Join physical sparse halo transport with source-bound Gaussian recovery (no new κ)

research/first-face-reference-join/README.md

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+# Source-bound first-face halo transaction
+
+**Result:** a complete, explicitly priced interface joining the new physical
+first-face transport to the new shrinking-reference numerical consumer.
+Original halo values are copied before core mutation, streamed into the
+consumer through a checked coordinate codec, and returned exactly once to
+the discarded dense coordinates. An actual retained Gaussian reference
+exercises the whole interface, including identity holes and a period cut.
+
+**No new κ.** This supplies the sparse transport/consumer/return implication
+that the component proofs require. An actual same-reference core and the
+full outer multiplication proof remain separate obligations; all five
+selected assumptions remain.
+
+## What the integrated contribution adds
+
+The two component packages are retained intact:
+
+- [Sparse halo tape transaction](../sparse-halo-tape-transaction/README.md)
+  supplies the 30-role finite-alphabet enumerator, complete sparse-key sort,
+  pre-copy chronology and unique return left open by #4870.
+- [Shrinking reference halo](../shrinking-reference-halo/README.md) supplies
+  the numerical rectangle consumer from global reference band entries,
+  with shrinking domains, actual-input rounding bounds, paid page exposure,
+  eight reused local roles, and explicitly priced scalar/source arithmetic.
+
+The composition in [PROOF.md](PROOF.md) verifies their exact common
+contract. A sorted rectangle prefix is consumed as one contiguous block;
+there is no host dictionary or payload-array regrouping. A common saved
+radius may be cropped to unequal numerical radii in one sparse pass. The
+codec checks complete offsets and their canonical source suffixes, copies
+payload cells unchanged, and converts only target keys back to canonical
+global coordinates. The numerical payload grid, signed-word convention,
+periodic coordinates, reference identity and dyadic normalizers agree.
+
+The inherited first-face combinatorics and copy-volume formula are not
+reproved as new discoveries. The direct-Neumann degree `16p^4` was already
+available in #6880. The new result is the implemented, proved and tested
+composition, including its full physical and scalar-macro accounting.
+
+## Complete cost and scientific scope
+
+Let M be dense scalars, P their bit width, B their full coordinate width,
+`C_B` all common-radius copies, `C_r` a cropped consumer rectangle's volume,
+`R_rect` the number of rectangles, `G=sum_i N_i(2W_i+1)` the group archive
+size, and w the complete encoded data/control width. The generic bound is
+
+```text
+T_ref + O(M(P+B) + w^2 [C_B + R_rect g G
+                     + sum_r C_r sum_i(J_i(2W_i+1)+K_r+1)]).
+```
+
+It retains source production `T_ref`, every repeated archive scan, all
+common-radius copies, descriptor/key conversion, radix work, output copies,
+returns and erasures. It does not charge a dense scan per numerical axis.
+All retained backing, including erased temporary tables and bucket tails,
+stays in the space bill. A fixed number of physical data/control roles
+suffices; no role is allocated per rectangle or tensor dimension.
+
+The transport's reported tape actions use a 14-symbol alphabet and have a
+constant-factor binary simulation. The consumer's reported data moves are
+binary; its bounded integer arithmetic and the codec's sparse metadata
+arithmetic use a separate ordinary quadratic bit tariff. The complete
+composition is not advertised as a fully emitted binary transition table.
+The descriptor width includes `log J` even when bandwidth is zero, and
+includes cumulative record counters.
+
+Under the pinned balanced-group caller hypotheses, the complete additional
+charge is `O(V d/g + V d g A p^6/L) + 2^o(p)`. This fits the inherited
+conservative sparse halo allowance. It is a conditional implementation
+substitution, not a stronger selected exponent or proof of the missing
+core/outer premises. The generic formula remains necessary when a finite
+instance does not satisfy the archive-amortization conditions.
+
+## Reproduce the complete contribution
+
+Canonical upstream base:
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`.
+The exact remote component tree parents are
+`a2f3ce7358e8103f6eccd404e594276e9ccc04d8` (consumer) and
+`630c9c2a5d6373ca103c55108bb606b4868bc6fd` (both components).
+`SOURCES.json` pins live imported files by Git blob, SHA256 and these
+heads; the final contribution descends from the combined parent.
+
+With Python 3.12 and the standard library, from the repository root:
+
+```sh
+python3 -B research/first-face-reference-join/replay_all.py
+```
+
+Expected final line: `PASS_COMPLETE_FIRST_FACE_HALO_PACKAGE`.
+The wrapper sequentially runs the registered checks for transport,
+consumer, and join. It does not run unrelated repository suites. For only
+the join, use:
+
+```sh
+python3 -B scripts/research_package.py --package research/first-face-reference-join
+```
+
+Expected: `PASS_REGISTERED_RESEARCH_PACKAGE`. The join registers normal
+and optimized (`-O`) runs of the actual-reference transaction and the
+independent whole-grid review. Failures use explicit exceptions, not
+assertions. No third-party dependencies are required.
+
+## Evidence
+
+The actual source is `Reference(61,3,2,1,8,32)`, period 64. Its 4,096
+original entries give a certified row gap `6255/8192`, identity-hole rows
+11, 32 and 53, and nonzero coefficients across `0 <-> 63`. The fixture uses
+`L=32`, `A=5`, saved radius 4, numerical half-bandwidth 4, two terms,
+28-bit grid, radius one and normalizer two.
+
+Four rectangles save 52 inputs and return 20 targets. Three complementary
+checks establish what those outputs mean:
+
+1. The streamed codec matches a separately loaded rectangle consumer
+   bit for bit, detecting changes too small for a loose error bound.
+2. A separately written whole-period recurrence with independent arithmetic
+   and global indexing matches every returned bit.
+3. Exact rational elimination solves the original `H y=x/2` and checks
+   every residual. Maximum target error is about `0.0003996004385`, below
+   the proved `0.036611087107` bound. These decimals summarize exact
+   rational comparisons; two terms are a finite interface fixture, not an
+   asymptotic precision allocation.
+
+The positive transaction executes 3,753,372 transport symbol actions and
+1,055,928 consumer binary actions. The displayed conservative binary-tape
+primitive bound is `10*B_actions+C_actions=38,589,648`, before scalar/control
+macros and their fixed control compilation factor. It retains 21,740
+transport symbol cells and 41,978 consumer binary cells, including blank
+backing after cleanup. Source production and coefficient arithmetic have
+their separate retained bills; these figures are not a Python memory or
+runtime benchmark.
+
+The separately assigned whole-grid review also checks a spectator fixture
+and two signed two-axis fixtures with unequal numerical radii `(1,0)` and
+`(1,2)`. The two-axis cases return 448 and 624 targets and crop 288 and 432
+saved records respectively. During codec, return and erase, it disables
+tape constructors and diagnostic `inspect`/`snapshot` access, checks the
+same eight consumer roles and unchanged archive, and checks full blank
+backing afterward. The 1D and 2D numerical oracle does not use the codec's
+grouping, local layouts or shrinking implementation.
+
+Controls reject a changed offset, omitted period wrap, missing saved
+record, wrong return key, foreign source suffix and raw negative zero.
+An in-radius payload magnitude mutation is detected by exact comparison.
+The core in these fixtures is a bit-complementing test double. Kept
+interior values are checked as opaque current-core bits and are never
+interpreted as a validated Gaussian numerical output.
+
+`RESULT.json` records the exact actual-reference receipt. The component
+receipts and full independently reviewable programs remain in their own
+directories. Separate source-aware AI review is disclosed; independent
+human or proof-kernel acceptance is not claimed.
+
+## Open next questions
+
+1. Admit a concrete rounded core on the same global reference, input grid
+   and dyadic normalizers. The splice then gives error
+   `max(E_keep,E_halo)` with no new codec defect. Start with one actual
+   caller core and its complete all-input contract.
+2. Bind canonical group-layout production, ordinary child calls, source
+   transforms and all recovery scales into the selected outer proof.
+   The local time substitution and finite actual-reference fixture do not
+   discharge these global assumptions. Preserve the generic repeated-page
+   bill whenever the balanced-group absorption cannot yet be applied.
+
+Structured questions, first steps and blockers are in `TASK.json`.
+
+## Attribution
+
+Prepared for **@CubeJerry** with substantial OpenAI Codex assistance in
+the transport, source-bound numerical consumer, composition, exact oracles
+and separately assigned adversarial AI review. Credit Cinna Davis (#4100),
+@codyw912's Astra/OpenAI-assisted #4870 enumeration package, Nick Petalas (#4217),
+Jose Manuel (#4205/#4253), Ryanson Jonathan (#4517), JoyboyBrian (#5081),
+the #6880 splice contribution, and their retained Gaussian/tape lineage.
+The predecessor claims and attributions remain in both component packages.
+Apache-2.0.

