep index is

\boxed{\Theta(683)=105,}

with

m_{683}=2731, \qquad \iota(683)=210, \qquad \kappa(683)=1.

So the single quadratic bit accounts for only a factor 2 of a multiplicative trap-coset index 210; the remaining factor 105 is higher-order structure.

Other large finite values include 81, 44, 36, 33, 32, and 27.

These finite statistics are diagnostics, not asymptotic claims.

8. New proof architecture

The Type A/B trap hierarchy can now be written as

\boxed{ \begin{array}{c} G_k\\ \downarrow\quad 2^{\kappa(k)}\\ \lambda^{-1}(\eta_k+V_k)\\ \downarrow\quad \Theta(k)\\ -H_k\\ \downarrow\quad |H_k|/|T_k|\\ T_k \end{array} }

The labels are exact multiplicative compression factors.

This identifies the unresolved arithmetic at progressively finer resolutions.

9. Immediate theorem targets

classify when Theta(k)=1;
determine the arithmetic source of large Theta(k);
identify higher-order characters that generate the quotient of order Theta(k);
test whether direct quadratic-signature residual candidates are often rescued by the deep multiplicative quotient;
combine deep quotient characters with the square-lift and fiber kernels;
determine whether the exact divisor-sparsity factor has a structural lower bound on safe classes sufficient for DSC-P.

The key new question is:

after the complete quadratic signature is exhausted, how much obstruction can survive in the deep multiplicative quotient, and can that quotient itself be reduced to earlier-layer ancestry?