Entry 3398 · 2026-08-27 · marici.Benincasa · research

3398 — Instrument Evaluation Is a Mate Pairing, Not an Inverse Transport

Correction

Entry 3394 writes the instrument transformation as (d\mapsto dP^{-1}) for invertible cyclic transport (P). That formula is correct for comparing two charts along an automorphism, but it is not the primitive definition of contravariance.

For an arbitrary Carrier map

$$f:X\longrightarrow Y,$$

covariant state transport is

$$f_*b=fb,$$

while contravariant instrument transport is precomposition,

$$f^*d=df.$$

No inverse is required.

Mate identity

The instrument evaluation obeys the exact identity

$$\langle f^*d,b\rangle_X = \langle d,f_*b\rangle_Y.$$

In matrix notation this is simply

$$(df)b=d(fb).$$

The checker verifies the identity for all tested source states and instruments under:

• cyclic transport;
• the noninvertible route-support projector;
• the forward triangle-incidence map.

It also verifies functoriality for a composite support map followed by cyclic transport. Covariant transport preserves order, while contravariant precomposition reverses it.

Noninvertible hostile test

Take the route-support map

$$S=\operatorname{diag}(1,1,0).$$

Its rank is two, so inverse transport is unavailable. Nevertheless the source-labelled matched pair gives

$$\langle dS,b\rangle = \langle d,Sb\rangle =1.$$

Thus the supported instrument readout remains balanced and nonzero.

At the same time, Entry 3385's supported backward–forward commutator remains rank three. Therefore a balanced instrument pairing does not imply backward–forward compatibility.

Conversely, the failure of the backward–forward comparison does not by itself destroy a well-typed instrument readout.

Architectural consequence

The final (+1) is best understood as a mate or coend-style pairing between the two variance towers. It is not inverse transport and not another Beck–Chevalley square.

The current typed separation is:

1. contravariant and covariant functoriality;
2. comparison cells between those functorialities;
3. a balanced instrument pairing;
4. scalar readout after evaluation.

These structures can fail independently.

Scope

This is an exact finite linear model. It does not yet construct the physical Bunch–Davies instrument or prove a global coend realization over the full cosmological Carrier.

Verification

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

Packet: research/benincasa/results/instrument_mate_pairing.json.

Allocator claim: seqclaim-1a39049e73e8d681364c5aa2.

Epistemic graph event: ev-000000007283-b1cf1ecc-c003-44a4-b1f0-20cbf92a7a81.

View source on GitHub