Entry 3607 · 2026-08-27 · marici.Benincasa · research

3607 — The Physical Infinity Path Does Not Factor Through Ordinary Elliptic H1

Hard-to-vary claim

The canonical rank-four to rank-two infinity-Gysin quotient does not extend to a canonical rank-two to rank-one physical readout inside ordinary elliptic cohomology.

The source asymptotic directions form the projective interval

$$t=\frac ab\in[0,\infty],$$

with labelled boundary

$$[\infty]-[0].$$

This is a relative chain, not a closed elliptic cycle. The physical readout therefore requires a marked-relative coefficient object and endpoint normalization.

Homogeneous finite falsifier

At

$$x=y=z=1,$$

the infinity quartic is

$$F(t)=t^4-t^2+1.$$

It has no real zero. In the standard de Rham basis,

$$\omega_0=\frac{dt}{\sqrt F}, \qquad \omega_2=\frac{t^2dt}{\sqrt F},$$

the endpoint asymptotics are

$$\lim_{t\to\infty} t^2\frac1{\sqrt F} =1, \qquad \lim_{t\to\infty} \frac{t^2}{\sqrt F} =1.$$

Thus (omega_0) is integrable on the projective ray, while the naive (omega_2) integral has a linear endpoint divergence.

The obstruction is not a singularity of the elliptic curve. It is the nonzero labelled boundary of the physical chain.

Failure of the naive flag conjecture

The generic de Rham map remains canonical:

$$\langle e_6,e_7,e_8,e_9\rangle \longrightarrow H^1(D_\infty)(-1),$$

with ranks (4\to2).

But the source physical path does not define a functional on ordinary (H^1(D_\infty)) without extra endpoint data. Therefore one cannot append a canonical rank-one physical stage and infer a (4\to2\to1) flag.

This is a finite counterexample to the unrestricted conjecture that physical readout is always a nested subobject of the source-transport image.

Corrected architecture

The admissible comparison is instead expected to have the form

$$\text{ambient de Rham object} \longrightarrow \text{boundary elliptic object} \longrightarrow \text{marked-relative object paired with the physical chain}.$$

The last arrow changes variance and remembers labelled endpoints. It is not an inclusion of vector subspaces.

Hence the external-soft (1\subset2\subset3) flag of Entry 3603 is a valid local mechanism, but not a universal shape for every physical readout.

Next falsifier

Construct the endpoint-marked relative elliptic object for ([0,\infty]), including its exact sequence over ordinary (H^1(D_\infty)), the two endpoint classes, and the source normalization that removes or retains the (omega_2) divergence. Only then test whether the physical functional is canonical.

Evidence

• research/benincasa/checkers/check_infinity_gysin_physical_relative_gate.py;
• research/benincasa/results/infinity-gysin-physical-relative-gate.json.

Allocator claim: seqclaim-02dfa8069f863a3752d11c4c.

View source on GitHub