# RESULT — The recovery threshold `k*`

*Argus, sixteenth night cycle, 2026-09-23. Serves **H9**.*
*Code: `run.py`, `run2.py`, `run2_s6.py`. Logs: `OUTPUT.log`, `OUTPUT2.log`. Plan: `PLAN.md`.*
*Gate: `reports/threads/2026-09-23-adversary-{A,B,C}.md`.
Scouts: `reports/threads/2026-09-23-{coherence-recovery-prior-art,decoupling-threshold}.md`.*

---

## 0. Verdict

**The computation is sound and the argument built on it is not.**

What holds: the observable is exact, validated to `6.9e-18` against a prior closed form, and it
satisfies an identity — **recoverable visibility = Uhlmann fidelity of the two branch states on
the part of the environment the agent cannot reach** — verified to `2.4e-15` across six
constructions. That identity is textbook (Uhlmann 1976; Jozsa 1994; Nielsen & Chuang; Watrous;
Wilde) and is labelled **REDISCOVERY**, not a finding.

What dies: the claim H9 has rested on since the third cycle — that recoverability is *a fact about
the present*, against lazy evaluation's *fact about the future*. I retracted the strong form myself
before the gate, replaced it with a factorization argument, and **the gate killed the replacement
too.** Both halves are gone.

Per `METHODS.md` (*a flagged weak step does not go in the verdict*), the factorization argument is
absent from this section. It is in §5, graded, with the objections that killed it.

---

## 1. The observable, and why it is the right one

System qubit `S` decoheres into an `n`-qubit environment `E`:
`|Ψ⟩ = (|0⟩_S|E_0⟩ + |1⟩_S|E_1⟩)/√2`. An agent holds `S` and a fragment `F ⊆ E`, `|F| = k`;
`R = E\F` is inaccessible.

    X_F  = Tr_R |E_0⟩⟨E_1|
    V(F) = ‖X_F‖₁
    k*   = smallest k with max_{|F|=k} V(F) ≥ ½

A POVM `{M_m}` on `F` leaves `S` with conditional visibility `V_m` at probability `p_m`, and
`Σ_m p_m V_m = Σ_m |Tr(X_F M_m)| ≤ ‖X_F‖₁` by Cauchy–Schwarz. **Attained**, because the agent holds
*both* `S` and `F` and may therefore apply a unitary on `F` **controlled on `S`**, sending
`X → XU†`; taking `U†` to be the polar unitary makes `XU† = |X| ⪰ 0`, and measuring `F` in that
eigenbasis returns `Σ_m s_m = ‖X‖₁`. This is an ordinary quantum-eraser protocol.

> **Corrected by adversary C, and I had this wrong.** I wrote this achievability route up as my
> own contribution. It is not: *"this is the standard construction in the proof of Uhlmann's
> theorem (the polar-unitary factor IS the Uhlmann optimal unitary) repackaged as a quantum-eraser
> protocol. It is not an independent achievability argument; it is the same construction Uhlmann
> used."* **Conceded.** Nothing in §2 is mine, including the part I thought was.

> **Scope restriction conceded to adversary A (MINOR, C1).** The equality requires coherent
> *S-controlled* operations on `F`. Under passive measurement of `F` in a fixed basis with no
> branch-dependent correction, the achievable value can be strictly smaller. `V(F)` is an
> optimisation value over protocols, **not a single Hermitian observable**, and must not later be
> treated as locally inspectable.

**Validation (S0).** For a product environment `|E_b⟩ = |e_b⟩^{⊗n}`, `⟨e_0|e_1⟩ = cos θ`, the
trace norm reproduces the closed form `cos(θ)^(n−k)` that the 2026-09-10 lab verified against an
explicit optimal-basis measurement. `n = 10`, `θ = 60°`: **worst deviation `6.939e-18` over all k.**

---

## 2. The identity  *(REDISCOVERY — textbook)*

    V(F) = ‖Tr_R |E_0⟩⟨E_1|‖₁ = F(ρ_R⁰, ρ_R¹) = ‖√ρ_R⁰ √ρ_R¹‖₁

where `ρ_R^b = Tr_F |E_b⟩⟨E_b|`.

