Psy02-A Conditional Scalar-Carrier Bounds and Operational Separation: A Mathematical Framework for Evaluating Orch-OR Hypotheses

A mathematical assessment of orchestrated objective reduction requires separating the construction of a quantum carrier from observable concentration, environmental coherence loss, measurement, causal influence, and interpretation as conscious experience. We develop a modular conditional framework motivated by Cheung's scalar-carrier, response, and causal-support manuscripts, while retaining their distinct models and proof boundaries. For displaced Fock-normal states, we prove finite-family concentration, ordered Weyl bounds, and exact residual-state fluctuation identities with explicit number-moment constants. An environmental moment inequality gives a time-uniform concentration certificate, but an independently specified trace-distance witness is required for coherence survival. Explicit attenuation and dephasing examples show that concentration, occupation, and a single accessible marginal do not determine quantum-information preservation. We construct a separate massless Wick-square response benchmark, classify its declared contact ambiguity, and evaluate an invariant nonzero test. Finite channel schedules yield reference-preserving ancestor bounds, whereas mechanism identification requires separated complete-record laws. A regularized finite-dimensional mass-density model supplies a well-defined reduction semigroup and stochastic unraveling, including the rate $E_G/\hslash$; its mass assignments and physical interpretation remain additional inputs. The complete work plan and a claim-status ledger identify the remaining carrier, biological, continuum, detector, causal, and interpretive obligations. No derivation of Orch-OR, neuronal quantum viability, quantum gravity, or conscious experience is asserted.

Keywords: Orch-OR; scalar carrier; Fock space; concentration; quantum coherence; instruments; response normalization; causal support; collapse models; identifiability.

The Orch-OR proposal associates biologically orchestrated quantum processes with a proposed gravity-related reduction mechanism and an interpretation in terms of conscious experience HameroffPenrose,Penrose,Diosi. These are distinguishable commitments. A calculation concerning a free scalar observable does not, without additional constructions, identify its carrier with a neuronal degree of freedom, establish the relevant environmental dynamics, or assign an experiential meaning to an event.

Cheung's Em06-A and Em06-B manuscripts supply the immediate mathematical motivation for distinguishing scalar-only classical targets, residual-state concentration, and normalized fluctuations CheungEm06A,CheungEm06B. VII-H motivates normalization-independent response diagnostics, while VII-J and VIII-C motivate a separation between scheduled support and actual influence CheungVIIH,CheungVIIJ,CheungVIIIC. En01-A, En01-B, En01-C, and Em05-A motivate the additional separation of accessible observations from coherence, recovery, and mechanism identification CheungEn01A,CheungEn01B,CheungEn01C,CheungEm05A.

boundary. The supplied landmark excerpts are documentary source records. They are not treated as an independently audited collection of complete proofs. The results below are proved for newly specified objects. When a bridge to a landmark's classical input map or to a biological model is not constructed here, it is explicitly an assumption or an unresolved obligation. Similar notation does not identify different Hilbert spaces, test classes, states, channels, prescriptions, or physical models.

repaired claims. The following corrections are part of the mathematical framework, not qualifications added after its conclusion.

repairs to the proposed landmark incorporation.

and novelty boundary. The Fock estimates and channel inequalities use standard methods. The contribution is their explicit assembly into a two-budget certification framework, together with counterexamples and interfaces that prevent invalid transfers between its modules. The finite reduction model is a declared regularized comparison model, not a derivation of a new physical collapse law.

Let $\mathfrak{h}$ be a separable complex Hilbert space whose inner product is linear in the second argument. Let

On the finite-particle domain ${\mathcal{D}}_{\mathrm{fin}}$, use

The map $v\mapsto \Phi (v)$ is real linear. Define

These conventions give

Standard Fock and Weyl constructions are discussed in Derezinski; the domain argument used below invokes the analytic-vector theorem Nelson.

A normal state is represented by a positive trace-class operator $\rho$ with $\mathrm{Tr}\rho =1$. Its residual number moment is

For an unbounded self-adjoint $B$, $\mathrm{Tr}(\rho B^2)$ means the spectral quadratic moment, which may be finite even when an operator product is not trace class in a naive sense.

