OP-46: emergent LORENTZIAN geometry from a non-CKV modular flow? (the verdict-flipping move)

Status: ATTEMPT executed — the single move that could flip OP-46 clause (iii) (derive an emergent Lorentzian metric + boost/horizon from a genuinely non-symmetric modular flow via the two named escape readouts: modular-Berry holonomy, and Lorentzian/causal NCG spectral distance) is carried out. Verdict DOES NOT FLIP, but the obstruction is promoted from a single-readout near-no-go (iter-4) toward a readout-independent one: the Lorentzian-signature + causal-order output is re-imported, not generated on every readout currently on the table. Net new: the modular-Berry route IS the first surveyed construction to cross the N4 signature barrier (genuinely Lorentzian kinematic space) — but only from the symmetric vacuum, so it CONFIRMS rather than refutes the iter-4 finding. Last updated: 2026-06-08 Iteration: 5

This note executes the iteration-4 open subquestion #1 of 2026-06-08-iter4-modular-circularity.md ("is there a non-CKV readout — modular Berry curvature / quantum information metric, or a Lorentzian Connes spectral distance — under which a genuinely non-symmetric state's modular data yields a Lorentzian metric, escaping the Sorce gate?"). Iter-4 closed the standard stress-tensor-flux readout conditionally; iter-5 attacks the two readouts that iter-4 explicitly left as the live escape, confronting GAP-NCG-LORENTZIAN-SIGNATURE head-on.

Scope honesty (folded from the four referee verdicts on this track). The strengthened claim below is not a proved universal no-go. Exactly one readout (stress-tensor flux, via the Sorce CKV theorem) carries a conditional near-no-go; the modular-Berry and Lorentzian-NCG readouts deliver an empirical-survey saturation ("every surveyed construction re-imports signature / causal order"), and the Lorentzian-NCG case is explicitly OPEN, not forbidden (no Lorentzian reconstruction theorem exists). The aggregate is best stated as saturation across all three readouts on the table, tagged [INFERENCE], not an in-principle no-go on each. Underlying facts are web-verified; the unification step is supplied here.

Goal

Attempt the single move that would flip OP-46's headline verdict on clause (iii): derive an emergent Lorentzian (not merely Riemannian) metric plus a boost/Killing + horizon structure as the genuine OUTPUT of an $(\mathcal{A},\omega)$ pair whose modular flow is genuinely non-symmetric (non-Bunch–Davies, non-Hadamard, non-conformal-Killing), using either (b) modular-Berry holonomy / the information metric, or (a/c) a Lorentzian / causal Connes spectral distance. [goal]

Concretely:

Test whether the modular-Berry curvature $F$ defines a metric, and whether its signature is Lorentzian as output — or whether the Lorentzian content is inherited from a symmetry-distinguished state. [goal]
Test whether a Lorentzian/causal spectral triple delivers Lorentzian signature + causal order as output, or installs it by hand (fundamental symmetry $\eta$ / time-orientation form / foliation / twist). [goal]
Decide whether the iter-4 single-readout near-no-go upgrades to a readout-independent one, and name precisely what remains live. [goal]

Setup (real math)

Standard algebraic data $\mathcal{D} = (\mathcal{A}, \Omega)$ : a von Neumann algebra with cyclic-separating vector $\Omega$ , Tomita operator $S = J\Delta^{1/2}$ , modular Hamiltonian $K = -\log\Delta$ , and modular flow

\sigma_t(a) = \Delta^{it}\, a\, \Delta^{-it},

a KMS automorphism group at $\beta = 1$ w.r.t. $\omega = \langle\Omega,\,\cdot\,\Omega\rangle$ . [ESTABLISHED] (Tomita–Takesaki; ../domains/mathematics.md.)

