The Principia Machina/TPS/2.3 The Translation Manifold

Instruction artifact, machine-optimized text: TPS, section 2.3 The Translation Manifold. Part of TPS 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). TPS 0.1.1, which changed only the status lines, was declared final on 2026-09-26. This page is revision 6 of the tree, proposed as version 0.3.0 until H-DECL declares it; revision 5, proposed as 0.2.0, stays undeclared and is carried whole. The text below is this section of the TPS artifact at 0.3.0 (proposed), 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 and which parts are MECHANICAL; the
         header and entry schemas; the inventory the TrM maps onto
governs: 2.3 | version: 0.3.0 (proposed) | class: instruction artifact
audience: operator, agent

2.3.1  The TrM is the second zone of the TFK. Its identification fields
       - the header (2.3.10) and, per entry, NAME, DOF, DOMAIN,
       ENVIRONMENT_QUANTITY, the identification, FALSIFIER,
       FALSIFIER_CLASS, BEARS and DIMENSION (2.3.11) - 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] [D39.1] [D98.1]
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.
       "Restricted" reads as the TPK's DOF less the TARGETs of REMOVE
       constraints: a FIXed DOF stays in the inventory the TrM maps onto
       (2.3.19). [B-02] [D107.1]
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. As the TrM is entered the gap is held;
       at the final pass the TMS's IC2_POLICY disposes of it: GAP admits
       the TrM with the gap named, REJECT returns it (2.4.5, 5.5.2).
       [B-02] [I-124] [D107.1]
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 by IC2_POLICY REJECT. [I-057]
       [B-02] [I-162] [D85.2]
       EXECUTABLE, in TPC, both residences, over the fields of 2.3.11
       and 2.3.18: IC1 whole (2.3.12); IC2 whole, over the inventory of
       2.3.19, its disposition at the final pass; IC3 whole, in both
       directions (2.3.12); IC4 (b) (2.3.13); the necessary part of IC4
       (c), and IC4 (a) syntactically where a SIGNATURE carries FORM
       and IMAGE_FORM (2.3.14); CODOMAIN (2.3.19). They run on a TPK
       that passes T1 and a TrM that passes its structural check;
       otherwise NOT_RUN. [D106.1] [D107.1] [D132.1]
       INSPECTION: equivalence under a TRANSFORMATION (2.3.20); IC4 (a)
       beyond the syntactic renaming, that the image holds in the
       environment; the sufficiency of IC4 (c); IC5. OWED: none of the
       bodies above; C-01 stands for the parts a future declared form
       could carry. [DI-4.1] [D132.1]
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:
         NAME                  identifier (2.2.7), unique among entries;
                               what an M record names as its QUANTITY
                               (SR3)
         DOF                   a TPK DOF NAME
         DOMAIN                prose; not blank
         ENVIRONMENT_QUANTITY  a member of the ENVIRONMENT INVENTORY;
                               the structured half of IDENTIFICATION
         IDENTIFICATION        prose
         FALSIFIER             prose; not blank
         FALSIFIER_CLASS       F1, F2 or F3; F0 is inadmissible (guide
                               C4); what PROMOTION_FLOOR is tested
                               against (3.2.12)
         BEARS                 a set of TPK LAW and CONSTRAINT NAMEs
         DIMENSION             a dimension in the form of 2.5.12
       The name is ENVIRONMENT_QUANTITY, not QUANTITY: QUANTITY is the
       defined noun for the TrM entry an M record's VALUE belongs to,
       and one word does not carry two referents. [D39.1] [D98.1]
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, executable: IC1 TOTAL, every
       environment quantity of the ENVIRONMENT INVENTORY is some entry's
       ENVIRONMENT_QUANTITY, failure REJECT; IC2 SURJECTIVE, every DOF
       of the inventory of 2.3.19 is some entry's DOF, failure a NAMED
       GAP "IC2: DOF v is untied" disposed of under IC2_POLICY (2.3.6);
