Finite-Window Escape and Hidden Fibers

Abstract

Finite prime windows escape; finite readings retain a nonzero kernel difference.

Theorem 1.1 (Finite prime windows escape and finite readings retain hidden differences).

$$\forall S{\subset }_{\mathrm{fin}}\mathbb{N},\left(\forall p\in S,\mathrm{Prime}(p)\right)\Rightarrow P_S=\prod _{r\in S}r,E_S=P_S+1,\left(\left(\forall p\in S,E_S\equiv 1[\mathrm{mod}p]\right)\wedge \neg (E_S\in S)\wedge \left(\exists q\in \mathbb{N},\mathrm{Prime}(q)\wedge q\mid E_S\wedge \neg (q\in S)\right)\wedge \left(\forall q\in \mathbb{N},\mathrm{Prime}(q)\wedge q\mid E_S\Rightarrow \neg (q\in S)\right)\wedge \forall G\text{ infinite additive group},\forall A\text{ finite additive group},\forall R:G\to A\text{ additive},\exists x,y\in G,x\ne y\wedge R(x)=R(y)\wedge x-y\ne 0\wedge R(x-y)=0\right)$$

Proof. Machine-checked in Lean as D5/S3/Arith/FiniteWindowEscape.finite_window_escape_and_hidden_fiber (✓ std3). ∎

Source. Repository-derived.

Commentary.

Fix a finite set S of natural primes, write P_S for its product, and set E_S = P_S + 1. The declaration proves that E_S is congruent to one modulo every member of S, that E_S itself is outside S, that E_S has a prime divisor outside S, and that every prime divisor of E_S is outside S. Thus every finite prime window leaves an external prime direction: the prime-axis tail persists and the window is not closed under the product-plus-one escape construction.

For the finite-reading clause, G is any infinite additive group, A is any finite additive group, and R is any additive homomorphism from G to A. The witnesses x and y are distinct but have equal readings; their difference is explicitly nonzero and R maps it to zero. The formal statement therefore realizes the kernel branch of the hidden-difference alternative directly. It introduces no narrative ledger object and makes no claim about ledger custody.

The proof combines the classical product-plus-one argument with the finite pigeonhole principle. Divisibility of P_S by every window prime gives the modular escape and excludes every prime divisor of E_S from S; existence of a prime divisor supplies the persistent tail. Finiteness of A and infinitude of G force a repeated reading, while additivity places the resulting nonzero difference in the kernel. The exact conjunction and its packaging as one declaration are repository-derived, and the result has no numerical certificate.

References

• Truth anchor: D5/S3/Arith/FiniteWindowEscape.finite_window_escape_and_hidden_fiber