Hard-class union shadows and the five-of-nine C1 3-adic fiber theorem

Status: proved exact row-domination and conditioned-fiber theorem

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem applies to the stated Mordell-hard residue classes and the stated C1 residual row pattern. It proves the exact 3-adic extension lower bound for that pattern. It does not prove full C1, universal DSC-P, López universal coverage, or the Erdős-Straus conjecture.

Read with:

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

1. Hard residue facts

Let

Every h in H satisfies

Modulo 21, the six hard classes occupy only

These two elementary facts are enough to turn several apparently independent residual rows into exact union shadows.

2. The j=520 row is union-shadowed by j=3 and j=25

The relevant moduli are

The trap sets needed below are

and

Since

a trap at layer 520 can occur on a Mordell-hard integer only if its residue modulo 21 is one of 1,4,16.

The complete hard-compatible portion of T_520 is:

There are no other T_520 residues in the three hard-compatible classes modulo 21.

Therefore:

Theorem 1: hard union shadow at j=520

For every integer x with

Thus, on every Mordell-hard progression, once layers 3 and 25 are avoided, layer 520 is automatically avoided.

This is stronger than the previously observed projected relation between the j=25 and j=520 parameter fibers. It is an exact two-anchor union shadow in x-space.

3. Exact one-class fiber at j=25

The hard condition gives

The only traps in T_25 compatible with that condition are

Their residues modulo 11 are

They are distinct.

Hence:

Lemma 2

After fixing the 11-coordinate of a hard-compatible parameter class, layer j=25 can contribute at most one compatible trap.

In the common C1 residual regime

the affine pullback is a coordinatewise bijection. Therefore, after fixing the parameter modulo 11, the j=25 row forbids at most

residue modulo 3.

Equivalently, when lifted to modulo 9, it removes at most three of the nine classes.

4. Exact one-class fiber at j=88

Now

Again hard compatibility requires trap residue 1 mod 3.

The complete compatible trap fiber is

Their residues modulo 13 are respectively

all distinct.

Therefore:

Lemma 3

After fixing the 13-coordinate of a hard-compatible parameter class, layer j=88 can contribute at most one compatible trap.

In the recurring C1 regime

the affine pullback preserves this fiber multiplicity. Thus after fixing the parameter modulo 13, layer j=88 forbids at most

residue modulo 9.

5. Other apparent 3-adic rows are exact redundancies

The recurring {3,11,13} residual systems also display rows

They contribute no independent 3-adic obstruction once the anchor rows are safe:

j=205

By the divisor-child theorem,

j=790

Again the divisor-child theorem gives

So the fine 3^4 row at j=790 is completely redundant after layer 10 is avoided.

j=322

The exact hard-compatible union-shadow theorem already proved in C1-ANCHOR-ROW-DOMINATION.md gives

Hence j=322 is redundant after the unary anchors j=3 and j=10 are avoided.

j=520

Theorem 1 above removes j=520 after j=3 and j=25 are avoided.

Thus the entire 3-adic stage of this row pattern reduces exactly to

6. Five-of-nine theorem

Fix a Mordell-hard progression in the recurring C1 row pattern and suppose the relevant 11- and 13-coordinates have already been chosen to avoid their unary anchors and the preceding pair stage.

By Lemma 2, j=25 forbids at most one residue modulo 3. Lifted to modulo 9, this removes at most

classes.

By Lemma 3, j=88 forbids at most one additional residue modulo 9.

All other 3-adic rows are redundant by Section 5.

Therefore the total number of forbidden residue classes modulo 9 is at most

Hence:

Theorem 4: conditioned five-of-nine extension

For every Mordell-hard C1 residual system whose 3-coupled rows are the recurring pattern

or any subset thereof, after the anchor-safe 11/13 assignment is fixed,

In particular a 3-adic extension always exists.

If the residual 3-coordinate is actually modulo

the only row reaching that precision is j=790, already dominated by j=10. Therefore every safe mod-9 class lifts freely to all nine classes modulo 81 above it, giving at least

safe full 3-adic values.

7. This exactly explains the finite conditioned-fiber spectrum

The frozen k<=1500 {3,11,13} family exhibited safe full 3-fiber sizes

The theorem explains the structure:

• the base values 5,6,8,9 are the possible surviving counts modulo 9;
• the values 45,54,72,81 are exactly ninefold lifts when the coordinate extends to modulo 81;
• the universal lower bound within this row pattern is 5.

Thus the empirically observed minimum 5 is no longer merely a finite statistic. It follows from exact trap geometry plus exact ancestry/union-shadow domination.

8. A second hard union shadow: j=465

The same mechanism explains the fine 13^2 pair row.

Here

For any hard integer x, we again have x=1 mod 3. Since m_465 contains both 11 and 13, a trap residue t in T_465 determines x mod11 and x mod13; together with x=1 mod3, the latter determines x mod39.

Direct enumeration of the 15 exact traps gives:

The only traps not already caught by j=36 or j=3 have 13-residues whose hard-compatible CRT lift modulo 39 is one of

all belonging to T_10.

Therefore j=465 is an exact three-anchor union shadow after j=3, j=10, and j=36 are avoided.

This proves, rather than merely observes, why the fine 13^2 row introduces no new obstruction in the recurring conditioned-fiber family.

9. Where the remaining C1 work now sits

Two large chunks of the conditioned-fiber certificate have moved from computation into theorem:

1. the fine 13^2 row j=465 is anchor-dominated;
2. the entire 3-adic stage has a proved five-of-nine lower bound.

The unresolved recurring pair-stage is therefore concentrated mainly in the coarse rows

after unary 11- and 13-adic constraints are imposed.

This is the next proof target.