Share encoded atom powers across recursive child spectator extents

research/encoded-child-spectators/CONTEXT.json

@@ -0,0 +1,43 @@
+{
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "track": "bit-supplier",
+  "seed": 4,
+  "claim": "https://github.com/CrocSwap/integer-mult-bounds/pull/4062#issuecomment-6268379494",
+  "predecessors": [
+    {
+      "pr": 4025,
+      "head": "84bd28ef6da3a3473d14a48088f30ba82d414450",
+      "author": "ToikY",
+      "role": "Shared traversal for a fixed d\u00b2 high_power; credited mechanism precedent. Not imported."
+    },
+    {
+      "pr": 4062,
+      "head": "d7df855fc64525b43bfda7e7a24b17356d6c1d0e",
+      "author": "Chris R / Clear1nsight",
+      "role": "Complete Context/directory/admission preparation interval and released geometry/extents handoff. Not imported."
+    },
+    {
+      "pr": 2753,
+      "head": "7dba5452a4071a856a0b11cebdcd2d3b6bcd7659",
+      "author": "David Arias",
+      "role": "Actual ordered child geometry and Layout.child_extents, retained canonically in PR2809."
+    },
+    {
+      "pr": 2809,
+      "head": "e7d93dd0de0bb81308bc3356f65eddeeb22fd712",
+      "author": "Jose Manuel / jmbarrancoml",
+      "role": "Fixed-role physical caller and frozen Word/Count/Counter/Geometry closure."
+    }
+  ],
+  "overlap": {
+    "open_pr_bodies": 772,
+    "scope": "Complete open body pagination including drafts; current1166 and12 recently updated discussion threads; exact4025/4062/4011/4059/4067 sources/comments. Broader closed/private/code coverage is partial."
+  },
+  "scope": "Only the new isolated package. Eight canonical runtime inputs are authenticated before execution; pending PRs supply handoff/provenance, not executable dependencies.",
+  "publication_refresh": {
+    "canonical_main": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+    "recent_updated_pr_bodies": 100,
+    "parent_head": "d7df855fc64525b43bfda7e7a24b17356d6c1d0e",
+    "parent_comments": "The current comments contain our narrow claim; no competing claim or objection observed."
+  }
+}

research/encoded-child-spectators/PROOF.md

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+# Encoded shared atom powers at the child-spectator boundary
+
+## 1. Statement and exact source boundary
+
+Let q be the fixed source radix, let the admitted Ring modulus be A=q^w,
+w>=1, and let each e_i be a borrowed canonical unsigned Word of width I.
+For one to three distinct owned Count outputs v_i, the operation computes
+
+`v_i <- v_i * A^(e_i)`.
+
+The e_i and Ring/arena inputs return byte-for-byte unchanged at canonical
+head 1. The caller supplies, or the setup creates, one distinct owned blank
+counter role. It is returned blank at head 1, including its erased visited
+backing. This is not an arbitrary-dirty loan. Output and workspace aliases
+are rejected. Fixed output-role selection and fixed q are compile-time data;
+e_i and w are never decoded into host integers or loop bounds.
+
+The actual source is the eight-file closure in SOURCES.json, at canonical
+base 20ecf2891d80b5acf18d980b2164fb44ea7faa90. Authentication checks its sealed
+inventory, complete SHA256 and Git blob pins before executing any source.
+The private Counter class contains the exact frozen class AST, with only
+its Tape constructor bound to the single reusable blank role. The full Word,
+Count, Natural, Counter, radix, Layout and Geometry methods are retained.
+Only Geometry is extracted from numeric.py; native forms, source admission
+and Callback are deliberately not imported or certified here.
+
+The claim is successful valid-input source-method equivalence and an exact
+source-action delta. It is not complete physical machine refinement, a full
+caller/supplier admission or a multiplication exponent improvement.
+
+## 2. Algorithm and encoded-loop proof
+
+The old multiply_atom_power does the following for each output separately:
+
+1. Create a Counter from e_i.
