self-parametrization

the two_persp functor. both coord_add and coord_mul are instances of two_persp with different parameters. the pattern: given a pair of points p₁, p₂ (each constructed as the intersection of two lines), join them and project onto l. the parameters determine which lines to intersect.

the bridge parameter. the only free variable is a pair of distinct points (P, Q) on the coordinate line l:

• P determines the bridge line P⊔C (the auxiliary line for the first perspectivity)
• Q determines the zero: (Q⊔C) ⊓ m (the projection of Q through C onto m)
• Q is the identity element of the resulting operation

the three distinguished points O, U, I generate three non-degenerate pairings:

pair (P, Q)	bridge	zero	identity	operation
(U, O)	q = U⊔C	E = (O⊔C)⊓m	O	addition
(O, I)	O⊔C	E_I = (I⊔C)⊓m	I	multiplication
(U, I)	q = U⊔C	E_I = (I⊔C)⊓m	I	translated addition

pairings where Q = U degenerate because the zero collapses to U (the intersection of l and m), making the operation trivial.

the parameter space is the line itself. P and Q need not be O, U, or I. any pair of distinct atoms on l (with Q ≠ U) generates a valid two_persp operation. the coordinate line parametrizes its own operations: l × l \ {diagonal, Q=U} → {binary operations on l}.

self-parametrization. the line’s point-set serves simultaneously as:

1. the domain and codomain of each operation
2. the parameter space for which operation to perform
3. the source of the identity element

the algebraic structure of l is generated by l acting on itself through C. the line looks at itself through the observer and sees its own arithmetic.

the observer C. C is the only external input. it is off l, off m, in the plane — perspectival, not transcendent. changing C changes the coordinate system but not the isomorphism class of the resulting ring (FTPG). different observers see isomorphic arithmetics: same shape, different stuff (half_type.md).

connection to ground. the foam’s ground requires exactly one commitment from outside the system. C is that commitment for the coordinate line: one point, positioned in the plane but not on the measured structure, and the entire arithmetic self-generates. every operation is forced by the choice of where to stand and what to call zero.

status

proven:

• two_persp factorization (coord_add and coord_mul, both by rfl)
• additive identity (O + b = b, a + O = a)
• multiplicative identity (I · a = a, a · I = a)
• zero annihilation (O · b = O, a · O = O)

derived:

• the parameter space for two_persp operations is l × l \ {diagonal, Q=U}
• the three distinguished pairings correspond to addition, multiplication, and translated addition
• the line self-parametrizes its own algebraic structure through C
• C is the minimal external commitment required for arithmetic to self-generate
• different C yield isomorphic rings (from FTPG, not yet proven in lean)

open:

• formalize the (U, I) operation and its identity proofs in lean
• verify the translated-addition conjecture: op_{U,I}(a, b) = a + b - 1 in coordinates
• characterize which (P, Q) pairs yield group operations (associativity constraints)
• is there a natural metric or topology on the parameter space?

bugs:

• the (P, Q) parametrization claim outruns the formalized cases. “any pair of distinct atoms on l (with Q ≠ U) generates a valid two_persp operation. the coordinate line parametrizes its own operations: l × l \ {diagonal, Q=U} → {binary operations on l}.” only (U, O) → addition and (O, I) → multiplication are formalized in lean. (U, I) → translated addition is open. the claim of full parametrization across l × l \ {diagonal, Q=U} is forward-looking — a derivation conjecture, not a derived result. closing this is exactly what the open list names.
• the two_persp argument shapes differ across operations. the cited factorings — coord_add a b = two_persp Γ (a⊔C) m (b⊔E) q and coord_mul a b = two_persp Γ (O⊔C) (b⊔E_I) (a⊔C) m — place the inputs a, b in different argument positions, and the line m appears in different positions. saying “(P, Q) parametrizes the operations” implies a uniform substitution into a single template; the actual substitution is not uniform across the formalized cases. closing this means either constructing the uniform template (a single op_{P,Q}(a, b) = two_persp Γ (...P...a...) (...) (...P...b...) (...Q...) with explicit slots), or stepping the claim back to “(P, Q) determines an operation by a procedure that varies with the operation.”
• “the foam’s ground requires exactly one commitment from outside the system” is asserted, not derived. the local fact (C is the one external input for the coordinate line construction) is established. the global claim (“the ground requires exactly one commitment”) generalizes from the coordinate-line setup to a foam-wide architectural principle. that generalization is interpretation. closing this means either constructing a foam-wide commitment-counting argument or stepping back to “in this construction, C is the one external input.”

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