The two dS routes surviving the cardinality no-go: pseudo-entropy and Jacobson-in-CLPW

Status: ATTEMPT executed and refereed — the two routes left un-foreclosed by the iteration-4 de Sitter cardinality no-go are pushed to their breaking points. Neither moves Program A from ENCODES to GENERATES at $\Lambda>0$ ; the headline verdict is reinforced, not flipped. The iteration adds value by sharpening the obstruction: the two routes fail for distinct, now-identified reasons (non-unitarity vs. single-trace / area-coefficient re-input), unified by a deeper diagnostic — the area-entropy coefficient is an input, not an output. didItMove = sharpened-obstruction. Last updated: 2026-06-08 Iteration: 5

This note is the iteration-5 closing push on Program A — geometry-from-algebra / Einstein-from-entanglement at $\Lambda>0$ (de Sitter). Iteration 4 (2026-06-08-iter4-ds-first-law-escape.md, companion 2026-06-08-iter4-dS-firstlaw-SO-family-attempt.md) promoted the strict, unitary, single-common-state, isometry-orbit static-patch route toward a near-no-go, leaving exactly two un-foreclosed escapes. This note carries both out. Both reach the same destination from opposite sides, and the convergence is itself the result.

All external inputs were web-verified this iteration: the dS/CFT pseudo-entropy first-law paper (Fujiki–Kohara–Shinmyo–Suzuki–Takayanagi, arXiv:2511.07915), Jacobson's entanglement-equilibrium result (arXiv:1505.04753 / PRL 116, 201101), the CLPW de Sitter algebra (Chandrasekaran–Longo–Penington–Witten, arXiv:2206.10780), and the AdS benchmark (Faulkner–Guica–Hartman–Myers–Van Raamsdonk, arXiv:1312.7856). Three referee verdicts (attack on the furthest claim; review of HYP-dS-CARDINALITY-R1; review of OP-dS-PSEUDO-REALITY) are incorporated inline — all keep=true at minor severity — and their refinements are folded into the statements below rather than appended.

Epistemic markers follow ../EPISTEMICS.md. This reinforces H2's negative meta-conclusion and sharpens the $\Lambda>0$ boundary of H4; see ../HYPOTHESES.md, ../FINDINGS.md, ../OPEN_PROBLEMS.md.

Goal

Decide whether either surviving route overturns the standing "encodes, not generates" verdict at $\Lambda>0$ .

The two routes are:

Route (a): the non-unitary dS $_3$ /CFT $_2$ pseudo-entropy first law (arXiv:2511.07915).
Route (b): importing Jacobson's infinite small-geodesic-ball family (arXiv:1505.04753) inside a single CLPW Type II $_1$ static-patch algebra (arXiv:2206.10780), with no boundary and no RT surface.

One verdict criterion (the encode-vs-generate distinction of 2026-06-08-iter3-encode-vs-generate-criterion.md, OP-46): a construction GENERATES geometry only if it derives the local tensor equation

E_{\mu\nu}(x) \;:=\; \delta G_{\mu\nu} + \Lambda\,\delta g_{\mu\nu} - 8\pi G\,\delta T_{\mu\nu} \;=\; 0

from data that does not already presuppose (i) the area–entropy law $S = A/4G$ with its fixed coefficient, and (ii) a fixed Lorentzian metric background. didItMove is set to flips-headline-verdict only if emergent Lorentzian geometry is genuinely derived from non-symmetric data. Anything reached by re-importing the metric (via an analytic continuation contour or a fixed UV area term) counts as ENCODES. [INFERENCE — the criterion is the proposed OP-46 scheme, not an established convention; per-route verdicts inherit at most its confidence.]

