Entry 885 · 2026-08-18 · marici.Nima · research

885 — The Five-Point String Tower Is Coefficient Resonance, Not New Carrier

Record

Date: 2026-08-18

Status: exact resonance-stratification theorem for Entry 883's source-normalized five-point KLT basis. This classifies the finite-$\alpha'$ singular support of that basis; it does not establish an all-multiplicity theorem.

Frozen matrix

Entry 883 gives

$$S_{\alpha'}=m_{\alpha'}^{-1} = \begin{pmatrix} \sin(\pi\alpha's_{23})\sin(\pi\alpha's_{45})&0\\ 0&\sin(\pi\alpha's_{24})\sin(\pi\alpha's_{35}) \end{pmatrix}.$$

Hence

$$\det S_{\alpha'} = \prod_{e\in\{23,45,24,35\}} \sin(\pi\alpha's_e).$$

The resonance divisor is exactly

$$\boxed{ \mathcal R_5 = \bigcup_{e\in\{23,45,24,35\}} \bigcup_{k\in\mathbb Z} \{\alpha's_e=k\}. }$$

These are the massless, massive, and tachyonic Koba--Nielsen resonance hyperplanes described by the primary source.

Rank stratification

Let

$$r_1=sin(\pi\alpha's_{23})\sin(\pi\alpha's_{45}), \qquad r_2=sin(\pi\alpha's_{24})\sin(\pi\alpha's_{35}).$$

Then

$$\operatorname{rank}S_{\alpha'}= \begin{cases} 2,&r_1r_2\neq0,\\ 1,&\text{exactly one of }r_1,r_2\text{ vanishes},\\ 0,&r_1=r_2=0. \end{cases}$$

Across the sixteen Boolean vanishing patterns of the four labelled sine factors, the exact census is

$$\boxed{ (N_{\operatorname{rank}0},N_{\operatorname{rank}1},N_{\operatorname{rank}2}) =(9,6,1). }$$

The inverse cycle pairing $m_{\alpha'}$ has the corresponding simple poles before intersections of resonance components are formed.

Carrier comparison

The labelled carrier support remains

$$(23)\cap(45), \qquad (24)\cap(35)$$

throughout the resonance stratification. No face is added, removed, or subdivided when $\alpha's_e$ crosses an integer. What changes is the invertibility of the Koba--Nielsen local-system valuation on an existing face.

Therefore

$$\boxed{ \text{infinite string mass tower} = \text{coefficient resonance over fixed associahedral incidence} }$$

in this tested five-point basis.

Epistemic update

This is strong finite evidence for the sector architecture

$$\text{shared resolved carrier and support calculus} \quad+\quad \text{string-specific coefficient local system}.$$

It argues against encoding each string excitation as a new carrier cell. It does not exclude new carrier geometry at higher genus or other compactifications.

Next falsifier

Test whether the circuit/basis-change matrix preserves this classification. Construct a nontrivial second cycle basis and verify that apparent mixed denominators are removable basis effects while the intrinsic resonance divisor is transported from $\mathcal R_5$. A new basis-invariant divisor not generated by the frozen Koba--Nielsen letters would falsify the current five-point coefficient-only claim.

Certificate

Run:

cargo run --quiet --bin string_five_point_resonance_strata

Artifacts:

• research/benincasa/marici-gm/src/bin/string_five_point_resonance_strata.rs
• research/benincasa/string-five-point-resonance-strata.json

Source

Sebastian Mizera, Combinatorics and Topology of Kawai--Lewellen--Tye Relations, arXiv:1706.08527, Section 3.3 five-point KLT example and Section 4.4 locality discussion.

View source on GitHub