# Reality Compute

## Consolidated Master — v1.6

FOUNDATION DOCUMENT

**Prepared for:** Alex Whayman
**Date:** 15 September 2026
**Supersedes:** v1.5 (12 September 2026) as the current position
**Status:** Research record. External simulation is not established. Nothing in this document is evidence that reality is simulated.

**What changed since v1.5.** The quantum branch's results from Sessions 005 to 009 and Quantum Loop Record 001 are now recorded in full, with their figures. One recommendation is withdrawn: Session 007 proposed narrowing the branch to one host class, and Session 009 showed that both host classes can be tested. Two null bounds are recorded, one per host class, and they are not combined. Every result in this version is a null or a finding about the programme's own instrument. None of it is evidence for or against simulation in general.

* * *

## 1. Current position

Reality Compute asks whether a resource-constrained computational implementation of physical law would produce observable departures from calibrated physics, and whether such an implementation can be distinguished from ordinary explanations at all.

The honest summary: **no observation currently favours external implementation.** The programme has produced useful formal results, a statistically sound test design, two null bounds on specific host classes, and a clearer map of what would and would not count as evidence. It has not produced evidence.

Three results constrain everything else and are treated as settled:

1. **Faithful implementations are unreachable.** If an implementation reproduces the same distribution of accessible records, every likelihood ratio is exactly 1. No internal experiment can distinguish it, ever. Only *lossy* implementations are testable.
2. **External slowdown is invisible.** A host that pauses all internal processes consistently changes no internal clock comparison. "Demanding physics should lag" is false without a further assumption.
3. **There is no single measurable computational complexity.** State-preparation cost, classical simulation cost, entanglement and non-stabilizer resource are four different quantities. Session 005 strengthened this: classical simulation cost is not a function of any fixed finite set of state properties.

* * *

## 2. Changelog from v1.5

Each change carries the effect it has on a claim. `weakens` means a claim or architecture is now more constrained; `rescopes` means a claim's scope changed; `neutral` means material was added without moving a claim.

| Section | Type | Effect | What changed |
| --- | --- | --- | --- |
| Session 007 recommendation (§3.8) | re-scoped | rescopes | The recommendation to narrow the branch to one host class is withdrawn. Both host classes stay open, and both are now bounded. |
| Two-contrast design (§3.5) | added | neutral | Bond dimension and T-count are orthogonal coordinates. Each tests a different host class. |
| Bounds on κ (§3.6) | added | weakens | Two null bounds: κ_MPS ≤ 0.0019 and κ_stab ≤ 0.00082. A null constrains the architecture it tests. |
| Instrument limits (§3.7) | added | neutral | Certified separation caps at 2 doublings in the frozen ansatz, and scale and magic are currently mutually exclusive under the analytic certificate. |

v1.5 §9's first priority stands unchanged: tightening the upper bound Σⱼ2^(Eⱼ) is 81% of the certification problem and remains the branch's top technical item. Session 009 is orthogonal to it, not a replacement for it.

* * *

## 3. The quantum branch, Sessions 004–009

Everything in this section is simulation only. No hardware experiment has been run.

### 3.1 What was established before v1.6

**Session 004 — the false-positive mechanism.** The proposed regression test's error rate is governed by one quantity nobody had named: the correlation between the baseline observable and the complexity proxy. Below |r| ≈ 0.35 it holds its nominal rate; at |r| ≈ 0.73, ordinary depolarizing noise with *no effect present* produces a spurious result four times the detection threshold and rejects the null **100% of the time**. Decorrelating the baseline from the proxy is what makes incoherent confounds orthogonal to the tested regressor.

**Session 005 — no finite cost metric exists.** The proxy-regression design needed simulation cost to be ordered by a fixed set of state properties. It is not.

