Projective rays here describe field coefficients. Particle positions remain ordinary spatial coordinates in the infinite-dimensional containing space.

Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: A normal-ordered fermionic scalar interaction yields the auxiliary chiral polarization coefficient under Gaussian closure, while an exact one-particle counterexample fixes the limits of that bridge.

Written scalar term versus the attractive auxiliary energy

The self-contact module writes L_sc=−kappa J²/2 and J=bar(Psi)Psi. This derivative-free term contributes +kappa J²/2 to the Hamiltonian. In the fixed positive-energy mode of the current-projection note, J=c n with c=m/E. Absorb c² and a positive spatial mode-overlap factor into kappa_eff; this note writes kappa for that effective coefficient. The source coefficient's sign must be retained.

For positive kappa, the commuting classical field energy +kappa n²/2 is not the bare attractive term −3g²n²/(4mu) obtained by eliminating the bivector in the chiral selection model. Directly equating these two drops an energy sign. If the microscopic coefficient is negative, that is a different case requiring density stability checks, not a sign inferred from this note.

Exact exchange decomposition under a specified closure

There is a more structured route when the field is fermionically quantized. In a finite four-mode regulator, let a_i satisfy canonical anticommutation, rho_ij=<a_j dagger a_i>, and J-hat=a dagger C a. Normal ordering gives :J-hat²:=J-hat²−a dagger C² a. For a number-conserving Gaussian fermion state, the exact expectation is

EG(ρ)=κ2[(tr⁡Cρ)2−tr⁡(CρCρ)]. E_{\rm G}(\rho)=\frac{\kappa}{2} \left[(\operatorname{tr}C\rho)^2- \operatorname{tr}(C\rho C\rho)\right].

This is checked directly in the full sixteen-dimensional Fock space. The mode regulator avoids undefined products of continuum fields at a point; it does not specify the continuum regularization or renormalized coupling. Fermionic quantization, normal ordering, and Gaussian closure are additional structure not established merely by the written classical Dirac equation.

For C=I after the stated coefficient absorption and n=tr rho,

∥J(ρ)∥2=4tr⁡ρ2−n2,EG=3κ8n2−κ8∥J(ρ)∥2. \|J(\rho)\|^2=4\operatorname{tr}\rho^2-n^2, \qquad E_{\rm G}=\frac{3\kappa}{8}n^2-\frac{\kappa}{8}\|J(\rho)\|^2.

The negative polarization term follows from exchange while kappa stays positive. It has the exact auxiliary representation

EG=3κ8n2+min⁡B{μ∥B∥2−g⟨B,J(ρ)⟩},g2μ=κ2. E_{\rm G}=\frac{3\kappa}{8}n^2+ \min_B\{\mu\|B\|^2-g\langle B,J(\rho)\rangle\}, \qquad \frac{g^2}{\mu}=\frac{\kappa}{2}.

Thus the coefficient combination in the chiral selection model can be matched to a projected scalar exchange channel without arbitrarily reversing the master sign. The direct density term is essential. It was constant in the earlier fixed-normalization orientation problem, but cannot be discarded in a spatial density calculation. Algebraic auxiliary elimination supplies no independent B inertia or new propagating mode.

For a pure density matrix rho=n zz dagger, the positive density term cancels the negative exchange term. This is consistent with a normal-ordered two-body contact having no self-interaction in an exact one-particle sector. The auxiliary CP3 family remains the pure-ray family of that representation; this alone does not establish physical self-binding.

Exact limitation: the closure cannot be treated as universal

In the same finite Fock model, H=kappa N(N−1)/2 vanishes throughout N=1. An equal classical mixture of the four one-particle states has rho=I4/4 and exact interaction energy zero. Substituting that rho into the Gaussian formula instead gives 3kappa/8. That mixed state is not Gaussian; the Gaussian state with the same rho has particle-number fluctuations. The discrepancy is an exact counterexample, not roundoff or a failed numerical search.

Consequently the exchange expression cannot be used to claim that the exact scalar interaction purifies a mixed single electron. Its apparent preference for a rank-one rho depends on the class of states over which the energy is varied. Also, its self-consistent one-body matrix kappa(nI−rho) commutes with rho, so this scalar channel by itself produces no polarization torque.

If V2 requires nonlinear one-particle self-contact to generate its mass operator, the normal-ordered two-body replacement above is not automatically an equivalent formulation of that self-contact. It must be justified rather than silently adopted. The mass-formula depth assignment is kept fixed; this note tests a proposed interaction bridge, not that assignment.

Settled result and remaining decision

The coefficient match is proved within the stated fermionic Gaussian reduction. The direct classical sign mismatch and the exact mixed-state counterexample are also proved. None invalidates the earlier effective CP3-selection theorem; they determine whether this particular master-level implementation is available. A derivation must specify V2's field statistics, current prescription, occupied-state approximation, and treatment of one-particle self-contact before this exchange route can be promoted to the physical source of the effective model.

Verification

master_chiral_bridge_checks.py checks canonical anticommutation, exact Gaussian Fock expectations for general Hermitian C, the mixed-density bivector identity, coefficient matching, the no-torque identity, and the exact one-particle closure counterexample. All verdicts use explicit runtime conditions, including under Python -O.