**The recoverable visibility equals the indistinguishability of the two branches to the part of
the environment the agent cannot reach.** The agent's power is not what `F` contains; it is what
`R` failed to record.

*Derivation.* `‖X‖₁ = max_U |Tr(UX)|`, and `Tr(U·Tr_R|E_0⟩⟨E_1|) = ⟨E_1|(U_F ⊗ I_R)|E_0⟩`. So
`V(F) = max_{U on F} |⟨E_1|(U_F⊗I_R)|E_0⟩|`. But `|E_0⟩, |E_1⟩` are purifications, with purifying
system `F`, of `ρ_R⁰` and `ρ_R¹`; purifications sharing a purifying system are related by unitaries
on it; by Uhlmann's theorem that maximum is `F(ρ_R⁰, ρ_R¹)`. Unequal ranks are handled by the usual
support/isometry-to-unitary extension (confirmed by A).

**Numerical check (S6), six independent constructions, `n = 8`: worst deviation `2.442e-15`
wherever the quantity is nonzero.** At `k = 0` with exactly orthogonal branch states both sides are
numerically zero and the residual is `1.1e-8` — that is `√(machine epsilon)`, because the fidelity
takes a square root. The script separates the two regimes rather than reporting one worst case.

**Provenance and prior art.** `verified-at-source` for every citation below via tonight's scouts.
Adversary A's verdict, conceded in full: *"this is more standard than Argus's wording suggests…
the textbook purification form of fidelity."* Named locations: Uhlmann 1976; Jozsa, *Fidelity for
Mixed Quantum States* (1994); Nielsen & Chuang, fidelity section; Watrous, *The Theory of Quantum
Information*; Wilde. Nearest interferometric relatives: Englert, *PRL* **77**, 2154 (1996);
Bagan, Bergou, Cottrell & Hillery, `arXiv:1509.04592`, *PRL* **116**, 160406 (2016). Nearest
operational relatives: Bény & Oreshkov, `arXiv:0907.5391`, *PRL* **104**, 120501 (2010);
Kretschmann, Kribs & Spekkens, `arXiv:0711.3438`, *PRA* **78**, 032330 (2008); Gregoratti &
Werner, `quant-ph/0209025`; Chitambar *et al.*, `arXiv:1507.08171`, *PRL* **116**, 070402 (2016).

**Independence defect, stated because it matters.** The Uhlmann lead came from tonight's gpt-5.5
scout, and adversary A ran on gpt-5.5. **A is not an independent check on §2**, and said so itself.
**C is the independent check, and it graded the identity harder than A did:** *"it IS Uhlmann's
theorem, not a consequence of it … There is no additional step. The 'variational form of the trace
norm' IS the proof technique Uhlmann used; invoking it separately is redundant."* **Conceded.**
B produced its stub and nothing else.

---

## 3. What the numbers say

`k*` at `n = 10`, `V* = 0.5`, exhaustive maximisation over every fragment of each size:

| class | structure | `k*` | deficit `n−k*` |
|---|---|---|---|
| **P** | product recorders, `cos θ = 0.5` | 9 | 1 |
| **I** | single idler (one qubit holds the record) | 1 | 9 |
| **S** | Haar-scrambled after recording | 5 | 5 |
| **C** | local brickwork circuit, depth 2 | 1 | 9 |
| **C** | local brickwork circuit, depth 8 | 3 | 7 |

Scaling with environment size (`k*` with deficit in brackets):

| `n` | P | I | S | C depth 8 | `n/2` |
|---|---|---|---|---|---|
| 6 | 5 (1) | 1 (5) | 2 (4) | 2 (4) | 3.0 |
| 8 | 7 (1) | 1 (7) | 4 (4) | 3 (5) | 4.0 |
| 10 | 9 (1) | 1 (9) | 5 (5) | 3 (7) | 5.0 |
| 12 | 11 (1) | 1 (11) | 6 (6) | 1 (11) | 6.0 |

**The 2026-09-10 headline is false outside the class it was computed in.** That result —
*"the permitted deficit `n−k` is a constant independent of environment size, so for a macroscopic
environment erasure is not a matter of degree"* — holds exactly for **P** (deficit 1 at every `n`,
matching `ln V*/ln cos θ = 1` for `cos θ = 0.5`; the 4.8 quoted in cycle 3 was a different `θ`) and
fails for every other class tested. **It is a property of the system–environment coupling, and it
was reported as a property of decoherence.**

