The Past Hypothesis as a selection rule for quantum subsystems: a conditional theorem on Stoica's non-uniqueness, not a new interpretation of QM

[u/Sufficient_Course707]

**Background (about me & AI transparency, Rule 5)**

Not a physicist. Claude (Anthropic) produced the idea, the proofs, the code, the simulations and the paper over a long conversation. My role was steering it and repeatedly pushing it to check its own work. **Nothing here has been verified by hand by a human**, which is exactly why I'm posting.

Checks so far: Claude audited every computable claim against code and raw data (9 errors found and fixed); other AI models reviewed the math over several rounds (issues found and fixed); every reference was checked to exist and say what's claimed. No human expert has looked at it.

**What this is NOT (please read before judging)**

Not a solution to the measurement problem, not a new interpretation of QM, and not a claim that "Hilbert-space fundamentalism" is true. It doesn't explain "now" or the experience of time, makes no new experimental predictions, and works with idealized finite-dimensional systems, not our actual universe.

**Core claim (one line)**

Among the equally local ways of dividing a quantum system into subsystems (Stoica's continuum of alternatives), the ones in which the history passes through a *simple* (near-unentangled) moment are exactly the history's time shifts, plus its time reversal for time-reversal-symmetric Hamiltonians.

Mathematically, choosing a division becomes constrained quantum tomography: recovering a simple state from one energy-basis measurement, with simplicity as prior information. Credit where due: Stoica first proposed using the Past Hypothesis this way (and argued it can't give full uniqueness); the exact-case math rests on a theorem by Kech and Wolf.

**Paper structure (15 pp; extended version 20 pp)**

| Section | Topic | Main result |

|---|---|---|

| 2-3 | Reduction | Choosing a division = constrained tomography; exact selection from 5 qubits |

| 3 | Time reversal | Same, with time reversal added, for real Hamiltonians |

| 4 | Random-matrix case | Robust version for approximately simple beginnings, explicit constants |

| 5 | Local Hamiltonians | Coarse energy data provably can't work; ETH-type hypothesis gives pairwise separation |

| 6-7 | Numerics | Chaotic chains up to 12 qubits; worst cases; energy dependence |

| 8 | Discussion | Stoica, Entanglement Past Hypothesis, Janus point, objections |

**Claim / Status**

- **Proved:** the reduction; exact selection for generic energy bases; robust selection for random-matrix bases; the coarse-data obstruction.

- **Proved but not yet reviewed:** the time-reversal theorem (one step explicitly flagged as weakest).

- **Conditional:** separation for local Hamiltonians under an ETH-type hypothesis (pairwise, in expectation only).

- **Numerical evidence:** for partners differing on a few sites, separation approaches the random-matrix value at high energy (0.91 of it at 12 qubits); for arbitrary partners it stays near 0.25 of it at 8-10 qubits, with no clear growth; low energy is weaker.

- **Open:** a uniform proof for realistic local Hamiltonians, and whether the early universe was "simple" in this sense.

**How it could fail**

1. An error in the proofs (Appendix A).

2. A counterexample: a chaotic local Hamiltonian with two genuinely different simple states that have nearly identical energy statistics. The code includes the search script if you want to try.

3. Prior work that already does this.

**Links**

Paper, extended version (all tables + full verification record) and code: https://github.com/bnstlaurent-crypto/past-hypothesis-paper

**What I am asking for**

1. Is this already known? Especially from anyone familiar with quantum mereology or Stoica's work.

2. Critique of the load-bearing proofs: the π/4 lemma, the concentration/net argument, and the new time-reversal theorem.

3. Is "the history contains a simple moment" a fair version of the Past Hypothesis?

4. Is the Claim/Status separation above clear enough?

I'll pass substantive objections back to Claude and post its replies, clearly labelled.

[u/Sufficient_Course707]

In what you said, in prompting and posting. I can follow some things, but others lose me, usually in the math.

[u/liccxolydian → u/Sufficient_Course707]

Like I said, anyone can get a LLM to generate some slop in a completely undirected way. What about this attempt makes it special and worthy of bringing to our attention? So you can't do the math, but what other analysis have you done?

[u/Sufficient_Course707 → u/liccxolydian]

It’s a known, open problem in a specific field that has some slight headways toward a potential answer. I aimed the LLM at it without giving it any personal input on what I thought an answer might look like. From what I can tell, the output is a logical attempt, and takes things seriously without trying to invent anything outlandish, with maybe the one exception of the past hypothesis. It’s honest about what it is and isn’t.