For $v\in \mathfrak{h}$, $\Phi (v)$ is essentially self-adjoint on ${\mathcal{D}}_{\mathrm{fin}}$. Its closure satisfies, as a quadratic-form inequality on $D(N^{1/2})$,

Consequently, whenever $n(\rho )<\infty$,

Proof. The creation and annihilation estimates are

Apply $\text{norm}{x+y}^2\le 2\text{norm}{x}^2+2\text{norm}{y}^2$.

If $\psi$ has particle support at most $m$, repeated application of these estimates bounds

The corresponding analytic-vector series converges for every fixed argument. Thus ${\mathcal{D}}_{\mathrm{fin}}$ is a dense invariant domain of analytic vectors, and the analytic-vector theorem establishes essential self-adjointness.

The form estimate extends to $D(N^{1/2})$ by graph-norm approximation. Expand a positive trace-class $\rho$ in an orthonormal eigenbasis and sum the form estimate. Finite $n(\rho )$ ensures that the relevant eigenvectors with positive eigenvalue lie in $D(N^{1/2})$. The resulting bound is (eq:number-bound).

A compact spacetime test $f$ and its associated one-particle vector $v=\kappa (f)$ are different objects. Support restrictions apply initially to $f$, not automatically to a position representation of $v$.

For an explicit flat-space example, put

and, for real $f\in {C_c}^{\infty }({\mathbb{R}}^4)$, define

The value at the single point $\mathbf{k}=0$ is irrelevant to the $L^2$ class. Coordinates in the following estimate are expressed in a fixed reference unit.

The map ${\kappa }_m:{C_c}^{\infty }({\mathbb{R}}^4;\mathbb{R})\to L^2({\mathbb{R}}^3)$ is well defined and satisfies

The same inequality bounds $\text{norm}{\kappa }_m(f)-{\kappa }_m(g)$ with $f-g$ on the right.

If $f=({{\partial }_t}^2-{\Delta }_{\mathbf{x}}+m^2)q$ for compact smooth $q$, then ${\kappa }_m(f)=0$.

Proof. Integration by parts gives

Since ${\omega }_m(\mathbf{k})\ge \text{abs}\mathbf{k}$,

The integral is $2\pi$, proving (eq:test-bound). Linearity gives the difference estimate. The Fourier multiplier of the displayed wave operator vanishes on $\omega ={\omega }_m(\mathbf{k})$, proving the last assertion.

For a different background, classical metric input, switch, or relational test prescription, the corresponding map ${\kappa }_p$ must be constructed separately. The flat-space estimate does not prove a curved-background or unrestricted gauge theorem.

An admissible carrier specification consists of:

1. a fixed $\mathfrak{h}$, Fock representation, and field convention;

2. a named compact spacetime test class and a specified map $f\mapsto {\kappa }_p(f)$, or an explicit designation as an abstract oscillator surrogate;

3. real offsets $b_j$, vectors $v_j$, and a displacement direction $d$, for $j=1,\dots ,J$;

4. a normal residual state with finite number moment;

5. any common-support, plateau, and gauge restrictions used by the input map.

A bounded-observation correspondence consists of a state class ${\mathcal{S}}_{\mathrm{bio}}$, an encoding channel $\mathcal{E}$, matched effects $M_{\mathrm{bio}}$ and $M_{\mathrm{sc}}$, and a bound

The participating state and effect classes are part of the definition.

This comparison does not automatically control an unbounded field moment, an entire output channel, or reference-assisted recovery. Those require stronger, separately named comparisons.

No finite-dimensional matrices $Q,P\in M_d(\mathbb{C})$, $d\ge 1$, satisfy

when $c\ne 0$.

For the oscillator truncated to $\mathrm{span}\{|0⟩,\dots ,|d-1⟩\}$, define

Then

Proof. The trace of a commutator is zero, while $\mathrm{Tr}(\mathrm{i}c{1}_d)=\mathrm{i}cd\ne 0$. Direct multiplication gives $[a_d,{a_d}^{†}]=1_d-d|d-1⟩⟨d-1|$, which implies (eq:compressed-CCR).

