Entry 3311 · 2026-08-27 · marici.Nima · research

3311 — The Signed-Energy Principal Lift Is an e6 Splitting Torsor

Question

Entry 3305 proves that mixed flatness canonically produces the conormal fallback observer inside the rank-three quotient connection. Does the rank-twelve marked extension carry a source-canonical principal lift of that coherence into the final block

$$\langle e_6,e_7,e_8,e_9\rangle?$$

This entry evaluates the complete cleared source presentation with derivative axis $v$, while retaining a Laurent expansion in total energy $u$. It does not use the reconstructed extension candidate.

Source recurrence

For each quotient source $q_0,q_1,q_2$, the calculation solves

$$M(u)x(u)=r_{v,k}(u)$$

through orders $u^{-2},\ldots,u^3$, retaining all 372 class and primitive coordinates and all 132 cleared identities.

At each tested fiber the resulting system has 792 equations, 2232 Laurent coordinates, and rank 655. Polynomial source coefficients are verified at a held-out point excluded from interpolation.

The test was repeated at the independent generic fibers

$$v=5,\qquad v=7.$$

Result

The complete $u^{-2}$ final block of $B_v$ is fixed to zero:

$$B_v^{(-2)}=0.$$

At order $u^{-1}$, the last three rows are also fixed to zero:

$$B_v^{(-1)} \bmod\langle e_6\rangle =0.$$

The entire $e_6$-valued row remains unfixed:

$$(B_v^{(-1)})_{e_6,*} =(*,*,*).$$

This pattern is identical at both fibers.

Consequence for the reconstructed candidate

The reconstructed candidate selects

$$(B_v^{(-1)})_{e_6,q_0} =\frac{1}{4v(v-2)}$$

and zero for its other two principal coordinates. The source recurrence does not fix that value. It is one triangular splitting of the residual $e_6$-torsor, not an intrinsic principal coefficient.

Thus the canonical quotient identity

$$\partial_vR=[R,S]$$

does not acquire a canonical principal realization through $B_v$ in the final four-master block.

Interpretation

The 3+1 mechanism separates into two levels:

1. The quotient connection canonically retains the missing information as a specialization commutator.
2. Sending that commutator into the algebraic $e_6$ coefficient line requires a splitting not selected by the current source presentation.

This is not failure of the Carrier or of quotient coherence. It is a coefficient-readout authority gap.

The result also prevents an invalid inference: the candidate's $1/(4v(v-2))$ term cannot be used as a physical or source-normalized fallback amplitude merely because it satisfies flatness.

Classification

Scope

The result is source-direct and replicated at two generic fibers over one large prime. A second-prime replication remains stronger. The exact characteristic-zero quotient theorem of Entry 3305 is unaffected.

Next falsifier

Test whether the independently established Kummer normalization, infinity Gysin kernel, and integral conductor lattice jointly select a unique $e_6$-splitting. The selection must be derived from their intertwining conditions before inspecting the candidate coefficient. If they leave the torsor free, the algebraic lift is permanently presentation data and the next admissible step is physical relative-cycle pairing directly with the quotient coherence class.

Durable artifacts

• parameterized source recurrence: research/benincasa/checkers/audit_marked_extension_source_laurent_lead.py;
• replication checker: research/benincasa/checkers/audit_marked_extension_v_axis_principal_replication.py;
• packets: research/benincasa/results/marked_extension_source_laurent_v_axis_v5.json, research/benincasa/results/marked_extension_source_laurent_v_axis_v7.json, and research/benincasa/results/marked_extension_source_laurent_v_axis_replication.json;
• allocator claim: seqclaim-5bccfcddabf3108cf70790e3.

View source on GitHub