Source-to-Tier-1 Realization

3.1 Role and Scope

The canonical source-to-readout route is

$$\widehat{\mathcal C}_{\rm phys}: \mathsf S_0\longrightarrow\mathsf{Result}(\mathsf{Comp}_{T1}), \qquad \mathsf S_{\rm adm} \xrightarrow{\operatorname{RealPrep}_{\Omega}} \mathsf{Real}_0 \xrightarrow{\operatorname{Read}_{\rm rel}} \mathsf Y_{\rm raw}^{\bot} \xrightarrow{\Gamma_{\rm phys}} \mathsf{Comp}_{T1}^{\bot}.$$

This chapter constructs the source-to-readout descent: prepared realization supports, realization-indexed raw sector construction, the redescription quotient and genuine coarse observation, sectorwise physical restriction with retained discrepancies, coupled fixed-point branches, and the conditional interface comparisons whose global coherence is the R9 problem.

3.2 Upstream Input from Chapter 2

Chapter 3 receives from Chapter 2 the admitted source category $\mathsf S_{\rm adm}=\mu\mathbb L_{\rm src}$ with strong morphisms $\mathcal K_{\rm adm}$, the marked deposition tokens $M_X$ with supports, strict pairs, lineages, invariants, and boundaries, and the sound derivation certificates $\operatorname{Der}_{\rm MinCl}(X)$. For every admitted $X$, Chapter 3 may form the realization fibre $\operatorname{RealPrep}_{\Omega}[X]$ and nothing upstream of it. No independently gated domain, target-conditioned predicate, or direct source-to-carrier map is received.

3.3 Realization and Source-to-Readout Descent

3.3.1 Prepared realization supports

An admitted source becomes physically relevant only through a selected region of its deposition potential. For $X\in\mathsf S_{\rm adm}$ and $m\in M_X$, define

$$\begin{aligned} J_{\rm real}(X,m) &= (X,m,S_m,\preceq_m,\ell_m,\mathsf{Inv}_m,\mathsf{Bnd}_m), \\ S_m&=\operatorname{supp}_X(m),\qquad \mathsf{Inv}_m=\mathsf{Inv}_X|_{S_m},\\ \mathsf{Bnd}_m&=\mathsf{Bnd}_X|_{S_m}. \end{aligned} \tag{3.1}$$

where $\preceq_m$, $\ell_m$, $\mathsf{Inv}_m$ and $\mathsf{Bnd}_m$ are respectively the source order, lineage, invariant data and intrinsic boundary data restricted to $S_m$. The prepared realization fibre is

$$\boxed{ \operatorname{RealPrep}_{\Omega}[X] = \{J_{\rm real}(X,m):m\in M_X\}. } \tag{3.2}$$

The subscript $\Omega$ labels the fixed, target-neutral source-local realization convention used throughout; it carries no Tier-1 target or physical-success predicate.

The tuple (3.1) contains no pre-existing metric, time coordinate, Hilbert space, field, action, gauge group, observed outcome or desired physical label. It is the finite, lineaged source support from which those structures may be constructed. Nonemptiness of (3.2) follows from source admission, but says only that a realization support can be formed—not that any physical sector succeeds.

3.3.2 Raw sector construction

Let $s$ range over the quantum, QFT, record, geometric, matter, cosmological, temporal and observer sectors. A sector construction is a relation

$$\mathscr K_s(Y) \subseteq \mathsf O_s^{\rm raw}\times\mathsf D_s, \qquad Y\in\mathsf{Real}_0, \tag{3.3}$$

where $\mathsf D_s$ is the space in which the relevant equation, covariance, anomaly, conservation or approximation discrepancy is evaluated. Retaining the discrepancy at this stage prevents the codomain from being defined to contain only successful physics.

For every interface $e:s\to t$, let

$$d_e\bigl(L_e(z_s),R_e(z_t)\bigr)\in\mathsf D_e \tag{3.4}$$

measure the difference between the two sector descriptions in their common comparison space. The raw coupled readouts supported by $Y$ are

$$\mathcal Z_Y^{\rm raw} = \left\{ (z,(\delta_s)_s,(\delta_e)_e): (z_s,\delta_s)\in\mathscr K_s(Y),\quad \delta_e=d_e(L_ez_s,R_ez_t) \right\}. \tag{3.5}$$

The corresponding source-indexed family is the disjoint union

$$\mathcal Z_X^{\rm raw} = \coprod_{Y\in\operatorname{RealPrep}_{\Omega}[X]} \mathcal Z_Y^{\rm raw}. \tag{3.6}$$

Thus every raw candidate retains its realization antecedent. Equation (3.6) is not a direct source-to-physics map; it factors through (3.2).

3.3.3 Redescription and genuine coarse equivalence

Let $\mathcal G_{\rm red}$ be the groupoid generated by admitted gauge, basis, label and presentation changes, together with diffeomorphisms on branches on which a manifold has already been reconstructed. First form the redescription quotient

$$\widetilde{\mathcal Z}_Y = \mathcal Z_Y^{\rm raw}/\mathcal G_{\rm red}. \tag{3.7}$$

A coarse protocol $\mathfrak c$ is fixed before an observable comparison. It consists of a finite or countable collection of measurable diagnostics $T_a$, fixed resolutions $\varepsilon_a>0$, and fixed origins $o_a$. Define the coarse signature

$$C_{\mathfrak c}([z]_{\rm red}) = \left( \left\lfloor \frac{T_a([z]_{\rm red})-o_a}{\varepsilon_a} \right\rfloor \right)_{a\in A_{\mathfrak c}} \in\mathsf Z_{\mathfrak c}. \tag{3.8}$$

Continuous or categorical diagnostics may replace the displayed binning provided that the map and its codomain are fixed in advance. Coarse equivalence is the kernel relation

$$\boxed{ [z]_{\rm red}\sim_{\mathfrak c}[z']_{\rm red} \quad\Longleftrightarrow\quad C_{\mathfrak c}([z]_{\rm red}) =C_{\mathfrak c}([z']_{\rm red}). } \tag{3.9}$$

Unlike threshold proximity, (3.9) is automatically reflexive, symmetric and transitive. Refinement of protocols is expressed by factorization: $\mathfrak c'$ refines $\mathfrak c$ when there is a map $r$ such that

$$C_{\mathfrak c}=r\circ C_{\mathfrak c'}. \tag{3.10}$$

The quotient fibre of (3.9) records the physical distinctions ignored at resolution $\mathfrak c$; the redescription fibre records only alternate presentations. Neither may be adjusted after the result is known.

