Dimensional Constriction and Hadronic Mass — Part 13

58. Investigation ledger: what has been tested and what remains

This section records the current state of the hadronic-constriction investigation so that the chain of reasoning is not dependent on chat history or transient calculations.

58.1 Supported structural chain

The following pieces currently survive the attempted falsifications.

The depth- $3$ --depth- $4$ coupling matrix obeys the rank-one identity $g_{34}^2=g_{33}g_{44}.$
Constraining the relaxed depth- $4$ variable by a Schur-complement calculation restores the depth- $3$ stiffness $g_{\rm con} =\frac{g_{34}^2}{g_{44}} =g_{33}.$
Conditional on the normal scalar block inheriting the depth- $3$ scalar-contact normalization, its physical energy unit is $\mu_3g_{33}.$
Canonical symmetric reciprocal normalization of the $4\to3$ phase-space carrier gives $\frac76,$ not $7/3$ .
Under the fixed-total-action constriction assumption, $\omega_3=\frac43\omega_4.$
Combining these gives the strongest presently supported band-blind constriction quantum $\boxed{ \varepsilon_{\rm constr} =\frac{14}{9}\mu_3g_{33} =154.808228\ {\rm MeV}. }$
The light-nucleon bulk contribution is numerically almost exactly twice this value: $2\varepsilon_{\rm constr} =309.616456\ {\rm MeV}.$

This numerical relation is retained as a clue, not promoted to a derivation.

58.2 Factor-two mechanisms explicitly tested and rejected as automatic consequences

The following possible explanations for the missing factor two have been checked and do not follow from the current V2 construction:

raw $8\to6$ phase-space dimension counting;
reciprocal zero-point trace interpreted as an oscillator gap;
counting coordinate and momentum quadratures separately;
adding virial kinetic and potential halves twice;
Gaussian self-contact curvature once exact self-consistent profiles are used;
exact cubic Schrödinger self-localization;
exact scalar Soler/Dirac self-localization;
Fubini--Study coherent-state contact curvature;
two Euler--Lagrange variations of one reciprocal interaction;
Dirac chirality;
particle/antiparticle doubling;
normal-connection multiplicity without a fixed independent norm;
baryon $K_3$ endpoint-incidence counting;
$K_3$ collective-mode multiplicity;
a second positive eigenvalue of the rank-one $3$ -- $4$ coupling matrix;
parity/even-level selection in the presently written V2 modules;
ordinary harmonic-oscillator quantization;
Dirac-oscillator spinor structure;
reinterpretation of the scalar mass operator as a mass-squared eigenvalue.

Several of these routes reproduce a factor two only by counting one physical interaction or one oscillator twice, so they are excluded on energy-bookkeeping grounds.

58.3 Genuine independent channel that survives

The codimension-one hypersurface Dirac decomposition contains a real lower-order extrinsic term,

$D_4 =\gamma(n)\left(\nabla_n+D_3-\frac32H\right),$

so the corresponding algebraic energy scale is

$E_H=\frac32\hbar c\,|H|.$

Unlike the rejected doubling mechanisms, this is a genuinely distinct operator contribution.

To provide exactly the missing light-nucleon contribution it would require

$\boxed{ |H_3| =\frac{28}{27\hbar c}\mu_3g_{33} \approx0.52301761\ {\rm fm}^{-1}. }$

This relation has not been derived from nucleon-independent V2 geometry.

58.4 Geometry-scale searches already performed

The following existing V2 structures were checked for an equation fixing this mean curvature:

tetrahedral/centred-simplex geometry;
the projective observer map;
cross-depth full-marginal normalization;
the sector projector definition;
derivatives of the zeroth-order projector;
normal connection and lower-order projected Dirac terms;
second-fundamental-form/extrinsic remainder terms;
the existing master-action corpus.

The result is consistent across these checks:

projective/tetrahedral geometry fixes dimensionless shape and angles but leaves the overall spatial scale free;
full cross-depth marginalization fixes normalized contact weight but not embedding radius or curvature;
$P_d$ is assumed to be a smooth zeroth-order projector and its derivatives may generate lower-order geometry, but no equation of motion for $P_d$ is currently supplied;
the files describe $P_d$ as dynamically defined, yet no written variational dynamics for $P_d$ , the embedding, $H_d$ , or the second fundamental form is present;
no Einstein--Hilbert, projector sigma-model, Nambu--Goto, Polyakov, Willmore, bending, or other geometric action was found in the current V2 corpus.

58.5 Exact open problem

The present investigation has therefore isolated a specific missing piece rather than merely an unexplained numerical coefficient:

$\boxed{ \text{V2 needs dynamics for the sector embedding/projector geometry.} }$

Those dynamics must determine the lower-order normal/extrinsic part of

$P_3D_\infty P_4$

without using the proton or neutron mass as an input.

The strongest V2-native route to test next is whether the already present projected Dirac/contact fields induce an effective geometric action

$S_{\rm eff}[P,g,\mathrm{II}],$

whose stationary point fixes

$H_3\propto\frac{\mu_3g_{33}}{\hbar c}.$

If such a derivation predicts the coefficient $28/27$ , the missing factor two would be closed by a genuinely independent extrinsic Dirac contribution. If it predicts another coefficient or no finite curvature scale, the near- $309.6$ MeV relation must be regarded as accidental or explained by some different new V2 structure.

58.6 Recording rule for the continuation

From this point onward, every substantive calculation in this investigation should be recorded in this file with:

the hypothesis being tested;
the mathematical derivation or numerical test;
whether the hypothesis survives or fails;
any assumptions required;
the next narrowed proof obligation.

Transient shell commands and search mechanics need not be preserved verbatim, but no scientific conclusion or failed derivation should exist only in chat history.

59. No induced geometric action is presently derived in V2

The V2 corpus was searched for an induced geometric action generated by the projected Dirac/contact sector. Search targets included

effective actions;
fermion determinants;
$\operatorname{Tr}\log D$ constructions;
one-loop or heat-kernel effective actions;
Seeley--DeWitt coefficients;
spectral actions;
induced gravity or induced curvature terms;
Lichnerowicz-based curvature expansions.

No such derivation is presently contained in the V2 files.

The only independent heat-kernel reference in the geometry modules concerns short-time heat-kernel behaviour as a diagnostic of spectral dimension. It does not construct a curvature-dependent effective action for the sector projector or embedding.

Therefore the proposed route

$\Psi_\infty \longrightarrow S_{\rm eff}[P,g,\mathrm{II}] \longrightarrow H_3$

remains a possible completion strategy, not an already existing result of V2.

In particular, the current files do not derive a coefficient that would imply

$H_3 =\frac{28}{27\hbar c}\mu_3g_{33}.$

The next test is external to the presently written V2 equations but still mathematically diagnostic: determine whether a generic Dirac/contact-induced effective action would fix such a coefficient uniquely from existing V2 scales, or whether local counterterm/renormalization freedom leaves the curvature coefficient arbitrary. If the latter occurs, an induced-action route cannot close the nucleon factor-two problem without an additional V2 renormalization or geometric principle.