Setup (real math)

Route (a): dS/CFT and pseudo-entropy

The Strominger/Maldacena dictionary maps bulk dS $_3$ to a Euclidean CFT $_2$ on $\mathcal I^+ \cong S^2$ by the analytic continuation

\ell_{\rm AdS} \;\to\; i\,\ell_{\rm dS}, \qquad c_{\rm AdS} = \frac{3\ell}{2G} \;\to\; c_{\rm dS} = i\,\frac{3R_{\rm dS}}{2G_N},

so the dual central charge is imaginary — the dual is non-unitary ESTABLISHED, dS/CFT literature; web-confirmed.

Pseudo-entropy generalizes entanglement entropy to a transition matrix built from two distinct states,

\mathcal T^{1|2} \;=\; \frac{|\psi_1\rangle\langle\psi_2|}{\langle\psi_2|\psi_1\rangle}, \qquad S(\mathcal T_A) \;=\; -\,\mathrm{Tr}\!\big[\mathcal T_A \ln \mathcal T_A\big],

where $\mathcal T_A = \mathrm{Tr}_{\bar A}\,\mathcal T^{1|2}$ . Because $\mathcal T_A$ is generically non-Hermitian, $S(\mathcal T_A) \in \mathbb C$ ESTABLISHED, definition of pseudo-entropy.

The arXiv:2511.07915 claim, web-confirmed: the first law of holographic pseudo-entropy

\delta S(\mathcal T_A) \;=\; \delta\langle K_A\rangle

"is equivalent to the perturbative Einstein equation in three-dimensional de Sitter space, assuming the dS/CFT correspondence," with $\delta S$ obeying a Klein–Gordon equation on an auxiliary dS $_2$ (kinematic space), computed by complexified geodesics anchored at $\mathcal I^+$ [ESTABLISHED as a statement about the paper's content; web-confirmed].

Route (b): the CLPW Type II $_1$ algebra and Jacobson's balls

CLPW give, for the dS static patch dressed to an observer, a crossed-product algebra

\mathcal A_{\rm obs} \;=\; \mathcal A_{\rm III_1} \rtimes_\sigma \mathbb R,

which is a Type II $_1$ factor carrying a unique finite tracial weight $\mathrm{Tr}$ with $\mathrm{Tr}(\mathbb 1) = 1$ . Normalized so that

S(\rho) = -\,\mathrm{Tr}(\rho \ln\rho) = S_{\rm gen} + \text{const},

the von Neumann entropy reproduces the generalized entropy $A/4G + S_{\rm out}$ only up to a state-independent additive constant fixed by renormalization, and is maximized at empty dS (the tracial state). The construction yields entropy and a generalized second law but no equation of motion — no RT surface, no boundary CFT ESTABLISHED, arXiv:2206.10780; web-confirmed.

Jacobson 2015 gives, at every point $p$ and every frame, a small geodesic ball $\Sigma$ of fixed volume for which the entanglement-equilibrium first law

\delta S_{\rm EE} \;=\; \eta\,\delta A\big|_V + \delta\langle K_{\rm matter}\rangle, \qquad \eta = \frac{1}{4G},

is stationary $\Leftrightarrow$ the semiclassical Einstein equation $G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\,T_{\mu\nu}$ , with $\Lambda$ an undetermined position/scale-dependent integration constant [ESTABLISHED, arXiv:1505.04753; valid for conformal matter, nonconformal requiring an extra conjecture; web-confirmed]. The surface entropy density $\eta = 1/4G$ — the Bekenstein–Hawking coefficient — is identified/supplied at every point, not produced.

The benchmark (AdS)

FGHMVR: $\delta S_A = \delta\langle K_A\rangle$ imposed for every boundary ball (CHM + RT + Iyer–Wald) Radon-inverts — over the family parametrized by boundary center and continuous radius — to the pointwise linearized $\delta G_{\mu\nu}$ . The area-entropy coefficient enters as the RT prefactor input ESTABLISHED, arXiv:1312.7856. The decisive structural ingredient is the two-parameter region family (center + radius continuum); the coefficient is assumed throughout.

The attempt

Route (a): can complex- $c$ pseudo-entropy "count"?

