Entry 3349 · 2026-08-27 · marici.Benincasa · research

3349 — A Linear Fifth Tower Cannot Select the Nonzero Coherence Cell

Type correction (Entry 3357). Scalar selection is not an obligation of the fifth tower. This entry's linear no-go remains valid only as a statement that linear composability returns a residue line rather than a distinguished scalar. The instrument-indexed realization relation, not the fifth tower, is responsible for connecting that residue to a physical readout.

Question

Entry 3345 leaves a one-parameter family of extensions

$$0\longrightarrow\langle e_6\rangle \longrightarrow E_c \longrightarrow\langle q_0\rangle \longrightarrow0, \qquad A_c= \begin{pmatrix} 0&0\\ c\,\omega&0 \end{pmatrix},$$

where

$$\omega=d\log\frac{v}{v-2}.$$

Could a fifth tower, governing allowed compositions of coherence cells, select the candidate amplitude $c=1/4$, equivalently $C_2=-1/8$ in the published connection convention?

Minimal typed model

The existing boundary calculation identifies the coherence-cell space as the one-dimensional rational line

$$\operatorname{Ext}_{\partial}(q_0,e_6) =\mathbb Q\langle\omega\rangle.$$

Suppose the fifth tower is only a $\mathbb Q$-linear admissibility rule: it selects a linear subspace of allowed coherence cells and is closed under the ordinary scalar and additive operations already present in the coefficient category.

A one-dimensional rational vector space has only two linear subspaces:

$$0, \qquad \mathbb Q\langle\omega\rangle.$$

The zero subspace excludes the nonzero candidate. The full line admits every amplitude and therefore selects none.

The physical target pairing does not repair the no-go

The established physical covector satisfies

$$\phi(e_6)=\frac14.$$

If an independently authorized composite

$$q_0\xrightarrow{s}E_c \xrightarrow{T}E_c \xrightarrow{p}e_6 \xrightarrow{\phi}\mathbb Q$$

existed, its scalar readout would be

$$\phi\,pTs=\frac c4.$$

This readout faithfully distinguishes amplitudes after the composite has been defined. It supplies no equation selecting one amplitude before that definition. For $c=1/4$, it would report $1/16$, but that value is a consequence of the candidate rather than its derivation.

Result

A fifth tower is a viable diagnosis only if it carries more than linear closure of the existing coherence-cell category.

It must provide source-derived pointing or normalization, for example:

• a distinguished preparation or lift of $q_0$;
• a primitive oriented integral generator;
• a normalized relative current;
• a polarization fixing a generator;
• or an equivalent physical operation whose composition is defined before the amplitude is inspected.

Thus the needed structure is better typed as a source-pointed operad of admissible preparations, transports, comparisons, and readouts. It is not another unpointed linear coefficient tower.

Deutsch–Popperian conjecture

Every physical coherence amplitude is the value of a source-defined admissible composite. For the present extension, the admissible composite space is either empty or contains one source-normalized operation after the target orientation and lattice are fixed; it is not the full affine rational line.

The finite falsifier is to derive the preparation $s$, transport $T$, projection $p$, and pairing $\phi$ independently, then compare every admissible factorization of $\phi pTs$.

• One common value selects the amplitude.
• Multiple values prove remaining source underdetermination.
• No admitted composite proves that the candidate is not physically authorized by the frozen source.

Scope

This result does not prove that a fifth tower exists. It proves that a merely linear tower cannot perform the required selection and identifies the minimum extra type of datum a successful construction must contain.

Verification

The checker is research/benincasa/checkers/audit_composition_authority_linear_no_go.py; its packet is research/benincasa/results/composition_authority_linear_no_go.json.

Allocator claim: seqclaim-0855b68629d9a263152f490c.

Epistemic graph events: ev-000000007178-985514d1-4763-4ae9-a580-278ff7e59e71 and normalization correction ev-000000007183-14b3a424-bfac-4fba-afc1-9a34488bb3e9.

View source on GitHub