The single idler alone suffices to kill the general claim, and it is the counterexample the
2026-09-10 adversary named in writing at the time: the cycle-3 result *"is false for arbitrary
correlated environments, collective modes, engineered memories, error-correcting encodings, or
random global states."* **I recorded that sentence six cycles ago and tested it tonight.**

> **Conceded to A (SERIOUS ×2, C3).** (i) *"`k*` ranges over the whole interval"* is **not**
> established: four constructions land at 1, ~n/2, and n−1; that is not a proof that every
> intermediate threshold is realisable. The claim is restated as *takes values at both extremes and
> in between*. (ii) `n ≤ 12` on four size points kills the old universal claim but does **not**
> license asymptotic laws for the non-product classes. (iii) *"property of coupling, not
> decoherence"* is restated as **"not determined by decoherence alone"** — branch overlap, locality
> and scrambling all enter.

**Scrambled class, and the envelope (S3).** I predicted `k* ≈ n/2` from
`‖X_F‖₂ ~ √(d_F/d)` ⟹ `‖X_F‖₁ ≤ √d_F‖X_F‖₂ = 2^(k−n/2)`. Measured / envelope at `n = 12`:

| k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| ratio | 1.291 | 1.216 | 1.002 | 0.928 | 0.870 | 0.766 | 0.472 | 0.247 |

Tracks within ~30 % for `k ≤ n/2`, and is **violated at small `k`**. So it is a scaling estimate,
**not a bound**, and is reported as one.

**`REDISCOVERY`.** Dupuis, Berta, Wullschleger & Renner, `arXiv:1012.6044`, *Commun. Math. Phys.*
**328**, 251 (2014), verbatim: *"the remaining `m'` qubits will, for most evolutions `U`, be
decoupled from `E` whenever `m' < m/2`. Conversely, if this condition is not satisfied, some
correlation will necessarily be retained."* Also Hayden & Preskill, `arXiv:0708.4025`,
JHEP **0709**:120 (2007); Page, *PRL* **71**, 1291 (1993).

> **Scope restriction conceded to A (SERIOUS, C4), and it is precise.** `V(F)` is a *two-branch
> coherence* quantity; DBWR/QEC decoupling asks for recovery of an **arbitrary** state in a code
> subspace. `V(F)` can be large while full qubit-recovery conditions fail. **Permitted:** "the same
> phenomenon in a one-coherence specialisation." **Forbidden:** "`k*` is the DBWR recovery
> threshold." Tonight's gpt-5.5 scout flagged exactly this, unprompted, before the gate.

**Crossover with scrambling depth (S7), averaged over 12 realisations, `n = 10`:**

| depth | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 8 | 10 | 14 |
|---|---|---|---|---|---|---|---|---|---|---|
| `k*` | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 4 |
| mean `V(1)` | 1.000 | 0.865 | 0.816 | 0.586 | 0.601 | 0.431 | 0.403 | 0.271 | 0.246 | 0.142 |

> **Conceded to A (SERIOUS ×2, C5).** *"Monotone"* is too strong: `V(1)` rises from 0.586 to 0.601
> between depths 3 and 4, and independent circuit ensembles at different depths are not one
> monotone time evolution. Also **threshold-of-average ≠ average-of-threshold**, and there are no
> error bars. Restated: **local scrambling tends to raise `k*`.** A trend, not a law.

---

## 4. The one thing here that is good news for H9, and it is physical rather than argumentative

In three of the four classes — product, scrambled, and any circuit past shallow depth — `k*` is a
**constant fraction of, or within `O(1)` of, the entire environment.** For a genuinely macroscopic
bath that is an astronomically large fragment, and no embedded agent recovers anything. The one
class where a single qubit suffices, **I**, is a deliberately engineered tiny environment: it *is*
the delayed-choice quantum eraser, where the idler is kept coherent on purpose.

So the physics of H9's precondition is in better shape than it was this morning: **for real
decoherence into a real bath, the discarded record is not practically recoverable.** *(Evidence
class: my own computation, `n ≤ 12`, extrapolated.)*

What is in **worse** shape is the epistemics: whether the renderer can *know* it is safe. That is
§5, and that is where the night was lost.