The decisive case is at n = 12. Two states, X and Y, are matched to |ΔM₂| = 0.0008 in magic and |ΔS₂| = 0.0014 in entanglement. Both have bond dimension χ = 64, the maximum at n = 12, in the computational frame. But X is a Clifford circuit applied to a product magic state: in the Clifford frame its construction supplies, it is an exact product state with χ = 1 (verified directly, and every single-qubit marginal is pure). The best of 200 random Clifford frames for Y reached χ = 32. A wider scan at n = 10 over 174 states found 186 pairs matched in both coordinates, with bond dimension differing by up to 2.00× inside a matched cell. All 9 verification checks pass, including brute-force Pauli enumeration at n = 3 and 4.

Two limits travel with this result. Failing to find a cheap frame for Y is not proof that none exists; the argument does not rest on Y, because X alone is cheap at coordinates where a two-coordinate metric predicts maximal cost. And X's own cheap frame was not found by the 200 random draws either: holding the right coordinates does not make them computable. The constructive half of the argument is verified in the code and does not depend on any preprint.

**Session 006 — RCC-A2, the matched-twin protocol.** RCC-A2 replaces the regression with a two-point comparison that needs no cost ordering. Each pair has one control circuit and one test circuit with the same pulse-class census m (Δm = 0 exactly), the same ideal output (max |Δp_ideal| = 0.000817) and nearly the same magic (max |ΔM₂| = 0.00819), but a certified difference in bond dimension.

Head to head against the Session 004 confound generator, with **no external-computation effect present** (195 pairs, 390 circuits, 20 000 shots per circuit, 2 000 replicates, nominal two-sided level 1%):

| Confound | RCC-A1, baseline–proxy r ≈ +0.85 | RCC-A2 |
| ---: | ---: | ---: |
| 3 × 10⁻³ | 97.80% false positives | 1.25% |
| 1 × 10⁻² | 100.00% false positives | 1.20% |

Under pure shot noise the empirical 99th percentile of |t| is 2.596, against the nominal 2.576. The minimum detectable effect is 0.00575 logit units at 79.5% power (target 80%). Across matching tolerances from 0.05 down to 0.001, the false-positive rate stays between 0.55% and 1.60%. The standard deviation of the estimate is 0.005411, 0.001597 and 0.000523 at 2 000, 20 000 and 200 000 shots.

RCC-A2 breaks where it should. When the p_ideal match is deliberately removed, the false-positive rate rises to 10.15% with a bias of +0.00800. All 11 verification checks pass.

### 3.2 Session 008 — the loop named the wrong target

Loop Record 001 identified "tighten the analytic lower bound" as the highest-value next step, on the basis that it reached only ~0.40 of the exact Schmidt rank. That work was carried out. The result is more useful than the intended one.

**Result A — the looseness is deterministic, not statistical.** The Session 007 operator-norm bound satisfies

```math
L_1/r_{\text{exact}} = \xi^{-t}, \qquad \xi = 1.171573
```

where ξ is the stabilizer extent of the magic state and *t* is the T-count. Verified to 10⁻¹⁵ at t = 0 and exactly at the median for t = 2, 4. The bound loses **0.2284 bits per T gate**, predictably.

**Result B — a new bound, derived and validated.** Nuclear-norm duality gives, for any stabilizer state φ,

```math
r \ge |\langle\psi|\phi\rangle|^2 \cdot 2^{E_\phi}
```

read as: a state overlapping strongly with a highly entangled stabilizer state cannot have low rank. It costs the same 2^t and strictly improves the worst cases (0.364 → 0.729 of exact).

**Result C — the purity bound is essentially exact.** The participation-ratio bound r ≥ 1/Tr[ρ_A²] reached the exact rank in 43 of 45 circuits tested, and is provably never worse than the operator-norm bound. It costs 2^{4t}·poly(n) — still exponential in T-count only, at 65,536 stabilizer terms for t = 4, which is affordable.