3.3.4 Sectorwise physical restriction and coupled branches

For each sector $s$, fix a nonnegative discrepancy $\delta_s$ testing its equation, covariance, anomaly, conservation and regularity conditions, together with a tolerance $\varepsilon_s\geq0$. Define

$$\mathsf P_s(\varepsilon_s) = \{p_s\in\mathsf O_s^{\rm raw}: \delta_s(p_s)\leq\varepsilon_s\}, \qquad \mathsf P_s(0)=\mathsf P_s . \tag{3.11}$$

The sectorwise physical restriction of a redescription class is the partial relation

$$\Gamma_{\rm phys}^{\rm sect}: \widetilde{\mathcal Z}_Y \dashrightarrow \prod_{s\in V}\mathsf P_s(\varepsilon_s). \tag{3.12}$$

The exact sector theory is the special case $\boldsymbol\varepsilon=\mathbf0$. Source and realization lineage remain attached to the antecedent $(X,Y,[z]_{\rm red})$; they are not components of $\Gamma_{\rm phys}^{\rm sect}([z]_{\rm red})$. For an interface $e:s\to t$, the comparison $(L_e,R_e)$ is evaluated only on the overlap of the successful $s$- and $t$-sector domains. Its descent is therefore conditional: it proves a pairwise interface statement for that overlap, not simultaneous membership in the global coherent equalizer. The coarse signature (3.8) is attached only after the relevant sectorwise tests. Coarse equivalence never promotes an unsuccessful representative merely because it shares an observational bin with a successful one. The sectorwise relation factors through $\widetilde{\mathcal Z}_Y/\!\sim_{\mathfrak c}$ only when its domain and output are proved constant on every coarse fibre.

When sector equations are cyclic—most importantly, when QFT depends on geometry while geometry depends on the QFT stress tensor—the coupled branch is a fixed point of a correspondence

$$\mathfrak F_{Y,b}:\mathcal Z_{Y,b}\rightrightarrows\mathcal Z_{Y,b}, \qquad z\in\mathfrak F_{Y,b}(z). \tag{3.13}$$

Existence may be established, for example, by the Kakutani–Fan–Glicksberg theorem [4, 3, 5] on a nonempty compact convex branch with upper-hemicontinuous nonempty compact convex values, or by the Banach theorem for a strict contraction on a complete metric branch. Proving a global fixed point, simultaneous path and loop coherence, and nonemptiness of the exact equalizer (1.6) is the R9 compatibility problem. None of these conclusions is built into the sectorwise restriction (3.12) or assigned to R2.

For a stochastic raw relation, let $K_{\rm raw}(d\widetilde z\mid Y)$ be its measure on redescription classes and let $\gamma_{\rm phys}^{\rm sect}(dp\mid\widetilde z)$ be a sub-Markov kernel supported on the sectorwise outputs related to $\widetilde z$ by (3.12). Its total mass is zero when the required sector tests fail and at most one otherwise. The joint sectorwise-physical and coarse measure is

$$K_{\rm phys}^{\rm sect}(A,B\mid Y) = \int \mathbf 1_{B}\!\left(C_{\mathfrak c}(\widetilde z)\right) \gamma_{\rm phys}^{\rm sect}(A\mid\widetilde z) K_{\rm raw}(d\widetilde z\mid Y), \qquad A\subseteq\prod_{s\in V}\mathsf P_s(\varepsilon_s),\ B\subseteq\mathsf Z_{\mathfrak c}. \tag{3.14}$$

The missing mass represents branches that do not yield the stated sectorwise physical event. It is not removed by renormalizing over the successful subset. This formulation also remains meaningful when (3.12) is genuinely set-valued.

3.3.5 R2 starting point

Realization and sector-descent theorem target (R2). Starting from an admitted source $X$, prove that (3.1)–(3.6) are natural under strong source morphisms, that the redescription quotient and registered coarse map obey (3.7)–(3.10), and that every sector relation descends to its exact or tolerance-controlled physical restriction (3.11)–(3.12). On the overlapping successful domains, prove that each declared pairwise interface map is well typed, redescription-natural and stable under the registered tolerances. Establish the corresponding statements for the sub-Markov construction (3.14), without conditioning away unsuccessful mass.

R2 does not assert a nonempty global compatible fibre. Simultaneous satisfaction of every interface, path and loop condition, nonemptiness of $\mathsf{Comp}_{T1}$, and existence of the global coupled fixed point (3.13) are reserved for R9.

3.4 From Realization to the Physical Sectors

The sector chapters that follow develop the raw constructions, physical restrictions and interface comparisons of this chapter for the quantum, field-theoretic, record, geometric, gravitational, matter, cosmological, temporal and observer sectors. Each sector chapter assumes a prepared realization $Y\in\operatorname{RealPrep}_{\Omega}[X]$ and returns its raw candidates and physical restrictions; mutual compatibility is assembled in Chapter 16, and the associated theorem programmes are collected in Chapter 17.