wip

Author: acharneski

essays/NAM/CAS.md

@@ -0,0 +1,193 @@
+---
+specifies: main.mac
+related:
+  - README.md
+---
+
+# CAS Companion Goals: Numbers as Machines (NAM)
+
+This document specifies goals for an agentic coding process tasked with writing a
+companion verification/experimentation script for the NAM essay. NAM is an
+_architecture_ essay rather than a theorem essay, so the companion's job is to build a
+**small, honest reference prototype** of the generator-VM substrate and to use it to
+**empirically confirm or falsify** the essay's structural claims.
+
+A CAS (Maxima) is useful for the symbolic/number-theoretic checks (periodicity,
+valuations, BBP modular structure, state-dimension recurrences). A systems language
+is not required for the prototype — a high-level language with exact bignums
+(Python+gmpy2, or Maxima's own Lisp layer) suffices to demonstrate every claim. The
+LLVM/ABI claims are validated _structurally_ (state layout, fork cost) rather than by
+actually emitting machine code.
+
+---
+
+## 0. Infrastructure / Harness Goals
+
+- **G0.1 — The step primitive.** Implement the core interface
+  `step : State → (digit, State)` as the single contract every number satisfies.
+  Provide a driver that pulls N digits and a `take(vm, N)` helper.
+- **G0.2 — Fork primitive with cost accounting.** Implement `fork(vm)` and instrument
+  it to report whether the fork was an O(1) value copy (automaton tier) or required a
+  deep copy of growing accumulators (series tier). Cost accounting is the point of
+  the two-tier ABI; make it measurable.
+- **G0.3 — Base/codec separation.** Implement `in_base(vm, b)` as a _projection_
+  wrapper, demonstrating that changing base does not change the underlying generator.
+
+---
+
+## 1. The MUX / Coalgebra Foundation
+
+- **G1.1 — MUX combinator.** Implement `MUX(selector, left, right)` as a generator
+  over generators and demonstrate it builds: a number (digit function), a vector
+  (MUX over numbers), and a matrix (MUX over vectors) — the uniform nesting claim.
+- **G1.2 — Final-coalgebra sanity.** Demonstrate the digit-stream functor framing by
+  showing two VMs are bisimilar iff they emit identical streams on a finite prefix
+  (necessarily prefix-only, since full equality is undecidable — see G7).
+
+---
+
+## 2. Signed-Digit Arithmetic (the carry problem)
+
+- **G2.1 — Carry non-locality demonstration.** Implement _naive_ standard-base addition
+  and exhibit the failure: show that the first digit of `0.4999... + 0.5000...` cannot
+  be committed from any finite prefix (the combinator stalls). This is the essay's
+  motivating pathology; reproduce it concretely.
+- **G2.2 — Avizienis signed-digit fix.** Implement signed-digit (redundant)
+  representation with digits in {−(b−1),...,b−1} and show that addition becomes
+  digit-local with bounded look-ahead — the same input now commits digits.
+- **G2.3 — Streaming arithmetic.** Confirm addition, subtraction, and multiplication
+  are genuinely streaming (online) under signed digits, with the codec layer
+  projecting back to a non-redundant alphabet only on emission.
+
+---
+
+## 3. p-Adics as Periodic Automata (the clean case)
+
+- **G3.1 — Periodic generator.** Implement a p-adic rational as a finite-state machine
+  and verify its digit expansion is ultimately periodic with a finite state orbit.
+- **G3.2 — Local digit commitment.** Demonstrate the essay's key contrast: p-adic
+  digit commitment is _local_ (no tail oracle, no interval refinement) because a term
+  of valuation v affects only positions ≤ v. Implement `valuation(a_n)` and show
+  low-order digits computed from a finite term subset.
+- **G3.3 — Metric as product automaton.** Implement the p-adic distance
+  `p^{−v_p(x−y)}` for two periodic generators as a product automaton over their state
+  spaces, and confirm it is itself a finite periodic machine.
+
+---
+
+## 4. The "Odd Primitives" (series tier)
+
+- **G4.1 — Series stepper + tail-bound oracle.** Implement `series_step` and
+  `tail_bound` for at least e (Σ1/n!) and one of {π via a known series, log 2}.
+  Verify the tail bound is rigorous: the emitted digit never gets revised once
+  committed.
+- **G4.2 — Interval refinement engine.** Implement `refine_interval` /
+  `emit_digit_if_safe` and demonstrate online digit extraction as the interval
+  collapses.
+- **G4.3 — Exact-boundary honesty.** Demonstrate the documented caveat: a value that
+  is exactly 0.5 in base 10 may _never_ commit a leading digit in finite time. Show
+  the engine correctly returns "pending" rather than lying — and that the
+  interval-based predicates (G7) handle it gracefully.
+
+---
+
+## 5. State-Dimension Complexity Tiers (the central conjecture)
+
+The essay's headline table claims minimal VM state dimension grows: rationals (1),
+quadratic irrationals (2), classical transcendentals (3–4), higher transcendentals
+(4–6+), inaccessible (unbounded).
+
+- **G5.1 — Minimal recurrence orders.** For rationals and quadratic irrationals, build