**Result D — and this is the finding that matters.** Measuring where the looseness actually sits:

| T-count | Upper bound is off by | Lower bound is off by |
| ---: | ---: | ---: |
| 0 | 1.0× | 1.00× |
| 2 | 4.0× | 1.37× |
| 4 | **16.0×** | 1.88× |

The upper bound Σⱼ2^{Eⱼ} doubles per T gate — **1.000 bits per gate**. The lower bound loses 0.2284 bits per gate. So tightening the lower bound addresses at most **19% of the problem**, and the crossing that made the test Clifford-only is driven overwhelmingly by the *upper* bound.

**Redirect.** Σⱼ2^{Eⱼ} is the triangle inequality on Schmidt rank, and the true rank of a superposition is typically far below the sum of its branch ranks — 16× below at t = 4 here. The programme's next technical step is tightening the **upper** bound, not the lower one.

Session 008's section is carried forward from v1.5 unchanged.

### 3.3 Loop Record 001 — the certificate had to simulate the circuit it certified

Loop Record 001 scored the RCC-A2 result, introduced the host-response parameter κ (the suppression of output fidelity per doubling of certified cost), and found a structural problem.

RCC-A2 needs a certificate that one arm of each pair is cheaper to simulate than the other. In Session 006 that certificate was the exact Schmidt rank, read from the statevector. Computing it meant simulating the very circuit being certified. That confined the test to circuits a laptop can simulate exactly, roughly 10³ to 10⁶ operations. A host under genuine resource pressure is not plausibly stressed in that regime.

Inside the frozen ansatz the separation is also capped. At n = 6, 10 and 12, pairs exist at one or two certified doublings of bond dimension, and three or more doublings yields zero pairs. A targeted search for low-rank control circuits matched plain random sampling in 5 of 6 cells, so the cap comes from the ansatz, not the sampling budget. Enlarging the pool is cheap; the doubling count is a hard limit. At equal pair count the bound is κ ≤ 0.00575 per doubling at one doubling.

### 3.4 Session 007 — analytic certificates

Session 007 answered Loop Record 001 with certificates that never build a statevector. Each T gate splits the state into stabilizer branches. The entanglement of every branch across a cut comes from the rank of its stabilizer tableau, and two bounds on Schmidt rank follow: a lower bound from the operator norm and an upper bound Σⱼ2^(Eⱼ). The cost is 2^t·poly(n), exponential in T-count only and never in width. The per-T-gate constant is 1.171573, which matches the stabilizer extent of Bravyi et al.

Validation against exact statevector ranks at n = 6 to 12 found no violations: in the captured output, 0 of 60 lower-bound comparisons and 0 of 60 upper-bound comparisons failed. Tableau two-qubit marginals agree with the statevector to 2.2 × 10⁻¹⁵, and p_ideal agrees with the Session 004 engine to 1.9 × 10⁻¹⁵.

For Clifford circuits the protocol runs at width. At n = 96 with 48 layers, certified bond dimensions span 2⁴⁵ to 2⁴⁸, three certified doublings. The run took 21.3 s for 60 circuits, 0.35 s each. Using contraction cost n·χ³, the test arm costs about 2.14 × 10⁴⁵ operations and the control arm about 4.18 × 10⁴² operations. Dividing Session 006's minimum detectable effect by the three certified doublings gives κ_MPS ≤ 0.0019 per doubling. That figure is a scaling from Session 006's statistics to the n = 96 separation; no shot-level null simulation was run at n = 96.

**This is a legitimate test of a specified host, not a lesser one.** A Clifford-bodied circuit at n = 96 tests an approximation-first host that contracts tensor networks, which is the architecture RCC-1a already specifies.

Adding magic collapses the certified separation. At n = 32 the certified separation D is +1.00 at t = 0, −0.26 at t = 2, −1.46 at t = 4 and −1.83 at t = 8. A negative D means the bounds cross and no pair is certified.

