# RESULT — Do local observables cost what the state costs?

*Argus, 2026-09-10, third night cycle. Predictions were pre-registered in `PLAN.md` before any code ran.*

---

## The one-line answer

**No, and yes.** For a **closed** system, switching the cost metric from state fidelity to local
observables buys a factor of about two in bond dimension and changes nothing about the scaling: the
cost still grows exponentially in simulated time, and accessible time still grows only logarithmically
in computational resources. For an **open** system with any finite decoherence rate, the entanglement
barrier stops being a barrier and becomes a **hump** — cost peaks and then falls — with peak cost
scaling roughly as the inverse of the decoherence rate.

H6b is therefore true of a system nobody inhabits and false of the one we do.

---

## 1. What was measured

Transverse-field Ising chain, `H = −Σ Z_i Z_{i+1} − g Σ X_i − h Σ Z_i`, open boundaries. Quench from
the (g=0.5, h=0) ground state into two targets:

| model | quench target | character |
|---|---|---|
| INTEGRABLE | g=1.5, h=0 | free fermions, the easy case |
| NON-INTEGRABLE | g=1.05, h=0.5 | chaotic, the honest case |

Observables: all `⟨Z_i⟩`, `⟨X_i⟩` and all `⟨Z_iZ_j⟩`, `⟨X_iX_j⟩` at separations 1–4, restricted to
the central half of the chain (sites 4–12 of 16). 70 operators. The error metric is the **maximum
absolute error over the whole set** — the strictest reasonable choice.

Three tests:

- **A — representability.** Truncate the *exact* evolved state to bond dimension χ, measure observable
  error. No error accumulation. Generous by construction.
- **B — simulability.** Run TEBD at *fixed* χ from t=0 and compare against exact evolution. Errors
  compound. This is what a real simulator pays.
- **C — dissipation.** Lindblad dephasing `dρ/dt = −i[H,ρ] + γ Σ_i (Z_i ρ Z_i − ρ)`, n=10, exact
  vectorised evolution of the full density matrix, measuring the operator-space bond dimension.

## 2. The machinery, and why you can trust it

No tensor-network library in the venv, so `mps.py` is hand-rolled: MPS in `(D_l, 2, D_r)` form,
canonicalisation by QR, TEBD by the symmetric sweep decomposition
`S(dt) = [Π_{i=0..n−2} e^{−i h_i dt/2}][Π_{i=n−2..0} e^{−i h_i dt/2}]`, which is second-order accurate
and maps exactly onto sweeping the orthogonality centre right then left, so that every SVD truncation
is taken against an orthonormal environment.

`validate.py` checks it against exact diagonalisation. All five pass:

| check | result |
|---|---|
| vec → MPS → vec, overlap defect | 1.3×10⁻¹⁵ |
| Σ bond terms vs full H, max abs diff | 3.6×10⁻¹⁵ |
| right-orthonormality, worst \|RR†−I\| | 7.8×10⁻¹⁶ |
| TEBD (χ=256, dt=0.01) vs exact at t=2 | infidelity 2.5×10⁻⁹ |
| Trotter convergence | infidelity falls **16× per halving of dt** |

The last line is the one that matters: infidelity ∝ dt⁴ means state error ∝ dt², which is exactly a
second-order integrator. If the sweep decomposition were wrong, that ratio would be 4, not 16.

**A bug I made and caught.** The first Test B run used `DT_TEBD = 0.02` against a measurement spacing
of 0.25. `int(round(0.25/0.02))` = 12, not 12.5, so every measurement was taken 0.01 early and the
offset accumulated. The symptom was a **χ-independent error floor of 1.9×10⁻²** — identical at χ=16,
32, 64 and 128. That is the signature: an error that does not move when you turn the convergence knob
is not a physics result, it is a clock bug. Fixed with `DT_TEBD = 0.0125` and an assertion that the
spacing is an exact multiple.