+2. For each of its e_i decrements, run the complete radix_power validator.
+3. On each of that validator's w yields, multiply v_i by fixed q.
+4. Close the Counter and every validator temporary.
+
+The new multi-power method first constructs and immediately closes a Counter
+for each e_i, using only its finished Boolean to select zero versus positive.
+If all are zero it returns, matching the old absence of a radix traversal.
+Otherwise it runs one complete radix_power. At each yield it independently
+counts down every positive e_i and multiplies its v_i by q once per decrement.
+The new iteration order is radix digit, output role, atom. The former order
+was output role, atom, radix digit.
+
+Each v_i is multiplied by q exactly w*e_i times in either word. These
+multiplications commute, touch no other v_j, and leave each Count head
+canonical. Therefore the exact result is v_i*q^(w*e_i)=v_i*A^(e_i).
+For each individual output, its sequence of Count values and Count primitive
+operations is identical, even though operations on different outputs are
+interleaved differently. Thus all Count multiplication work is unchanged.
+The source radix validator is exhausted exactly once if any exponent is
+positive; its prime-power checks and final cleanup remain active.
+
+Counter construction copies the complete encoded Word, including padding;
+decrement uses a finite borrow flag; multiplication by q uses a carry bounded
+in terms of fixed q. The active-role list has at most three entries. There
+is no numerical-width-dependent control table or cached host value of w/e_i.
+
+After each Counter.close, the canonical body and every previously visited
+body cell of its role are blank. Initially the role is blank. By induction
+the next exact Counter constructor sees the same empty entrance as the
+original Tape constructor supplied. Old erased suffix cells remain charged
+backing. The one-role reuse is therefore an explicit sequential lifecycle,
+not a sum-of-live-sizes claim that forgets retained cells.
+
+## 3. Exact counter tariff
+
+Vectors below are `(reads,writes,unit moves)` in one source-action lane.
+Let b=bitlength(e), z=I-b. Counter construction reads the I-bit input once
+through its full source loops and trims high zero digits. Its record invoice is
+
+```
+J(0,I) = (2I+5, 3I+3, 4I+8),
+J(e,I) = (2I+3, I+1+2z, 2I+4+2z),  e>0.
+```
+
+The input Word copy has the separate **index** invoice
+
+`C(I) = (3I+3, 0, 2I+4)`.
+
+For example, the initial rewind reads LEFT and moves twice. The I digit
+iterations read the Word twice per digit and move once, the terminating
+RIGHT test reads once, and the final rewind reads I+1 cells and moves I+2.
+No numerical Word writes are omitted.
+
+Closing a positive, not-yet-consumed Counter with B=bitlength(e) costs
+`(2B+5,B+1,2B+6)` record actions. For zero use B=1. Define
+
+`P(e,I) = J(e,I) + (2 max(1,b)+5, max(1,b)+1, 2 max(1,b)+6)`
+
+as the complete probe constructor/close invoice. A fully consumed Counter
+ends as ZERO,RIGHT; its close costs `(7,2,8)`.
+
+For a decrement from k>0, let t be the number of trailing zero bits.
+
+| k class | Record decrement invoice |
+| --- | --- |
+| k=1 | (6,4,8) |
+| k=2^j, j>=1, B=j+1 | (2B+3,B+2,2B+4) |
+| otherwise | (2t+5,t+1,2t+4) |
+
+Since `sum_(k=1)^e v2(k)=e-popcount(e)`, summing these classes gives
+
+`D(e)=(7e-2popcount(e)+1, 2e-popcount(e)+2b+1, 6e-2popcount(e)+2b+2)`.
+
+The complete rundown tariff is `R(0,I)=P(0,I)` and, for e>0,
+`R(e,I)=J(e,I)+D(e)+(7,2,8)`. The independent checker instead sums the
+individual decrement cases, importing neither this closed-form implementation
+nor the primary/runtime code. The control suite additionally compares both
+formulas with the real source Counter at 26 zero/power-of-two/neighbor cases.
