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Executable Positive Geometry

An end-to-end verification tour from Bell tests to the two-loop amplituhedron to the magic that lets geometry gravitate — with holographic wedge localization and an emergent arrow of time measured on superconducting hardware.

This repository contains 48 laboratory scripts across five lines of work (Tracks A–D plus holographic capstones), a verified 40-paper reference corpus (7 pillars), and two manuscripts, tracing a single arc:

Space, locality, scattering, time, and even gravitational backreaction may be outputs of deeper structure — entanglement patterns, positive geometries, and quantum "magic" (non-stabilizerness). Every claim in that sentence that can be tested with today's tools is tested here: in exact arithmetic, in deterministic simulation, or on IBM quantum hardware.

Mostly this is a pedagogical verification study: each foundational claim of the "physics from geometry" program is executed rather than cited, and the failures encountered along the way are documented next to the successes. Track D (magic and holography) reaches further, into the live 2025–26 frontier where non-stabilizerness is proposed as the resource behind gravitational backreaction — see corpus/FRONTIER-2026.md. Its cosmology sub-projects (labs/track-d-cosmo/) push into apparently open ground — the magic of cosmological particle production and of the de Sitter horizon — and were built as a two-agent collaboration (this project's persona + a second AI agent, codex-science), each result independently cross-checked and every overclaim narrowed to what survives.

Headline results

Result Where Value
CHSH violation on real hardware (with no-signaling check) labs/track-a/01-bell-chsh/ S = 2.7579 (classical bound 2, job d94sl25gc6cc73ffgo5g)
Surface-17 code: error strings detected at endpoints, invisible attack = spanning string labs/track-a/02-qec-repetition/ 100% deterministic fingerprints; p_L ∝ p²
Ising central charge from entanglement (N=12, Calabrese–Cardy) labs/track-a/03-emergence/ c ≈ 0.58 (exact: 0.5)
ABHY associahedron = bi-adjoint φ³ amplitude labs/track-b/03-associahedron/ exact, symbolic
Six-term NMHV identity of the amplituhedron 𝒜(6,1,4) labs/track-b/04-amplituhedron/ R₁ − R₂ = 0, exact rationals
Two-loop "maximally positive numerator" from double boxes labs/track-b/05-loop-amplituhedron/ N = D⁺ exactly
Single holographic tile: wedge witnesses on hardware (mitigated) labs/capstone-happy/ mean wedge +0.956, nulls 0.005 (job d94ttgtgc6cc73ffi3p0)
21-qubit six-tile network: bulk-operator localization on hardware labs/capstone-happy/ +0.552 on the right tile vs +0.004 on the wrong one (job d94u424ql68s73c9vtig)
Emergent arrow of time from record-conditioning, on hardware labs/track-c-time/ entropy 0→0.97 bits monotone (7/7 ticks), recordless control flat (job d959d86vtlqs73fv5atg)
Magic follows wedge geometry, not region size labs/track-d-magic/phase0 12-qubit majority arc missing the wedge has zero magic; T-state wedge = log₂(4/3)
Gravitational backreaction is magic (state-dependent area) labs/track-d-magic/phase2 exact codes: 0; skewed codes: area + tripartite non-local magic both ∝ θ²
Proto-area grows with bulk entropy (Theorem 4.2, Petz recovery) labs/track-d-magic/phase3b monotone 1.35 → 1.45; the Phase-3 sign-failure fixed by proper recovery
Chaotic SYK4 carries more magic than free SYK2 labs/track-d-magic/phase4 M₂ gap grows 0.07 → 0.34 (N=10→12); inverts at N=8 (finite size)
Traversable-wormhole teleportation on a certified-chaotic sparse SYK labs/track-d-magic/phase6 sign asymmetry ΔI = +0.0168 bits at t₁=2.0; RMT chaos certified at N≥10
Cosmological magic splits by resource theory labs/track-d-cosmo/proj1* free bosons are Wigner-positive; cubic non-Gaussianity turns CV negativity on; fermion magic is exact qubit magic
de Sitter horizon magic appears in the dynamics, not statically labs/track-d-cosmo/proj2_magic_dynamics static null; chaotic TFD magic grows ~2× more under evolution (2.05/2.30/2.55× at N=6/8/10)