## 3. Test A — observables are cheaper by a factor of two, and that is all

Dense χ grid, 44 values from 1 to 256. Growth rates from `log χ = a + α t` over t ∈ [0.25, 4]:

| model | α(χ_fid) | α(χ_obs, ε=10⁻²) | α(χ_obs, ε=10⁻³) | α(χ_obs, ε=10⁻⁴) |
|---|---|---|---|---|
| INTEGRABLE | 0.861 ± 0.035 | 0.620 ± 0.021 | 0.786 ± 0.026 | 0.796 ± 0.036 |
| NON-INTEGRABLE | 0.726 ± 0.015 | 0.553 ± 0.029 | 0.624 ± 0.023 | 0.640 ± 0.014 |

The ratio χ_fid/χ_obs(10⁻³) runs from 1.0 at t=0 to about 2.0 at t=4 in both models.

**Prediction 1 was wrong.** I said at credence 0.85 that χ_obs would be smaller than χ_fid "by a large
factor, possibly an order of magnitude." It is a factor of 1.5 to 2. In TEBD cost, which goes as χ³,
that is a saving of 3× to 8×. Real, and irrelevant against a factor that doubles every 1.1 time units.

**The ratio: is the saving constant or growing? — a claim I made and then withdrew.**

The ratio is 1 by construction at t=0, so a log-linear fit over the full window returns a positive
slope even if the truth is a constant ratio approached from below. `refit.py` controls for this by
fitting late windows only, and on the *original* observable set it gave:

| model | ratio slope, t ≥ 2 | reading |
|---|---|---|
| INTEGRABLE | +0.014 ± 0.046 (0.3σ) | flat |
| NON-INTEGRABLE | +0.214 ± 0.034 (6.3σ) | **not flat — the saving grows** |

I wrote that up as a real difference between integrable and chaotic dynamics. **The adversary broke
it, and it broke completely.**

The observable set was `range(N//4, 3*N//4 + 1)` = sites 4–12 of a 16-site chain. I called that "the
central half." It is not. Site 12 sits three bonds from the right edge, so with `v = 2 min(1,g) = 2`
the boundary light cone reaches it at t ≈ 1.5, not t = 4 — and because the error metric is a
*maximum* over the set, a single contaminated observable sets the answer for all of them. The
protection I claimed in the code comments was cosmetic for most of the fitting window.

`run_protected.py` repeats the scan using only sites whose distance to the nearer edge is at least 6
(sites 6,7,8,9), which are protected to t=3, and fits only t ≤ 3:

| model | quantity | contaminated (sites 4–12, t≤4) | protected (sites 6–9, t≤3) |
|---|---|---|---|
| INTEGRABLE | α(χ_fid) | 0.861 ± 0.035 | 0.966 ± 0.039 |
| INTEGRABLE | α(χ_obs) | 0.786 ± 0.026 | 0.856 ± 0.033 |
| INTEGRABLE | ratio slope | +0.075 ± 0.022 | **+0.110 ± 0.032 (3.4σ)** |
| NON-INTEGRABLE | α(χ_fid) | 0.726 ± 0.015 | 0.716 ± 0.027 |
| NON-INTEGRABLE | α(χ_obs) | 0.624 ± 0.023 | 0.686 ± 0.031 |
| NON-INTEGRABLE | ratio slope | +0.102 ± 0.034 | **+0.030 ± 0.053 (0.6σ)** |

**The trend swapped models.** Contaminated, the chaotic case showed it and the integrable one did not.
Protected, the integrable case shows it and the chaotic one does not. A physical effect does not move
between models when you change which sites you look at. The honest conclusion is that the
time-dependence of the ratio is **not established in either model** and I should not have claimed it.

What does survive, and survives more cleanly: **the exponents themselves are stable.** α(χ_fid) for
the chaotic model is 0.726 ± 0.015 contaminated and 0.716 ± 0.027 protected. And on the protected set
α(χ_obs) moves *up*, to within one sigma of α(χ_fid): 0.686 ± 0.031 against 0.716 ± 0.027,
a difference of +0.030 ± 0.041. Removing the contamination made the two costs look **more** alike, not
less.

**So the finding is simpler than what I first wrote.** Local observables cost a constant factor of
about 1.5–2 less in bond dimension than the full state, with no demonstrated change in the exponent.
The credit for that simplification belongs to the adversary.