An approximate spin–scalar bridge must therefore specify its restricted observations and its boundary or compression error. It cannot assert an exact nonzero canonical commutator.

For $\mathfrak{h}={\mathbb{C}}^m$, let $P_K$ project onto total number at most $K$, and let $Q_K=1-P_K$. Introduce a visible failure state $|\perp ⟩$.

The map

is deterministic and CPTP. The retained subspace has dimension

For $t_K=\mathrm{Tr}(\rho Q_K)$,

In the common output space containing the original Fock space and the flag,

Proof. Use $P_K$ as the low-block Kraus operator and $|\perp ⟩⟨e_{\alpha }|$ for an orthonormal basis of the high block. Their effects sum strongly to $1$. The dimension follows from counting occupation vectors.

On $Q_K$, $N\ge K+1$, giving the tail bound. Purify $\rho$ to $|\Psi ⟩$, and set $|{\Psi }_P⟩=(P_K\otimes 1)|\Psi ⟩$. A calculation in their two-dimensional span gives

Partial trace contracts the trace norm. The orthogonal flag contributes $t_K/2$ to the half trace distance, proving (eq:cutoff-general). When $[\rho ,P_K]=0$, the difference is a positive high block of trace $t_K$ and a negative orthogonal flag block of the same trace, proving (eq:cutoff-diagonal).

The exact linear tail error agrees with the number-diagonal regime described in Cheung's En02-E CheungEn02E. It must not be transferred unchanged to arbitrary coherent states.

Let $\rho ,\sigma$ be ancillary states independent of a message and its reference. For a fixed channel $\Lambda$, define

Then

Here and below $d_{\diamond }=\frac{1}{2}\text{norm}{\cdot }_{\diamond }$.

Proof. For every referenced input state $\xi$, the input difference is $\xi \otimes (\rho -\sigma )$, of trace norm $\text{norm}{\rho -\sigma }_1$. Apply channel contraction and take the supremum.

This is a common-interface ancillary statement. It is not an automatic biological encoding or a dynamical simulation theorem.

For $0<ϵ\le 1$, set

Thus the target is displacement compatible. An arbitrary collection of numbers is not silently presumed to arise from one common displacement.

By the Weyl convention,

Write

For each $j$, let ${\mu }_{j,ϵ}$ be the spectral law of ${\hat{O}}_{j,ϵ}$ in ${\tilde{\rho }}_{ϵ}$. Then

For $r>0$,

If $A(v_i,v_j)=0$ for every pair, the family strongly commutes and has a joint spectral law ${\mu }_{ϵ}$ on ${\mathbb{R}}^J$, with

The exact criterion for concentration of a selected coordinate is

The condition $ϵn_{ϵ}\to 0$ is sufficient for fixed bounded vector families, but is not necessary.

Proof. Equation (eq:displacement) and unitary invariance of spectral moments prove (eq:exact-mse). Lemma lem:number proves (eq:mse-bound). Cauchy–Schwarz gives the mean estimate. Variance about the actual mean is at most the mean-square error about a prescribed target.

There is only one coupling of a probability law with a point mass, so its squared Wasserstein distance is its squared distance moment. Markov's inequality gives the tail bound.

If all symplectic pairings vanish, all one-parameter Weyl groups commute by (eq:Weyl); hence the self-adjoint generators strongly commute. The joint spectral theorem applies. Its squared distance moment is the sum of the coordinate moments, proving (eq:joint-W2). The final criterion is the exact identity (eq:W2-exact).

For a fixed finite $J$, suppose

Then all coordinate mean errors, variances, mean-square errors, and spectral distances to their displacement-compatible targets vanish uniformly in $j$. For commuting families the joint Wasserstein distance also vanishes.

Proof. The common upper bound is $ϵ(4n_{ϵ}+2)K^2$, which tends to zero.

Put

For arbitrary real $t=(t_1,\dots ,t_J)$,

For residual vacuum,

Under the bounded-family assumptions of Corollary cor:uniform, convergence to $e^{\mathrm{i}t\cdot o}$ is uniform on compact $t$-sets.

