From the Signature to the Ethics

Uptake and venture

Two operators carry everything. Uptake is gated on the taker’s third coordinate; venture zeroes the hidden slot and carries fidelity through.

^(A, B)  =  B₀ − A₀   if A₂ = 0,  else 0
@(A, V)  =  (A₀ + V,  0,  A₂)

The predicates, in terms of D1

Every predicate in the ethics is built from uptake and venture. Nothing else is primitive.

K A B   ≡  (A @ (A ^ B)) = B                  A knows B
g A B   ≡  p(b(A ^ B))                        B counts to A
W A B   ≡  g A B  ∧  ((A @ b(A ^ B)) = B)       A wants B
loss(A,B) ≡ DIST(B, A @ (A ^ B))               composed, not expanded
CAP x   ≡  ∀V.  x ^ (x @ V) = V                x can act

p is positivity, b amplitude, DIST distance. K ventures by the uptake itself; W ventures by its amplitude. That single difference is what separates knowing from wanting throughout.

The axioms stand in both carriers

Checked by calling the kit’s own check.run() on the adopted pair — the repaired sheaf and the unfused classical carrier.

RC1 is the expected failure: the rational sheaf is not real closed, so intuitionistic real closure is adopted on Palmgren’s authority and flagged rather than checked.

Capacity is the open gate

CAP x holds exactly when x₂ = 0.

Proof

By D1, x @ V = (x₀+V, 0, x₂). Applying uptake, x ^ (x₀+V, 0, x₂) is (x₀+V) − x₀ = V when x₂ = 0, and 0 otherwise. So the equation holds for every V exactly when x₂ = 0; when x₂ is non-zero it fails at every V but zero. □

Hypotheses: R is cancellative under addition, and the range of V contains a defined non-zero magnitude. The second is a property of the sample and it is load-bearing: CAP x is a universal over that range, and at a shut gate the left side is 0 for every V, so the equation fails only where some V is non-zero. Narrow the range to {0} and the lemma is false — capacity then holds at 8 of 8 gate-shut points in the sheaf and 12 of 12 in the classical, instead of 0 and 0. Lemmas 2–4 carry no such dependency.

Self-knowledge is the empty hidden slot

K x x holds exactly when x₁ = 0.

Proof

K x x says x @ (x ^ x) = x. Uptake of a point by itself is x₀ − x₀ = 0 when the gate is open and 0 by definition when it is shut — either way zero. And x @ 0 = (x₀, 0, x₂), which equals x exactly when x₁ = 0. □

The gate does not enter. This is why the two guards in §5 are independent conditions and not one condition stated twice.

The null venture is not the identity

x = x @ 0 holds exactly when x₁ = 0.

Proof

Venture zeroes the hidden slot for every magnitude, the null one included: x @ 0 = (x₀, 0, x₂). That equals x exactly when x₁ = 0. Venturing by nothing still discards what was hidden. □

A shut gate values nothing

If x₂ ≠ 0 then g x y fails for every y.

Proof

Uptake returns 0 whenever the taker’s gate is non-zero. Value is the positive amplitude of uptake, and the amplitude of 0 is not positive. So the entire evaluative vocabulary — g, and W which contains it — is empty at such a point, by the definition of uptake and not by anything about the sample. □

Knowing and losing nothing coincide

K A B and loss(A,B) = 0 agree at every pair.

Argument, and why it is not a derivation

Since loss is composed — loss(A,B) = DIST(B, A @ (A ^ B)) — a distance of zero and the round trip landing on its target are the same event. But K is an equality at P and loss = 0 an equality at R, and those differ intuitionistically in general. That they coincide here is checked, not derived, and is marked accordingly.

Which equivalences survive a change of standpoint

The speaker is a variable, not the origin. A claim holding only from where one happens to stand is a claim about the coordinate.

3 robust, 2 standpoint-bound. That being known and being able to act are the same predicate is a consequence of the speaker sitting at the origin, and breaks as soon as the speaker moves.

The order is exactly twelve covers

Every implication between named predicates holding in both carriers, transitively reduced, is exactly the twelve sentences below — no more and no fewer.

Construction

The 19 vocabulary entries collapse to 12 distinct behaviours once predicates sharing a behaviour are identified and the degenerate entry (someone values x, valid throughout, implied by everything) is excluded. All 132 ordered pairs were tested in both carriers, yielding 25 implications; transitive reduction leaves 12 covers.

Compared with the stated twelve as a set, not a count: 0 missing, 0 extra. Keying on behaviour rather than on the sentence is what makes this come out; keying on glyphs returns 31 covers, each duplicated once per naming.

The guards read as their names claim

Both guards were found by inspecting where failures fell and then testing a guess — a good way to find a hypothesis and a bad way to confirm one. So their readings were checked from every standpoint, not asserted at the origin.

CAP i     U V { { $ V } - { { o i { s i V } } : V } }     I can act
K i i     { K i i }                                     nothing of me is hidden

By L1 and L2 those two coordinates are exactly what capacity and self-knowledge are. The guards are therefore not ad hoc repairs: they name the two coordinates the operators already distinguish.