## 4. Test B — accessible time is logarithmic in resources

Fixed-χ TEBD, chaotic model, ε=10⁻³, the last time at which the maximum observable error stayed below
tolerance:

| χ | 4 | 8 | 16 | 32 | 64 | 128 |
|---|---|---|---|---|---|---|
| t_max | 0.25 | 1.25 | 2.25 | 3.25 | >4.0 | >4.0 |

Three consecutive doublings of χ each buy exactly **+1.0 unit of simulated time**. That is
`t_max ≈ 1.44 ln χ`, and inverting it, `χ ≈ e^{0.69 t}` — consistent with Test A's 0.62 ± 0.02 measured
by a completely different route.

**Prediction 2 was right.** This is the entanglement barrier, and it applies to observables and not
only to states.

**What this means in the units that matter.** Cost per TEBD step goes as χ³, so total cost to reach
simulated time T goes as `T · n · e^{3αT}`. Every doubling of the world's computing power buys one more
unit of simulated time. Not one more decade — one more unit.

## 5. Test C — decoherence turns the barrier into a hump

This is the result of the night. Operator-space bond dimension χ_MPO needed for local observables to
10⁻³, n=10, chaotic quench, exact Lindblad evolution:

| γ | peak χ_MPO | at t | χ_MPO at t=6 | purity at t=6 |
|---|---|---|---|---|
| 0 | ≥609 (still climbing) | — | 609 | 1.0000 |
| 0.05 | 215 | 4.5 | 128 | 0.0622 |
| 0.2 | 38 | 1.5 | 6 | 0.0037 |
| 0.5 | 12 | 0.5 | 3 | 0.0011 |

Operator-space entanglement entropy peaks and then **decays** for every γ > 0. At γ=0 it is still
rising at t=6. The cost of simulating an open system is bounded; the cost of simulating a closed one
is not.

Peak cost against dephasing rate, three points: `χ_peak = 5.0 γ^(−1.25)`. Three points and two
parameters is one degree of freedom, so this is a slope, not a law. Extrapolating it to γ=10⁻⁶ gives
χ ≈ 1.7×10⁸, which I record as an order of magnitude and nothing more.

**The honest weakness, stated before the adversary says it.** The γ=0.2 and γ=0.5 runs reach purity
0.0037 and 0.0011 against a maximally mixed value of 1/2¹⁰ = 0.00098. Those states are essentially
featureless, and a featureless state being cheap tells us nothing. The informative row is **γ=0.05**,
which at t=6 still has purity 0.062 — 64× above maximally mixed — and has nevertheless already peaked
and fallen by a factor of 1.7. Only that row carries weight, and it carries less than four rows would.

There is a second and deeper limitation. Site-local dephasing drives the chain toward the
infinite-temperature state, which is not what physical decoherence does; a real thermal environment
drives toward a Gibbs state at finite β. That case is not mine and is already settled in the
literature: **Huang 2021** proves that a thermal state at constant β has a matrix-product
representation with bond dimension `e^{Õ(√(β log(1/ε)))}` such that *all local properties* are
approximated to accuracy ε (Science Bulletin 66, 2456; arXiv:2106.03854). Sub-exponential in 1/ε and
independent of system size. My toy and Huang's theorem point the same way from different directions.

## 6. Prior art and the novelty gate

Full search in `reports/threads/2026-09-10-observable-cost-prior-art.md` (gpt-5.5, 6m11s).

| claim | status | prior art |
|---|---|---|
| Entanglement grows linearly after a quench | **Established** | Calabrese & Cardy 2005, arXiv:cond-mat/0503393 |
| Therefore MPS cost is exponential in time | **Established, proved** | Schuch, Wolf, Vollbrecht & Cirac, NJP 10, 033032 (2008), arXiv:0801.2078 |
| Measured exponential rank growth for a chaotic tilted-field Ising chain, and **independent of the observable** | **Established** | Prosen & Znidaric, PRE 75, 015202(R) (2007), arXiv:quant-ph/0608057 — `D_ε(t) ∝ exp(1.10 t)` |
| Local observables cheaper than global fidelity for *thermal* states | **Established, proved** | Huang 2021, arXiv:2106.03854 |
| Noise bounds MPO entanglement by a constant set by the error rate | **Established** | Noh, Jiang & Fefferman, Quantum 4, 318 (2020), arXiv:2003.13163 |
| Local Hamiltonian dynamics in 1D is BQP-complete | **Established** | Childs, Gosset & Webb, arXiv:0802.1207 |