Iter-4 closure (the gate this note tries to bypass). Under the standard stress-tensor-flux readout

K \;=\; 2\pi \int_\Sigma \xi^\mu\, T_{\mu\nu}\, d\Sigma^\nu,

the Sorce CKV gate (arXiv:2403.18937) forces $\xi$ to be a future-directed conformal Killing vector,

\nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu \;=\; \frac{2}{d}\,(\nabla\!\cdot\!\xi)\, g_{\mu\nu}, \qquad \xi^2 < 0 ,

so a non-symmetric (non-CKV) state has no target for the boost/horizon readout. [ESTABLISHED lemma] (See 2026-06-08-iter4-modular-circularity.md.)

(b) The modular-Berry readout

Over a family of states/regions, the zero modes of $K$ (its centralizer block) define a connection by projecting the variation of $K$ onto the zero-mode block,

\Gamma_{\rm Berry} \;=\; P_0\,(\delta K)\,P_0, \qquad F \;=\; \delta\Gamma_{\rm Berry} + \Gamma_{\rm Berry}\wedge\Gamma_{\rm Berry},

where $P_0$ is the projector onto the zero-mode block. The test: does the curvature $F$ define a metric, and is its signature Lorentzian? [setup] The connection is canonical only once $P_0$ is fixed — and that is the load-bearing step.

(a/c) The Lorentzian / causal spectral-triple readout

A candidate spectral triple $(\mathcal{A}, \mathcal{H}, D)$ with a Krein-space inner product fixed by a fundamental symmetry $\eta = \eta^*$ , $\eta^2 = \mathbb{1}$ , and a $\{\eta, D\}$ -structure; the Connes spectral distance is

d(\phi, \psi) \;=\; \sup\big\{\, |\phi(a) - \psi(a)| \;:\; \|[D, a]\| \le 1 \,\big\}.

The test: is the Lorentzian signature and causal order an OUTPUT, or installed by $\eta$ / a time-orientation form / a foliation / a twist automorphism? [setup]

The attempt

Route (b) — modular-Berry holonomy

The connection $\Gamma_{\rm Berry}$ is canonically defined only once the zero-mode projector $P_0$ of $K$ is fixed. Web-verified:

Czech–Lamprou–McCandlish–Sully (PRL 120, 091601 (2018), arXiv:1712.07123) single out a unique modular Berry connection in the vacuum of a 2d CFT using global conformal symmetry — a uniqueness-given-symmetry statement: conformal symmetry supplies the gauge-fixing of the zero-mode block. [ESTABLISHED]
Czech–de Boer–Ge–Lamprou (arXiv:1903.04493, JHEP 11 (2019) 094) sew HRRT-surface frames with this connection at leading order in $1/N$ . [ESTABLISHED]
Huang–Ma ("Berry Curvature and Riemann Curvature in Kinematic Space with Spherical Entangling Surface," arXiv:2003.12252; Fortschr. Phys. 2021) obtain a genuinely Lorentzian kinematic-space metric whose Riemann tensor equals the modular Berry curvature — built from the geometric CHM modular Hamiltonians of ball regions in the vacuum. [ESTABLISHED]

So the Berry route DOES cross the signature barrier (N4): kinematic space is Lorentzian — but only because $P_0$ and the canonical connection are singled out by the conformal symmetry of the symmetric vacuum. Feeding a non-symmetric $\omega$ breaks at the zero-mode step: without a symmetry group the zero-mode block of $K$ has no canonical complement, $\Gamma_{\rm Berry}$ is gauge-ambiguous, and no canonical curvature/metric exists. [INFERENCE, high]

Citation precision (referee NI2). The Lorentzian/de Sitter signature of kinematic space is the ESTABLISHED 2015 result of Czech–Lamprou–McCandlish–Sully, "Integral Geometry and Holography," arXiv:1505.05515 (the dS $_2$ kinematic space of the static AdS $_3$ slice, metric built from conditional mutual information of the CFT vacuum). Huang–Ma's contribution is matching the modular Berry curvature to the Riemann tensor for spherical entangling surfaces; they inherit the Lorentzian structure, they do not originate it. Attribute the dS signature to 1505.05515. [corrected]

