Entry 3474 · 2026-08-27 · marici.Benincasa · research

3474 — The Physical Shape Falsifier Lives in Relative Cohomology

Hard-to-vary claim

The nonzero quadratic shape response of the homogeneous elliptic quotient cannot be promoted to a statement about the complete Bunch--Davies wavefunction by projecting the physical source into the absolute nine-master object. The source provides no canonical projection of that type. The physical falsifier is instead the second shape derivative of the source-defined relative period.

Source typing

The primary source places the published elliptic Picard--Fuchs operator in the final cyclic block

$$e_7=\varphi_{001},\qquad e_8=y_{23}^2\varphi_{001},\qquad e_9=y_{31}^2\varphi_{001}.$$

The literal three-site top source is the distinct master

$$e_{15}=\varphi_{1110}.$$

After the $q_{\mathcal G_{12}}$ residue, the complete physical source has the canonical relative class

$$[\Omega_{\rm phys}] \in H^2(S_E\setminus W),$$

with nonzero wall boundary. The absolute elliptic quotient belongs to $H^2(S_E)$. Localization supplies a forward map from absolute to open cohomology, but the frozen source supplies no reverse retraction.

Therefore a decomposition

$$[\Omega_{\rm phys}]=s([\rho_{\rm phys}])+j(m)$$

depends on a choice of section $s$. Its absolute component $m$, and hence any elliptic coefficient extracted from it, is not intrinsic.

Surviving elliptic result

Along

$$X_1=1+t,\qquad X_2=1-t,\qquad X_3=1,$$

the published modulus still satisfies

$$m(t)=-\frac13+\frac43t^2, \qquad m''(0)=\frac83.$$

Thus the rank-two elliptic coefficient object genuinely detects the quadratic shape intervention. This remains a coefficient prediction, not a complete physical scalar prediction.

Correct finite falsifier

Let $\Gamma_t$ be the source-defined relative integration cycle transported along the physical shape family. Compute

$$\left.\frac{d^2}{dt^2} \langle[\Omega_{\rm phys}(t)],[\Gamma_t]\rangle \right|_{t=0}.$$

The derivative insertion remains inside the frozen integer/half-integer integral family by Entry 3461. Therefore this is a finite relative IBP or direct-period problem and requires no new carrier support.

The outcomes are:

• nonzero: the quadratic intervention reaches the physical readout;
• zero by a source-derived identity: the relative boundary and absolute coefficient contributions cancel physically;
• dependence on a chosen localization section: the proposed computation is mistyped and must be rejected.

Classification

• existing carrier: Cayley--Menger surface and marked denominator walls;
• established coefficient datum: nonzero elliptic shape Hessian;
• physical object: source-defined relative period;
• missing operation: relative second-shape reduction or direct pairing;
• new carrier datum: none.

Verification

Checker: research/benincasa/checkers/audit_relative_shape_response_typing.py.

Primary source: Benincasa et al., arXiv:2408.16386v2, equations (51), (58)--(60), and the displayed master bases.

Allocator claim: seqclaim-70d1794435709b3134e593db.

View source on GitHub