Proper Jacobi ancestor theorem for higher Type A/B signature codimension

Geometry

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Status: proved universal structural theorem

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem concerns the quadratic-signature envelope of Type A/B traps. It does not prove exact Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. Literature priority for this formulation remains under review.

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1. Setup

Fix a layer k and write

m=4k-1=\prod_{i=1}^r p_i^{a_i}.

Let

\lambda_m:(\mathbb Z/m\mathbb Z)^\times\to\mathbb F_2^r

be the vector of local Legendre signs, let

\eta=\lambda_m(-1),

and let

V_k=\operatorname{span}\{\lambda_m(\ell):\ell\mid k,\ \ell\text{ prime}\}.

The exact quadratic-signature trap theorem gives

\lambda_m(T_k)=\eta+V_k.

Define

\kappa(k)=r-\dim V_k=\dim V_k^\perp.

Let

\alpha=(a_i\bmod2)_{i=1}^r.

This is the exponent-parity vector defining the Jacobi character modulo m.

2. The Jacobi vector lies in the annihilator

For every prime divisor ell|k, the divisor-Jacobi theorem gives

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

Therefore

\alpha\cdot\lambda_m(\ell)=0 \pmod2

for every generator of V_k, hence

\boxed{\alpha\in V_k^\perp.}

Since m=3 mod 4,

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

\boxed{\alpha\cdot\eta=1.}

Thus the usual Jacobi character is one nonzero affine annihilator of the trap-signature space.

3. Proper Jacobi ancestor theorem

Theorem

\boxed{\kappa(k)\ge2,}

then there exists a squarefree divisor

\boxed{d\mid\operatorname{rad}(m)}

such that

\boxed{1<d<m,\qquad d\equiv3\pmod4,}

and every Type A/B trap at layer k is Jacobi-negative modulo d:

\boxed{ \left(\frac{t}{d}\right)=-1 \qquad(t\in T_k). }

Consequently

\boxed{d=4s-1}

for an earlier depth

\boxed{s=(d+1)/4<k.}

We call s a proper Jacobi ancestor of the higher-codimension signature layer k.

Proof

Because kappa(k)>=2, the annihilator V_k^perp has dimension at least two. Choose

u\in V_k^\perp

independent of alpha.

u\cdot\eta=1,

set w=u. Otherwise set

w=u+\alpha.

Then in either case

\boxed{w\in V_k^\perp,\qquad w\cdot\eta=1.}

Also w is nonzero and

\boxed{w\ne\alpha.}

Define the squarefree divisor

d=\prod_{i:w_i=1}p_i.

Because w dot eta = 1, an odd number of the selected primes are 3 mod 4, so

\boxed{d\equiv3\pmod4.}

Clearly

d\mid\operatorname{rad}(m).

We claim d<m. Since d<=rad(m)<=m, equality could occur only if m were squarefree and w selected every prime factor of m. But for squarefree m, the all-prime vector is exactly alpha, contradicting w!=alpha. Hence

\boxed{d<m.}

Now let t in T_k. Its local signature has the form

\lambda_m(t)=\eta+v, \qquad v\in V_k.

Since w in V_k^perp,

w\cdot\lambda_m(t) =w\cdot\eta+w\cdot v =1.

But d is squarefree and supported exactly on the coordinates selected by w, so

\left(\frac{t}{d}\right) =(-1)^{w\cdot\lambda_m(t)} =-1.

Finally d=3 mod 4 gives

d=4s-1

for an integer s, and d<m=4k-1 gives s<k. QED.

4. Meaning

Every signature layer with more than one independent quadratic restriction is already contained in the Jacobi-negative half-space of a strictly earlier modulus of the same 4s-1 form.

Thus higher quadratic codimension is not primitive.

At quadratic-signature resolution, the only layers that can be primitive are

\boxed{\kappa(k)=1.}

For those layers the unique nonzero annihilator is the Jacobi character itself, so their trap-signature envelope is exactly the full Jacobi-negative hyperplane.

This creates a sharp dichotomy:

\boxed{ \begin{array}{ll} \kappa(k)=1 &: \text{primitive quadratic layer;}\\[1mm] \kappa(k)\ge2 &: \text{has a strict earlier Jacobi ancestor.} \end{array} }

5. Relation to signature shadowing

The theorem is a genuine ancestor/shadow statement at the character-envelope level:

\boxed{ \lambda_m(T_k) \subseteq \left\{x:\left(\frac{x}{d}\right)=-1\right\} }

for some earlier modulus d=4s-1.

It does not claim

T_k\subseteq T_s

as exact residue sets. The earlier exact trap T_s is generally much smaller than the entire Jacobi-negative half modulo d.

So this theorem explains redundancy in the quadratic envelope without overclaiming exact Direct-Shadow Completeness.

6. Why this is important for the proof program

The full local signature problem seemed to introduce increasingly complicated affine restrictions as the number of prime factors of 4k-1 grew.

The theorem reverses that intuition:

every genuinely higher-dimensional quadratic trap envelope descends to a simpler, strictly earlier scalar Jacobi obstruction.

Therefore the primitive quadratic skeleton is made only of kappa=1 layers.

The remaining proof problem is to understand how target-fixed signs interact with this strict descent. In particular:

if the proper Jacobi ancestor is target-positive, the higher layer is automatically defeated;
if the ancestor is target-negative but still has higher signature codimension, descent can continue;
a descent chain can terminate at a primitive kappa=1 layer;
if that terminal layer is fully fixed and negative, it is a direct signature obstruction;
otherwise its Jacobi sign supplies a free linear safety equation.

This is precisely the shape suggested by the zero-collective-obstruction finite replay through k<=1200.

7. Next theorem target

The immediate target is now stronger and more concrete:

Prove that proper-Jacobi-ancestor descent, together with character-shield saturation, implies quadratic-signature Direct-Shadow Completeness: if no single earlier layer is a direct signature obstruction, then all earlier quadratic-signature trap envelopes can be avoided simultaneously.

If proved, the entire elementary quadratic quotient would disappear from the unresolved exact DSC-P problem.

What would remain would be the genuinely higher-order arithmetic already isolated in SQUARE-LIFT-CORE.md: prime-power lifts and exact divisor-generated residues inside a fixed signature class.