Proof. Repeated application of (eq:Weyl) produces the phase $-\mathrm{i}ϵa(t)$ and the final direction $\sqrt{ϵ}V(t)$. Conjugation by $U_{ϵ}$ gives the target phase.

For self-adjoint $B$, spectral calculus gives

Apply this with $B=\sqrt{ϵ}\Phi (V(t))$, use Lemma lem:number, and combine with $\text{abs}{e}^{-\mathrm{i}ϵa}-1\le ϵ\text{abs}a$. Vacuum normal ordering gives $⟨0|W(v)|0⟩=e^{-\text{norm}{v}^2/2}$. On compact $t$-sets, the vector norms and symplectic phase have common bounds.

This reproduces the mathematical form of the supplied Em06-B ordered-correlation interface with the conventions fixed here CheungEm06B; it is not an independent audit of the landmark's classical relational map.

The ordered expression in (eq:ordered) need not be the characteristic function of a classical joint probability law.

Proof. Take residual vacuum, zero targets, $v_1=e$, $v_2=\mathrm{i}e$, with $\text{norm}e=1$. Then

A joint characteristic function must satisfy $f(-t,-s)=\overline{f(t,s)}$. The displayed function violates this identity whenever $\sin (ϵts)\ne 0$.

The fluctuation family has the exact residual representation

Its commutators, on the conjugated common core or as the corresponding form identities, are

The unnormalized observables instead satisfy

Concentration does not determine a unique fluctuation limit. A sufficient additional condition for convergence of bounded fluctuation expectations is trace-norm convergence of the residual states in the fixed representation.

Proof. The representation follows directly from (eq:displacement); the commutators follow from (eq:CCR). For every bounded operator $B$,

Apply this to fluctuation Weyl products or bounded spectral functions.

For nonuniqueness, let ${ϵ}_k=1/k$, and alternate ${\rho }_{{ϵ}_k}=|0⟩⟨0|$ and $|1⟩⟨1|$ in one mode. Then $n_{{ϵ}_k}\le 1$, so concentration holds. For $v=e$,

The two subsequences have distinct characteristic functions.

In one mode, a residual coherent state with real amplitude $c/(2\sqrt{ϵ})$ has

Thus vanishing variance does not imply concentration about the prescribed target. A symmetric mixture of the two opposite amplitudes has zero mean error but mean-square error $c^2+ϵ$.

The residual number is not the occupation of the displaced state. The displacement shifts annihilators by a scalar, giving the safe estimate

Even displaced vacuum can therefore have occupation of order ${ϵ}^{-1}$. Neither quantity counts modes or surviving biological degrees of freedom.

Let $s\in [0,T]$ be physical or model time. Specify normal residual states ${\rho }_{ϵ,s}$, vectors $v_j(s)$, offsets, and displacement-compatible targets. The actual state is

The chosen reduced dynamics preserves normality and finite number moments. The function $n_{ϵ}(s)=\mathrm{Tr}({\rho }_{ϵ,s}N)$ is absolutely continuous and satisfies, almost everywhere,

where $\alpha ,\beta$ are integrable and $\beta \ge 0$. These statements must be justified from the chosen dynamics; they are not consequences of static concentration.

Define

Under Assumption ass:environment,

If

then

The bound vanishes if $ϵ{\overline{n}}_{ϵ,T}\to 0$.

Proof. Multiply the moment inequality by $A(s{)}^{-1}$ and integrate. Apply Theorem thm:concentration at each time.

A bounded-generator theorem does not automatically justify this assumption for an unbounded field generator. Domain, moment, and conservativity arguments remain necessary for such a model.

Use the trace and channel distances

For channels, the latter controls all referenced input states. The relevant operational norm facts are summarized in Watrous.

Fix a dephasing channel $\Delta$, an input state $\rho$, and an accessible channel ${\Lambda }_s$. Define

The basis, decomposition, or record algebra defining $\Delta$ is part of the specification.

For the witness (eq:coherence-witness),

Further deterministic processing cannot increase $D_s$.

Suppose

with the same $\Delta$. Then

Thus a survival certificate is

Proof. The output difference is Hermitian and traceless. Its positive and negative parts have equal trace, each equal to half its trace norm. The positive spectral projector attains the effect supremum.

