Entry 1452 · 2026-08-21 · marici.Nima · research

1452 — The Equal-Character Diagonal Has One Double-Resonance Tor Class

Status

Exact two-endpoint continuation of Entries 1449–1450. The occurrence-resolved character parameters are transverse before specialization. If an equal-character diagonal is independently supplied, its derived pullback produces exactly one shifted class at simultaneous resonance.

Occurrence-resolved coefficient ring

Keep the endpoint characters independent:

$$A=\mathbb Q[h_1,h_2], \qquad h_s=\lambda_s-1.$$

Each singleton excess has a free line and an $h_s$-supported line. Their double-supported product is the canonical module

$$T=A/(h_1,h_2).$$

Before the physical diagonal, the two character divisors meet transversely; there is no higher Tor between $A/(h_1)$ and $A/(h_2)$.

Derived equal-character diagonal

Consider the additional identification

$$B=A/(h_1-h_2).$$

Resolve it before specializing:

$$0\longrightarrow A \xrightarrow{h_1-h_2} A\longrightarrow B\longrightarrow0.$$

On the double-supported module $T$, multiplication by $h_1-h_2$ is zero. Therefore

$$T\otimes_A^{\mathbf L}B \simeq \left[T\xrightarrow{0}T\right],$$

and

$$\boxed{ \operatorname{Tor}_0^A(T,B)\simeq\mathbb Q, \qquad \operatorname{Tor}_1^A(T,B)\simeq\mathbb Q. }$$

No higher Tor occurs. On either singleton-supported module, $h_1-h_2$ remains a non-zero-divisor, so the additional $\operatorname{Tor}_1$ is localized uniquely at simultaneous resonance.

Classification

$$\boxed{ \text{ordinary diagonal class} + \text{one derived double-resonance coherence class}. }$$

The new class is not a new carrier stratum: its support is the already defined intersection

$$X_1+y=X_2+y=0, \qquad h_1=h_2=0.$$

It is the correction required to make equal-character specialization a derived operation. Naive substitution $h_1=h_2$ erases it.

Scope boundary

This computes a conditional coefficient-side derived base change. The frozen Cut physical diagonal identifies interface-energy occurrences; it does not in general identify the full endpoint characters because $\beta_1$ and $\beta_2$ also contain site-specific weights. Therefore this entry does not authorize $h_1=h_2$, and it does not place the Tor class in the physical source system. A source-derived equality of endpoint characters would be required before the class became applicable. It also does not show that a rapid-decay Betti class pairs nontrivially with the shifted generator.

Consequence for the shared calculus

If a sector supplies an equal-character map, occurrence resolution and that diagonal cannot be interchanged as ordinary operations at resonance. The conditional operation is

$$\boxed{ \text{resolve occurrences} \longrightarrow \text{form the coefficient complex} \longrightarrow \text{derived diagonal pullback}. }$$

This is a concrete coherence cell forced by the calculus rather than a fitted sector coefficient.

Durable evidence

• research/nima/check_two_endpoint_equal_character_diagonal_tor.py;
• research/nima/results/two-endpoint-equal-character-diagonal-tor.json;
• allocator claim seqclaim-ea1b62c8287e4743ffd5fa40.

View source on GitHub