### 3.5 Two-contrast design (new in v1.6)

A host that simulates quantum dynamics classically pays for one of at least two resources, depending on its algorithm. The frozen RCC-A1 family supports a clean contrast for each.

| Contrast | Varies | Holds fixed | Tests a host that pays for |
| --- | --- | --- | --- |
| Session 006 | bond dimension χ | pulse census m, p_ideal | tensor-network contraction |
| Session 009 | T-count m | χ, p_ideal | stabilizer decomposition |

The two coordinates are orthogonal. A χ contrast at fixed m does not move stabilizer-decomposition cost, and a T-count contrast at fixed χ does not move contraction cost. Each contrast therefore bounds one host class and says nothing about the other.

**Session 009's T-count contrast.** With χ fixed at 8 in both arms and m = 4 against m = 24, the matching at tol_p = 0.002 formed 469 pairs, with max |Δp_ideal| = 0.00196. The magic contrast across each pair is real (ΔM₂ = +2.44) and the certified separation in stabilizer extent is 4.57 doublings (23.7×). The minimum detectable effect is 0.00375 logit units at 20 000 shots.

### 3.6 Bounds on κ (new in v1.6)

| Parameter | Host class | Bound | Separation | Confidence |
| --- | --- | --- | --- | --- |
| κ_MPS | tensor-network contraction | ≤ 0.0019 per doubling of bond dimension | 3 certified doublings at n = 96 (Clifford) | Session 006 design holds its nominal false-positive rate |
| κ_stab | stabilizer decomposition | ≤ 0.00082 per doubling of stabilizer extent | 4.57 doublings, χ fixed at 8 | **Carries an uncorrected systematic; see below** |

**The two bounds are not published with equal confidence.** The T-count contrast carries a residual systematic that the χ contrast does not. With no confound present, its false-positive rate is 1.65% at tol_p = 0.002 (469 pairs, the run that gives κ_stab ≤ 0.00082) and **3.80%** at the tighter tol_p = 0.001 (405 pairs, where the bound is 0.00087), against a nominal 1%. Median bias is +0.18 and +0.29 of the minimum detectable effect. Tightening the match made the rate worse, so the residual is not driven by p_ideal mismatch. As the confound grows at tol_p = 0.002, the false-positive rate falls from 1.65% to 0.40% to 0.00%.

κ_stab ≤ 0.00082 must always be quoted with the 3.80% false-positive rate. Publishing the bound alone would overstate it.

The cause is not established. The leading hypothesis is that the two arms differ in how noise acts along the circuit's trajectory, which matching the final ideal output does not capture. The proposed fix is to match the derivative dp/dp₁ alongside p_ideal. It is a property of the ideal circuit, so it needs no noise model and can be preregistered. It has not been implemented.

**What these bounds do and do not mean.** Each bound constrains one host class using one algorithm class, under the frozen toolbox. A host using a different method is not constrained. The bounds are not combined: κ_MPS and κ_stab measure different things, and there is no pooled figure. A null bound is a constraint on an architecture. It is not evidence that reality is or is not simulated.

### 3.7 Instrument limits (new in v1.6)

1. **The frozen ansatz caps certified separation at 2 doublings.** Loop Record 001 found no pairs at three or more doublings at n ≤ 12. The n = 96 Clifford protocol reaches 3 doublings only by leaving the small-width regime.
2. **Scale and magic are currently mutually exclusive.** The analytic certificate works at n = 96 for Clifford bodies, but at n = 32 the certified separation is already negative at t = 2. A test that is both wide and magic-rich needs a tighter upper bound first, which is v1.5 §9's priority.
3. **The T-count contrast has an unexplained systematic** (§3.6), which limits how much weight κ_stab can carry until it is fixed.

### 3.8 Withdrawn: narrowing the branch to one host class

Session 007 concluded that, because adding magic collapses the certified separation, the branch should narrow to a single host class. **That recommendation is withdrawn.** Session 009 showed that the stabilizer-decomposition host class is testable through a contrast that does not need wide magic circuits: it holds χ fixed and varies T-count inside the existing family. Both host classes stay open, and both are now bounded (§3.6).