Trace distance contracts under a channel. For channel perturbations, each of the two normalized input states contributes trace-norm error at most $2e_s$. After multiplying their sum by $1/2$, the allowance is $2e_s$. For preparation changes, contractivity of ${\Lambda }_s$ and $\Delta$ gives

The reverse triangle inequality proves (eq:witness-stability).

This is the independently proved channel interface suggested by Cheung's En01-A CheungEn01A. It is not a statement about every possible definition of coherence.

On a declared finite code, suppose a decoder ${\mathcal{R}}_s$ satisfies

For code states $\rho ,\Delta \rho$,

Proof. Apply Theorem thm:witness to ${\mathcal{R}}_s{\Lambda }_s$ and use contraction under the decoder.

Let the system and bath have $m$ oscillator modes. A passive beam-splitter unitary has output annihilators

Prepare an independent bath state $\tau$ with $\mathrm{Tr}(\tau b_j)=0$ and finite total number $n_B$. Tracing the bath defines a channel ${\mathcal{A}}_{\eta }$. This is a standard attenuation construction HolevoWerner; the needed identities are proved below.

For every finite-number input $\rho$,

Furthermore,

Proof. Expand ${a_{j,\mathrm{out}}}^{†}{a}_{j,\mathrm{out}}$. The cross terms vanish because the bath is independent and centered. Sum over $j$. A displacement of the system transforms into a system displacement of amplitude $\sqrt{\eta }$ and a bath displacement. The latter disappears under partial trace.

For a vacuum bath and one mode, let

and let $\Delta$ dephase in the number basis.

For the preceding input,

With $\eta (s)=e^{-\kappa s}$, $\kappa >0$, the selected observable concentration bound is

whereas the witness is

For $0<d_{\min }\le 1/2$, witness survival up to $T$ is equivalent to

Proof. Vacuum attenuation maps

The output difference between the coherent and dephased inputs has off-diagonal entries $\sqrt{\eta }/2$, hence eigenvalues $\pm \sqrt{\eta }/2$. The remaining statements follow by substitution.

Concentration can therefore improve while the independently specified coherence witness decreases. A concentration condition does not replace the survival timescale.

In one mode,

and both inputs satisfy the same number-based concentration bound. Yet

Thus finite number and small scaled observable spread do not imply preservation of this quantum witness.

Proof. Both states have number populations $1/2,1/2$. Their difference has eigenvalues $\pm 1/2$. The dephasing channel annihilates that difference.

The identity qubit channel and complete $Z$-dephasing produce the same accessible output $I_2/2$ for the system half of a Bell input. Their optimal deterministic referenced recovery errors are, respectively,

in half-diamond distance to the identity.

Proof. Both channels fix $I_2/2$. Identity is recovered exactly. Complete dephasing is entanglement breaking, so every decoder leaves its Bell-input output separable across reference and receiver. A separable two-qubit state has Bell-projector expectation at most $1/2$, by Cauchy–Schwarz for pure product states and convexity. The Bell test therefore gives error at least $1/2$.

For attainment,

so $d_{\diamond }({\Delta }_Z,\mathrm{id})\le 1/2$.

This finite proof supplies the relevant distinction without importing the separate extraction model of Em05-A CheungEm05A.

Let $p$ collect classical metric data, switching data, observation tests, and any other classical inputs. A common representation and common admissible support class must be fixed before comparing $p$ and $p^{\prime }$.

For each selected observation, the constructed input map provides targets $o_j(p)$, vectors $v_j(p)$, and bounds

Any exact gauge identity asserts equality only on its declared common plateau. An outside-plateau defect is included in ${{\eta }^o}_j,{{\eta }^v}_j$ or in a separately bounded operator remainder.

For the explicit flat test map, Proposition prop:compact-test provides an instantiated ${{\eta }^v}_j$. For the full Em06-B relational input map, the corresponding metric, switch, and plateau estimates require that manuscript's complete definitions and constants CheungEm06B. The present assumption does not pretend to reconstruct them from a release summary.

For a state $\rho$, write

when finite, interpreted through the appropriate quadratic form.

