The gate was written before the run.
Postulate 3 — the Withdrawal — says a system cannot hold new entanglement with a distant partner without shedding its entanglement with the vacuum around it. This is the axiom the entire theory stands on: Folio II's derivation of the field equations begins from it. House rules require the falsification to be written before the narrative. It was. Verbatim, from the script header, committed before execution:
If, as A:B entanglement rises, A's correlation with the intervening vacuum region M does not fall, the withdrawal picture of Postulate 3 is falsified in its simplest arena and the theory is dead.
The import.
The arena: the ground state of a harmonic chain — 240 sites, periodic, IR mass m = 10⁻³ — the standard lattice vacuum, handled exactly via the covariance methods of Peschel (2003). Plain numpy; every number below reproduces in seconds on a laptop.
A naive test would squeeze two chain sites into entanglement — but an active squeeze mints currency by injecting energy, and proves nothing about budgets. So the protocol imports instead:
Prepare imported entanglement, off the books.
An external pair of ancilla modes in a two-mode-squeezed state (squeezing s = 1.5): 5.23 nats of mutual information, initially uncorrelated with the chain.
Couple it in through a tunable, passive door.
Ancilla α meets chain site a = 60, ancilla β meets site b = 180 (120 sites apart), each through a beam splitter of transmissivity τ. Beam splitters are passive — they cannot create entanglement, only relocate it. At τ = 0 nothing happens; at τ = 1 the imported pair fully replaces the local vacuum modes.
Trace out the ancillas and read every account.
As τ runs 0 → 1: the imported matter–matter account I(A:B) and its genuine entanglement EN(A:B); A's account with the intervening vacuum region M (sites 110–130); A's total budget 2S(A); the energy bill; and M's own entropy as a tamper seal.
The account drains to exactly zero.
| τ | I(A:B) | EN(A:B) | I(A:M) | S(A) | S(M) |
|---|---|---|---|---|---|
| 0.000 | 0.332 | 0.000 | 0.491 | 1.296 | 2.22242 |
| 0.146 | 0.611 | 0.000 | 0.317 | 1.707 | 2.22242 |
| 0.371 | 1.158 | 0.000 | 0.179 | 2.063 | 2.22242 |
| 0.500 | 1.497 | 0.193 | 0.126 | 2.211 | 2.22242 |
| 0.750 | 2.336 | 0.826 | 0.052 | 2.437 | 2.22242 |
| 0.933 | 3.574 | 1.869 | 0.012 | 2.570 | 2.22242 |
| 1.000 | 5.229 | 3.000 | 0.000 | 2.615 | 2.22242 |
Three facts, monotone across thirteen sampled couplings:
The import succeeds. I(A:B) rises 0.33 → 5.23; genuine entanglement EN switches on near τ ≈ ½ and reaches 3.00 — two distant sites of the chain end up holding a hot, pure entangled pair.
The vacuum account drains — monotonically, to exactly zero. I(A:M) falls 0.491 → 0.000, every step downward, hitting zero precisely when the import completes. To host the imported account, A liquidated its vacuum holdings entirely.
Nobody else's books moved. S(M) is constant to machine precision across the entire protocol (drift = 0.0): the operation touches A's relationships, never M's own holdings. Locality of bookkeeping, exhibited.
The vacancy is the law; the tenant is incidental.
A second run with vacuum ancillas (s = 0) — same door, nothing imported. Result: I(A:B) does not rise (it falls to zero — even the vacuum's own A–B correlation is evicted), yet I(A:M) drains identically to zero.
An auditor must read this correctly, so the house says it plainly: the drain is not caused by the arriving entanglement. It is caused by the eviction of the vacuum mode — monogamy demands the vacancy regardless of what moves in. The two arms together give the sharp statement:
Only importing entanglement converts the vacated account into a matter–matter balance.
The kill-gate condition — A:B rising while A:M holds — occurred in neither arm, in no regime, at no coupling. The gate stands passed.
Where every nat sat, at every moment.
The finest measurement of the run: keep the ancillas on the books and split A's entire budget 2S(A) into its pairwise accounts — with B, with the whole vacuum, and with the departing ancillas (the funds in escrow):
| τ | 2S(A) — budget | with B | with vacuum | in escrow | multipartite gap |
|---|---|---|---|---|---|
| 0.000 | 2.593 | 0.332 | 2.261 | 0.000 | 0.000 |
| 0.146 | 3.414 | 0.611 | 0.820 | 0.706 | 1.277 |
| 0.500 | 4.422 | 1.497 | 0.253 | 0.918 | 1.755 |
| 0.854 | 5.030 | 2.883 | 0.052 | 0.595 | 1.499 |
| 1.000 | 5.229 | 5.229 | 0.000 | 0.000 | 0.000 |
Read the first and last rows: the books open balanced and close balanced — at settlement, 5.229 = 5.229 to four decimals, every nat of A's budget sitting with B and not one elsewhere. But read the middle rows too, because they contain the run's one genuinely new lesson:
The run is yours to check.
One file, plain numpy, no other dependencies. The kill-gate is in the header; the three arms, the verdict, and the JSON dumps are in main. Runtime: seconds.
Source: n2_withdrawal.py · results: n2_results.json · portfolio: n2_portfolio.json
- Scope: one vacuum (free boson), one region pair, single-site accounts, Gaussian states throughout. The interacting-theory withdrawal is an open debt.
- The τ = 1 endpoint is guaranteed by construction (a full swap replaces the mode). The evidence lives in the continuous trade-off curve, the control comparison, and the portfolio decomposition — not in the endpoint.
- Mutual information, not distilled entanglement, is the account unit here (EN reported alongside). The distillable version of the same measurement is part of milestone A5.
- What this folio establishes: in the simplest arena that could have killed it, the withdrawal happened — monotone, total, and exactly settled. Postulate 3 survives its first audit, and Folio II's derivation keeps its foundation.