Scalar Master Geometry and the Cyclic Lift

Purpose

Marici investigates whether NLSM, Yang–Mills, and gravity arise from distinct intrinsic operations on one scalar master geometry. The active problem is not amplitude reconstruction; it is the existence of a canonical off-shell cyclic/BV lift of the pure-Einstein state projector.

Established within this project

The working branches are

[ \mathrm{NLSM}=\operatorname{gr}_{\mathrm{rank\ jump}}(\mathrm{Scalar})\to\mathrm{QTDS}, ]

[ \mathrm{YM}=H_{\mathrm{gauge}}(J^1\mathrm{Scalar}), ]

and

[ \mathrm{GR}_{\mathrm{tree}}=\operatorname{PrimSym}g^2(J^1\mathrm{Scalar}) \quad\text{paired by}\quad I{\mathrm{scalar}}^{-1}. ]

The internal Einstein projector and its projected coevaluation close arbitrary generalized cuts on pure graviton states.

Strong inference

The modular completion of the primitive-symmetric retract defines pure-Einstein generalized-cut data at all loops.

Active chain-level question

Let

[ C_W=C_{\mathrm{YM}}^L\otimes C_{\mathrm{YM}}^R. ]

Determine whether the cohomological projector

[ P_E=\frac12(1+\tau)-\frac1{D-2}\operatorname{coev}\circ\operatorname{ev} ]

admits a canonical cyclic idempotent lift satisfying

[ [Q_W,P_E]=0, \qquad \omega_W(P_Ex,y)=\omega_W(x,P_Ey). ]

The graded swap has a canonical chain lift. The transverse trace pairing is canonical on physical cohomology, but its strict chain representative may require noncanonical contraction or gauge-fixing data. The full Maxwell/BV complex has vanishing Euler characteristic, so its categorical coevaluation cannot simply be normalized by the physical dimension (D-2).

Complement

The ambient complement contains physical dilaton and antisymmetric-tensor cohomology. It is therefore not BRST-contractible in the doubled complex and is not generally an interaction ideal.

Epistemic perimeter

Do not claim that external projection commutes with ambient NS–NS quantization, that unwanted states are BRST exact, or that a canonical Einstein loop integrand has already been selected. Generalized-cut closure and a canonical off-shell representative are separate results.

Next falsification tests

  1. Construct the Maxwell/YM detour or BV complex and its cyclic pairing explicitly.
  2. Classify cyclic representatives of the physical transverse evaluation map.
  3. Test whether scalar first-jet geometry selects contraction data ((i,p,h)) canonically.
  4. If it does, transfer brackets and derive the projected propagator.
  5. If it does not, identify the first obstruction in cyclic mapping, Hochschild, BV, or modular-operadic cohomology.