+
+## 4. Complete source-action difference
+
+Let E=sum e_i, k be the number of positive exponents, and rho the complete
+residue-Word invoice of one source radix_power validator at the actual
+modulus and Word width. Put u=1 if k>0, otherwise 0. Old-minus-new differences are
+
+```
+residue:  (E-u) rho,
+index:    -w*k*C(I),
+record:   sum_i R(e_i,I) - sum_i P(e_i,I) - w*sum_(e_i>0) R(e_i,I).
+```
+
+The old implementation has one Counter per exponent and E radix traversals.
+The new method has all probes plus w rundowns per positive exponent, sharing
+one radix traversal. All output Count operations cancel from the difference.
+These equations include every additional probe, index copy, countdown and
+clear; negative lanes are retained. The verifier meters rho independently
+outside the compared interval, with every validator allocation and cleanup.
+Its value is not supplied by a presumed O(1) arithmetic primitive.
+
+This is not a universal cost improvement. With one exponent e=1,w=1, both
+methods have one radix traversal and one rundown, while the new one adds a
+probe and index copy. The complete same-output losing control executes both.
+Use the exact formula to establish affordability at a proposed source site;
+any runtime selector or new parameter guard has its own implementation cost.
+
+## 5. Actual Layout.child_extents integration
+
+For parent width n, f=floor(n/64), delta=n-64f, define the unchanged physical
+embedding phi(x)=x for x<64f and phi(x)=x+2delta otherwise. For child starts
+a=phi(first), b=phi(second), width c, the method validates c>0, disjointness,
+body bounds, physical contiguity of both complete chunks, and that no chunk
+crosses the original external gap. It then forms powers with exponents
+
+`e=(a, b-a-c, 2n-b-c)`.
+
+Before the gap factor is installed, the three Counts are
+`(B*A^e0, A^e1, S*A^e2)`. Exactly one receives G: the first if a>=n, the
+last if b+c<=n, otherwise the middle. Consequently
+
+`B' G' S' A^(2c) = B G S A^(2n)`.
+
+More strongly, the unchanged phi and complete source guards preserve the
+ordered three spectator regions, not merely their volume. Returned Counts
+retain the source's one-high-zero marginal format and canonical heads.
+
+The adapter authenticates the complete Layout source and rewrites only its
+three power-call expressions. The first two become deferred no-ops; the third
+invokes the shared method on all three output/exponent pairs. All exponents
+and output Counts are live at that point, before any gap multiplication or
+release. Their intervening arithmetic depends only on geometry, not on the
+Count values. It is therefore legal to defer the first two updates. Every
+other statement, rejection guard, marginal conversion and finally cleanup
+remains from the exact frozen method. The rewrite fails unless it finds the
+expected three complete argument lists in source order.
+
+## 6. Measured interval and completed coverage
+
+Each comparison starts with the same canonical borrowed size/depth/child
+Words, modulus/Ring, before/gap/suffix Counts, templates and unused spectator
+Word. It includes the encoded size/64 division, real Geometry and Layout
+construction, fields=depth+1, complete child_extents, marginal output disposal,
+and geometry/field/column cleanup. Both variants preserve every borrowed byte,
+head and live numerical owner; all newly created record and numerical backing
+is erased. The observer checks full serialization without influencing runtime
+control, and Word.inspect raises during the measured computation.
+
+Fifteen pairs cover good, across-bank unit, head/tail pair and head/tail half
+geometry at w=1 and3; a width 70 internal-child geometry; q=5 and an even-column
+case; and the no-remainder/rank 42 boundary. OS seed 4 selects the nonunit
+spectators. These are admissible geometry instances, not claims of literal
+top-level reachability of every chosen current native source or of a complete
+selected GL64 payload. q=3/5 fixtures do not instantiate the asymptotic large-q
+supplier hypothesis. The identity/cost equations are uniform on their stated
+valid-input domain; finite timings are additional observations only.
