Why Non-Linear Source Geometry Does Not Imply Superluminal Signaling: A TCGS-SEQUENTION Response to Gisin-Polchinski

1. The Gisin-Polchinski Argument

The core signaling concern for non-linear quantum mechanics was articulated in the original discussions of Weinberg’s framework and its EPR implications [1,2,3,4].

1.1. Setup

Consider an entangled pair of qubits shared between Alice (A) and Bob (B):

$$|\mathsf{\Psi }\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left({|0\rangle }_A{|0\rangle }_B+{|1\rangle }_A{|1\rangle }_B\right)$$

(1)

Bob’s reduced density matrix is:

$${\rho }_B={\displaystyle \frac{1}{2}}\left(|0\rangle \langle 0|+|1\rangle \langle 1|\right)={\displaystyle \frac{1}{2}}\mathbb{I}$$

(2)

1.2. The Standard Argument

Step 1: Suppose quantum evolution is state-dependent (non-linear), meaning the evolution operator $U\left[\rho \right]$ depends on $\rho$ itself.

Step 2: The maximally mixed state ${\rho }_B={\displaystyle \frac{1}{2}}\mathbb{I}$ admits multiple decompositions:

• As mixture of $\left\{\right|0\rangle ,|1\rangle \}$: the ensemble $\{\left(\right|0\rangle ,{\displaystyle \frac{1}{2}}),\left(\right|1\rangle ,{\displaystyle \frac{1}{2}}\left)\right\}$
• As mixture of $\left\{\right|+\rangle ,|-\rangle \}$: the ensemble $\{\left(\right|+\rangle ,{\displaystyle \frac{1}{2}}),\left(\right|-\rangle ,{\displaystyle \frac{1}{2}}\left)\right\}$

Step 3: If Alice measures in the $\left\{\right|0\rangle ,|1\rangle \}$ basis, she “collapses” Bob’s state into the first ensemble. If she measures in the $\left\{\right|+\rangle ,|-\rangle \}$ basis, she collapses it into the second.

Step 4: Under non-linear evolution $U\left[\rho \right]$, these two ensembles evolve differently, even though they represent the same density matrix.

Step 5: Bob can therefore detect which basis Alice chose → superluminal signaling → contradiction.

1.3. The Implicit Assumptions

The GP argument requires:

• Ontic time: There exists a global time parameter t such that Alice’s measurement at $t_1$ precedes Bob’s detection at $t_2$.
• Causal collapse: Alice’s measurement causes a real, physical change in Bob’s state.
• Density matrix completeness: The density matrix $\rho$ is the complete physical description; different ensemble decompositions represent the same physical state.

2. The TCGS Dissolution

2.1. Core Axioms (Recap)

Axiom 1

(Whole Content — A1). There exists a smooth 4D counterspace $(\mathcal{C},G,\mathsf{\Psi })$ containing the full content of all “time stages” simultaneously.

Axiom 2

(Identity of Source — A2). All correlated phenomena derive from a unique geometric origin $p_0\in \mathcal{C}$.

Axiom 3

(Shadow Realization — A3). The observable 3D world is embedded by $X:\Sigma \to \mathcal{C}$; observables are pullbacks $(g,\psi )=(X^{*}G,X^{*}\mathsf{\Psi })$. Time has no ontic status.

2.2. Entanglement as Identity of Source

In TCGS, the “entangled pair” is not two particles mysteriously correlated across space. It is one geometric object in the 4D counterspace $\mathcal{C}$, viewed from two projection points.

Let $\mathsf{\Psi }\in \mathcal{C}$ be the static 4D content field encoding the entangled system. Define:

• $X_A:{\Sigma }_A\to \mathcal{C}$ — the projection at Alice’s location
• $X_B:{\Sigma }_B\to \mathcal{C}$ — the projection at Bob’s location

The observables are:

$${\psi }_A={X_A}^{*}\mathsf{\Psi },{\psi }_B={X_B}^{*}\mathsf{\Psi }$$

(3)

The “entanglement correlation” is the geometric constraint that both are pullbacks of the same $\mathsf{\Psi }$.