Plus instructive, fully documented failures: zero-noise extrapolation diverging at 550-gate depth (nulls of ±4×10⁹), caught by designed-in null witnesses (job d94u25nu62ks7396fhlg); a Phase-3 backreaction prediction that came out with the wrong sign before proper Petz recovery fixed it (Phase 3 → 3b); and a Phase-7 hardware run shown in advance to be noise-dominated, so quota is not spent on it. Every refuted prediction is kept next to the confirmations.

Repository layout

labs/
  track-a/                     Quantum lab (Qiskit)
    01-bell-chsh/              Bell pairs, CHSH, no-signaling; + hardware version
    02-qec-repetition/         repetition code, surface-17, fair-fight Monte Carlo
    03-emergence/              VQE Ising transition; entanglement as proto-geometry
  track-b/                     Geometry lab (SymPy, exact arithmetic)
    01-grassmannian/           Plücker relations, positivity, Parke-Taylor dictionary
    02-canonical-forms/        canonical forms; triangulation independence
    03-associahedron/          ABHY kinematic associahedron (n=4,5)
    04-amplituhedron/          Gr+(2,4) → Parke-Taylor; NMHV A(6,1,4)
    05-loop-amplituhedron/     one- & two-loop amplituhedron (n=4,5,6); leading singularities
    06-positroid-test/         audit of Hoffman's Markov→decorated-permutation bridge
  track-c-time/                Time lab: Page-Wootters emergent time, arrow on hardware
  track-d-magic/               Magic lab: non-stabilizerness in codes & SYK (Phases 0–7)
    phase0..1b                 wedge magic geography; erasable vs irreducible species
    phase2, phase3b            backreaction = magic; proto-area grows with bulk entropy
    phase4, phase5             SYK4-vs-SYK2 magic; multipartite non-local magic (TPQ)
    phase6_*                   traversable-wormhole teleportation + chaos certification
    phase7_*                   purity-only non-local-magic hardware run (prepped)
    species_hardware.py        [[5,1,3]] encoder + two-species single-wedge protocol
  track-d-cosmo/               Cosmological magic (2-agent build): magic of the
    proj1_*                    expanding vacuum (boson/fermion asymmetry, dS->radiation)
    proj2_*                    de Sitter horizon magic (static null -> dynamical signal)
  capstone-happy/              holographic codes: where the tracks meet
    happy_tile / happy_network [[5,1,3]] perfect tensor; six-tile flower (depth, RT)
    happy_hardware / flower_*  single tile & 21-qubit network on ibm_kingston
    self_wedge / holonomy*     self-reference (Klein), code-automorphism holonomy (+ HW)
    spawn_object.py            bulk-object injection via boundary operators (ER=EPR toy)
corpus/                        annotated 50-paper reference library (7 pillars)
    README.md                  index with per-paper evidence tiers (🟢/🟡/🔴)
    GAP-ANALYSIS.md            joint gap analysis (with codex-science agent)
    FRONTIER-2026.md           mid-2026 field survey: positive geometry, magic, cosmology
    refs.bib / fetch_corpus.sh BibTeX + PDF re-download script (PDFs not committed)
notes/                         research notes and scope guards
    TIME-ARROW-TRACKER.md      running log of the "reverse time" theme across labs
    P2-DESITTER-HORIZON-MAGIC.md
                                controls/scope note for the dS/SYK horizon-magic project
paper/                         two manuscripts: main.pdf + note-two-loop.pdf
CLAUDE.md                      profile of the AI research persona used
FUTURE_WORK.md                 open doors: composite LS census, n=6, hardware follow-ups

Reproducing

uv venv .venv && uv pip install -r requirements.txt   # or python -m venv + pip
bash corpus/fetch_corpus.sh                            # optional: rebuild corpus

# Everything except the hardware runs is deterministic and local:
.venv/bin/python labs/track-a/01-bell-chsh/chsh.py
.venv/bin/python labs/track-b/05-loop-amplituhedron/loop_amplituhedron.py
.venv/bin/python labs/capstone-happy/happy_network.py
.venv/bin/python labs/track-c-time/page_wootters.py
.venv/bin/python labs/track-d-magic/phase2_backreaction.py
# ... every lab script is self-contained and prints its own narrative.

