The dual pole

A window reads f at bounded lookahead iff f is Lipschitz at cell scale in the window's own metric and the image of each deep input cell fits one output cell. Cell scale is load-bearing: the readable class contains below-the-top digit permutations whose pointwise ratios are unbounded; at ball windows cell scale and pointwise coincide. The fitting clause is where windows differ, and it follows from the cell family's Lebesgue number:

At the trailing end readability is Lipschitz, verbatim: reading at lookahead c is the statement d_b(f n, f n′) ≤ b^c d_b(n, n′), the minimal lookahead exactly max(0, ⌈log_b Lip_b(f)⌉) — multiplication is nonexpanding, free at c = 0, and ⌊n/v⌋ is Lipschitz iff rad v | rad b: the trailing division gate is a b-adic Lipschitz fact. No digit set changes this end — a redundant prefix's completions fill a full coset, cosets are equal or disjoint — so redundancy is a leading-end-only door. At the leading end every scaling is a log-metric isometry, so the whole gate is alignment: non-redundant, ⌊(u/v)n⌋ is window-local iff rad u | rad b with the denominator free (long division emits digits MSB-first from bounded remainder state); redundant digits {−a..a}, 2a + 1 > b, make the cells an overlapping cover and both radical gates dissolve — every rational slope reads at bounded delay, within one digit of the Lebesgue bound. The lookahead unifies as log_b(Lipschitz constant) plus a window margin: 0 at ultrametric windows, log_b(2a/(2a − b + 1)) at redundant covers, and infinite — a wall — at misaligned non-redundant reads. Catastrophic cancellation is the criterion's negative space: subtraction's log-metric blowup at its zero locus.

Scope. The four regimes verified under the one statement at stated scopes. Component tiers: the trailing instance proved, elementary; the coset rigidity a two-line proof; the numerator criterion a rule (the boundary-preimage method); the rational-slope delay law a rule at scanned digit sets (2,1), (4,2), (10,6).

verifiers: explore_dual_lipschitz.py, explore_dual_locality.py, explore_dual_redundant.py

A product's exponent is read from the operand exponents up to 1, but equality, comparison of nearby values, and the zero-test of a difference are unreadable at every finite precision: one deep fiber realizes difference exponents {0..5}, both signs, and zero. The finite pole's crown capability — the exact zero-test, a residue is zero or it is not, in every window — is exactly the dual pole's blind spot, and the dual pole's native reads — size, sign, order — are exactly what the tower's deletion removes: the two poles exchange crown capability and blind spot. The redundant purchase does not close the gap; it pays with it: behind the cover, sign is non-local at every lookahead, comparison blurs across 3 cells against the non-redundant window's 1, and the window cannot read its own exponent. Redundancy is the archimedean deletion performed a second time, inside the archimedean window — what it buys is arithmetic, what it spends is the archimedean data itself.

Scope. Property with witnesses at stated scopes. The redundant order reads (sign, order blur, normalization) are rules, the order-blur law an iff, at both scanned redundant systems.

verifiers: explore_dual_pole.py, explore_dual_redundant.py

Base b's trailing digits read exactly the finite places p | b; its leading digits perfectly hide exactly the same places — the bias of [n ≡ r (mod p)] on a deep leading fiber is exactly zero iff p | b, and otherwise nonzero on every deep fiber, with a derived attained ceiling. One numeration, two ends: the same primes exactly readable at one end, exactly invisible at the other. The same radical condition runs the arithmetic in exchange — trailing multiplication free, division gated by rad v | rad b; leading division free, multiplication gated by rad u | rad b — one condition on opposite parts of the fraction: each end freely reads the operation whose carry flow points away from it. And where a price survives, each pole prices in its own metric's arithmetic. Dividing by v costs the trailing end max_p ⌈v_p(v)/v_p(b)⌉ digits — a maximum over places — while multiplying by u costs the redundant leading end ⌈Σ_p v_p(u) log_b p + margin⌉ — a sum over places: the product formula surfacing as the two poles' price sheet.

Scope. The hiding law, the two-ends law, and the exchange are rules at stated scopes; the pricing is an identity, measured.

verifiers: explore_dual_pole.py, explore_dual_locality.py, explore_dual_lipschitz.py