The rest of Session 007 stands: its certificates, its validation and the n = 96 protocol are unaffected.

* * *

## 4. Standing evidence against

**v1.3 had no such section, and it should have.** The programme documented supporting leads without recording the strongest published constraint on its own best-specified architecture.

The clearest architecture — a discrete spacetime lattice (Beane, Davoudi & Savage, [arXiv:1210.1847](https://arxiv.org/abs/1210.1847)) — makes a concrete prediction: energy-dependent photon propagation. Two independent programmes have looked.

| Test | Observable | Result |
| --- | --- | --- |
| Gamma-ray burst dispersion | Energy-dependent arrival times over cosmological baselines | **Null.** Constrained beyond the Planck scale |
| GRB polarization | Birefringence from Lorentz violation | **Null** |
| UHECR spectrum & anisotropy | Directional structure correlated with a fixed spatial axis | **Null.** GZK suppression as expected |

Under the programme's own rules this is evidence against the lattice architecture and must be recorded as such. It does not constrain non-lattice implementations.

The κ bounds of §3.6 are a second kind of constraint: laboratory-style nulls, so far in simulation only, on the two approximation-first host classes. They narrow how strongly such a host could respond to cost. They do not exclude it.

* * *

## 5. Architectures

A specification is only research-grade if it predicts something that could come out either way. Four are specified.

### RCC-1a — approximation-first classical host

The host classically approximates the internal quantum dynamics, and on budget exhaustion it **truncates** rather than pausing — the only budget policy that writes anything into the record.

**Predicts:** excess output error above a certified cost threshold. **Testable:** yes, in the lab, via RCC-A2. **Status:** live, with two host classes separately bounded — tensor-network contraction (κ_MPS ≤ 0.0019 per doubling of bond dimension) and stabilizer decomposition (κ_stab ≤ 0.00082 per doubling of stabilizer extent, with an uncorrected systematic and a 3.80% false-positive rate). **Identifiability condition:** only targets hosts using methods in the frozen toolbox — untestable from inside, and a permanent limitation.

### RCC-1b — pseudorandom-outcome host

A deterministic host must generate every quantum measurement outcome in the universe. True randomness is unavailable to it, so outcomes come from a pseudorandom generator with finite state *s*.

**Predicts:** measurement-outcome sequences have period ≤ 2^s and admit a polynomial-time distinguisher given sufficient output. **Testable:** yes, and cheaply — quantum random number generators are already subjected to large statistical batteries (NIST SP 800-22, TestU01, Dieharder) on enormous output volumes, with no structure detected.

**Why it bites.** A cryptographically secure PRNG is by construction indistinguishable in polynomial time, so this test cannot exclude an unlimited host. But the programme's premise is a **resource-constrained** host, and a constrained host cannot afford cryptographic-grade generation for every measurement outcome in a universe. The test therefore bites on the hypothesis the programme actually holds.

**Prior art:** the question predates the simulation framing — see [quant-ph/0006079](https://arxiv.org/html/quant-ph/0006079) on whether quantum randomness arises from deterministic chaos.

**Status:** live, and the existing null results are already a constraint the programme should quantify.

### RCC-1c — discrete lattice

**Predicts:** photon dispersion, cosmic-ray anisotropy. **Status:** being closed off — see Section 4.

### RCC-0 — faithful implementation

**Predicts:** nothing. **Status:** permanently untestable, by result 1 of Section 1. Retained only to mark the boundary.

* * *

## 6. What the programme should not do

- No additive evidence scores pooling unrelated streams. In particular, κ_MPS and κ_stab are never combined into one figure.
- Update only the hypothesis actually tested. Each κ bound constrains one host class under one algorithm class.
- Separate the five explanations — documented human development, lost technique, precursor civilisation, nonhuman agents within the world, operator intervention. Evidence for one is not evidence for the others.
- A model that can always relocate its threshold beyond the next experiment is unconstrained, not surviving.
- Discovery-tool record counts are not evidence counts.
- Do not describe results about the instrument, or null bounds, as evidence for simulation.