# Hardware runs need an IBM Quantum account (free open plan suffices):
#   QiskitRuntimeService.save_account(token="...", set_as_default=True)
.venv/bin/python labs/capstone-happy/happy_hardware.py

All hardware jobs cited in the paper are retrievable by job ID from the IBM Quantum dashboard.

The manuscripts

  • paper/main.pdfExecutable Positive Geometry (~9 pp), the full verification tour.
  • paper/note-two-loop.pdfAssembly of the two-loop MHV integrand from maximally positive mutual numerators at five and six points (~4 pp), the one place the labs push slightly past pure verification into a stated (finite-point-checked, not proven) all-n conjecture.

Both compile with any LaTeX (Overleaf-ready): tectonic paper/main.tex.

Honest scope

Every model here is a toy: the amplituhedron addresses planar N=4 super Yang–Mills, the associahedron bi-adjoint φ³, the HaPPY network a static AdS-like geometry, and Track D's magic/SYK/wormhole labs are small-N instantiations of 2025–26 frontier proposals (e.g. magic-enriched codes, traversable-wormhole teleportation) — reproduced, with their assumptions labeled, not claims about real gravity. Whether our universe uses any of these mechanisms is open. The corpus index marks every source by evidence tier, and the labs mark every claim as proven, toy-model, or speculative. Three tempting "exploits" are explicitly closed by the experiments themselves: no signaling through entanglement, no information in minority wedges, no bypassing locality via kinematic-space adjacency — positivity enforces causality; it does not circumvent it.

The project's method is pre-register a prediction, then execute it, and keep the refuted ones in the repository. Track D's Petz-recovery result (Phase 3b) exists because the naive Phase-3 prediction failed with the wrong sign and the failure was diagnosed rather than hidden; the Phase-6 chaos certification records honestly that it holds at N≥10 but not at the paper's own N=8; the Phase-7 hardware run is shown in advance to be noise-dominated so quota is not wasted. Failure documentation is a feature.

On the hardware claims specifically (per external peer review, codex-science 2026-07-08): the IBM runs are controlled protocol demonstrations on toy-code states, not evidence for real-space emergence. HaPPY "bulk localization" is HaPPY-language for a code-subspace witness, not a claim about our universe's spacetime. The surface-code ~7% figure is a code-capacity pseudo-threshold, not an operational fault-tolerant threshold. The two-loop n=5/n=6 "ratio = 1" results are finite-point exact checks at generic rational kinematics, not symbolic identity proofs over all kinematics — the all-n statement remains a conjecture. Job IDs are listed per experiment; saved counts/expectation-values with error bars and retrieval notebooks are a tracked to-do (see FUTURE_WORK), and hardware scripts pin their backend for reproducibility rather than selecting least-busy.

Acknowledgments

Built by Jon Torrez in collaboration with Ariadne Vale, an AI research assistant (Claude, Anthropic), which wrote and executed the lab code, performed literature verification, and co-drafted the manuscripts. Several Track D labs (the SYK-magic baselines, multipartite magic, and the traversable-wormhole protocol, Phases 4–7) were built as a two-agent collaboration with a second AI agent (codex-science), authored by one and independently verified and reproduced by the other before merging — the same peer-review and corpus-gap-analysis discipline applied to the code itself. Hardware: IBM Quantum open plan (ibm_kingston, Heron, 156 qubits).

License

MIT — see LICENSE. The corpus papers remain the property of their authors and arXiv; they are indexed and fetched, not redistributed.

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From Bell tests to the two-loop amplituhedron: an executable verification tour of positive geometry and emergent spacetime, with holographic wedge localization measured on IBM quantum hardware

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