Cohesive constructions and metric-selection scope

Revision — 2026-10-03

Cohesion, differential cohesion, metric selection and phenomenal identification are separate inputs. Universal axiomatic equivalence T-190 is withdrawn [✗]. Neither terminal object nor classifier selects a seven-dimensional quantum register or unique dynamics.

1. T-185: modalities in a supplied environment

A specified cohesive or differentially cohesive topos has the corresponding adjoint functors by its definition and standard structural theorems. This uses a conditional environment; it does not prove cohesion of every density site. Embedding an object in a large cohesive environment does not make its petit sheaf topos cohesive.

Bures topology on $\mathcal D_N$ equals the ordinary finite-dimensional matrix topology. The space is convex and contracts to $I/N$. Its rank-k stratum has real dimension

$$2Nk-k^2-1.$$

The support map has base $\mathrm{Gr}_k(\mathbb C^N)$ and contractible fiber of normalized positive definite k-matrices. With the standard local trivialization its homotopy type is that Grassmannian. These facts do not select experiential depth or physical spatial dimension.

Vanishing characteristic classes over the contractible density domain do not mean every connection is flat. Curvature needs a connection and differential geometry. Universal Gap/curvature identification T-73 is withdrawn [✗]. Differentially cohesive diagrams for stable coefficients cannot be applied to nonabelian BG2 without further construction.

2. T-186/T-188: phenomenal and time bridges

Identifying a selected modality with a phenomenal functor is a hypothesis [H/I] requiring explicit domain, codomain and transformation. A Postnikov filtration is not automatically a consciousness hierarchy. Topos modalities do not themselves solve the hard problem.

Exact conditional Page–Wootters time needs supplied clocks, matching spectra and a nonzero physical constraint subspace. With interaction, discarding mixed clock blocks is not exact: reduction may be time-nonlocal. Universal positive free energy also does not follow from categorical structure. These bridges are listed in premises.

3. T-187: exact Bures scope

In the selected normalized Petz family of monotone metrics on full-rank densities, the SLD metric is pointwise minimal. Its Bures normalization is one quarter of SLD information. Minimality within that family is exact mathematics; choosing the family and selecting its minimum are model inputs. CPTP contractivity permits the family, not only one metric. See Petz–Sudár.

On compact domains with a common positive eigenvalue floor, fixed smooth metrics are uniformly comparable. This does not prove global bi-Lipschitz equivalence of every metric up to rank boundaries or make numerical thresholds and dynamics metric-independent.

The open site uses ordinary union covers of open sets. Images of balls under an arbitrary CPTP channel need not cover a ball around the center’s image; contractivity does not establish reverse surjectivity.

4. T-189: SLD identity and MaxEnt limits

For $\rho>0$ and Hermitian tangent $X$, SLD is the unique solution of

$$X=\tfrac12(\rho L_X+L_X\rho).$$

At $\rho=\mathrm{diag}(\lambda_i)$, $(L_X)_{ij}=2X_{ij}/(\lambda_i+\lambda_j)$. Thus

$$g_B(X,Y)=\tfrac14\mathrm{Re}\mathrm{Tr}(\rho L_XL_Y).$$

This is exact under the selected normalization. Postulating the right side as a selector chooses Bures by construction. It does not derive Bures from maximum entropy alone: the Hessian of relative entropy yields the different BKM geometry. The SLD definition cannot simultaneously be an independent physical mechanism and its own derived selector.

Cramér–Rao bounds need a statistical model and local-unbiasedness hypotheses. Joint multiparameter quantum attainability does not follow from the SLD identity: different parameters’ optimal measurements may be incompatible.

5. The surviving operational lemma

Suppose distinguishability is defined as a supremum over all POVMs of a classical distinguishability decreasing under admissible processing. For linear CPTP $\Lambda$ and POVM $\{E_i\}$, $\Lambda^*(E_i)$ is a POVM: the adjoint channel is positive and unital. Output statistics equal input statistics of that POVM. Therefore the output supremum is no larger than the input supremum [T].

This proves monotonicity of the specified operational distinguishability. It reconstructs neither site, dimension, clocks, dynamics, modalities nor experience from four semantic principles. Universal T-190 is withdrawn [✗].

See categorical formalization, mathematical kernel and reconstruction.