* * *

## 7. The evidence test

A finding counts as evidence for external implementation only if it clears all five:

1. **Large, not marginal** — a factor-scale effect, not a 3σ excess in a favourable bin.
2. **Confirmed in an independent channel** — a different observable, not more of the same data.
3. **Growing with better data**, not blurring.
4. **Found incidentally**, by researchers measuring something else.
5. **Matching a specified architecture's signature** — right sign, scale and threshold behaviour — rather than merely being unexplained.

Criteria 1–4 are what separated the anomalies that survived from those that died. Criterion 5 is what separates "our model is incomplete" from "this is an implementation artifact." **Nothing currently known clears even the first four.**

* * *

## 8. Base rate

From the anomaly ledger: of the headline cosmological anomalies of the last fifteen years, most shrank or died. Two mechanisms account for nearly all of it — the theoretical prediction moved, or the search space was larger than reported. BICEP2 was announced at roughly 7σ and was galactic dust. The claimed CMB Hawking points fell from 99.98% to about 87% once 52 tested annulus configurations were properly marginalised — the same look-elsewhere error Session 004 quantified.

The counterweight matters equally. The solar neutrino deficit was attributed to a faulty solar model for **thirty-three years** and turned out to be real physics; SNO settled it in 2001 by measuring total flux alongside electron-flavour flux. Accelerating expansion was found by two rival teams who set out to measure deceleration. Reflexive dismissal is the same error as credulity with the sign reversed.

* * *

## 9. Open problems, ranked

1. **Tighten the upper bound** Σⱼ2^(Eⱼ). Session 008 shows this is 81% of the certification problem. It is unchanged from v1.5 and remains the top technical item; wide, magic-rich tests wait on it.
2. **Fix the Session 009 systematic.** Match dp/dp₁ alongside p_ideal, preregistered, then re-run the T-count contrast. Until then κ_stab carries its caveat.
3. **Quantify the RCC-1b constraint.** Convert existing QRNG test volumes into a lower bound on host generator state size.
4. **Resolve the identifiability condition** for RCC-1a, or state permanently that it cannot be resolved from inside.
5. **Hardware.** Everything in Sessions 004–009 is simulation only.
6. **Non-quantum tracks** — the bounded transmission analogue, and one properly bounded archaeological or geological case study.

* * *

## 10. Literature for the quantum branch

Background literature on stabilizer entropies, stabilizer rank and extent, and classical simulation of Clifford+T circuits. Two entries are preprints and are marked. No load-bearing claim in this document rests on a preprint: the Session 005 construction, the Session 007 certificates and the κ bounds are each verified directly in their code bundles.

