# AI provenance: The Mirror Clause of the Geometric Selection Conjecture Is Empty

Authors: Deep Bhattacharjee and Ushashi Bhattacharya. No AI system is an author or contributor.

## How the work was produced

Deep Bhattacharjee asked for an unconditional proof of the Geometric Selection Conjecture (Bhattacharjee, Bhattacharya; Theorem 22 of the strings paper, Hexagon 2610.00446v1) as a separate paper. Claude (Anthropic), through Claude Code, established that the conjecture as stated cannot be closed, because its viability conditions (C7) to (C9) are fixed by comparison with measurement and no published heterotic model meets all three, and then proved the one thing that can be proved from the definitions: the mirror-fixed clause of the equivalent form is empty. It drafted the note, wrote the check and recorded the attempt, with its failures, in the Anaxia engine (creelie/eBLLM, module construct/gsc_mirror_fixed.py). The authors directed the work and reviewed the text.

## How the results are checked

The proof is by hand and three steps long: a vacuum fixed by the mirror involution is isomorphic to its mirror, so h^{1,1} = h^{2,1} and chi(X) = 0; its component contains a standard embedding and Chern classes and Euler numbers are constant on a component, so c_3(V) = chi = 0; Hirzebruch-Riemann-Roch on a Calabi-Yau threefold with c_1(V) = 0 gives chi(X, V) = c_3(V)/2 = 0, so the vacuum has no net families and fails (C2). `code/mirror_clause.py` (Python, standard library) carries the index formula, the mirror map and the fixed-point test on discrete data and checks them on self-mirror pairs (19,19), (11,11), (3,3), (1,1), (2,2), (5,5), three-family pairs (6,9), (1,4) and others, with free quotients of order 2, 3, 5 and with refuters (a would-be fixed point with nonzero c_3 is caught; an off-standard vacuum with three families on a self-mirror threefold is outside the lemma): `511 checks passed, 0 failed`.

## What is not claimed

The Geometric Selection Conjecture is not proved. No viable vacuum is exhibited; the note reduces the conjecture to the existence of a viable vacuum on a component of the landscape carrying no standard embedding and leaves that existence open.

## Labels given on the Hexagon form

- Text of the contribution: a mix of human-written and AI-generated.
- Human understanding at submission: some parts, but not all, checked and understood by a human.

## Where everything is

Release v1.0.0: https://github.com/creelie/geometric-selection-conjecture/releases/tag/v1.0.0 (note, source zip, check). The strings paper: https://hexagonmath.org/2610.00446v1.