Mathematics Crosswalk
Status
Wiki state: intake
Source posture: public-safe crosswalk from published or mirrored GCAT / BCAT equation forms into Transition Table roles
Authority boundary: this page explains mappings. It does not prove the equations, validate parameters, replace source papers, or create commit-time authority.
Purpose
The Mathematics Crosswalk identifies selected mathematical objects from GCAT / BCAT-related papers and maps them into Transition Table roles.
The crosswalk exists so mathematical terms can be related to state-transition governance without silently turning analogy into equivalence.
Core Equation Crosswalk
The expressions below are written in MDX-safe plain text so the site renderer does not evaluate mathematical braces as JavaScript.
| Equation or object | Source framing | Transition Table role | Translation record link |
|---|---|---|---|
| state vector x = tuple(g, c, a, t) within unit interval to the fourth power | Four-dimensional governed-autonomy state | Prior state or resulting state | dtg-physics-state-v0-1, dtg-gcat-capacity-margin-v0-1 |
| Lambda of x = K times g^alpha times c^beta times t^gamma | Legitimacy / governance capacity function | Constraint reference | dtg-gcat-capacity-margin-v0-1 |
| margin m of x = Lambda of x minus a | Admissibility margin | Admissibility question | dtg-gcat-capacity-margin-v0-1 |
| admissible region A = states where margin is greater than or equal to zero | Admissible region | Boundary condition / admissible state set | dtg-gcat-capacity-margin-v0-1, dtg-bcat-boundary-v0-1 |
| time derivative of x = f of x plus G of x times control u | Control-affine system evolution | Candidate transition / transition dynamics | dtg-physics-interaction-v0-1 |
| time derivative of margin under x and u is bounded below by negative kappa times margin | Barrier condition preserving admissibility | Constraint reference / policy gate analogue | dtg-gcat-capacity-margin-v0-1, dtg-runtime-fail-closed-v0-1 |
| normalized x = x divided by L1 norm of x, projected into Delta cubed | Normalized simplex projection | State normalization for recoverability review | dtg-physics-state-v0-1 |
| face distance of normalized x = minimum component of normalized x | Distance to nearest collapse face | Boundary distance / recoverability evidence | dtg-bcat-boundary-v0-1 |
| Rigel number R_i of normalized x = face distance divided by face distance plus Delta distance plus delta | Rigel number | Recoverability posture | dtg-bcat-boundary-v0-1 |
| total lag tau = observation lag plus decision lag plus action lag | Total lag | Drift / time validity condition | dtg-runtime-fail-closed-v0-1 |
| robust recoverability under lag R_rob of tau | Robust recoverability under lag | Commit-time validity input | dtg-runtime-fail-closed-v0-1 |
| ideal ALLOW predicate for u, x, and tau holds when Phi of x and u is in robust recoverability under tau | Ideal commit admissibility condition | Decision result and commit-time validity | dtg-runtime-fail-closed-v0-1 |
Physics Translation Boundary
Some mathematical objects resemble physics language: state, dynamics, interaction, horizon, lag, collapse, and recoverability.
This crosswalk allows those words to be mapped into transition roles only under a non-claim boundary:
A mathematical object may be translated into a Transition Table role.
That translation does not prove the source discipline claim.
That translation does not make governance mathematics identical to physics.
That translation does not create a new physical theory.
First-Pass Interpretation
state equation -> prior/resulting state
capacity equation -> constraint reference
margin equation -> admissibility question
barrier inequality -> policy-gate analogue
simplex distance -> boundary/recoverability evidence
lag equation -> drift/time-validity input
commit predicate -> decision result and commit-time validity
Required Evidence Before Promotion
A crosswalk row should remain intake until it has:
- a source equation or object;
- a public-safe source reference;
- a Transition Table role;
- at least one translation record ID;
- a non-claim boundary; and
- a review posture.
Next Build Target
The next build target is a machine-readable mathematics crosswalk JSON artifact and a validator connecting crosswalk rows to existing translation record IDs.
Non-Claims
This page does not validate GCAT / BCAT equations, prove physical interpretation, establish source-discipline equivalence, or authorize any transition. It is a public-safe crosswalk for review and later validation.