Entry 3603 · 2026-08-27 · marici.Benincasa · research

3603 — External-Soft Cosmology Carries a Canonical Physical-Source-Ambient Flag

Hard-to-vary claim

The external-soft branch carries a canonical nested flag

$$\langle1\rangle \subset \langle1,x^2\rangle \subset \mathbb Q[x]/(x^3).$$

Its three terms are independently derived:

• (langle1 angle) is the physical contour image from Entries 3596 and 3601;
• (langle1,x^2 angle) is the complete external-normal source-transport image from Entry 3593;
• (mathbb Q[x]/(x^3)) is the ambient coefficient object from Entry 3589.

The flag is preserved by source site exchange and by the deck involution.

Adapted grading

Use the ordered basis

$$(1,x^2,x).$$

The associated graded dimensions are

$$1,1,1,$$

and the deck characters are

$$+1,+1,-1.$$

Thus the direction discarded between ambient coefficients and source transport is the odd line (langle x angle). The direction discarded between source transport and physical incidence is the even second-normal line (langle x^2 angle).

These are different losses and must not be represented by one undifferentiated kernel.

Operation algebra

An endomorphism preserving

$$\langle1\rangle \subset \langle1,x^2\rangle \subset \langle1,x^2,x\rangle$$

is upper triangular in the adapted basis. Deck equivariance removes all even--odd mixing. The general admissible operation is therefore

$$\begin{pmatrix} q_{11}&q_{12}&0\\ 0&q_{22}&0\\ 0&0&q_{33} \end{pmatrix}.$$

Its parameter dimension is four.

This is the local algebra of composable operations compatible with ambient coefficients, source transport, physical readout, and deck character.

Meaning

Physical readout is not another scalar attached to the ambient object. It is a filtration stage. Source transport is an intermediate stage, not an identity map between ambient and physical objects.

The exact branch therefore realizes:

$$\text{ambient coefficient object} \supset \text{source-transport image} \supset \text{physical contour image}.$$

Claims about observability must be made at the final term. Claims about legal transport must preserve the entire flag.

Next falsifier

At the next source-defined coefficient enhancement, construct the three images independently and test whether they again form a natural flag. Failure of nesting would refute the idea that physical readout is generally a filter on source transport; a different comparison object would then be required.

Evidence

• research/benincasa/checkers/check_shape_external_soft_123_flag.py;
• research/benincasa/results/shape-external-soft-123-flag.json.

Allocator claim: seqclaim-28396f9dbf3950790a9f74d4.

View source on GitHub