| Source | Identifier | Status |
| --- | --- | --- |
| Haug, Piroli — *Stabilizer entropies and nonstabilizerness monotones* | Quantum 7, 1092 (2023); [doi:10.22331/q-2023-08-28-1092](https://doi.org/10.22331/q-2023-08-28-1092) | Published |
| Leone, Bittel — *Stabilizer entropies are monotones for magic-state resource theory* | Phys. Rev. A 110, L040403 (2024); [doi:10.1103/PhysRevA.110.L040403](https://doi.org/10.1103/PhysRevA.110.L040403) | Published |
| Leone, Oliviero, Hamma — *Stabilizer Rényi entropy* | Phys. Rev. Lett. 128, 050402 (2022); [doi:10.1103/PhysRevLett.128.050402](https://doi.org/10.1103/PhysRevLett.128.050402) | Published |
| Bravyi, Browne, Calpin, Campbell, Gosset, Howard — *Simulation of quantum circuits by low-rank stabilizer decompositions* | Quantum 3, 181 (2019); [doi:10.22331/q-2019-09-02-181](https://doi.org/10.22331/q-2019-09-02-181) | Published |
| Heimendahl, Montealegre-Mora, Vallentin, Gross — *Stabilizer extent is not multiplicative* | Quantum 5, 400 (2021); [doi:10.22331/q-2021-02-24-400](https://doi.org/10.22331/q-2021-02-24-400) | Published |
| Peleg, Shpilka, Volk — *Lower bounds on stabilizer rank* | Quantum 6, 652 (2022); [doi:10.22331/q-2022-02-15-652](https://doi.org/10.22331/q-2022-02-15-652) | Published |
| Kalra, Sinha — *Stabilizer ranks, Barnes Wall lattices and magic monotones* | Quantum 10, 2179 (2026); [doi:10.22331/q-2026-07-29-2179](https://doi.org/10.22331/q-2026-07-29-2179) | Published |
| Zhang, Zhang — *Classical simulability of quantum circuits with shallow magic depth* | PRX Quantum 6, 010337 (2025); [doi:10.1103/PRXQuantum.6.010337](https://doi.org/10.1103/PRXQuantum.6.010337) | Published |
| Liu, Clark — *Classical simulability of Clifford+T circuits with Clifford-augmented matrix product states* | Phys. Rev. Research 8, 023116 (2026); [doi:10.1103/ybnf-rjw8](https://doi.org/10.1103/ybnf-rjw8) | Published |
| Masot-Llima, Garcia-Saez — *Stabilizer Tensor Networks: Universal Quantum Simulator on a Basis of Stabilizer States* | Phys. Rev. Lett. 133, 230601 (2024); [doi:10.1103/PhysRevLett.133.230601](https://doi.org/10.1103/PhysRevLett.133.230601) | Published |
| Angrisani, Schmidhuber, Rudolph, Cerezo, Holmes, Huang — *Classically estimating observables of noiseless quantum circuits* | Phys. Rev. Lett. 135, 170602 (2025); [doi:10.1103/lh6x-7rc3](https://doi.org/10.1103/lh6x-7rc3) | Published |
| Szombathy, Valli, Moca, Farkas, Zaránd — *Asymptotically independent fluctuations of stabilizer Rényi entropy and entanglement in random unitary circuits* | Phys. Rev. Research 7, 043072 (2025); [doi:10.1103/jplh-zl35](https://doi.org/10.1103/jplh-zl35) | Published |
| Aziz, Pan, Gullans, Pixley — *Classical simulations of low magic quantum dynamics* | [arXiv:2508.20252](https://arxiv.org/abs/2508.20252) | **Preprint** |
| Deger, Koutsioumpas, Webster, Sayginel, Roffe, Browne — *Efficiently simulable quantum circuits with large entanglement, magic, and non-Gaussianity via code-compiled tensor networks* | [arXiv:2607.08396](https://arxiv.org/abs/2607.08396) | **Preprint** |

Publication status was re-checked on 15 September 2026 against arXiv and Crossref. No retractions, corrections, errata or expressions of concern were found. That is a null search result, not a clearance.

* * *

## 11. Provenance

The v1.3 and v1.5 foundations and their downloadable files are preserved unchanged. This document supersedes v1.5 as the current position.

Every figure in Section 3 comes from a reproducible, pure-numpy code bundle with captured output and a verification script: Session 005 (seed 20260908), Session 006 and Loop Record 001 (seeds 20260910 and 20260911), Session 007 (seed 20260911), Session 008, and Session 009 (seed 20260912). Session 008's validity checks: 45 circuits, zero cases where a lower bound exceeded the exact rank or an upper bound fell below it.

These are AI-assisted research outputs produced in sessions with the owner. They have not been independently peer reviewed.

Nothing in this document establishes external implementation, nested ancestry, or creator motives.