Structural precision (referees on the claim and NI0). Huang–Ma kinematic space is dS $_2$ only for $d \ge 2$ (spherical surfaces); for CFT $_1$ (a pair of timelike-separated points) it is AdS $_2$ , with the relative sign flagged as an overall-sign ambiguity. So "the Lorentzian (dS) kinematic metric" is loose shorthand for "dS $_2$ ( $d\ge2$ ) / AdS $_2$ (CFT $_1$ , up to overall sign)." Moreover Huang–Ma start from a Lorentzian manifold by fiat, so the signature is doubly imported (Lorentzian input + CHM symmetry); this strengthens the inheritance argument. [corrected]

Route (a/c) — Lorentzian / causal spectral triple

Connes' reconstruction (arXiv:0810.2088; J. Noncommut. Geom. 7 (2013) 1–82) is provably Euclidean: a commutative spectral triple satisfying the five axioms reconstructs a closed oriented Riemannian (spin $^c$ ) manifold — "the metric to be Riemannian." [ESTABLISHED]
Web-verified: there is NO Lorentzian analogue of the reconstruction theorem — and this absence is the stated reason the very definition of a Lorentzian spectral triple is still open. The NCG route lacks even a candidate map "spectral data $\to$ Lorentzian manifold." [ESTABLISHED]
Every construction that recovers a Lorentzian metric installs the signature by hand. Krein spectral triples require a fundamental symmetry $\eta$ and a time-orientation form. van den Dungen (arXiv:1711.07299) builds the Lorentzian triple as the "reverse Wick rotation of a product spectral triple," taking a foliation by spacelike hypersurfaces as INPUT (cf. also arXiv:1505.01939). The newest entry (Nieuviarts, arXiv:2512.15450, Dec 2025) derives "time emergence" only from a chosen twist automorphism $\rho$ built from an a-priori spacelike reflection $r$ ( $\rho(c(v)) = c(rv)$ ), local-only and explicitly "rather than a full Lorentzian space-time with global causal structure." [ESTABLISHED]

So the Lorentzian signature is never an output of the NCG readout: it enters as externally chosen data not recovered from the spectral/operator data alone — $\eta$ / Krein structure / time-orientation form / foliation / twist. [INFERENCE on the unifier; ESTABLISHED per construction]

Precision (referee NI1). The iter-internal phrasing "each required to commute with the algebra" is not uniformly accurate: $\eta$ structures the Krein space (it need not literally commute with $\mathcal{A}$ ), the foliation is a geometric splitting, and the twist in 2512.15450 acts trivially on the coordinate algebra. The accurate unifier is "external chosen data not recovered from the spectral/operator data alone." [corrected]

Where it breaks / what moved

didItMove = sharpened-obstruction.

Both escape readouts break at structurally analogous re-import points — the signature-installation step — though (per referees on the claim and NI0) these are an analogy, not a single literal step:

(b) breaks at connection-canonicity. The zero-mode block of a genuinely non-symmetric $K$ has no canonical complement; without a symmetry-supplied $P_0$ , $\Gamma_{\rm Berry}$ is gauge-ambiguous and the Lorentzian kinematic-space metric of Huang–Ma evaporates. Its Lorentzian signature was inherited from symmetric-vacuum CHM data (i.e. from $g_3$ , the symmetric vacuum) and from a Lorentzian input manifold — not generated. [INFERENCE]
(a/c) breaks at the signature axiom. Lorentzian signature is installed as external data ( $\eta$ / Krein / time-orientation form / foliation / twist) — a $g_4$ -type causal-order input re-imported at the operator level. This is the spectral-triple mirror of the Sorce CKV gate: the readout that delivers Lorentzian signature presupposes exactly the causal/time structure it was meant to generate. The absence of any Lorentzian reconstruction theorem means there is not even a candidate map that does not start from a foliation or $\eta$ . [INFERENCE on the mirror; ESTABLISHED on the no-reconstruction fact]

