Exact anchor-row domination inside the C1 residual core

Status: proved row-domination lemmas; hard-class union-shadow lemma for j=322

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: these lemmas remove specific earlier rows from any simultaneous residual system once their anchors are avoided. They do not prove full C1, universal DSC-P, López coverage, or Erdős-Straus.

Read with:

• ANCESTRY-DIVISOR-CHILD-THEOREM.md
• C1-CONDITIONED-FIBER-31113.md
• CLASS-C-CENSUS-K1500.md

1. Pullback-domination principle

Let a<b be earlier Type A/B layers with moduli

Suppose

and

Then for every integer x,

Therefore in any target progression x=r+Ls, once layer a is avoided, layer b is automatically avoided.

At parameter level the child forbidden pullback is consequently dominated by the ancestor pullback after projection to the common parameter line.

This is an exact logical implication, independent of probability, fiber bounds, or a finite search range.

2. The j=179 row is dominated by j=36

We have

This is ancestry shift s=1, and the child depth

is prime.

By the prime-child theorem,

Hence every exact j=179 hit is already a j=36 hit.

So after the j=36 mixed 11/13 row is avoided, j=179 contributes no new restriction.

This explains its complete redundancy in the two {11,13} C1 systems and throughout the recurring {3,11,13} family.

3. The j=205 row is dominated by j=10

We have

Here the ancestry shift is s=5 because

Also

with 41 prime, and

The divisor-child theorem therefore gives

Thus the j=205 3/13 row is exactly redundant once the unary j=10 row is avoided.

This is why it contributes no surviving conditioned 3-adic restriction in the {3,11,13} systems.

4. The j=790 row is dominated by j=10

Likewise

Now

so the ancestry shift is s=20.

The child depth factors as

with 79 prime, and

The divisor-child theorem gives

Therefore the apparently finer row whose residual 3-part reaches

is completely dominated by the j=10 anchor.

Once j=10 is avoided, j=790 can introduce no new 3-adic obstruction at any depth or target progression where both rows are present.

This converts the finite observation

the fine 3^4 row adds zero new hits

into an exact ancestry theorem.

5. Hard-compatible union domination of j=322

The common triple row is

It is not fully shadowed by either j=3 or j=10 individually.

However, the Mordell-hard target classes satisfy

Because

any hard-compatible j=322 trap must therefore satisfy

Exact trap table

The 15 distinct residues in T_322 are:

t	t mod 3	t mod 11 in T_3	t mod 39 in T_10
643	1	no	yes
965	2	yes	yes
1103	2	no	no
1126	1	no	yes
1195	1	yes	no
1231	1	yes	no
1241	2	no	no
1259	2	no	no
1264	1	yes	no
1273	1	yes	no
1279	1	no	yes
1280	2	no	no
1283	2	yes	yes
1285	1	no	yes
1286	2	yes	yes

Every residue in T_322 that is 1 mod 3 is caught by at least one of the two anchors:

The only residues missed by both anchors are

and all four are

Hard-class union-shadow theorem

For every integer x with

Therefore, once the unary anchor rows j=3 and j=10 are avoided, the triple row j=322 is automatically avoided for every Mordell-hard-compatible progression.

This is a genuine two-anchor union shadow, not a single-layer direct shadow.

6. Consequence for the {3,11,13} conditioned fiber

Before these lemmas, the recurring 3-coordinate appeared to receive restrictions from rows including

The exact domination theorems remove:

once the unary anchors are safe.

The effective 3-adic rows are therefore reduced to

inside the frozen {3,11,13} C1 family.

This explains a substantial part of the observed overlap:

• j=205 contributes no new restriction because it is a divisor-child of j=10;
• j=790 contributes no fine 3^4 restriction because it is another divisor-child of j=10;
• j=322 contributes no restriction on hard-compatible unary-safe pairs because its compatible trap fiber is union-shadowed by j=3 and j=10.

7. The remaining 3-adic question is tiny

The entire finite mod-3/mod-9 extension problem is now concentrated in three rows:

1. j=25, residual support 3·11;
2. j=88, residual support 3^2·13;
3. j=520, residual support 3^2·11.

The finite conditioned-fiber certificate shows:

• j=25 supplies at most one forbidden residue modulo 3 after conditioning on the 11-coordinate;
• j=520 never introduces a mod-3 class outside the class already forbidden by j=25;
• j=88 can add at most one further residue modulo 9.

Therefore at most four of the nine mod-9 classes are removed, leaving at least five.

The first and third bounds are small exact fiber statements. The middle containment is now the highest-value remaining row relation to prove universally in the relevant hard-compatible residual geometry.

8. New theorem target: projected ancestry domination

Although j=520 is not fully shadowed by j=25, their moduli satisfy

The finite C1 data indicate a weaker projection law:

after the shared fixed factors are removed, every compatible j=520 restriction on the mod-3 coordinate lies inside the mod-3 class already forbidden by j=25.

This is a projected ancestry shadow rather than full trap shadowing.

Proving that relation would turn the entire 3-adic conditioned-fiber lower bound of 5/9 from finite evidence into a theorem for this residual row pattern.

That is the next local algebra target.