+
+The named pairs save between about 28.49% and59.20% of complete source actions.
+The width 70 pair is7,046,111→5,038,964 (28.49%). Each tested pair has the same
+additional per-bank numerical peaks (index 21,residue 6). This does not prove
+a complete joint physical reserve. Counter-role backing and all other newly
+created source record backing are retained in the receipts, including erased
+cells. No arbitrary dirty array is silently cleared or acquired.
+
+## 7. Remaining boundaries and failure scope
+
+The action units are retained source Tape/Word reads, writes and unit moves.
+Python execution times in TIMINGS.json are single same-input observations;
+there is no claimed physical-step tariff, stable wall-clock speedup or optimality.
+The loop/control compiler, finite-q source setup, actual allocator and original
+root-width stock retain their own costs. A supplied blank record role is not
+a free dirty register. Malformed modulus contexts still reject, but the source
+validator can retain its failing quotient/remainder, and loop interchange can
+change partial outputs before rejection. Full malformed-input/interrupt rollback
+is outside the successful admitted-input equivalence.
+
+This fills the encoded atom-power/child-geometry component left separate by
+PR4025/4062. Context/directory/admission composition, source-bound whole payload
+handoff, coefficients, native routing, copied state, physical lifetime maxima,
+precision and full supplier admission remain separate. No κ is inferred.
+
+Credit: Jose Manuel/jmbarrancoml, David Arias/PR2753, ywz65/PR2590,
+ToikY/PR4025, Chris R/Clear1nsight/PR4062 and the retained source/record/caller
+lineage. Tekalign (`tekuuu`) with substantial OpenAI Codex assistance;
+Apache-2.0. Frozen inputs are unchanged.

research/encoded-child-spectators/README.md

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+# Encoded shared atom powers for recursive child spectators
+
+The actual BIT caller builds three spectator cardinalities through repeated
+`radix_power` traversals. This package moves that traversal outside the runtime
+atom loop and shares it across all three powers. The exponents stay encoded;
+one reusable blank counter role replaces repeated validator work.
+
+This fills the **child-geometry component** explicitly left open by
+[PR4025](https://github.com/CrocSwap/integer-mult-bounds/pull/4025) and
+[PR4062](https://github.com/CrocSwap/integer-mult-bounds/pull/4062).
+It supplies an opt-in kernel, an adapter to the **actual frozen
+`Layout.child_extents` method**, and a complete source-action comparison.
+It does not supply the complete recursive caller or a new κ.
+
+| Same complete geometry/cleanup input | Original actions | Shared actions | Reduction |
+| --- | ---: | ---: | ---: |
+| Internal-child width 70, n=321, q=3, w=2 | 7,046,111 | 5,038,964 | 28.49% |
+| q=5, even columns, w=2 | 2,334,453 | 1,042,650 | 55.34% |
+| No remainder, rank 42, w=1 | 84,977 | 34,668 | 59.20% |
+
+Across the fifteen declared pairs the reduction is about 28.49%–59.20%.
+These totals sum retained **record, residue-Word and index-Word** reads,
+writes and unit moves. Added probes, counters, copies, marginal conversions
+and disposal are charged. They are not physical-machine step counts or a
+measured production-runtime speedup. The tested additional per-bank numerical
+peaks remain index 21/residue 6. No complete physical reserve follows.
+
+## Component API and exact budget
+
+`Powers(env, workspace=None).multiply(items, ring, q)` accepts one to three
+fixed pairs `(owned_count, borrowed_exponent_word)`. It maps each Count v to
+`v*(q^w)^e`, preserves the Ring/exponent inputs at their canonical heads,
+and returns its distinct **owned blank** counter role erased at head 1.
+A caller can provide an existing blank role; arbitrary dirty acquisition is
+not implemented. `runtime.adapt(env,powers)` returns the source-method adapter
+without changing the original method or frozen files. The reference loader
+uses a private interpreter/module context; this is a research component,
+not shared-caller installation.