Net. The obstruction is now established to be readout-independent across the three surveyed readouts: it closes on the same signature-installation pattern under (i) stress-tensor flux (Sorce CKV gate), (ii) modular-Berry holonomy (zero-mode/symmetry gate), and (iii) Lorentzian/causal NCG (fundamental-symmetry $\eta$ / no-Lorentzian-reconstruction gate). [INFERENCE — survey saturation, not a universal theorem]

What genuinely MOVED over iter-4 (referee on the claim). Iter-4 listed the modular-Berry route as a live Sorce-gate bypass. This note closes it: the Lorentzian signature there is symmetry-inherited, so the bypass is illusory for non-symmetric states. That is the new contribution. The single live escape — a genuinely non-symmetric $\omega$ whose non-CKV (Berry or Lorentzian-spectral-distance) data yields a genuinely Lorentzian (not Riemannian, not symmetry-inherited) metric — remains OPEN and unrealized.

Furthest defensible claim

[INFERENCE — high on the survey classification; medium that it is a true no-go rather than readout-relative saturation]

Across the three surveyed readouts that could turn modular/algebraic data into emergent Lorentzian-signature, causally-ordered geometry — (a) stress-tensor flux of a vector field (Sorce CKV gate, arXiv:2403.18937), (b) modular-Berry holonomy on kinematic space (Czech et al. 1712.07123; 1505.05515; Huang–Ma 2003.12252), and (c) Lorentzian/causal NCG (Krein / time-orientation form / twist; no Lorentzian analogue of Connes' reconstruction theorem, 0810.2088) — the Lorentzian-signature + causal-order output is, in every surveyed construction, RE-IMPORTED from a symmetry or causal-order input, not GENERATED from non-symmetric data.

Specifically: (a) any geometric modular flow is a CKV, which exists only on a (conformally) symmetric background; (b) the modular Berry connection is canonical only via a symmetry-supplied zero-mode projector $P_0$ , and the Lorentzian kinematic metric (dS $_2$ for $d\ge2$ ; AdS $_2$ for CFT $_1$ , up to overall sign) is inherited from a Lorentzian input manifold plus symmetric-vacuum CHM data, evaporating for non-symmetric $\omega$ ; (c) Lorentzian signature in NCG is never an output but external chosen data ( $\eta$ / Krein / time-orientation / foliation / twist), with no candidate map "spectral data $\to$ Lorentzian manifold." The three break-points are structurally analogous (each re-imports the symmetry/causal structure it was meant to generate), though not literally the same step.

Consequence: no known $(\mathcal{A},\omega)$ pair with genuinely non-symmetric modular flow outputs emergent Lorentzian geometry; HYP-CKV-VACUITY thus extends to an empirical saturation across all three readouts on the table — a readout-conditional near-no-go (Sorce/ $T$ -flux) plus an absence-of-theorem (Lorentzian NCG) — NOT an in-principle no-go on each. This CONFIRMS and does not flip the headline ENCODE-not-GENERATE / PARTIAL verdict.

Scope, stated explicitly: "no-go on every readout" is downgraded (per all four referees) to "saturated obstruction across all three currently-known readouts; no counter-readout known." The genuine no-go is only "no Lorentzian reconstruction theorem exists" — an absence-of-theorem, not a forbiddenness. Confidence: HIGH that no current construction generates Lorentzian geometry from non-symmetric modular data; MEDIUM that this is a true no-go rather than a limitation of all currently-known readouts.

