The Type A/B reciprocity matrix — Free Computation Foundation

Geometry

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Status: proved structural theorem and active proof direction

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove QDSC, universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. It packages the quadratic-signature structure of one Type A/B layer into a binary matrix with canonical left and right conservation laws.

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1. Matrix definition

Fix a depth k>=1 and write

m=4k-1.

Let the distinct prime factors of m be

p_1,\ldots,p_r

and the distinct prime factors of k be

\ell_1,\ldots,\ell_s.

Because gcd(k,m)=1, every Legendre symbol below is defined.

Define the binary Type A/B reciprocity matrix

\boxed{ A_k=(a_{ij})\in M_{r\times s}(\mathbb F_2), }

where

\left(\frac{\ell_j}{p_i}\right)=(-1)^{a_{ij}}.

Thus column j is exactly the local quadratic-signature vector

\lambda_m(\ell_j).

Consequently

\boxed{ \operatorname{col}(A_k)=V_k, }

where V_k is the divisor-signature space from QUADRATIC-SIGNATURE-QUOTIENT.md.

In particular,

\boxed{ \kappa(k)=r-\operatorname{rank}(A_k). }

So the quadratic quotient dimension is the left nullity of the reciprocity matrix.

2. Canonical left null vector

Factor

m=\prod_{i=1}^r p_i^{\alpha_i}.

Let

\boxed{ c_k=(\alpha_i\bmod2)_{i=1}^r\in\mathbb F_2^r.}

This is the squarefree-kernel parity vector of m.

Because m=3 mod 4, m is not a perfect square, so

\boxed{c_k\ne0.}

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

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

Expanding the Jacobi symbol over the prime powers of m gives

\prod_i \left(\frac{\ell_j}{p_i}\right)^{\alpha_i}=+1.

In bit form,

\sum_i(\alpha_i\bmod2)a_{ij}=0.

Therefore

\boxed{ c_k^T A_k=0.}

Consequence

Every reciprocity matrix has nontrivial left kernel, so

\boxed{\kappa(k)\ge1.}

This recovers the universal Jacobi quotient as the canonical first left-null direction.

3. Canonical right null vector

Factor

k=\prod_{j=1}^s\ell_j^{\beta_j}

and put

\boxed{ b_k=(\beta_j\bmod2)_{j=1}^s\in\mathbb F_2^s.}

Modulo every prime p_i|m,

4k\equiv1\pmod{p_i},

hence

k\equiv4^{-1}\pmod{p_i}.

Since 4^{-1} is a square modulo p_i,

\left(\frac{k}{p_i}\right)=+1.

By multiplicativity,

\prod_j \left(\frac{\ell_j}{p_i}\right)^{\beta_j}=+1.

In bit form,

\sum_j(\beta_j\bmod2)a_{ij}=0.

Therefore

\boxed{ A_k b_k=0. }

When k is not a perfect square, b_k is nonzero. Thus every nonsquare depth has a canonical nontrivial right-kernel relation among the prime-factor signature columns.

This is the layer-level version of the square-lift defect conservation law.

4. Double conservation theorem

Combining the previous sections gives:

Theorem

For every Type A/B depth k, the reciprocity matrix satisfies

\boxed{ c_k^T A_k=0, \qquad A_k b_k=0.}

The left relation is always nontrivial. The right relation is nontrivial whenever k is not a square.

Thus the matrix is never an arbitrary binary matrix. Its row and column spaces are constrained simultaneously by the defining identity

4k-1=m.

5. Rank bounds

Let

r=\omega(m), \qquad s=\omega(k).

The left conservation law gives

\operatorname{rank}(A_k)\le r-1.

If k is not a square, the nonzero right conservation law also gives

\operatorname{rank}(A_k)\le s-1.

Therefore:

Corollary for nonsquare k

\boxed{ \operatorname{rank}(A_k) \le \min(r-1,s-1), }

and hence

\boxed{ \kappa(k) =r-\operatorname{rank}(A_k) \ge \max(1,r-s+1). }

Corollary for square k

The canonical right parity vector vanishes, but the left law remains, so

\boxed{ \operatorname{rank}(A_k)\le\min(r-1,s), }

and therefore

\boxed{ \kappa(k)\ge\max(1,r-s). }

These are elementary but useful candidate-independent lower bounds on the amount of quadratic quotient information available at a layer.