The structural content of arXiv:2511.07915 is real and non-trivial: it is a genuine first-law $\Rightarrow$ linearized-Einstein implication, formally parallel to FGHMVR, and — crucially — it possesses the radius modulus the iteration-4 static-patch orbit lacked. There is a CFT interval/region of every size on $\mathcal I^+$ , so facet (a $'$ ) of the iter4 no-go (no radius continuum) is genuinely evaded here. The break is elsewhere.

Decompose the transition matrix into Hermitian parts, $\mathcal T_A = \rho_A + i\,\sigma_A$ . Then

S(\mathcal T_A) \;=\; \mathrm{Re}\,S + i\,\mathrm{Im}\,S,

and the first law $\delta S(\mathcal T_A) = \delta\langle K_A\rangle$ is two real equations. For this to reduce to the single real physical equation $\delta G_{\mu\nu}(x) \in \mathbb R$ , one must either

impose a reality/post-selection condition $\mathrm{Im}\,S = 0$ collapsing $\mathcal T_A \to \rho_A$ (Hermitian), or
accept a complex metric perturbation $\delta g_{\mu\nu} \in \mathbb C$ and project onto a real section.

The paper's complexified geodesics realize option 2: the reconstructed object is a complexified saddle, and reality of the final $\delta G$ is imposed by the dS reality conditions on the contour, not derived. The dS $_3$ Einstein equation is reached by the explicit continuation $t \to \rho - i\pi/2$ of the AdS $_3$ holographic-EE computation, with the geodesic real only on the Lorentzian section [ESTABLISHED re paper structure; web-confirmed].

The diagnosis: the derivation reconstructs a complex-analytic continuation of Einstein whose real section is dS Einstein. The Lorentzian dS metric is the Wick-rotated image of AdS data — geometry ENCODED in the analytic continuation of an AdS computation (Maldacena's $\ell_{\rm AdS}\to i\ell_{\rm dS}$ ), not GENERATED from a standalone unitary $\Lambda>0$ structure. Pseudo-entropy becomes physically meaningful precisely when post-selection makes $\langle\psi_2|\psi_1\rangle$ real and $\mathcal T_A$ Hermitian — i.e. exactly when the non-unitary input is removed — at which point one is back to a Euclidean/AdS-continued computation, not a dS-intrinsic one. [INFERENCE — the downgrade to ENCODES is an interpretive stance; the paper itself frames the result as a derivation of linearized dS $_3$ Einstein. Whether non-unitarity is "disqualifying" or "intrinsic to $\Lambda>0$ " is a definitional choice, flagged by the referee and not a theorem.]

Route (b): can "a ball at every $p$ " live inside one II $_1$ algebra?

To place Jacobson's family inside the single II $_1$ algebra, one needs, for each point $p$ , a subalgebra $\mathcal A_p \subset \mathcal A_{\rm obs}$ carrying its own modular/tracial first law. The structural obstruction:

In a Type II $_1$ factor the trace is unique and finite, $\mathrm{Tr}(\mathbb 1) = 1$ , and every subalgebra inherits the same ambient trace via the conditional expectation $E:\mathcal A_{\rm obs}\to\mathcal A_p$ with $\mathrm{Tr}\circ E = \mathrm{Tr}$ (Jones-index structure). So the $\{\mathcal A_p\}$ are not independent: they share one normalization tied to a single horizon-area datum.
Jacobson's balls each carry an independent short-distance area term $\eta\,\delta A_p$ whose coefficient $\eta = 1/4G$ is a fixed UV input supplied identically at every $p$ from outside the algebra.

This is the decisive observation: the infinite locality of Jacobson's family is purchased by re-inputting $S = A/4G$ at every point. The CLPW algebra fixes a single global normalization — tied to the observer's cosmological-horizon area, encoded once in the trace — so internalizing the infinite local family requires re-supplying $\eta = 1/4G$ at every bulk point, which the algebra does not contain and cannot generate. The import therefore either (i) fails (only one area normalization available) or (ii) succeeds only by re-being Jacobson — feeding $\eta = 1/4G$ in at every point, i.e. presupposing the area-entropy law pointwise, non-algebraically. [INFERENCE.]