2.3. Why “Signaling” Does Not Occur

Claim 1.

Alice’s “measurement choice” does not change Ψ; it selects a foliation of $\mathcal{C}$.

Argument:

In standard QM, Alice’s choice of measurement basis is modeled as an intervention that collapses the wave function. The GP argument treats this collapse as a real, time-indexed event that changes the physical state.

In TCGS:

• The 4D content $\mathsf{\Psi }$ is static (Axiom A1). It does not “change” because time is not fundamental.
• Alice’s “measurement” corresponds to selecting a specific foliation (slicing) of $\mathcal{C}$. Different basis choices correspond to different ways of parameterizing the projection $X_A$.
• Bob’s observable ${\psi }_B={X_B}^{*}\mathsf{\Psi }$ depends on:
  • His projection map $X_B$ (fixed by his spacetime location)
  • The static field $\mathsf{\Psi }$ (unchanged by Alice’s foliation choice)
• Crucially: The pullback operation ${X_B}^{*}$ commutes with Alice’s foliation choice. Bob’s observable is determined by $\mathsf{\Psi }$ and $X_B$, not by how Alice parameterizes her slice.

2.4. Formal Statement

Theorem 2

(No-Signaling from Foliation Invariance). Let $(\mathcal{C},G,\mathsf{\Psi })$ be a static 4D counterspace. Let $X_A,X_B:\Sigma \to \mathcal{C}$ be spacelike-separated projections. Let ${\mathcal{F}}_A$ denote Alice’s choice of foliation.

Then Bob’s observable density matrix:

$${\rho }_B={\mathrm{Tr}}_A\left[X^{*}\mathsf{\Psi }\otimes X^{*}\mathsf{\Psi }\right]$$

(4)

is independent of ${\mathcal{F}}_A$.

Proof sketch. The pullback ${X_B}^{*}$ is a geometric operation determined by the embedding $X_B$ and the field $\mathsf{\Psi }$. Alice’s foliation choice ${\mathcal{F}}_A$ affects only the parameterization of her projection $X_A$, not the value of $\mathsf{\Psi }$ itself. Since $\mathsf{\Psi }$ is static and $X_B$ is independent of Alice’s operations, ${\rho }_B$ is foliation-invariant. □

2.5. The Decomposition “Problem” Dissolved

The GP argument hinges on different ensemble decompositions of ${\rho }_B$ evolving differently under non-linear dynamics.

In TCGS, this is a category error. The different decompositions are not different physical preparations—they are different descriptions of the same geometric projection. The 4D source $\mathsf{\Psi }$ determines:

• The correlations (Identity of Source)
• The evolution law (as a constitutive relation on $\mathcal{C}$)

The density matrix ${\rho }_B$ is an incomplete summary of the 3D shadow. It does not capture the full geometric information encoded in $\mathsf{\Psi }$. Non-linear evolution at the source level operates on $\mathsf{\Psi }$, not on ${\rho }_B$. The projection ensures that what Bob observes is consistent with no-signaling.

3. Analogy: ER = EPR

The TCGS resolution is structurally analogous to the ER = EPR correspondence (Maldacena-Susskind) [7]:

Table 1. Structural analogy between ER = EPR and TCGS interpretations of entanglement.

ER = EPR	TCGS
Entangled particles connected by wormhole	Entangled systems as single 4D geometric object
Wormhole is non-traversable	Projection constraint prevents signaling
Correlation without causation	Identity of Source without time-propagation
Information preserved at horizon	Observables are gauge-invariant pullbacks

Table 1. Structural analogy between ER = EPR and TCGS interpretations of entanglement.

ER = EPR	TCGS
Entangled particles connected by wormhole	Entangled systems as single 4D geometric object
Wormhole is non-traversable	Projection constraint prevents signaling
Correlation without causation	Identity of Source without time-propagation
Information preserved at horizon	Observables are gauge-invariant pullbacks

In both frameworks, the apparent “spooky action at a distance” is reinterpreted as a geometric unity that, by construction, does not permit superluminal information transfer.

4. Non-Linearity Preserved at the Source

A key point: TCGS does not require linearity.

