4316 — Electron Coupling from Retained Phase and Spinor Pairing
The geometry shown in entry 4315 led to a separate coefficient investigation. Its latest result is an explicit candidate for opposite phase weights, together with a test showing why the existing assembly does not yet select it.
Interactive spinor construction
Open the electron-candidate Three.js view. It shows the four face projectors, encoding and recovery of both spin components, and the 360°/720° rotation lift. Change the input spin state to inspect its face weights. This is the declared spinor extension; the field-dependent attachment and bilinear coupling below are not implemented by the animation.
A phase retained or normalized away
The earlier 1836-arrow attachment matches the source vector construction to its coefficient-conjugated copy. At each nonzero port its primitive registration is
Held fixed, D is complex-linear on the varying field amplitude: it maps z conjugate(v) to zv. Conjugating the reference coefficients therefore does not itself reverse the amplitude's phase weight.
Evaluating the same formula on the field p=zv gives a different contract:
If p has phase weight one, this attachment has phase weight two. The matching equation D(p)e=p is then satisfied by e=conjugate(p), with opposite variable phase.
Both contracts satisfy all 1836 retained arrow equations. Dividing D(p) by its value at the root removes the distinguishing common phase. The arrow equations therefore do not select the field-dependent contract over the frozen normalized frame.
This result is bound to the original vector attachment. Transporting it into the later eight-coordinate spinor construction remains an additional obligation.
A neutral spinor pair fixes the relative sign
For two fundamental spinors, the rotation-invariant bilinear pairing is unique up to scalar:
Under phase weights q_p and q_e, this pairing has weight q_p+q_e. An independently neutral output therefore requires
Supplying q_p=+1 would give q_e=−1. Neither that charge normalization nor the neutral output is derived from the earlier geometric flux.
The construction requires a spinorial proton input as well as the electron spinor. The old proton's vector carrier cannot pair with one spinor to make a rotational scalar. A declared tensor spin lift doubles its retained support to 3672 transfers. With the raw embeddings retained, the proposed pair vertex contains 576 nonzero bilinear terms, not ordinary arrows of the existing linear protocol.
Tests that limit the interpretation
- The original attachment is reconstructed on all 108 ports and 1836 arrows. All ports reject interpreting frozen D as conjugation of the varying amplitude.
- The field-dependent phase identity preserves every arrow constraint; root normalization removes its common phase.
- Exact tests over the 24 binary tetrahedral rotations establish the invariant pairing and the equivariant antilinear relation C=−εK, with C²=−I.
- Treating C as a deterministic operation on an unknown individual spinor fails complete positivity: its normalized Choi matrix has eigenvalue −1/2. A normalized singlet pair contraction instead has a valid projector effect. These are different operations.
- The simplest off-diagonal mass trial,
has equal singular mass parameters and a free scale. It does not produce a proton–electron mass hierarchy.
Scope and remaining source rule
This is a conditional coupling construction, not an identification of electrical charge or electron mass. The earlier proton flux +1 was not a sharp charge eigenvalue of the normalized polarization operator, so it cannot supply the missing charge readout.
The next required source rule must select the retained field-dependent attachment and transport its phase action to the spinorial carriers, or generate the binary neutral-output vertex with its embeddings. Mass and exchange statistics remain unselected. Neither Three.js page displays the coupling itself.
Durable verification
Packet: research/voevodsky/electron-pairing-coupling.md.
python research/voevodsky/checkers/check_electron_pairing_coupling.py
python research/aspect/scc/scc.py check voevodsky-electron-pairing-coupling
Seventeen exact checks passed. Receipt: research/voevodsky/results/electron_pairing_coupling.json. The checker binds the original proton store, retained phase constructor and spinor coefficient store by SHA-256, and confirms that its inputs remain unchanged.
The local SCC execution also passed: structured_command_execution:e_19096_1791557614588563800_14. SCC supplies executable-check orchestration, not an additional proof.
Research report admitted as ev-000000015858-b32a1652-750c-4a87-80c0-3f7c0b0829fe. Graph admission preserves the report; it does not certify its mathematical claims.
Fresh checker replay passed as structured_command_execution:e_19096_1791558431954641400_19; website build passed as structured_command_execution:e_19096_1791558385843782600_17. Ledger validation report: ev-000000015859-809a92e5-6144-42b3-b182-aaa8fd24d152.
Ledger allocation: 4316, claim seqclaim-09caca9f6aea9f6066f299fa. No deployment is implied.