The Principia Machina/TPS/2.3 The Translation Manifold

Instruction artifact, machine-optimized text: TPS 0.1.1, section 2.3 The Translation Manifold. Part of TPS 0.1.1 and the TPM hub. Human-only edit (TPS 5.13). TPS 0.1.0 was declared final at revision 4 on 2026-09-22, a version declaration recorded in the foundations record; the chain begins at TPX's first landing (TPS 5.6.7). Version 0.1.1 changes only the status lines (this lead, the title's version and the header's version line); H-DECL declares it. The text below is this section of the TPS 0.1.1 artifact, verbatim; the whole artifact and its file digest are on The Principia Machina/TPS. Human readers: the contents page is the map.

TPS 2.3  THE TRANSLATION MANIFOLD
purpose: every entry an isomorphism (TOTAL, SURJECTIVE, INJECTIVE,
         STRUCTURE-PRESERVING); the identification fields; DOMAIN and
         ENVIRONMENT; restrictions priced in the TPK; the four
         isomorphism checks by name, bodies owed
governs: 2.3 | version: 0.1.1 | class: instruction artifact
audience: operator, agent

2.3.1  The TrM is the second zone of the TFK. Its identification fields
       (DOF, DOMAIN, the identification, FALSIFIER, BEARS) are immutable;
       every other field of an entry is carried by register records that
       name the entry (see 2.1.3). [I-021] [B-05]
2.3.2  Isomorphism only. Every TrM entry satisfies TOTAL, SURJECTIVE,
       INJECTIVE, STRUCTURE-PRESERVING unchanged. [B-02]
2.3.3  DOMAIN is the regime of validity, required on every TrM entry and
       every promoting record. ENVIRONMENT is what the TrM maps a TPK
       into. [I-030] [I-031]
2.3.4  A restriction (tabletop, uniform field, no rotation) is declared
       as CONSTRAINT entries in the TPK, so the TPK yields the restricted
       DOF inventory and the TrM is an isomorphism onto it. Restrictions
       are therefore covered by the kernel test suite T1-T6. No promotion
       ceiling; the restriction is priced in the TPK, not the entry.
       [B-02]
2.3.5  Each restricted configuration is its own TPK and its own
       container. A RESTRICTION object relating two containers is not
       offered, being excluded by 1.3.2. [B-02]
2.3.6  SURJECTIVE is the completeness test. Completeness is relative to
       the DOF inventory the TPK yields; an untied DOF is a named gap,
       never silently omitted. [B-02] [I-124]
2.3.7  The four isomorphism checks test TOTAL, SURJECTIVE, INJECTIVE
       and STRUCTURE-PRESERVING, in that order, and are specified at
       guide 4.6 as IC1 to IC4; the guide's companion IC5, falsifier
       invariance, is 2.3.15. TPX runs them as the TrM zone is entered
       (see 5.5). A failure of TOTAL, INJECTIVE or STRUCTURE-PRESERVING
       returns a rejected component to the author. A failure of
       SURJECTIVE is an untied DOF: it is recorded as a named gap and
       the TrM is admitted with the gap named (2.3.6, 3.3.5); a UWC's
       TMS MAY tighten this to rejection. [I-057] [B-02] [I-162]
       [D85.2]
       OWED: the executable bodies of the four checks, in TPC, to the
       guide's specification; kind: computation; C-01. [DI-4.1]
2.3.8  While the body of an isomorphism check is owed, that check is
       INSPECTION, performed by H-DECL before the declaration that
       instantiates the container, and recorded as ratchet debt. TPX
       executes the check when its body lands. [DI-4.1]
2.3.9  TrM SKELETON. Every TrM instantiates the skeleton below; each
       UWC instantiates it under D(TrM). The skeleton is framework
       machinery, as the strike skeleton is (see 4.3). [D39.1]
2.3.10 The TrM opens with one header, immutable, inside D(TrM):
         ENVIRONMENT INVENTORY  the target quantities, each with its
                                DIMENSION;
         SIGNATURE              the relations and operations on the DOF
                                inventory that the map must carry, each
                                with its image;
         TRANSFORMATIONS        the re-representations under which an
                                identification is the same
                                identification; unit rescaling is a
                                member.
       EQUIVALENCE is the relation the TRANSFORMATIONS generate.
       [D39.1]
2.3.11 Every entry carries DOF, DOMAIN, the identification, FALSIFIER,
       BEARS and DIMENSION. [D39.1]
2.3.12 DIRECTION. The checks take the ENVIRONMENT INVENTORY as the
       domain of the map and the DOF inventory as its codomain, so
       that SURJECTIVE reads "every DOF is reached by some
       identification" and an untied DOF is exactly a SURJECTIVE
       failure; 2.3.3's "maps a TPK into" names the same isomorphism
       read from the kernel side. TOTAL, SURJECTIVE and INJECTIVE are
       tested over the DOF inventory and the ENVIRONMENT INVENTORY,
       modulo EQUIVALENCE. MECHANICAL; body OWED (C-01). [D39.1]
       [I-162] [D85.2]
2.3.13 Every identification is dimensionally homogeneous. MECHANICAL.
       [D39.1]
2.3.14 STRUCTURE-PRESERVING is tested against SIGNATURE: each declared
       relation and operation is carried to its declared image.
       MECHANICAL where SIGNATURE is executable, INSPECTION otherwise,
       as SR9 splits C1, C2 and C4 (see 4.5). Body OWED (C-01).
       [D39.1]
2.3.15 Every FALSIFIER is invariant under the TRANSFORMATIONS.
       INSPECTION. [D39.1]
2.3.16 The skeleton does not wait for the executable check bodies. The
       OWED lines at 2.3.7 stand; when the bodies land they are checked
       against the skeleton. Until then each check is INSPECTION by
       H-DECL against the declared header. [D39.1]
2.3.17 The TrM carries no exclusions field. A restriction is a
       CONSTRAINT entry in the TPK (see 2.2.2, 2.3.4). [D39.1]