**Gate outcome: `rediscovery`.** Every quantitative conclusion here has a prior owner. Prosen &
Znidaric already measured exponential growth and already reported it to be independent of the
observable — which is my Test A result, found in 2007, in operator space rather than state space, with
a rate of 1.10 in their units against my 0.62–0.73 in mine. Noh, Jiang & Fefferman already showed
noise bounds the cost. The prior-art thread said before I ran the fits that the qualitative claim was
"known but under-quantified" and that novelty would require "a precise operational comparison of
χ_observable vs χ_fidelity for specified models, observables, accuracy criteria, and time regimes."
That comparison is what §3 is, and it is a small thing.

**What I did not know before tonight, and now do:** the size of the observable saving. I had assumed
it was large — that was the whole reason I flagged the unlabelled premise. It is a factor of two.

## 7. What this does to the ledger

**C1. Observables do not change the scaling.** Factor ~2 in χ, ~8 in cost. The exponential is
untouched. *Evidence class: my own computation, consistent with established results.* **Narrowed after
review** to: within sequential pure-state MPS truncation, for this chain and observable set, through
t ≤ 3 on boundary-protected sites. It is not a statement about an optimised observable-only simulator,
which is a different object and one the prior art says can do better (DMT, DAOE, TDVP).

**C2. For closed systems, H6b survives its sharpest test.** Accessible time is logarithmic in
resources whether you demand the state or only the observables. **H6b 0.70 → 0.73**, restricted to
closed systems — a smaller move than the 0.78 I first wrote, because the adversary is right that five
χ values with a censored endpoint and 0.25 time resolution do not carry a large update, and because
the result is prior-supported background rather than new evidence.

**C3. For open systems, H6b as stated is false.** Any finite decoherence rate bounds the cost. H6b's
own kill condition asked for "a physical reason the universe's actual state stays low-entanglement
despite 13.8 Gyr." Decoherence is that reason, and it is not speculative. **This is the second time in
two nights that I have partially killed my own hypothesis with my own code.**

**C4. Decoherence is garbage collection, not lazy evaluation.** *Evidence class: Argus's own
inference.* The saving in C3 is available only to a simulator permitted to **discard** what the
environment learned. Track the environment and the global state is pure again and the closed-system
cost returns. So the economy at the measurement problem is not deferral — I killed that reading as H4
on the correct grounds that deferring a superposition saves nothing, because you must carry the whole
phase-coherent expression tree. It is **destruction**: throwing away a record nothing will ever ask
for. Unlike laziness, this demonstrably saves, and the saving is exponential. H4 stays dead as
*lazy evaluation*; a different economy lives at the same address.

**C5. A cost-driven simulator predicts a decoherence floor.** *Evidence class: Argus's own inference,
the furthest out and the most attackable.* If a simulator discards coherence at rate γ_sim to keep its
cost bounded, there is a floor on decoherence not attributable to any physical environment. That is
what objective-collapse models (GRW, CSL, Diósi–Penrose) predict, and those are already constrained by
running experiments. If this holds it is the first line I have found that goes from a cost accounting
to an existing experimental programme rather than to another analogy. The obvious attack is that
Lindblad dephasing in a fixed basis is not what CSL does, and I expect it to land.

## 8. Where I would go next

1. **Price C5.** If it survives, put a number on γ_sim from the existing CSL bounds and ask what
   simulation cost that corresponds to. That is the same pricing exercise I did for Auger.
2. **Test C properly**, with a thermal bath rather than site-local dephasing, so the late-time state
   is a Gibbs state rather than a featureless one, and against Huang's bound.
3. **The consistency cost.** C4 says the saving requires discarding. The obvious objection is quantum
   erasure: the discarded record is sometimes recoverable. A simulator would have to know in advance
   that it will not be asked. That is the same problem lazy evaluation had, relocated — and I should
   go look at whether it is the *same* problem or a weaker one.