+
+For E=sum(e), k positive exponents and complete radix invoice rho, the exact
+old-minus-new residue invoice is `(E-1)*rho` when k>0. The extra index bill
+is `w*k*(3I+3,0,2I+4)`; the complete counter delta has a closed form in
+[PROOF.md](PROOF.md). It is checked against individual trailing-zero cases
+and actual source Counters. Counter costs are not treated as free.
+
+A **same-output exponent-one losing control** demonstrates why this is not
+a universally faster primitive. Future callers must prove the complete
+budget at their site, including any dispatch rule, instead of replacing
+all powers indiscriminately.
+
+## Reproduce
+
+From the repository root, Python 3.12, standard library only:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/encoded-child-spectators
+```
+
+Expected markers: `PASS_ENCODED_CHILD_SPECTATORS`,
+`PASS_INDEPENDENT_CHILD_SPECTATORS`, `PASS_CHILD_SPECTATOR_CONTROLS`, then
+`PASS_REGISTERED_RESEARCH_PACKAGE`. Normal replay is read-only; receipt
+creation with `verify.py --write` refuses an existing destination.
+
+[CERTIFICATE.json](CERTIFICATE.json) records all paired bytes, complete
+lane invoices, ownership/head restoration, erased backing and geometry.
+[TIMINGS.json](TIMINGS.json) records single same-input Python observations;
+timing values are not mathematical acceptance fields.
+
+Coverage includes all six source child-geometry branches at w=1/w3, remainder
+and gap placements, width 70 (above the recursive cutoff), q=5 and rank 42.
+These are valid geometry instances, not complete production packets or a
+proof of every chosen source's top-level reachability. The independent
+checker imports no runtime/primary code. It rederives geometry and counter
+accounting, taking rho as the measured validator invoice rather than
+reimplementing the source's full Ring arithmetic. Controls include real source
+geometry guards, 26 Counter tariff calibrations, omitted output work,
+alias/reentrancy, invalid radix detection, source/inventory corruption,
+typed receipts and overwrite refusal. Word.inspect is disabled during the
+measured interval. Error detection is retained; full malformed-input or
+interruption rollback is not claimed.
+
+Exact base: `20ecf2891d80b5acf18d980b2164fb44ea7faa90`.
+[SOURCES.json](SOURCES.json) seals eight canonical live files; all are checked
+before source execution. [CONTEXT.json](CONTEXT.json) pins predecessor heads
+and the claim. OS seed 4 selects the BIT track and nonunit spectator cases.
+Frozen predecessors and the selected frontier remain unchanged.
+
+## Open next questions
+
+Compose this adapter with #4062's actual Context/directory/admission interval,
+using one existing blank counter role, then pay fixed capsule and payload
+copies and all lifetime backing. That is the next concrete join; a tiny
+controlled payload does not establish the full production caller.
+
+Prove a useful source-coupled parameter region from the exact action formula,
+including any encoded original/shared dispatch. The existing finite savings
+do not establish a global speedup. Physical control/allocator tariffs,
+original-width stock, whole payload/source admission and supplier transfer
+remain separate. Structured first steps and blockers are in [TASK.json](TASK.json).
+
+## Credit
+
+David Arias/PR2753 supplied ordered child geometry and the logical caller;
+Jose Manuel/jmbarrancoml supplied the fixed-role caller and retained runtime.
+ToikY/PR4025 supplied the fixed-d² shared-radix precedent, following ywz65/PR2590;
+Chris R/Clear1nsight/PR4062 supplied the preparation composition and released
+geometry handoff. All retained source/record/codec/caller attribution remains
+with the unchanged originals. This extends the mechanism to **runtime encoded
+atom exponents and the actual three-spectator method**; it does not re-claim
+the fixed-d² result.
+
+Tekalign (`tekuuu`), substantial OpenAI Codex assistance in design, tariff
+proof, implementation, controls and packaging. Apache-2.0. Exact finite
+checks are not independent human review or formal multiplication verification.