       IC3 INJECTIVE, no DOF is some entries' DOF with two inequivalent
       environment quantities and, the isomorphism being invertible, no
       environment quantity is identified with two DOF, failure REJECT
       naming the entries. [D39.1] [I-162] [D85.2] [D106.1] [D107.1]
2.3.13 Every identification is dimensionally homogeneous. MECHANICAL,
       executable as IC4 (b): each entry's DIMENSION equals, canonically,
       its DOF's and its environment quantity's; failure REJECT naming
       the side. [D39.1] [D106.1]
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). Executable parts: IC4 (c),
       the necessary part - every DOF named in the REFS of a LAW an
       entry BEARS, and the TARGET of every CONSTRAINT it BEARS, is
       identified by some entry, since a law cannot be carried into the
       environment through entries that do not exist, failure REJECT;
       and IC4 (a) syntactically, where a SIGNATURE carries the optional
       pair FORM and IMAGE_FORM (both or neither; 2.2.12, 2.2.13):
       IMAGE_FORM equals FORM with every DOF of OVER replaced by its
       entry's ENVIRONMENT_QUANTITY, the names rule holding both ways on
       OVER against FORM and IMAGE_FORM's identifiers being inventory
       members; a DOF in OVER that no entry identifies leaves the
       signature undecided, IC2's gap and not IC4's failure. INSPECTION:
       whether an identified law is in fact expressible, that the image
       holds in the environment, the sufficiency of (c), and the
       dimensional homogeneity of a FORM itself. [D39.1] [D106.1]
       [D132.1]
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
       bodies of 2.3.7 are checked against the skeleton; the parts 2.3.7
       lists as INSPECTION are judged by H-DECL against the declared
       header. [D39.1] [D106.1]
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]
2.3.18 THE HEADER SCHEMA. A TrM is an object with two parts, HEADER and
       ENTRIES, both required. HEADER has three parts, all required:
         ENVIRONMENT_INVENTORY  entries {ENVIRONMENT_QUANTITY,
                                DIMENSION}; ENVIRONMENT_QUANTITY an
                                ASCII identifier, unique in the
                                inventory
         SIGNATURE              entries {NAME, OVER, IMAGE, FORM?,
                                IMAGE_FORM?}; OVER a set of TPK DOF
                                NAMEs, IMAGE prose, FORM and IMAGE_FORM
                                the optional declared pair of 2.3.14
         TRANSFORMATIONS        entries {NAME, STATEMENT}
       Structural checks run as the zone is entered and return a
       rejected component to its author; whether the entries form an
       isomorphism is 2.3.7's. [D98.1] [D132.1]
2.3.19 THE INVENTORY THE TrM MAPS ONTO is the TPK's DOF less the
       TARGETs of REMOVE constraints. A FIXed DOF stays in it: IC2
       requires it identified, else IC2 names the gap, and its FIX may
       be borne by its entry. An entry whose DOF a REMOVE constraint
       targets maps outside the inventory and is refused (CODOMAIN).
       Under this rule an entry can never bear a REMOVE constraint or a
       law naming a removed DOF. DIVERGENCE, recorded against the guide
       under its 2.5: guide 3.1 defines the DOF INVENTORY as what the
       laws minus the constraints leave free; the specification governs
       the project and the guide's text is unchanged. In T1-T6 "the DOF
       inventory" is the TPK's DOF kind, constraints included, so T5 (b)
       requires a constraint's TARGET to be in it. [D107.1]
2.3.20 EQUIVALENCE by name. TRANSFORMATIONS are prose, so the checks
       treat distinct ENVIRONMENT_QUANTITY names as inequivalent. This
       errs toward rejection - IC1 may ask an entry for each of two
       equivalent names; IC3 refuses two equivalent names on one DOF -
       and never toward acceptance. Equivalence under a TRANSFORMATION
       stays INSPECTION, as does 2.3.15. [D106.1]