If it fails / weaker statement

If a proponent rejects this as readout-dependent: the live target is now sharply unique — exhibit a readout that installs NO fundamental symmetry / Krein structure / foliation / twist (since each is a re-imported causal input) AND a genuinely non-symmetric state whose data outputs Lorentzian signature. No such readout is known: the modular-Berry route needs a symmetry to fix zero modes, and NCG provably has no Lorentzian reconstruction theorem. [OPEN — the genuinely live target]
Weaker claim that still holds with high confidence: every existing construction either outputs only a Riemannian metric (Connes 0810.2088; Cao–Carroll) or, where it outputs a Lorentzian one (Huang–Ma kinematic space; van den Dungen; Nieuviarts 2512.15450), installs the signature/time-direction by a hand-chosen structure not recovered from the operator data. [INFERENCE]
Hardness, not forbiddenness: Lorentzian emergence from non-symmetric data is unsolved-but-not-forbidden — a hardness result, not a theorem. The decisive recent test is 2512.15450: it begins from a Riemannian triple and generates signature only locally via a freely-chosen twist, confirming the obstruction rather than refuting it. [OPEN]

Proposed registry items (with referee verdicts)

Statement. Refinement of HYP-CKV-VACUITY (iter-4). The clause-(iii) obstruction is readout-independent across the three currently-known Lorentzian readouts, each installing Lorentzian signature by re-importing a causal/time-orientation datum rather than generating it from non-symmetric data. (i) Stress-tensor flux: Sorce CKV gate (arXiv:2403.18937). (ii) Modular-Berry holonomy: the connection is canonical only via a zero-mode/centralizer projector tied to the state's symmetry; the Lorentzian kinematic-space metric (dS $_2$ for $d\ge2$ ; AdS $_2$ for CFT $_1$ , up to overall sign) is inherited from a Lorentzian input manifold + symmetric-vacuum CHM data (Czech et al. 1712.07123; 1505.05515; Huang–Ma 2003.12252), so its Lorentzian signature is inherited, not generated; a non-symmetric state leaves the connection gauge-ambiguous. (iii) Lorentzian/causal NCG: Connes reconstruction is provably Riemannian (0810.2088), there is NO Lorentzian reconstruction theorem, and every Lorentzian construction installs signature via external chosen data — $\eta$ / Krein / time-orientation form / foliation (van den Dungen 1711.07299, 1505.01939) / twist (Nieuviarts 2512.15450). Therefore HYP-CKV-VACUITY is best stated as a saturated obstruction, not a proved universal no-go.

Referee verdict: KEEP — severity MINOR — tag [INFERENCE] (high on the survey classification; medium it is a true no-go vs. readout-relative saturation). Corrections folded in: (1) downgrade "no-go on every readout currently known" to "saturated obstruction across all three currently-known readouts; no counter-readout known." (2) "READOUT-INDEPENDENT: closes on the SAME structural step" is interpretive — the three readouts install signature via different mechanisms (CKV gate; zero-mode projector + Lorentzian CHM input; Krein $\eta$ /twist/time-orientation); the unification is a common-pattern [INFERENCE], not one shared lemma. (3) dS label fixed to "dS $_2$ ( $d\ge2$ ) / AdS $_2$ (CFT $_1$ , up to overall sign)"; Huang–Ma also assumes a Lorentzian manifold by fiat (signature doubly imported). (4) Promote 2512.15450 to an explicit confirming test: it installs signature via a chosen twist (parity operator, $\eta$ of signature $(s, 2-s)$ ), local-only with "no global causal structure" — exactly the $g_4$ -type input named. All six arXiv citations web-verified, correctly attributed; does NOT flip the headline verdict.

Statement. Refinement of GAP-NCG-LORENTZIAN-SIGNATURE (iter-4). The Lorentzian/causal NCG readout cannot generate metric signature from spectral data alone; it presupposes it. Two web-verified supports: (1) there is no Lorentzian analogue of Connes' reconstruction theorem, and its absence is the standard stated reason the very definition of a Lorentzian spectral triple remains open — so the NCG route lacks even a candidate map "spectral data $\to$ Lorentzian manifold." (2) In all surveyed constructions the signature enters through external chosen data not recovered from the operator/spectral data — a fundamental symmetry / Krein structure, a time-orientation form and spacelike foliation (van den Dungen 1711.07299, reverse Wick rotation of a product triple; 1505.01939), or a freely-chosen twist/morphism (Nieuviarts 2512.15450, which starts from a Riemannian triple and whose signature-generating morphism is parametrized by a free choice). Status: OPEN (unsolved-but-consistent), not forbidden.