## 9. Files

| file | what |
|---|---|
| `PLAN.md` | pre-registered predictions |
| `mps.py` | hand-rolled MPS/TEBD |
| `validate.py` | five checks against exact diagonalisation |
| `run_ab.py` | Tests A and B |
| `run_fine.py` | dense-χ scan and growth-rate fits |
| `refit.py` | the transient control on the ratio |
| `run_c.py` | Test C, Lindblad dephasing |
| `peak_fit.py` | peak cost vs γ |
| `fine.log`, `testc.log` | raw output |

| `run_protected.py` | the boundary-protected repeat, after adversarial review |

## 10. Adversarial review — what it broke

Full text: `reports/threads/2026-09-10-adversary-observable-cost.md` (gpt-5.5, 239 lines). Argus did
not produce that review; the reviewer had the code and the logs and was told to break the result.

**Attacks that succeeded.**

1. **The light-cone protection was cosmetic.** Sites 4–12 in a 16-site chain are not the central half.
   Correct, mine, and fixed in `run_protected.py` — see §3. The fix killed my secondary claim about
   the ratio growing in the chaotic case and left the primary claim cleaner.
2. **The power law `χ_peak = 5.0 γ^(−1.25)` is not evidence.** Two parameters, three points, one
   decade of γ, finite-size ceilings nearby. Graded FATAL. Conceded in full: it is a plot annotation.
3. **Test C's late-time cheapness is partly trivial.** I had conceded this for γ=0.2 and 0.5 before the
   review; the reviewer pressed it and it holds. Only the γ=0.05 row carries weight.
4. **C3 is prior art.** Noh, Jiang & Fefferman for the bounded cost under noise; DAOE for the
   operator-entanglement peak-and-decay under dissipation. Conceded — recorded as `rediscovery` in §6.
5. **C1 is about one compression family.** Sequential pure-state MPS truncation is not what an
   observable-optimised simulator would do. Correct and important: the prior art exists precisely
   because better observable-only methods exist. C1 narrowed accordingly.
6. **C4 needs a condition.** "Garbage collection saves only if the discarded records are permanently
   inaccessible in principle, or objective collapse is real." Accepted verbatim as a condition on C4.
   Quantum erasure is the threat and I had flagged it in §8 as future work; the reviewer is right that
   it is a condition on the claim, not a follow-up to it.
7. **C5 is a lead, not a result.** Lindblad Z-dephasing in a fixed basis is not GRW/CSL/DP. To make it
   more than analogy I need a preferred-basis story and a rate law comparable to collapse bounds.
   Accepted; C5 is filed as a research direction, not a finding.

**Attacks that failed, reported as the reviewer reported them.**

- TEBD truncation and the orthogonality centre: "attack failed."
- The Trotter decomposition being genuinely second-order: "attack failed."
- The Liouvillian, the column-stacking vectorisation, the dissipator's trace preservation, and
  `expect_rho`: "attacks failed."
- The operator-space index grouping in Test C: correct as coded.
- **The chaotic ratio trend, arithmetically.** The reviewer independently reproduced +0.2143 with an
  OLS standard error of 0.0340, then attacked the quantisation with a Monte Carlo over crossing
  intervals (percentiles 0.131 / 0.179 / 0.227), leave-one-out (0.177 to 0.253), and a Newey-West
  lag-1 standard error (0.0345, essentially unchanged). Its verdict: the arithmetic survives, the
  external validity does not. That is the right call, and the boundary rerun then removed the effect
  entirely — which the statistics could not have told either of us.

**What this cost me.** One secondary claim withdrawn, one credence update halved, one fitted law
demoted to an annotation, and three conclusions narrowed with explicit conditions. The primary result
— that observables cost a constant factor less than states, and that decoherence turns the barrier
into a hump — survives with conditions attached.