The 4D content field $\mathsf{\Psi }$ and the constitutive law governing it can be arbitrarily non-linear:

$${\nabla }_{\mathcal{C}}·\left[\mu \left(|{\nabla }_{\mathcal{C}}\mathsf{\Psi }|\right){\nabla }_{\mathcal{C}}\mathsf{\Psi }\right]=\mathrm{sources}$$

(5)

The constraint is that the projection mechanism $(X:\Sigma \to \mathcal{C})$ and the pullback structure ensure that 3D observables respect operational no-signaling.

This is directly analogous to how General Relativity is highly non-linear, yet the causal structure (light cones) prevents superluminal signaling. The non-linearity of the field equations does not imply violation of causality—the metric structure enforces it.

5. Addressing Gisin’s Operational Protocol Directly

Gisin (1990) [1] provides a concrete signaling protocol that deserves explicit treatment.

5.1. The Protocol

• Alice measures her particle in either z-basis or u-basis (45° apart)
• Bob applies the non-linear Hamiltonian:

  $$h(\psi ,{\psi }^{*})={\displaystyle \frac{\psi {\sigma }_z{\psi }^2}{\psi \psi }}$$

  (6)
• Evolution equation:

  $$\psi t=-2i{\sigma }_z{\sigma }_z\psi$$

  (7)

Gisin’s key observation:

• z-eigenstates are stationary: ${\sigma }_y$ remains zero
• u-eigenstates rotate: after quarter-period, ${\sigma }_y>0$ for both $|u+\rangle$ and $|u-\rangle$

Thus Bob can distinguish Alice’s basis choice by measuring ${\sigma }_y$ over many particles.

5.2. Gisin’s Own Identification of the Critical Assumption

Gisin writes [1]:

“The most delicate point in the above example is probably the instantaneous preparation at a distance”

This is exactly what TCGS denies exists.

5.3. The TCGS Response

Question: When Bob applies his non-linear Hamiltonian, what state does he apply it to?

Standard QM answer: $|z\pm \rangle$ or $|u\pm \rangle$, depending on which Alice “prepared” by her measurement.

TCGS answer: Bob applies it to ${X_B}^{*}\mathsf{\Psi }$ — his projection of the static 4D field.

The decisive point: Is ${X_B}^{*}\mathsf{\Psi }$ different depending on Alice’s measurement choice?

In TCGS: No.

Alice’s measurement basis is a foliation choice — it determines how she parameterizes her projection, but does not change:

• The 4D field $\mathsf{\Psi }$ (which is static by Axiom A1)
• Bob’s projection map $X_B$ (which is fixed by his spacetime location)
• Therefore, Bob’s observable ${X_B}^{*}\mathsf{\Psi }$

The “instantaneous preparation at a distance” that Gisin’s argument requires is precisely what the timeless ontology eliminates. There is no moment at which Bob’s particle “becomes” $|z+\rangle$ rather than $|u+\rangle$. Both are descriptions of foliations, not ontological facts about the source.

5.4. What Bob Actually Has

In TCGS, Bob’s particle is not in a pure state that depends on Alice’s choice. It is in the state determined by its geometric relationship to the source $\mathsf{\Psi }$.

The correlation — that if Alice finds “up” then Bob finds “down” — is explained by Identity of Source (Axiom A2): both measurements are projections of the same 4D geometric structure. But this correlation is built into $\mathsf{\Psi }$; it is not created by Alice’s measurement.

When Bob applies any evolution (linear or non-linear) to his particle, he evolves ${X_B}^{*}\mathsf{\Psi }$. This quantity is independent of Alice’s foliation choice.

6. Addressing Other Specific Points

6.1. Local Applicability assumes 3D separation is fundamental

Response: Correct. TCGS rejects this assumption. What appears as spatial separation in $\Sigma$ is a projection artifact of a unified structure in $\mathcal{C}$. Entanglement reveals this: “distance” is a user-interface feature, not ontological.

6.2. The density matrix determines outcomes

Response: In standard QM, yes. In TCGS, the density matrix is an incomplete 3D summary. The complete description is $\mathsf{\Psi }\in \mathcal{C}$. Different ensemble decompositions of $\rho$ correspond to different foliations of the same $\mathsf{\Psi }$, not different physical states.