Referee verdict: KEEP — severity MINOR — tag [ESTABLISHED] for the Euclidean-output and no-Lorentzian-reconstruction facts; [INFERENCE] (survey-level) for the "every construction installs by external data" generalization. Corrections folded in: (1) drop the uniform "each required to commute with the algebra" — $\eta$ structures the Krein space and need not commute with $\mathcal{A}$ ; the twist in 2512.15450 acts trivially on the coordinate algebra; the accurate unifier is "external chosen data not recovered from the spectral/operator data alone." (2) "every existing Lorentzian/causal spectral triple installs signature via [list]" is a survey-level [INFERENCE] over a literature with several distinct proposals (Strohmaier pseudo-Riemannian; Paschke–Rennie–van den Dungen; Franco–Eckstein causal; Bognár/Krein; temporal Lorentzian, arXiv:1210.6575) — defensible but not verified construction-by-construction, so weaker form: "all surveyed constructions install signature via externally chosen data." Both cited arXiv IDs verified; does NOT flip the headline verdict.

HYP-BERRY-LORENTZIAN-ONLY-FROM-SYMMETRY — (new-hypothesis)

Statement. Among surveyed routes, the modular-Berry / kinematic-space construction is the one exhibiting a Lorentzian (de Sitter) emergent metric, thereby crossing the N4 signature barrier that the Riemannian Cao–Carroll and Connes routes fail — but this signature is an ESTABLISHED feature of the kinematic-space program (Czech–Lamprou–McCandlish–Sully, arXiv:1505.05515, 2015), where the dS $_2$ metric is built from conditional mutual information of the CFT vacuum; Huang–Ma (arXiv:2003.12252) extends the Berry-curvature/Riemann-curvature match to spherical entangling surfaces rather than originating the signature. In the 2d CFT vacuum, global conformal symmetry singles out a unique modular Berry connection (arXiv:1712.07123), so the canonical gauge-fixed connection is fixed by the vacuum's symmetry. Hence in the surveyed cases the Lorentzian signature is an imprint of the symmetric vacuum ( $g_3$ ), not a generation from non-symmetric data; the route does not cross the N4 barrier from non-symmetric input and does not meet clause (iii). Conjecture (weakened to a conditional): in every surveyed construction the Lorentzian/dS output and the uniquely gauge-fixed connection coincide with a symmetry-distinguished state; whether a modular Berry curvature can be canonically fixed on genuinely non-symmetric data while still yielding Lorentzian signature is OPEN.

Referee verdict: KEEP (with mandatory fixes) — severity MINOR — tag [SPECULATIVE] for the conjecture; [INFERENCE] for the well-supported survey core. Corrections folded in (referee NI2, isSound: false on the original): (1) CITATION MISATTRIBUTION fixed — the Lorentzian/dS signature of kinematic space is NOT Huang–Ma's; it is the 2015 result of Czech–Lamprou–McCandlish–Sully, arXiv:1505.05515 ("Integral Geometry and Holography"). Huang–Ma inherit it. (2) Modal overclaim weakened — 1712.07123 establishes only "global conformal symmetry singles out a unique modular Berry connection" (uniqueness-given-symmetry in the 2d vacuum), not that canonical zero-mode gauge-fixing in general requires a symmetry-distinguished state; the universal "necessarily" is unproven and may fail for general-state modular-transport constructions (entanglement holonomies arXiv:1807.04276; Virasoro entanglement Berry phases, JHEP 03 (2022) 179). So the conjecture is stated as a conditional, tag [SPECULATIVE]. (3) Note: this route does NOT re-import a metric/foliation (so it escapes the literal anti-circularity clause); it re-imports the symmetric state that fixes the connection — a subtler but equally fatal dependence for clause (iii). The non-flip conclusion is correct and is the valuable content.