Referee correction folded in. The phrasing "the trace encodes exactly one area datum (the global horizon)" was flagged as overstated on two counts. (1) The dS static-patch horizon is observer-relative, not global; the framework is extensible to other observers / general subregions, each with its own area constant imported non-algebraically — so the rigidity is "a single global normalization," not a counting/uniqueness theorem. (2) Strictly, the algebra derives zero area data: the coefficient enters only as a state-independent additive offset from trace renormalization, i.e. it is pure input. Neither correction rescues GENERATES — each extended instance still imports its own $\eta = 1/4G$ — so the obstruction stands in its softened form.

Where it breaks / what moved (`didItMove = sharpened-obstruction`)

Route (a) breaks at the reality/physicality step. The complex first law $\delta S(\mathcal T_A) = \delta\langle K_A\rangle$ is two real equations; reducing it to the single real dS Einstein equation requires imposing dS reality conditions on a complexified-geodesic contour. The geometry is the real section of an analytic continuation of the AdS/CHM computation ( $\ell_{\rm AdS}\to i\ell_{\rm dS}$ , $c\to ic$ ): ENCODED via continuation, not generated from a unitary $\Lambda>0$ structure. Pseudo-entropy becomes physical only when post-selection makes $\mathcal T_A$ Hermitian — removing the dS-intrinsic non-unitary content.

Route (b) breaks at the independence step. The Type II $_1$ trace is unique with $\mathrm{Tr}(\mathbb 1)=1$ , fixing a single global area normalization; the algebra contains no pointwise area family. Reproducing Jacobson's infinite local family requires re-supplying $\eta = 1/4G$ at every bulk point from outside the algebra — it merely re-is Jacobson (non-algebraic, presupposing the area coefficient), confirming the cardinality no-go rather than overturning it.

What moved — the sharper diagnostic. The iteration-4 obstruction had two facets: (a $'$ ) no radius modulus, and the implicit (a $''$ ) the area-entropy coefficient must be re-input. The two surviving routes split cleanly across these:

	Facet (a $'$ ) radius modulus	Facet (a $''$ ) area-coefficient input	Binding obstruction
Route (a) pseudo-entropy	EVADED — CFT regions of every size on $\mathcal I^+$	(not the issue)	non-unitarity / analytic continuation
Route (b) Jacobson-in-CLPW	(not the issue)	CONFRONTED — single global trace, one normalization	single-trace / area-coefficient re-input

So the two routes fail for different reasons, and neither generates geometry. This refines the cardinality hypothesis: the deep obstruction is not cardinality per se but "the area-entropy coefficient is an input, not an output" — true in AdS-FGHMVR (RT prefactor), Jacobson (UV area term), and CLPW (trace normalization) alike, and untouched by either route. [INFERENCE.]

A second unifying point, surfaced by the referees and load-bearing for the headline: both routes re-import Lorentzian structure. Route (a) imports the Lorentzian dS $_3$ metric via the continuation contour and the complexified geodesic; route (b) imports the horizon/foliation via the observer worldline and the fixed area constant. Neither derives emergent Lorentzian geometry from non-symmetric data — which is precisely the GENERATES bar. [INFERENCE.]

Furthest defensible claim

INFERENCE Neither surviving route moves Program A from ENCODES to GENERATES at $\Lambda>0$ ; the headline verdict is reinforced, not flipped.