6.3. Non-linear evolution allows distinguishing mixtures

Response: Only if the evolution acts on the 3D ensemble descriptions. In TCGS, evolution is a constitutive law on $\mathsf{\Psi }$ in $\mathcal{C}$. The projection ensures that Bob’s accessible observables—the pullback ${X_B}^{*}\mathsf{\Psi }$—do not depend on Alice’s foliation choice.

7. Empirical Distinguishability

The TCGS framework makes the same operational predictions as standard QM for the GP setup: no signaling. The difference is ontological, not empirical (for this experiment).

However, TCGS makes distinct predictions in other regimes:

• Weak-field gravitational dynamics (the $\mu$-law replacing dark matter halos)
• Cosmological slice-invariants
• Convergent biological patterns (SEQUENTION)

The GP experiment does not falsify TCGS because TCGS predicts exactly what is observed: no superluminal signaling, despite non-linear source geometry.

8. Conclusions

The Gisin-Polchinski argument is valid within its ontological assumptions: ontic time, causal collapse, and density matrix completeness. In the TCGS framework, where time is a foliation artifact and observables are pullbacks of a static 4D geometry, the argument does not apply.

Non-linearity at the 4D source level is compatible with operational no-signaling at the 3D shadow level. The “problem” of different ensemble decompositions is dissolved: they are different descriptions of the same geometric projection, not different physical preparations.

The critique that “non-linear QM violates relativity” targets a different ontology than TCGS proposes. Within TCGS, the Identity of Source (Axiom A2) explains entanglement correlations, and the projection structure (Axiom A3) guarantees no-signaling—without requiring linearity.

Appendix A.1. Ensemble Equivalence and the GP Lever

Let ${\rho }_{AB}$ be a bipartite state shared between Alice (A) and Bob (B), and let

$${\rho }_B:={\mathrm{Tr}}_A\left({\rho }_{AB}\right)$$

(A1)

be Bob’s reduced state. A textbook fact is that Alice can prepare (by measuring A in different bases) different ensembles on Bob’s side that correspond to the same${\rho }_B$. Schematically,

$${\rho }_B=\sum _i{p}_i|{\psi }_i\rangle \langle {\psi }_i|=\sum _j{q}_j|{\varphi }_j\rangle \langle {\varphi }_j|,$$

(A2)

where the decompositions $(p_i,|{\psi }_i\rangle )$ and $(q_j,|{\varphi }_j\rangle )$ may be operationally realized by different measurement choices on A.

In ordinary (linear) quantum mechanics, Bob cannot distinguish which decomposition was used, because every local expectation value is a linear functional of ${\rho }_B$:

$$\langle O_B\rangle =\left({\rho }_B{O}_B\right).$$

(A3)

The entire no-signaling theorem, at the laboratory level, is the statement that for all local procedures available to Bob, the statistics depend on ${\rho }_B$ and not on the choice of remote ensemble.

The GP argument identifies the following failure mode. Suppose Bob has access to a local dynamical rule $\mathcal{E}$ that maps inputs to outputs, and suppose this rule is not well-defined on ${\rho }_B$ alone, but (implicitly or explicitly) depends on the specific ensemble used to realize ${\rho }_B$. Then Alice can choose between two decompositions of the same ${\rho }_B$ and thereby change Bob’s output statistics. This is superluminal signaling.

Appendix A.2. Affine Linearity from Local Randomization

The minimal operational closure property one expects of any laboratory theory is closure under classical randomization: Bob can flip a classical coin and decide to prepare ${\rho }_1$ with probability $\lambda$ and ${\rho }_2$ with probability $1-\lambda$. Operationally, this is a legitimate local preparation procedure, hence it must be represented by the mixed state

$${\rho }_{\lambda }=\lambda {\rho }_1+(1-\lambda ){\rho }_2.$$

(A4)

If Bob now applies a physically admissible local transformation $\mathcal{E}$, the output should be independent of whether the randomization occurred “before” or “after” the transformation. This forces

$$\mathcal{E}\left(\lambda {\rho }_1+(1-\lambda ){\rho }_2\right)=\lambda \mathcal{E}\left({\rho }_1\right)+(1-\lambda )\mathcal{E}\left({\rho }_2\right),0\le \lambda \le 1.$$

(A5)

Equation (A5) is affine linearity. Any non-affine $\mathcal{E}$ allows Bob to operationally distinguish different decompositions of the same $\rho$ (because mixing is itself a preparation operation), and thus recreates the GP signaling mechanism.