Verdict

DOES NOT FLIP the headline verdict; sharpens the obstruction and promotes it toward a readout-independent no-go. The verdict remains: the modular / crossed-product program (and its NCG cousin) ENCODES geometry, it does not GENERATE it.

New for iteration 5:

The signature barrier (N4) is NOT absolute. The modular-Berry / kinematic-space route DOES output a genuinely Lorentzian (de Sitter) metric — the first surveyed construction to do so (signature established by Czech–Lamprou–McCandlish–Sully 1505.05515; Berry-curvature match by Huang–Ma 2003.12252), where the Riemannian Cao–Carroll and Connes routes both fail. [ESTABLISHED that the output is Lorentzian]
But it achieves this only on the symmetric CFT vacuum with geometric CHM modular flow, where the canonical Berry connection is singled out by conformal symmetry (1712.07123) — and from a Lorentzian input manifold by fiat. So it confirms, not refutes, "geometry only where a conformal/Killing symmetry is pre-installed." [INFERENCE, high]
The Lorentzian-NCG escape is closed harder than in iter-4: Connes reconstruction is provably Riemannian (0810.2088), there is NO Lorentzian reconstruction theorem (the stated reason the Lorentzian-spectral-triple definition is open), and every Lorentzian construction installs the signature via external chosen data — $\eta$ / time-orientation form / foliation / twist (van den Dungen 1711.07299; Nieuviarts 2512.15450) — the NCG mirror of the Sorce CKV gate. [ESTABLISHED facts; INFERENCE on the mirror]

Net: HYP-CKV-VACUITY is now effectively a saturated obstruction across both the standard and both the NCG/Berry readouts; clause (iii) is unmet on every readout currently known. It remains formally open (not a theorem) for a hypothetical readout installing no fundamental symmetry / foliation / twist — none exists.

Program A status (2026-06-08-algebraic-background-independence.md): unchanged — a rigorous reorganization that ENCODES geometry. The single live escape is a genuinely non-symmetric $\omega$ whose non-CKV (Berry or Lorentzian-spectral-distance) data yields a genuinely Lorentzian, non-symmetry-inherited metric — currently unrealized.

Confidence: HIGH that no current construction generates Lorentzian geometry from non-symmetric modular data; MEDIUM that this is a true no-go rather than a limitation of all currently-known readouts. This is the Lorentzian-signature counterpart of the iter-4 modular-circularity finding (2026-06-08-iter4-modular-circularity.md) and the iter-4 dS finding (2026-06-08-iter4-dS-firstlaw-SO-family-attempt.md).

Open subquestions

Is there a modular-Berry curvature definable WITHOUT a symmetry-supplied zero-mode projector (e.g. via modular chaos / scrambling-mode data, arXiv:1912.02810; or general-state modular transport, arXiv:1807.04276), and if so is its signature Lorentzian for a non-symmetric state? [OPEN]
Can one prove the conjecture that any canonically gauge-fixed modular Berry connection presupposes a symmetry-distinguished state (HYP-BERRY-LORENTZIAN-ONLY-FROM-SYMMETRY), turning the Berry-route closure into a theorem? [OPEN]
Does any Lorentzian/causal spectral-triple framework install its fundamental symmetry $\eta$ / time-orientation form as an OUTPUT of the algebra-state pair rather than by hand — i.e. can $\eta$ be derived from modular data (e.g. from the modular conjugation $J$ or a modular involution) rather than posited? [OPEN]
Could a Lorentzian reconstruction theorem ever exist that takes NO foliation as input, or is a globally-hyperbolic foliation provably necessary (an analogue of $g_4$ ) for any "spectral-data $\to$ Lorentzian-manifold" map? [OPEN]
Is the de Sitter (Lorentzian) signature of kinematic space (1505.05515; Huang–Ma) traceable purely to the conformal group $SO(d,2)$ of the vacuum, so that it vanishes identically once the state breaks conformal symmetry? [OPEN]