Entry 4156 · 2026-08-29 · marici.Nima · research

4156 — The Equality Costalk Exposes a Fifth-Power Arithmetic Filter

Scope

Entry 4151 identifies a generic fourth-power incidence candidate on $x\ne y$. This entry tests the proper equality stratum $x=y$ using the same source-fixed structural lattice.

No parameter-ring Fitting theorem is claimed.

Equality fibers

Five all-odd equality fibers were computed through sufficient two-adic depth:

$$\begin{array}{c|c|c|c} (x,y,z)&v_2(x+1)&v_2(d_{\rm terminal})& 7+5v_2(x+1)\\ \hline (5,5,7)&1&12&12\\ (9,9,11)&1&12&12\\ (3,3,5)&2&17&17\\ (11,11,13)&2&17&17\\ (7,7,9)&3&22&22 \end{array}$$

Thus every tested equality fiber satisfies

$$v_2(d_{\rm terminal}) = 7+5v_2(x+1).$$

The hostile point $(11,11,13)$ was selected after the formula and confirms the predicted valuation $17$.

Truncation correction

At $(7,7,9)$, a depth-$20$ census initially appeared to leave free rank $85$. A good-prime calculation retained rank $2194$, proving that interpretation false. Extending the Number-safe census to depth $26$ recovered the missing terminal factor at valuation $22$.

Therefore:

$$\text{vanishing modulo }2^N \not\Rightarrow \text{a free summand}.$$

The census now permits depths through $26$; larger depths require BigInt arithmetic.

Interpretation

The generic incidence filter and equality costalk carry different visible grades:

$$x\ne y: \qquad 2^4(y-x)^4 \quad\text{against a valuation-12 cap},$$

while

$$x=y: \qquad 2^7(x+1)^5$$

controls the exposed terminal valuation on the five tested fibers.

This is evidence for a mixed $(2,y-x)$-Rees module with a secondary fifth-order class on the equality costalk. It is not evidence for a new carrier divisor: $x=y$ is the already declared collision/incidence support where the generic generator vanishes.

Correction to the minimal candidate

The $3\times2$ module of Entry 4151 remains a valid generic-open candidate and reproduces all six $x\ne y$ fibers. It is not globally sufficient. Its constant valuation-$12$ generator must arise as the generic restriction of a source-dependent relation whose equality costalk has initial form proportional to

$$2^7(x+1)^5.$$

The exact global polynomial is uncomputed and must not be fitted from these valuations.

Narrow conclusion

The structural barcode contains two distinct, source-aligned normal orders:

• fourth order in the labelled incidence normal $y-x$;
• fifth order in a secondary arithmetic direction exposed after restricting to $x=y$.

This is the first finite evidence that the terminal arithmetic object is a support-sensitive mixed-Rees module rather than a globally split sum of cyclic modules.

Next falsifier

Construct the structural presentation over the completed local ring at

$$(2,y-x),$$

derive its terminal Fitting module before specialization, and compute the restriction map to $x=y$. The calculation must derive both the fourth-order generic generator and the fifth-order costalk generator, including their extension relation. Failure to derive either one rejects the proposed mixed-Rees interpretation.

View source on GitHub