+  the minimal-state generator and confirm the logical register count (1 and 2
+  respectively) and O(log n) bit-width growth. Symbolically derive the order-2
+  recurrence for √2, φ, √3.
+- **G5.2 — Transcendental state tier.** Build generators for π and e and _measure_
+  live state (number of coupled accumulators + index/error state). Report the observed
+  dimension. The script should explicitly flag that this is an _upper witness_, not a
+  proven lower bound (the essay is honest that minimality is an open problem).
+- **G5.3 — Non-minimal engine cross-check (ties to PI_RCC).** Implement the x+sin(x)
+  cubic engine from the PI_RCC companion _as a generator VM_ and confirm it carries
+  MORE live state than the 3–4 minimal figure for π — directly illustrating the
+  essay's "the table tracks the floor; specific engines sit on or above it" remark.
+  This is the key cross-essay consistency check.
+- **G5.4 — Inaccessible = O(n).** Demonstrate that a "random"/encrypted stream
+  generator requires state growing with digits emitted (O(n)), contrasting with the
+  O(log n) bit-width of accessible numbers.
+
+---
+
+## 6. BBP / Skip-Ahead (automaton-codec resonance)
+
+- **G6.1 — Base-16 π digit extraction.** Implement the BBP formula and a `skip(n)`
+  primitive that computes the n-th hex digit of π via repeated-squaring modular
+  exponentiation, _without_ computing predecessors. Verify against a reference π in
+  base 16.
+- **G6.2 — Fast-forward is O(log n).** Confirm skip cost scales as O(log n) (modular
+  exponentiation), not O(n), by timing/operation-counting against n.
+- **G6.3 — Codec dependence.** Demonstrate the essay's "property of the projection, not
+  the number" claim: show the base-16 generator admits skip while the base-10
+  generator does not (no _known_ fast-forward). Frame the base-10 absence honestly as
+  conjectural, per the essay.
+- **G6.4 — Genuine periodicity vs fast-forwardable.** Contrast a truly periodic case
+  (a rational, or a periodic p-adic) where skip is exact finite-orbit jumping, against
+  π-in-base-16 where skip is the repeated-squaring structure — confirming the essay's
+  flagged distinction that "periodicity" is shorthand that overstates the symmetry in
+  the transcendental case.
+
+---
+
+## 7. Equality / Comparison (interval-honest, undecidable)
+
+- **G7.1 — Predicate family.** Implement `definitely_less_than`,
+  `agrees_with(digits=N)`, and `compare(max_digits=N) → {Less|Greater|Indistinguishable}`.
+  Show they never claim exact equality.
+- **G7.2 — Undecidability witness.** Demonstrate concretely (e.g. two different
+  generators for the same constant) that `agrees_with` can only ever confirm a finite
+  prefix, never decide true equality — the Rice's-theorem boundary made operational.
+
+---
+
+## 8. Randomness as Inaccessible Determinism
+
+- **G8.1 — PRNG-as-number.** Wrap a small deterministic PRNG (e.g. a counter through a
+  hash) as a `step`-conforming generator and show it is a perfectly valid number whose
+  generator is small but _cryptographically inaccessible_ from the output.
+- **G8.2 — Accessibility classification.** Provide a function/report that classifies a
+  generator as accessible / cryptographically-inaccessible / computably-inaccessible,
+  and emphasize (per the essay) that the distinction is epistemic, not ontological —
+  the digit-driver behaves identically for all three.
+
+---
+
+## 9. The Two-Tier ABI (structural validation)
+
+- **G9.1 — Struct-layout mirror.** Mirror the essay's `AutomatonVM` (fixed inline
+  state[4]) and `SeriesVM` (immutable spec + mutable accumulators) as data structures,
+  and verify the fork-cost claims structurally: AutomatonVM fork is a pure value copy;
+  SeriesVM fork must deep-copy the accumulators.
+- **G9.2 — Codec-in-payload.** Verify that the base lives in a payload/flag field and
+  that swapping it does not alter the ABI or the underlying generator (re-confirms
+  G0.3 at the struct level).
+
+---
+
+## 10. Cross-Essay PNT Consistency
+
+- **G10.1** Confirm the same number-theoretic input `log lcm(1,...,M) = ψ(M) ~ M`
+  appears in NAM's series-tier bit-width accounting exactly as in the RCC and PI_RCC
+  companions. Run the shared ψ(M)/M → 1 check and assert the constant is consistent
+  across all three companion scripts. This is the "load-bearing number-theoretic
+  input recurs in all three" thread from the trilogy README.
+
+---
+
+## 11. Reporting Contract
+
+Emit a report that, per claim, states: (claim, prototype-verified? yes/no, measured
+quantity, honest caveat where the essay flags one). The headline verdicts are:
+signed-digit makes addition local (G2.2), p-adic commitment is local (G3.2), the
+state-dimension table holds as an _upper-witness_ hierarchy (G5), BBP skip is O(log n)
+and codec-dependent (G6), equality is only prefix-decidable (G7.2), and the two-tier
+fork-cost split is real (G9.1). Every "open problem" the essay flags (state
+minimality, base-10 BBP absence, randomness metaphysics) must be reported as
+_probed but not settled_, never as proven.

