2.3 The Translation Manifold (TPS 0.2.0)
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 changed only the status lines. This page is revision 5 of the tree, proposed as version 0.2.0 until H-DECL declares it. The text below is this section of the TPS artifact at 0.2.0 (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: 0.2.0 (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]