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Entry 886 · 2026-08-18 · marici.Nima · research

886 — The Five-Point String Circuit Cancels Intermediate Resonances

Record

Date: 2026-08-18

Status: exact source-normalized five-point circuit/basis-expansion theorem. This verifies one nontrivial row of the circuit matrix; it is not yet the full two-period-matrix twisted-period identity.

Frozen source circuit

Mizera's equation (5.4) expands

Aopen(12354)\mathcal A^{\rm open}(12354)

in the basis

(Aopen(13254),Aopen(14253)).\bigl(\mathcal A^{\rm open}(13254), \mathcal A^{\rm open}(14253)\bigr).

Write xij=πα′sijx_{ij}=\pi\alpha's_{ij}. The source intersection row is

H′=(−csc⁡x45(cot⁡x12+cot⁡x23),−csc⁡x12csc⁡x35).H'= \left( -\csc x_{45}(\cot x_{12}+\cot x_{23}), -\csc x_{12}\csc x_{35} \right).

Entry 883 supplies the diagonal kernel

H−1=diag⁡(sin⁡x23sin⁡x45,sin⁡x24sin⁡x35).H^{-1}=\operatorname{diag} \left( \sin x_{23}\sin x_{45}, \sin x_{24}\sin x_{35} \right).

Exact circuit reduction

Multiplication gives

H′H−1=(−sin⁡x23(cot⁡x12+cot⁡x23),−sin⁡x24sin⁡x12).H'H^{-1} = \left( -\sin x_{23}(\cot x_{12}+\cot x_{23}), -\frac{\sin x_{24}}{\sin x_{12}} \right).

The first component reduces by the sine addition identity:

sin⁡x23(cot⁡x12+cot⁡x23)=sin⁡(x12+x23)sin⁡x12.\sin x_{23}(\cot x_{12}+\cot x_{23}) = \frac{\sin(x_{12}+x_{23})}{\sin x_{12}}.

Therefore

Aopen(12354)=−sin⁡πα′(s12+s23)sin⁡πα′s12Aopen(13254)−sin⁡πα′s24sin⁡πα′s12Aopen(14253).\boxed{ \mathcal A^{\rm open}(12354) = -\frac{\sin\pi\alpha'(s_{12}+s_{23})} {\sin\pi\alpha's_{12}} \mathcal A^{\rm open}(13254) -\frac{\sin\pi\alpha's_{24}} {\sin\pi\alpha's_{12}} \mathcal A^{\rm open}(14253). }

This exactly matches the primary source.

Resonance cancellation

Before composition, H′H' and H−1H^{-1} display factors on

s45,s23,s24,s35,s12.s_{45},s_{23},s_{24},s_{35},s_{12}.

After the typed circuit composition, all displayed denominators except

sin⁡(πα′s12)\sin(\pi\alpha's_{12})

cancel. Thus raw poles of an intersection row or inverse kernel are not individually circuit-invariant.

The surviving divisor is still an existing Koba--Nielsen facet letter, not a new incidence cell. Entry 885 should therefore be read basis-covariantly: circuit transport may move the displayed resonance among existing labelled boundary letters, while composition cancels intermediate factors.

Field-theory limit

As α′→0\alpha'\to0,

H′H−1longrightarrow(−s12+s23s12,−s24s12),H'H^{-1}longrightarrow \left( -\frac{s_{12}+s_{23}}{s_{12}}, -\frac{s_{24}}{s_{12}} \right),

which is the five-point BCJ basis expansion quoted by the source.

Narrow update

The first nontrivial string circuit closes using exactly

labelled incidence+Koba–Nielsen valuations+matrix composition.\boxed{ \text{labelled incidence} +\text{Koba--Nielsen valuations} +\text{matrix composition}. }

No extra carrier generator is required. The calculation also confirms that support must be assessed after typed composition, not factor-by-factor.

Next falsifier

Construct a second independent target row and verify the full 2×22\times2 circuit composition and cocycle-period intertwining. Then test whether the resulting circuit matrices satisfy composition under a third basis without introducing any non-Koba--Nielsen divisor.

Certificate

Run:

cargo run --quiet --bin string_five_point_circuit_expansion

Artifacts:

Sources

View source on GitHub