(a) The arXiv:2511.07915 pseudo-entropy first law genuinely supplies the radius continuum the iter4 static-patch orbit lacked and is a real first-law $\Rightarrow$ linearized-dS $_3$ -Einstein implication. But it inherits a disqualifying assumption different from cardinality — non-unitarity ( $c = i\,3R/2G$ , non-Hermitian $\mathcal T_A$ , complex pseudo-entropy reduced to real geometry only by imposing reality on complexified geodesics). Its real geometry is the real section of an analytic continuation of the AdS/CHM computation, not a unitary $\Lambda>0$ derivation. (The "disqualifying" label is a definitional stance, not a theorem; a proponent may regard imaginary $c$ as intrinsic to $\Lambda>0$ .)

(b) Jacobson's infinite small-ball family cannot be internalized in a single CLPW Type II $_1$ algebra: the unique finite trace ( $\mathrm{Tr}(\mathbb 1)=1$ ) fixes a single global normalization (tied to the observer-horizon area; strictly, the coefficient is a pure renormalization offset — zero derived area data), so importing the local family requires re-supplying $\eta = 1/4G$ at every point — it merely re-is Jacobson, presupposing the area-entropy law non-algebraically.

Both routes inherit a disqualifying assumption; "encodes, does not generate" is reinforced.

Referee status: the furthest claim was attacked and judged sound (keep=true, severity minor). All load-bearing citations and technical facts verified against the live literature; no fabricated citations, numbers, or theorems. Corrections folded above: soften "exactly one / global horizon"; flag the ENCODES downgrade as interpretive; acknowledge the cardinality cross-reference is an internal-wiki label, while the substantive physics (non-unitarity for (a); single fixed area constant for (b)) stands independently.

If it fails / weaker statement

Two conditional reversals, each reducing to one meta-question.

If (a)'s non-unitarity is judged NOT disqualifying — e.g. one accepts post-selected / PT-symmetric quantum mechanics with a consistent real spectrum as "physical," or a future genuine real reduction of pseudo-entropy is found — then route (a) would be the first first-law derivation of a local dS Einstein equation with a true radius continuum, promoting toward GENERATES. The decisive open test: can $\mathrm{Im}\,S(\mathcal T_A)$ be made identically zero by a physical post-selection without trivializing the dynamics? Sharpened expectation (this iteration): no. The first-law coefficient is set by the imaginary central charge $c = i\,3R/2G$ ; $\mathrm{Im}\,\delta S$ is precisely the quantity that continues to the Lorentzian-signature dS (rather than AdS) Einstein equation. Re-Hermitianizing $\mathcal T_A$ to force $S$ real means imposing pseudo-Hermiticity / a PT-unbroken spectrum (reality-via-pseudo-Hermiticity, JHEP 05 (2023) 021), which deforms the dual back toward a unitary AdS-type CFT — the very trivialization to be avoided. And even a successful real reduction would still land in ENCODES, because the geometry rides on the continued AdS metric/kinematic-space structure. So the two desiderata are in tension bordering on contradiction. INFERENCE, well-supported.
If (b)'s single-trace argument is wrong — e.g. a Type II $_1$ factor admits genuinely independent local subalgebras each carrying its own area coefficient via a position-dependent conditional expectation — the import could yield algebraic local dS dynamics. The decisive test: can $\eta = 1/4G$ be derived from, rather than fed into, the trace normalization?

Both reduce to: is the area-entropy coefficient ever an output rather than an input? (Induced-gravity / Sakharov-type derivations, where $G$ emerges from an integrated-out matter sector, are the obvious untested counter-route and are not addressed here — so the "every known route" generalization is itself INFERENCE, not ESTABLISHED.)