Two clarifications align this with the TCGS posture:

• Affine linearity is a shadow-level constraint: it is imposed on the operational state space $\mathcal{S}$ (represented by density operators). It does not preclude a nonlinear law on the underlying 4D source configuration space $\mathcal{C}$.
• What GP rules out is non-affine operational maps. The TCGS claim is that source-level nonlinearity, when projected through the quotient map defining the shadow state, induces an affine (indeed CPTP) evolution on $\rho$ and therefore respects the no-signaling closure required by experiments.

Appendix D.1. Remote Preparation of Inequivalent Ensembles with the Same ρ B

If Alice measures in the computational basis $\left\{\right|0\rangle ,|1\rangle \}$, Bob is prepared in the ensemble

$${\rho }_B={\displaystyle \frac{1}{2}}|0\rangle \langle 0|+{\displaystyle \frac{1}{2}}|1\rangle \langle 1|.$$

(A9)

If instead Alice measures in the Hadamard basis $\left\{\right|+\rangle ,|-\rangle \}$, with $|\pm \rangle =\left(\right|0\rangle \pm |1\rangle )/\sqrt{2}$, then Bob is prepared in the ensemble

$${\rho }_B={\displaystyle \frac{1}{2}}|+\rangle \langle +|+{\displaystyle \frac{1}{2}}|-\rangle \langle -|.$$

(A10)

Both ensembles yield the same density matrix $I/2$.

In linear quantum mechanics, any subsequent local processing by Bob is described by a CPTP map $\mathcal{E}$, hence

$$\mathcal{E}\left({\displaystyle \frac{I}{2}}\right)={\displaystyle \frac{1}{2}}\mathcal{E}\left(\right|0\rangle \langle 0\left|\right)+{\displaystyle \frac{1}{2}}\mathcal{E}\left(\right|1\rangle \langle 1\left|\right)={\displaystyle \frac{1}{2}}\mathcal{E}\left(\right|+\rangle \langle +\left|\right)+{\displaystyle \frac{1}{2}}\mathcal{E}\left(\right|-\rangle \langle -\left|\right),$$

(A11)

and no signaling is possible.

Appendix D.2. Non-Affine Evolution Generates Distinguishable Output Ensembles

Now suppose, instead, that Bob postulates a “local” evolution rule on pure states that depends nonlinearly on the state. One schematic form (sufficient for the logical point) is a deterministic, norm-preserving map $|\psi \rangle \mapsto |{\psi }^{\prime }\rangle$ that depends on the amplitudes through a nonlinear functional. In Weinberg-type constructions this arises when the effective Hamiltonian depends on expectation values or other state functionals [3,4]. Gisin’s observation is that if such a rule is applied to the elements of an ensemble, then the image of a mixed state depends on its decomposition [1].

In Weinberg-type constructions, state dependence typically arises because the effective Hamiltonian or generator depends on expectation values or other functionals of the state [3,4]. The GP point is that this dependence is benign only if it is not promoted to a decomposition-sensitive local rule on operational states.