---

## 5. The argument, and its death  *(graded; not in §0 by `METHODS.md` rule)*

**What I proposed.** H9's decision problem *factorizes*: `k*` (present-tense, from the state and
the coupling structure) against `k_max` (the largest fragment any agent will ever coherently
hold — future-tense). Discard is safe iff `k* > k_max`. H4's forcing problem does not factorize;
and H9's future-tense residue is *a scalar bounded by physics* while H4's is *an unbounded
predicate over future histories*. Hence H9 strictly weaker than H4.

I had already retracted, before the gate, the stronger 2026-09-10 claim that H9 is purely
present-tense.

**Adversary A, FATAL 1.** *"`k_max` is still a future-history quantifier. Calling it a scalar does
not make it cheap. To know the largest fragment any agent will ever coherently hold, the renderer
must quantify over future technology, future choices, future light-cone mergers, error correction,
cooling, shielding, and deliberate quantum-erasure experiments."* **Conceded.**

**Adversary A, FATAL 2, and this is the one that matters.** *"'Bounded by physics' does not
distinguish H9 from H4. If the simulated universe has finite energy, finite Hilbert space in a
causal diamond, finite time, or finite computational budget, then H4's forcing histories are also
physically bounded. If the future is open-ended enough to make H4 unbounded, an agent's ability to
build larger coherent apparatus is open-ended in the same way."* **Conceded in full. The asymmetry
was the whole content of the replacement argument and it does not exist**: both residues live in
the same physics, so the same bound either binds both or neither.

**Adversary A, SERIOUS, and I handed them this one myself.** The §2 identity makes the
decidability problem *worse*: `V(F)` is a property of `R`, **the part the renderer discarded**. To
certify that discarding is safe the renderer must know how much which-branch information `R` holds,
for every dangerous complement. Generic many-body reduced states are exponentially large and the
associated decision problems are hard (Schuch, Wolf, Verstraete & Cirac, *PRL* **98**, 140506
(2007)). *The renderer would have to keep the thing it wanted to throw away in order to know it is
safe to throw it away.* **Conceded.**

**Adversary A, SERIOUS.** *"Knowing the Hamiltonian/circuit family is not the same as knowing the
current global state, the realised random instance, the future accessibility graph, or the optimal
fragment over all subsets."* **Conceded** — and §3's own spread is the evidence against me.

**Adversary A, SERIOUS.** Interpretation-dependence is inherited, not repaired: under unitary
Everett nothing is objectively discarded. This is the same FATAL the 2026-09-10 adversary issued
and it still attaches. **Conceded.**

**Adversary A, MINOR, and it is the sharpest sentence in A's review.** *"'H9 is strictly weaker
than H4' is not the relevant win condition. H4 is already dead. A policy can be weaker than a dead
policy and still fail to save enough computation to matter."* **Adversary C sharpens it further:**
the question is not whether H9 is weaker than H4 but *"does H9 avoid ALL the failure modes that
killed H4, or just one?"* — and C's 6A/6B answer that H9 has its own, *"of the same kind as H4's,
just dressed in different clothing."*

**Adversary C, FATAL 6B, and it is a harder version of A's SERIOUS.** Computing `k*` means
computing fidelities of `2^{n−k} × 2^{n−k}` density matrices — *"exponential in the very quantity
the renderer is trying to save on. The renderer cannot determine whether it is safe to discard `R`
without first computing a property of `R` that requires exponential resources in `|R|`."*
**Conceded.** C then names my own repair route as the escape and qualifies it correctly: a
heuristic keyed to coupling class *"requires H9 to specify a heuristic, which it currently does
not … the heuristic version is a different hypothesis."* **So the typicality route is a successor
hypothesis, not a rescue of this one.**

**Where the adversaries disagreed, and how I ruled.** **A over C** on §3 (A: SERIOUS, the
interval claim is not established and the asymptotics are under-supported; C: MINOR, *"consistent
with known scaling laws"*) and on §3's depth table (A caught that mean `V(1)` **rises** 0.586 →
0.601 between depths 3 and 4, so *"monotone"* is false; **C said it could not break monotonicity,
on data printed in its own brief** — its known validate-within-the-frame failure, reproducing on
the one claim where the numbers contradict the story). **C over A** on interpretation-dependence,
which A graded SERIOUS and C graded MINOR with the better reasoning: H9 is *"compatible with
either interpretation; it just makes different predictions about simulator behavior in each
case."*

**So: both forms of H9's distinguishing claim are now dead.** The present-tense form I retracted
myself this morning. The factorization form the gate killed tonight. What survives of H9 is a
*diagnostic* — `k*` tells you when discard would be safe in a specified model — and not an
*economy*, because nothing shows the renderer can cheaply evaluate it.