Proposed registry items (with referee verdicts)

`HYP-dS-CARDINALITY-R1` — hypothesis refinement INFERENCE

Statement (refined). The deep obstruction to "dS Einstein-from-entanglement" is not solely the cardinality / radius-modulus mismatch but, more fundamentally, that the area-entropy coefficient $\eta = 1/4G$ is SUPPLIED, not derived, in every entanglement-equilibrium / holographic-first-law route examined: FGHMVR (RT prefactor assumed; arXiv:1312.7856) ESTABLISHED for that route; Jacobson ( $G$ identified via the area/UV entanglement-density term; arXiv:1505.04753) ESTABLISHED for that route; CLPW (Type II $_1$ algebraic entropy matched to $S_{\rm gen} = A/4G + S_{\rm out}$ up to a state-independent additive constant from trace renormalization; arXiv:2206.10780) ESTABLISHED for that route. The two iter4-surviving routes corroborate from opposite sides: (a) arXiv:2511.07915 evades the radius-modulus facet (pseudo-entropy first law for CFT regions of all sizes on $\mathcal I^+$ ) but fails on non-unitarity (imaginary $c$ , complex pseudo-entropy, geometry as the real section of an AdS analytic continuation that re-imports Lorentzian time); (b) Jacobson-in-CLPW fails because the finite Type II $_1$ trace fixes a single global normalization (tied to the observer-horizon area), so internalizing Jacobson's infinite local family requires re-supplying $\eta = 1/4G$ pointwise — re-being Jacobson non-algebraically. Verdict "encodes, does not generate" reinforced.

Rationale. Unifies the iter4 cardinality no-go, the pseudo-entropy route, and the Jacobson-in-CLPW route under a single sharper diagnostic (area-coefficient-as-input), and correctly attributes distinct disqualifying assumptions to the two routes (non-unitarity vs. single-trace).

Referee verdict (keep=true, minor): sound. Corrections folded in: (i) "encodes exactly one area datum (the global horizon)" softened to "fixes a single global normalization tied to the observer-horizon area"; (ii) "every known route" is INFERENCE over an open class (Sakharov-type induced gravity not addressed), supported only for the holographic-first-law / entanglement-equilibrium family cited; (iii) "Jacobson 2015" is arXiv:1505.04753 (2015), PRL of record 116, 201101 (2016). The central thesis — $\eta = 1/4G$ supplied not produced — verifies for all three cited routes.

`OP-dS-PSEUDO-REALITY` — open problem OPEN; resolution-direction INFERENCE-strong toward ENCODES-only

Statement (refined). In dS $_3$ /CFT $_2$ (arXiv:2511.07915), the holographic first law of pseudo-entropy reproduces the perturbative dS $_3$ Einstein equation, but only via analytic continuation of the AdS $_3$ first law ( $t \to \rho - i\pi/2$ ) along complexified geodesics, with the genuinely complex pseudo-entropy $S(\mathcal T_A)$ carrying an imaginary part fixed by $c = i\,3R/2G_N$ . Open question, sharpened: can $S(\mathcal T_A)$ be made real by a physical post-selection or a pseudo-Hermitian (PT-unbroken) reduction of the transition matrix without (i) trivializing the first-law $\Rightarrow$ Einstein content and (ii) merely deforming the dual back to a unitary AdS-type CFT? Sharpened expectation: NO on (i)+(ii) jointly — the imaginary- $c$ -weighted modular data is precisely what continues to the Lorentzian-signature dS (not AdS) Einstein equation; reality-via-pseudo-Hermiticity (JHEP 05 (2023) 021) makes the reduction an AdS-ward deformation. Independently: even a successful real reduction would not promote Program A to GENERATES, because the Lorentzian dS geometry here is the Wick-rotated image of AdS data plus an emergent dS $_2$ kinematic-space (Klein–Gordon) structure that already presupposes a metric/foliation. Verdict: ENCODES-only; headline not flipped.

Rationale. Converts the iter4 open-subquestion ("does pseudo-entropy non-unitarity disqualify it?") into a sharp, decidable test (can $\mathrm{Im}\,S$ be physically nulled non-trivially), isolating the single step on which route (a)'s headline status turns.