To exhibit the mechanism with complete transparency, pick a toy but explicit state-dependent unitary of the form

$$U_{\psi }:=exp\left(-i\theta \left({\langle {\sigma }_z\rangle }_{\psi }\right){\sigma }_y\right),\theta \left(x\right):=\kappa x,$$

(A12)

where $\kappa$ is a fixed (small) constant and ${\langle {\sigma }_z\rangle }_{\psi }:=\langle \psi |{\sigma }_z|\psi \rangle$. This is a rotation about the y-axis by an angle proportional to the z-polarization of the input state. For the four states relevant to EPR steering one has

$${\langle {\sigma }_z\rangle }_{|0\rangle }=+1,{\langle {\sigma }_z\rangle }_{|1\rangle }=-1,{\langle {\sigma }_z\rangle }_{|+\rangle }=0,{\langle {\sigma }_z\rangle }_{|-\rangle }=0,$$

(A13)

hence

$$U_{|0\rangle }=e^{-i\kappa {\sigma }_y},U_{|1\rangle }=e^{+i\kappa {\sigma }_y},U_{|+\rangle }=U_{|-\rangle }=I.$$

(A14)

If one then defines the action on mixtures by acting on ensemble elements, the Z-prepared mixture evolves into a convex sum of two rotated pure states, whereas the X-prepared mixture is left invariant. Consequently, for $\kappa \ne 0$ the two output density matrices differ and Bob can detect Alice’s basis choice by a local measurement (e.g., of ${\sigma }_x$), completing the signaling protocol [1,2].

The point of writing this explicit toy model is not to advocate (A12) as a physical law; it is to display the structural fact that any decomposition-sensitive non-affine “local” rule recreates the GP lever.

Concretely, suppose the rule acts as

$$\mathcal{N}\left(|\psi \rangle \langle \psi |\right)=U_{\psi }|\psi \rangle \langle \psi |U_{\psi }^{†},$$

(A15)

where $U_{\psi }$ is a unitary that depends on $|\psi \rangle$ (for example through a phase rotation by an angle that is a nonlinear function of ${\langle {\sigma }_z\rangle }_{\psi }$). The resulting action on a mixed state is defined by acting on the ensemble elements:

$$\rho =\sum _i{p}_i|{\psi }_i\rangle \langle {\psi }_i|⇝{\rho }^{\prime }=\sum _i{p}_i\mathcal{N}\left(|{\psi }_i\rangle \langle {\psi }_i|\right).$$

(A16)

Unless $\mathcal{N}$ is affine in $\rho$, ${\rho }^{\prime }$ depends on the chosen decomposition. In the present EPR scenario, the output mixture

$${\rho }_Z^{\prime }={\displaystyle \frac{1}{2}}\mathcal{N}\left(\right|0\rangle \langle 0\left|\right)+{\displaystyle \frac{1}{2}}\mathcal{N}\left(\right|1\rangle \langle 1\left|\right)$$

(A17)

will generically differ from

$${\rho }_X^{\prime }={\displaystyle \frac{1}{2}}\mathcal{N}\left(\right|+\rangle \langle +\left|\right)+{\displaystyle \frac{1}{2}}\mathcal{N}\left(\right|-\rangle \langle -\left|\right),$$

(A18)

because the state-dependent unitaries $U_{\psi }$ differ between $|0\rangle ,|1\rangle$ and $|+\rangle ,|-\rangle$. Bob can therefore measure an observable $O_B$ for which $\left({\rho }_Z^{\prime }{O}_B\right)\ne \left({\rho }_X^{\prime }{O}_B\right)$, and infer Alice’s measurement choice faster than light.

This is the operational core of the GP theorem: if one treats a pure-state dependent, non-affine rule as a “local” operation, remote parties can steer ensembles and thereby signal. Polchinski refines this point by showing that to prevent such EPR signaling, observables for a separated system must depend only on the reduced density matrix [2].

Appendix D.3. Why This Does Not Target TCGS-SEQUENTION

In TCGS, Bob is never permitted to apply a map defined on fiber representatives $|\psi \rangle$ in the above sense, because those representatives are not operational inputs. Bob’s operational input is the equivalence class (density matrix) $s_B\sim {\rho }_B$, and any admissible shadow-level map must therefore satisfy the affine constraint (A5) and, in the entangled setting, the complete-positivity constraint described in Appendix B. Source-level nonlinearities live in $F_t$ on $\mathcal{C}$, but the induced shadow map ${\Phi }_t$ is defined on $\mathcal{S}$ and is required to be well-defined on equivalence classes. Consequently, the decomposition dependence exploited above cannot be operationally accessed, and the GP signaling channel is absent by construction.