essays/NAM/README.md

@@ -362,13 +362,25 @@ a **reformulation rather than a discovery**.
 What the generator-VM model contributes is a _first-class library primitive_ that lifts this phenomenon out of one-off
 optimizations and into the ABI:
 
-**A BBP-type formula exists exactly when the generator automaton has a periodic orbit under the chosen codec.**
+**A BBP-type formula corresponds to the generator automaton admitting an O(log n) fast-forward under the chosen codec —
+in the cleanest cases, a periodic orbit; more generally, a sub-automaton whose n-th state is reachable by repeated
+squaring (modular exponentiation) rather than by stepping through all n predecessors.**
+
+The honest form of the claim is a correspondence, not a strict equivalence: the existence of a known BBP-type extraction
+formula is what licenses installing a `skip` opcode, and that opcode is implemented by the modular-exponentiation
+structure underlying the formula. Calling this "a periodic orbit" is exact only in the genuinely periodic cases
+(rationals, periodic p-adics); for π in base 16 the precise mechanism is the repeated-squaring fast-forward of a partial
+sum modulo a moving denominator. We use "periodicity" below as a convenient shorthand for this broader fast-forwardable
+structure, and flag here that the shorthand overstates the symmetry in the transcendental case.
 
 In base 16, the π-generator VM has a periodic sub-automaton. The codec (base 16) aligns with that periodic structure,
-exposing a fast-forward opcode: you can jump ahead in the automaton's orbit in O(log n) time rather than stepping
-through all n digits.
+exposing a fast-forward opcode: you can jump ahead in the automaton's state in O(log n) time — via repeated squaring of
+the relevant modular quantities — rather than stepping through all n digits. (Strictly, this is the fast-forwardable
+structure described above, of which a true finite periodic orbit is the special case.)
 
 In base 10, the same automaton has no such periodic orbit under that codec. No fast-forward is possible.
+(More carefully: no such O(log n) fast-forward is _known_; the absence of a base-10 BBP formula for π is an empirical
+and conjectural state of affairs, not a proven impossibility.)
 
 This is not a property of π. It is a property of the π-generator under a specific projection. The base is a lens on the
 automaton's structure. Some lenses expose periodicity; others do not.
@@ -415,7 +427,10 @@ struct size does not grow as you emit more digits.
 
 **Quadratic irrationals** (√2, φ, √3) require two coupled registers — a second-order recurrence. The field count is
 fixed; the bit-width grows as O(log n) for the n-th digit. These are algebraic of degree 2, and the generator reflects
-that exactly.
+that exactly. (Note that the `AutomatonVM` struct over-provisions with `state[4]` so that a single fixed layout covers
+algebraic numbers up to degree 4; the _logical_ register count for a degree-d algebraic number is d, even though the
+_physical_ struct reserves a uniform four-word slot. The complexity metric below counts logical, minimal registers, not
+the padded struct width.)
 
 **Transcendentals** (π, e, log(2), ζ(3)) require more coupled accumulators, error-control state, and index tracking.
 Still O(log n) bit-width growth, but with a strictly richer invariant structure — more fields, more intertwined
@@ -441,6 +456,13 @@ ascending sequence of irreducible generator-complexity classes, each requiring g
 correctness over an unbounded digit stream. Whether the sequence is strictly ascending in a provable sense is a question
 for computable analysis complexity theory; the architecture is committed to making the question well-posed and
 instrumentable, not to settling it.
+A subtlety worth flagging, since it connects to the companion essays: the state-dimension figures above are claims about
+the _minimal_ generator for a constant, not about any particular generator one might choose to run. The x + sin(x) cubic
+engine for π analyzed in the companion PI_RCC essay, for instance, carries more live state than the 3–4 register figure
+— it maintains a growing rational seed plus Taylor-truncation accumulators — yet it computes the same π. This is not a
+contradiction: a constant's complexity class is the infimum of state dimension over all correct generators, and a
+deliberately non-minimal engine (chosen for its cubic outer convergence rather than for state economy) sits above that
+infimum. The table tracks the floor; specific engines sit on or above it.
 
 ---