Inputs: the Translation Manifold (TrM)

The Principia Machina: map
Reading the diagrams Structure key Process key Definition of terms Roles
Reading by audience Start here: the walk through The operator's guide Tutorial: create a UUID Workspace Container (UWC) Public Recorder inputs Sponsors: your first agent
The specification Contents: parts, tables, diagrams TPS 0.5.0 (declared 2026-10-03, revision 8; machine-optimized text) Decisions (the foundations record, one line each)

Public Recorder inputs: a help set for the people who put things into The Principia Machina - authors of a new UWC, operators, and the declaring authority. It explains; the specification governs. Start at the index; for a guided walk from a blank file to a declared UWC, follow the tutorial. Agents read the agent-facing page instead.

Purpose

What you write in the second zone: the header, then the entries that tie each degree of freedom to a quantity of the environment.

Explains: TPS 0.1.0, revision 4 (declared final 2026-09-22); the zone schemas the declaring authority selected on 2026-09-22 (records D92.1, D98.1, D99.1, D100.1, and for units D103.1, D104.1, D105.1; schema 0.3.0), whose text enters TPS at revision 5, and the kernel tests and isomorphism checks selected the same day (D106.1, D107.1, D108.1; tpm_kernel_tests 0.2.0), whose text follows at the next TPS revision; TPC-PY as built 2026-09-22 (tpc_zone 0.3.0, tpc_units 0.1.0, tpc_tests 0.1.0; no TPC-PY version is declared yet) with check_draft 0.3.0; Generic Specification Guide 0.1.0, sections 4.2 to 4.6 and C4. Where this page and the specification differ, the specification governs.

What the Translation Manifold is

The Translation Manifold (TrM) says what each DOF of the kernel is in the environment the kernel describes - which measurable quantity x is, for example. Every entry must be an isomorphism: nothing lost, nothing merged, structure kept. Its header and every entry are immutable once declared (TPS 2.1 and 2.3).

Two parts: HEADER and ENTRIES

Both are required. The header is entered first.

The header

ENVIRONMENT_INVENTORY: The quantities of the environment, each an object with ENVIRONMENT_QUANTITY (an identifier, unique in the inventory) and DIMENSION.
SIGNATURE: The relations and operations on the DOF that the map must carry, each with NAME, OVER (the list of DOF it acts on) and IMAGE (prose: what it becomes in the environment).
TRANSFORMATIONS: The re-representations under which an identification stays the same identification - a change of units, a shift of the time origin - each with NAME and STATEMENT. Two identifications related by a transformation count as one.

Each entry

All nine fields are required.

Field	What it holds
NAME	an identifier, unique among entries. A measured register record names this entry as its QUANTITY.
DOF	the kernel DOF this entry identifies
DOMAIN	prose: where the identification holds. Not blank.
ENVIRONMENT_QUANTITY	the inventory quantity the DOF is
IDENTIFICATION	prose: the identification itself
FALSIFIER	prose: what observation or derivation would show the identification false. Not blank.
FALSIFIER_CLASS	how severe that falsifier is: F1, F2 or F3 (below)
BEARS	the kernel LAW and CONSTRAINT names the entry carries into the environment
DIMENSION	the dimension of the identified quantity

The field is called ENVIRONMENT_QUANTITY, not QUANTITY, because in TPM "QUANTITY" already names something else: the TrM entry a measured record's value belongs to. One word, one meaning.

Falsifier classes

From the Generic Specification Guide, C4:

F0, no falsifier: not admitted. The entry is malformed.
F1, nominal: shown false only by an error in the author's own computation. Re-derivation is the test.
F2, independent: a second party can evaluate it from the record's fields alone, without the author's steps. Independent reproduction is the test.
F3, external: it names an external quantity with stated uncertainty that a measurement or an outside source could contradict.

The class also matters later: a result is promoted only when its falsifier reaches the UWC's promotion floor, which the Measure Set declares (Measure Set).

The checks

The structure check runs first: parts present, fields present and of the right type, every DOF in an entry or a SIGNATURE is a kernel DOF, every ENVIRONMENT_QUANTITY is in the inventory, every BEARS name is a kernel LAW or CONSTRAINT, no duplicate names.

If the manifold and the kernel both pass their structure checks, the four isomorphism checks run as the zone is entered (TPS 2.3; tpm_kernel_tests 0.2.0). They map onto the kernel's DOF less those a REMOVE constraint takes out (record D107.1). A DOF that a FIX constraint pins stays in: it still needs an entry, and that entry may bear the constraint. A removed DOF must have no entry.