6. Exact interpretation of kappa=1

The condition

\kappa(k)=1

means the canonical Jacobi vector c_k spans the entire left kernel:

\boxed{ \ker(A_k^T)=\langle c_k\rangle. }

Equivalently, every linear local quadratic character that is trivial on all divisor signatures is generated by the ordinary Jacobi character.

So the kappa=1 layers are precisely those for which there is no hidden quadratic separator beyond Jacobi.

This gives a matrix formulation of the automatic square-lift signature theorem:

if a squarefree ancestor has left nullity one, every square-lift divisor signature lies in the ancestor column space because reciprocity already places it in the Jacobi kernel.

7. Exact interpretation of the reciprocity defect quotient

For a squarefree ancestor a, let d=4a-1. Then the vector c_a is the all-ones vector on the prime factors of d, because every exponent in squarefree d is odd.

The defect quotient from RECIPROCITY-DEFECT-QUOTIENT.md is

\mathcal R_a=\ker(c_a^T)/\operatorname{col}(A_a).

Its dimension is

\boxed{ \dim\mathcal R_a =\kappa(a)-1. }

Thus the defect quotient is exactly the part of the left-null geometry that remains after the canonical Jacobi direction has been removed.

A square-lift prime factor contributes a new signature obstruction only through this cokernel.

8. Right-kernel conservation as defect cancellation

For a square-lift depth

j=\prod_t q_t^{e_t},

the parity vector

(e_t\bmod2)_t

is the canonical right-kernel vector for the signature matrix measured against the squarefree ancestor primes.

Passing to the ancestor defect quotient gives

\boxed{ \sum_t(e_t\bmod2)\delta_a(q_t)=0. }

So the defect conservation theorem is not an isolated identity. It is the right-null law of the same reciprocity matrix whose left-null law is the Jacobi character.

This gives a symmetric picture:

\boxed{ \begin{array}{ccc} \text{Jacobi parity of }m &\longrightarrow& \ker(A_k^T)\\ &&\\[-2mm] A_k&&\\[-2mm] &&\\ \ker(A_k)&\longleftarrow&\text{prime-exponent parity of }k. \end{array}}

The Type A/B relation m=4k-1 constrains both sides at once.

9. Why this matters

The full quadratic-signature problem had appeared to involve many unrelated Legendre bits. The reciprocity matrix shows that those bits are organized by a highly constrained binary incidence object.

Three previously separate observations become one structure:

every Type A/B trap has Jacobi sign -1;
higher local quadratic quotient dimensions are measured by the left nullity beyond the canonical Jacobi vector;
square-lift defect classes must cancel according to the exponent parity of the lifted depth.

The proof search for QDSC can therefore be reframed as a rank-and-support problem for a family of reciprocity matrices rather than an unconstrained Boolean covering problem.

10. New theorem targets

classify when the rank bounds above are sharp;
characterize kappa(k)=1 arithmetically without constructing the full matrix;
classify the possible low-dimensional defect quotients when kappa(k)=2,3;
prove that the norm-form constraint

j=(1+d s^2)/4

forces every nonzero defect configuration into a shadowed support pattern;

relate modulus ancestry maps to linear maps between the corresponding reciprocity matrices;
determine whether the five currently known layer-level quadratic-signature exceptions through k<=3000 arise from one common matrix normal form.

The fifth and sixth targets are particularly close to a universal QDSC theorem.

11. Novelty boundary

Legendre symbols, quadratic reciprocity, binary matrices, null spaces, and rank-nullity are classical. The candidate contribution is the two-sided reciprocity-matrix organization of the López Type A/B minimal-depth/shadow system, including its identification with the quadratic quotient and square-lift defect conservation structures.

Publication priority remains subject to broader literature review and independent mathematical scrutiny.