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. At cell scale means the Lipschitz ratio is measured between cells — an image cell's diameter against its input cell's — never between points, and the coarseness is load-bearing: the readable class contains below-the-top digit permutations whose pointwise ratios are unbounded; at ball windows — windows whose cells are metric balls, as the trailing end's cosets are — 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⌋ reads at bounded lookahead 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. 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: the trailing pair at bases 2, 6 and 10 with collision witnesses below each threshold; the leading pair by the alignment sweep at the same bases and the redundant-cover scans. Component tiers: the trailing instance proved, elementary; the coset rigidity a two-line proof; the numerator criterion a rule (the boundary-preimage method: a monotone map is exception-free at a scale iff every input point where it crosses an output-grid line is itself grid-aligned there); the rational-slope delay is priced exactly on Redundant arithmetic.

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 blind spot is the log's pole at zero, one wall at both poles: a window reading size, sign and leading digits is a log coordinate, its chart excludes zero, and a difference is the operation that reaches it; the finite pole's index coordinate is the same log shape and affords the pole exactly — log 0 = −∞ realized, the exact zero-test kept — where this pole's vanishing is a limit, a difference's exponent unbounded below. The redundant purchase — reading through an overlapping digit set rather than a partition — does not close the gap; it pays with it. What it pays: comparison blurs across 3 cells against the non-redundant window's 1, and the window cannot read its own exponent. What it does not pay for, being gone already at a balanced non-redundant window — symmetric digits with 2a + 1 = b, one representation per value — is sign: the standard rung stores sign as its own coordinate — what makes it a native read — but a symmetric digit set folds it into the digits, and computed from digits it is non-local at every lookahead on either side of the purchase. 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 it still had.

Scope. Property with witnesses, exhaustive at bases 2, 3 and 10 at precisions up to 4; the difference witness is one deep fiber of length 64 at base 2. The redundant order reads (sign, order blur, normalization) are rules, the order-blur law an iff, at both scanned redundant systems. Sign's non-locality on either side of the purchase is a rule exhaustive at the widths run, over seven redundant digit sets and two balanced non-redundant ones.

verifiers: explore_dual_pole.py, explore_dual_redundant.py, explore_order_wall_shape.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, written in the valuations v_p — v_p(m) the exponent of p in m. 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 is derived, its ceiling attained at the scanned depths (bases 2, 3 and 10); the exchange is derived and swept at bases 2, 6 and 10; the pricing is an identity, measured.

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

Over any b^Λ consecutive class-interior fibers at depth j the number of exceptional ones is exactly d ⌊a(b^j − 1)/(b^jd)⌋ with d = gcd(a, b^Λ) — independent of where the period starts, and equal to a − d on deep fibers (b^jd > a). The count vanishes iff d = a, which is exactly rad a | rad b together with Λ ≥ max_p ⌈v_p(a)/v_p(b)⌉ — jointly the statement a | b^Λ: the numerator gate, with Λ clearing exactly the max-over-places quantity the two-ends law prices the trailing end with. So the gate is the vanishing locus of one density formula: the classification and the grading are not two facts, the classification being where the grading is zero.

Mixing the two carry classes with their counting weights cancels α. On deep fibers, for a coprime to b, the per-exponent exception density under counting measure is exactly (1 − 1/a)/b^l: the base drops out of the prefactor and so does the multiplier's mantissa, leaving only the multiplier's reciprocal, and b survives solely as the factor per unit of lookahead. Two operands multiplied together give a density converging to 2(1 − b^−j)/b^c at their common lookahead c, in the unsaturated regime 2b^1−c ≤ 1 — which excludes c = 1 at every base, where the image interval can cover a whole grid period and the density saturates instead. That limit is base-free, each operand contributing b^−c and the two ADDING, so lookahead spends as an additive budget.

Scope. Rules, exhaustive over 219 scopes — bases 2, 6 and 10, multipliers coprime, mixed and radical, lookaheads 1 to 3, depths 1, 2 and 4, both carry classes, three period phases each — with all 48 zero-locus cells coinciding with the criterion. The collapse to (1 − 1/a)/b^l is a DEEP-fiber statement and is stated as one: a shallow fiber (b^j < a) sits at its own mixed law, which the period formula still covers. The pair limit is measured, to ~10⁻⁴ at the precisions swept.

verifier: explore_dual_measure.py

The mantissa space of the dual rung is the scale circle R_>0/b^Z, a compact group, and log-uniform mass — Benford's law — is its Haar measure, the only scale-invariant one. That makes the mirror between the poles exact one level below the arithmetic. At the finite pole the ambient group is additive Z, the rung is Z/N, Haar is counting measure, and the reading cells are cosets — every cell of equal Haar, which is why no density at the finite pole ever needed a measure named. Here the reading cells are digit windows and they are not cosets: cell m carries Haar log_b((m + 1)/m). Benford's law is that grid/Haar mismatch, and the finite pole's measure-blindness is not simplicity but the degenerate luxury of a coset reading partition. The same mismatch at the continued-fraction window is the Gauss–Kuzmin law (Reading).

Read under its own measure the pole reports what counting cannot. The density of the same exception sets becomes (1 − 1/a) R(x)/b^l, where x = {log_b a} and R(x) = b^x−1(1 + (b − 1)(1 − x)) — a hump with R(0) = R(1) = 1 and R > 1 strictly between, and a multiplier coprime to b can never land on those endpoints, so Benford sees strictly more exceptions and carries the multiplier's mantissa exactly where the counting prefactor forgot it. That same x runs the carry layer: under the pole's own measure the two carry classes weigh 1 − x and x, so the θ above occurs with probability x exactly, where counting gives a rational function of a and b. The carry structure flattens for two operands too: their mantissa product carries with probability exactly 1/2 at every base, the class boundary being a straight line in log coordinates about which the sum of the two log-mantissas is symmetric, where under counting the same probability is base-entangled and transcendental. And the two-operand Benford density is the counting one times K_b = (b − 1)²/(b ln² b) — which is the mean of the hump R over x uniform: two derivations of one constant.

Scope. Property; the Haar statement itself is classical and the mirror is what is claimed. Scale invariance is checked in exact rational arithmetic with no logarithms — the preimage of a window cell under y ↦ qy is a single arc, two chart pieces at most, whose endpoint ratios telescope to (m + 1)/m — and counting measure fails that same check exactly (mass 10/27 against 1/9 for the leading-digit-1 cell under tripling), which is the control. The densities and both carry laws are rules, the closed forms derived and confirmed across the swept cases.

verifier: explore_dual_measure.py