IC1 TOTAL: every environment quantity is identified by some entry. Failure refuses the manifold.
IC2 SURJECTIVE: every DOF, counted as above, is some entry's DOF. A DOF no entry reaches is an untied DOF. It is held as a named gap, and at the final pass the Measure Set's IC2_POLICY decides: GAP, the default, admits the manifold with the gap named; REJECT refuses it.
IC3 INJECTIVE: no DOF is identified with two environment quantities, and no environment quantity with two DOF. Failure refuses.
IC4 STRUCTURE-PRESERVING, the parts a machine can read: each entry's DIMENSION equals its DOF's and its environment quantity's; and every DOF named by a law an entry bears, and the TARGET of every constraint it bears, is identified by some entry. Failure refuses.
CODOMAIN: an entry on a removed DOF is refused.

The checks take environment quantities by name: two names are two quantities, even where a transformation relates them. This can refuse too much, never admit too much.

What stays with the declaring authority, by inspection (TPS 2.3):

whether two names are the same quantity under one of the TRANSFORMATIONS;
IC4, the rest: whether each SIGNATURE relation is carried to its image, and whether each borne law can in fact be expressed in the environment;
IC5 falsifier invariance: no transformation changes what would falsify an entry.

The Generic Specification Guide describes the inventory as what the laws, minus the constraints, leave free. TPM keeps fixed DOF in it, so that a fixed DOF is still tied to something measurable; that difference is recorded as a divergence against the guide.

Non-goals

The manifold has no exclusions field and no per-entry ceiling. To restrict a kernel, add a CONSTRAINT to the TPK (TPS 2.3).

Worked example

The spring manifold: three environment quantities, one signature relation (the time derivative takes position to velocity), one transformation (a shift of the time origin), and one entry per DOF.

{
  "TrM": {
    "HEADER": {
      "ENVIRONMENT_INVENTORY": [
        {"ENVIRONMENT_QUANTITY": "position", "DIMENSION": "L1"},
        {"ENVIRONMENT_QUANTITY": "velocity", "DIMENSION": "L1 T-1"},
        {"ENVIRONMENT_QUANTITY": "height", "DIMENSION": "L1"}
      ],
      "SIGNATURE": [
        {
          "NAME": "time_derivative",
          "OVER": ["x", "v"],
          "IMAGE": "d/dt maps position to velocity"
        }
      ],
      "TRANSFORMATIONS": [
        {"NAME": "time_translation", "STATEMENT": "t -> t + t0"}
      ]
    },
    "ENTRIES": [
      {
        "NAME": "E_x",
        "DOF": "x",
        "DOMAIN": "|x| < x_max",
        "ENVIRONMENT_QUANTITY": "position",
        "IDENTIFICATION": "x is the displacement from equilibrium along the spring axis",
        "FALSIFIER": "a measured restoring force not proportional to x inside the domain",
        "FALSIFIER_CLASS": "F3",
        "BEARS": ["N2", "KIN"],
        "DIMENSION": "L1"
      },
      {
        "NAME": "E_v",
        "DOF": "v",
        "DOMAIN": "|x| < x_max",
        "ENVIRONMENT_QUANTITY": "velocity",
        "IDENTIFICATION": "v is the time derivative of x",
        "FALSIFIER": "a measured velocity differing from dx/dt beyond uncertainty",
        "FALSIFIER_CLASS": "F2",
        "BEARS": ["KIN"],
        "DIMENSION": "L1 T-1"
      },
      {
        "NAME": "E_z",
        "DOF": "z",
        "DOMAIN": "always",
        "ENVIRONMENT_QUANTITY": "height",
        "IDENTIFICATION": "z is the height above the track",
        "FALSIFIER": "any measured z different from 0",
        "FALSIFIER_CLASS": "F3",
        "BEARS": ["FREE_Z", "C_NOZ"],
        "DIMENSION": "L1"
      }
    ]
  }
}

E_x and E_z name outside measurements as their falsifiers, so they are F3. E_v is F2: anyone can check it from the recorded values alone. The checker reports TrM PASS and IC1-IC4 PASS: three quantities, three DOF, one entry each way. E_z identifies z, which C_NOZ fixes; z stays in the manifold, and E_z bears C_NOZ. Built step by step on tutorial step 3.

Governing sections: TPS 2.1, 2.3 and 5.5.

Previous: The Prime Kernel. Next: The Measure Set.