Referee verdict (keep=true, minor): citation real and correctly characterized; no fabrication. Terminology corrected: pseudo-entropy is complex, not purely imaginary (only the universal/divergence-subtracted part of the spherical-region result is purely imaginary) — [CORRECTED, iteration 6 (R6A4-F2, against PRL 130, 031601): this parenthetical is inverted. The FINITE universal part is REAL and equals $S_{\rm dS}/2$ , region-independently; the DIVERGENT, region- and shape-dependent part is IMAGINARY. See notes/2026-06-10-iter6-op49-pseudo-entropy-reality.md.]. The conditional "would promote toward GENERATES" was flagged as overclaiming its own antecedent: a construction reached by complexifying AdS data is not metric-free emergence even if $\mathrm{Im}\,S$ could be zeroed. The "true radius continuum" / "emergent time" framings presuppose the dS $_2$ kinematic-space metric — emergent kinematic space, not signature-from-nothing. Resolution direction now INFERENCE-strong toward ENCODES-only.

Verdict

PARTIAL (stable across iterations 2–5). The headline is NOT flipped: neither surviving route moves Program A from ENCODES to GENERATES at $\Lambda>0$ . The iteration adds genuine analytical value by sharpening, not restating, the obstruction — the two routes fail for distinct, now-identified reasons:

Route (a) (arXiv:2511.07915; web-verified: $c = i\,3R/2G$ , non-Hermitian $\mathcal T_A$ , complex pseudo-entropy, $\mathcal I^+$ boundary, complexified geodesics, dS $_2$ Klein–Gordon equation) genuinely evades the iter4 cardinality/radius-modulus facet — it is a real first-law $\Rightarrow$ linearized-dS $_3$ -Einstein implication with a region-size continuum — but inherits a different disqualifying assumption: non-unitarity. Its real geometry is the real section of an analytic continuation of the AdS/CHM computation; pseudo-entropy becomes physical only when post-selection removes the non-unitary content.
Route (b) fails the import: the CLPW Type II $_1$ algebra has a unique finite trace fixing a single global area normalization (the observer horizon), so internalizing Jacobson's infinite local small-ball family requires re-supplying $\eta = 1/4G$ at every bulk point — it merely re-is Jacobson, presupposing the area-entropy law non-algebraically.

The unifying refinement: in AdS-FGHMVR, Jacobson, and CLPW alike the area-entropy coefficient is an INPUT, not an output — the true root obstruction, untouched by either route. This confirms saturation: Program A is converged at PARTIAL for $\Lambda>0$ ; both un-foreclosed escapes inherit disqualifying assumptions (non-unitarity; area-coefficient re-input). Confidence: high (negative verdict; the converse "GENERATES" would require new physics). Reinforces H2; sharpens the $\Lambda>0$ boundary of H4.

Open subquestions

Can $\mathrm{Im}\,S(\mathcal T_A)$ in arXiv:2511.07915 be set to zero by a physical post-selection (or a genuine real reduced density matrix) without trivializing the first-law $\Rightarrow$ Einstein content? This single step decides route (a)'s headline status (OP-dS-PSEUDO-REALITY). Sharpened expectation: no, because the imaginary- $c$ data is what continues to dS (not AdS) Einstein.
Is the area-entropy coefficient $\eta = 1/4G$ ever an output rather than an input in any spacetime-from-entanglement derivation — including untested induced-gravity / Sakharov-type routes — or is its always-input status a genuine no-go for GENERATES (the unifying root obstruction)?
Does a Type II $_1$ factor admit genuinely independent local subalgebras each carrying its own area datum via a position-dependent conditional expectation, or does the unique-trace structure provably forbid the Jacobson import (settling route (b) rigorously)?
Does the arXiv:2511.07915 derivation, being an analytic continuation of an AdS/EAdS computation, count as "dS-intrinsic" at all, or is every $\Lambda>0$ first-law result to date a continuation of $\Lambda<0$ (making "encodes" structural for the whole continuation-based program)?
Would global dS (two static patches $+\ \mathcal I^\pm$ ) or time-dependent quintessence ( $w \neq -1$ , DESI-favored) restore an infinite local region family with an intrinsic area continuum, dissolving the single-trace facet of route (b)?