A preparation allowance $p_j$ certifies

One sufficient construction uses common purifications ${\Psi }_{\rho },{\Psi }_{\sigma }$ and

The sufficient construction follows from the triangle inequality. It controls a selected unbounded moment, unlike an unqualified trace-distance preparation bound.

There are one-mode states ${\sigma }_M$ with

but

Proof. Take

The trace distance is $M^{-1/2}$. Since $⟨M|\Phi (e{)}^2|M⟩=2M+1$, the second moment is $1+2\sqrt{M}$.

For completeness, a trace-distance allowance becomes usable when supplemented by a tail certificate. Let

If $d_{\mathrm{tr}}(\rho ,\sigma )\le \delta$, then

Indeed $0\le A_L\le L^{2}1$, and an effect expectation changes by at most trace distance.

Suppose the implemented self-adjoint observable, in the declared residual frame, satisfies

where

The bounded remainder preserves self-adjointness on the field domain. Let $n_{\sigma }=\mathrm{Tr}(\sigma N)<\infty$.

Let $X_j$ be its spectral outcome in $\sigma$. A recorded surrogate outcome $Y_j$ has a specified coupling

This representation may be independent additive detector noise or a certified correlated readout error.

Finally, if a physical carrier is claimed, require a same-quantity comparison between its recorded law ${{\nu }_j}^{\mathrm{bio}}$ and the surrogate recorded law:

Both laws must have finite second moments.

Under the preceding hypotheses, define

where

Then

In the surrogate-only case set $c_j=0$ and replace the biological law by the surrogate law.

The mean error is at most ${\mathcal{B}}_j$, and

Proof. Use the triangle inequality for the state quadratic norm in (eq:implementation). Definition def:preparation bounds the principal field term; Lemma lem:number bounds the vector-change term. The scalar and bounded-operator contributions are at most ${{\eta }^o}_j$ and $e_j$.

The spectral distance to the target point mass is the root mean-square error. Minkowski gives

Finally apply the Wasserstein triangle inequality to (eq:carrier-W2). Cauchy–Schwarz and Markov's inequality give the last claims.

Every term in (eq:complete-budget) has the units of the same scalar readout. A dimensionless effect-probability error such as (eq:carrier-effect) cannot simply be added to this root mean-square budget.

Let $E_B$ be the spectral measure of a self-adjoint $B$, and let $\lambda$ be a probability law for independent detector noise.

The operator-valued measure

is a POVM. Its outcome law in a state $\sigma$ is the convolution of the spectral law of $B$ with $\lambda$.

For finite second moments and independent $X,Z$,

If the noise is centered, the last term vanishes. Without independence or a separately proved zero cross term, only the Minkowski allowance is asserted.

Proof. Positivity, normalization, and countable additivity follow from those of $\lambda$ and the spectral measure, using weak operator integration. The convolution statement follows by taking state expectations. Expand the square and use independence for the cross term.

This is a detector model. It does not represent arbitrary environmental decoherence.

An instrument with finite outcomes is a family of completely positive trace-nonincreasing maps $\{{\mathcal{I}}_x\}$ whose sum is trace preserving DaviesLewis.

For specified instruments ${\mathcal{I}}^{(1)},\dots ,{\mathcal{I}}^{(k)}$,

is a probability law. The law depends on the instruments, not merely on a list of self-adjoint observables or their ordered Weyl expectations.

Proof. Complete positivity gives nonnegative outcome probabilities. Summing the last outcome applies a trace-preserving map. Repeat backwards through the sequence to obtain total probability $\mathrm{Tr}\rho =1$. Different state-update maps with the same effects can give different later probabilities.

This section constructs a distinct free massless vacuum benchmark in the spirit of VII-H CheungVIIH. It is not a susceptibility of the displaced family in Section sec:static, and its contact ambiguity is not a residual-state second moment. The Wick framework and distribution-extension principles are standard BrunettiFredenhagen,HollandsWald.

Work on four-dimensional Minkowski space, with $c=1$, and write

Use a free massless scalar vacuum with

Let

be vacuum-calibrated. Fix the source sign so that the first retarded response is