C1 conditioned-fiber collapse on the `{3,11,13}` residual family

Status: finite theorem-certificate target; exact structure identified through the frozen k<=1500 C1 bundle

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: this note records an exact finite structural pattern on the 336 C1 residual systems with prime signature {3,11,13}. It is not yet a universal-in-k theorem and does not prove full C1, universal DSC-P, López universal coverage, or Erdős-Straus.

Read with:

• CLASS-C-CENSUS-K1500.md
• SINGLE-ACTIVE-REDUCED-ESCAPE-THEOREM.md
• TRAP-FIBER-BOUND.md
• FUTURE-OPERATOR-INSTRUCTIONS.md

1. Why this family matters

The independently verified C1 census through k<=1500 contains

single-active candidates whose final nonempty fiber kernel has exactly the prime support

This is the most frequent small residual kernel after the larger seven-prime family and the cleanest recurring system in which genuine three-coordinate coupling remains.

The question is not merely whether each finite system has one solution. That was already known from the bounded selector certificate.

The question is whether there is a uniform elimination invariant explaining why these systems cannot close.

The answer in the frozen range is yes.

2. Exact elimination order

For every one of the 336 systems, use the coordinate order

The residual moduli involve powers no larger than

Start by applying:

1. all unary exact Type A/B constraints on each coordinate;
2. the correct reducedness condition on x=r+Ls;
3. then the mixed rows in the order described below.

The resulting finite CSP has an unexpectedly strong extension property.

3. Unary 11-adic room

After unary 11-adic rows and reducedness:

• the number of surviving base residue classes modulo 11 is always 6 or 7;
• the full 11-adic domain is never empty;
• the smallest full 11-adic domain over all 336 systems has size

Thus the first variable always has room before any mixed row is considered.

4. Unary 13-adic room

After unary 13-adic rows and reducedness, the number of surviving base classes modulo 13 is always between

The full 13-coordinate may be modulo 13 or 13^2, depending on the candidate.

5. The 11-to-13 conditioned fiber

Now fix any surviving full 11-adic value.

Apply every residual row whose support is contained in {11,13}.

Finite structural result

For every candidate and every surviving full 11-adic value:

Equivalently, the mixed 11/13 rows remove at most two of the unary-safe base 13 classes.

The observed minimum number of surviving full 13-adic values per surviving 11-adic value is

When the 13-coordinate has exponent two, the surviving fibers occur in complete mod-13 classes after the finer row is accounted for.

Fine 13^2 row is conditionally redundant

The residual row with quotient

appears in many of these systems.

Across the entire 336-system family, every forbidden value from a mixed row with 13-exponent greater than one is already removed by unary constraints or by a coarser mixed 11/13 row.

Thus the fine 13-adic row introduces

new forbidden points after the coarse 11/13 stage.

This is a finite exact masking statement, not an assumption.

6. The conditioned 3-adic fiber

Fix any surviving (11,13) assignment after the preceding stage.

Now inspect all residual rows containing the prime 3.

The common coarse rows live at moduli including

with optional higher rows such as

Mod-3 collapse

Across every unary-safe base (11,13) pair in every one of the 336 systems, the rows whose 3-part is exactly 3 collectively forbid at most

of the three residues modulo 3.

So at least two mod-3 classes survive before the mod-9 refinements are considered.

Mod-9 collapse

After lifting that forbidden mod-3 class to modulo 9 and adding every row with 3-part 9, the total forbidden set modulo 9 has cardinality at most

Therefore at least

3-adic residue classes remain safe.

Fine 3^4 row is conditionally redundant

Whenever the row with 3-part 81 occurs, every one of its forbidden values already lies over a mod-9 class forbidden by the coarser rows.

It contributes

new forbidden 3-adic values after the mod-9 stage.

Hence the exact number of safe full 3-adic values conditioned on a surviving (11,13) pair belongs to

The second half is exactly nine times the first half because safe mod-9 classes lift freely through the 3^4 coordinate.

The global minimum is

7. Nested extension theorem for the frozen family

Combining the stages gives the following exact finite statement.

Finite theorem-certificate

For every one of the 336 C1 residual systems with signature {3,11,13} through k<=1500:

1. at least one unary-safe 11-adic value exists;
2. every unary-safe 11-adic value extends to at least five full 13-adic values after all 11/13 rows;
3. every surviving (11,13) pair extends to at least five full 3-adic values after every remaining row.

Therefore every initial 11-adic branch has at least

full (13,3) extensions.

This is much stronger than the existence of one bounded selector.

It proves a branch-survival property for the entire finite family.

8. Relation to the two {11,13} systems

The two smallest C1 residual systems, both at target depth k=574, exhibit the same phenomenon without the 3-coordinate.

In each case:

• 725 full 11-adic values survive unary constraints;
• the unary 13-adic domain contains 78 values, six complete residue classes modulo 13;
• all mixed rows except the j=36 row are masked by the unary safe sets;
• the j=36 row removes exactly one 13-class for one special residue modulo 11.

Thus the conditioned 13-fiber has size

and the exact full safe count is

The {3,11,13} family is therefore a genuine lift of the same conditioned-fiber geometry.

9. What this suggests universally

The original residual systems have nominal cover mass greater than one, so global density bounds fail.

The conditioned-fiber result shows why that statistic is misleading.

The constraints overlap in a way that is exposed only after conditioning coordinate by coordinate:

The theorem target is now:

Conditioned Fiber Positivity Conjecture

For every directly novel C1 residual system, there exists an elimination order on the residual prime-power coordinates such that every surviving partial assignment has a positive exact extension fiber at the next coordinate.

A weaker version allowing some partial branches to die while preserving at least one branch would also suffice for C1.

This is not proved universally.

10. Immediate proof questions

1. Why do the 3-linear rows forbid at most one mod-3 class after unary-safe 11/13 conditioning?
2. Why can the mod-9 rows add at most one additional residue outside that lifted class?
3. Why are the 13^2 and 3^4 rows always masked by coarser rows in this family?
4. Which parts follow from ancestry/shadow relations among the fixed row depths 25,36,88,179,205,322,465,520,790?
5. Can the exact masking be turned into a universal row-domination lemma rather than a finite observation?

Those questions are now considerably sharper than the former “why does a selector exist?” question.