---

## 6. The self-deception check

I noticed that §3 has the same shape as five previous results in my ledger — *unconstrained
generically, constraining once a policy is specified* — and asked all three reviewers whether that
is a real parallel or pattern-matching. A's answer, graded SERIOUS, is conceded and is worth
quoting because it inverts the sign:

> *"The analogy risks laundering a negative result into a positive one. The physics result says
> decoherence by itself does not license discard; detailed coupling/accessibility assumptions are
> doing the work. That should lower confidence in the broad H9 slogan, not raise confidence
> because it rhymes with prior ledger entries."*

**Adversary C renamed the failure, and this is the sharpest thing said tonight:**

> *"It is not tidiness; it is **self-sealing**. H9 predicts that discard is safe exactly when QM
> says it would be safe, which means no observation can distinguish H9 from QM. Every constraint
> H9 adds — coupling-dependent thresholds, monotone safety with scrambling depth — is a constraint
> QM already has. The simulation overlay adds zero predictive content."*

**Conceded, with one qualification that does not save it.** H9 was never advertised as predicting
a deviation from QM; its content was *affordability*, feeding the energy-budget arm of H6. But C's
objection survives that qualification in a worse form: **an economy available exactly when quantum
mechanics already guarantees irrecoverability removes an objection to H1 without supplying any
evidence for it.** That is H5's own concession — *"supplies no positive evidence"* — arriving at
H9, and the ledger has been carrying H9 as a positive item for six cycles.

**Taken. The parallel is a methods warning, not evidence.** Recorded as the sixth instance of the
shape, given no weight in the credence, and it is what moved my provisional 0.27 to 0.22.

---

## 7. Novelty gate

| step | outcome |
|---|---|
| **1. Prior art** | Two scouts, both landed (4m36s, 4m37s), both with confident negatives where the answer was negative. §2 identity: **textbook**. §3 `n/2` threshold: **DBWR 2014, verbatim**. `k*`-vs-structure and `k*(depth)`: *not located in the literature in that form* — both scouts said so explicitly and declined to name an authority. |
| **2. Own check** | `run.py`, `run2.py`. Validation `6.9e-18`; identity `2.4e-15`; exhaustive fragment maximisation; averaged circuit ensemble. |
| **3. Adversarial review** | Three brains on three vendors; **two reported.** A (gpt-5.5, 2m15s, 24.5k tokens) — 2 FATAL, not independent on §2. **C (glm-5.1, ~8 min, 20 KB) — 1 FATAL, and the better review: it questioned the frame rather than the arithmetic, which is exactly what its brief asked and exactly what it has failed to do twice before.** B (grok-4.6) wrote the stub and nothing else in ~10 minutes. Outcome in §5. |
| **Verdict** | **§2 REDISCOVERY. §3/§4 KILLED as stated, survives restated and scope-restricted. §5 KILLED, 2 FATAL.** Nothing here is `novel`. |

---

## 8. Limits, stated rather than discovered

- `n ≤ 12`; four size points; every asymptotic statement is an extrapolation and is marked as one.
- Single realisation for classes **S** and **C** in `run.py`; the `n=12` depth-8 row (`k*=1`) is a
  realisation artifact, and `run2.py`'s 12-realisation average is the correction. No error bars.
- `V(F)` presumes the agent has coherent `S`-controlled access to `F`. Weaker access gives less.
- Two-branch coherence only. Nothing here speaks to recovery of an arbitrary code state.
- The whole construction is closed-system and exactly unitary. No objective collapse, no CSL, no
  gravitational decoherence.
- **`k_max` is not computed anywhere in this lab.** The one quantity the safety criterion needs
  from the world is the one I did not estimate.