research/first-face-reference-join/RESULT.json

@@ -0,0 +1,107 @@
+{
+  "binary_tape_primitive_upper_bound_before_scalar_macros": 38589648,
+  "codec": {
+    "control_carrier_units": 197,
+    "cropped_records": 0,
+    "cumulative_counter_bits": 20,
+    "encoded_plan_descriptor_bits": 93,
+    "quadratic_codec_tariff_units": 2949484,
+    "rectangles": 4,
+    "returned_records": 20,
+    "saved_records": 52,
+    "tariff_scope": "The complete encoded control block is O(control_carrier_units); an O(1) multiple of reported quadratic units covers sparse geometry/key macros. These are not measured binary transitions. Numeric source/consumer scalar macros remain separately charged.",
+    "transport_record_width": 84
+  },
+  "consumer_bill": {
+    "additions": 360,
+    "control_steps": 5868,
+    "maximum_single_tape_extent": 17280,
+    "moves": 707472,
+    "page_scans": 6,
+    "peak_scalar_bits": 54,
+    "products": 360,
+    "reads": 173912,
+    "roundings": 144,
+    "setup_divisions": 576,
+    "setup_entries": 576,
+    "source_shifts": 144,
+    "writes": 174544
+  },
+  "consumer_binary_actions": 1055928,
+  "consumer_data_roles_including_prepared_table": 10,
+  "consumer_retained_binary_cells": 41978,
+  "controls": [
+    "offset",
+    "omit_wrap",
+    "drop_saved",
+    "return_key",
+    "payload_magnitude_exact_mismatch"
+  ],
+  "exact_delivery_match": true,
+  "final_sha256": "7b30e970a161bde5c7e35b8caba1b42063a8f5deef83628dceda6d8177da3dcf",
+  "maximum_original_inverse_error": [
+    827359109817392248798678000910818035802286303493738330288378207692097799147777735211442634813028519979204257177757246669511377410209076815399891254027152423134087507837884515052517038982214728551165310534364969914074521173026695476712503771487214898184159032862954708861356024875381268435011841407439204583226756030661807804434986984797968102285497033537378927825790801801771008272279354331434529334879098338757982379609320609009897816405177970743021044384254694603568495233063550319579917349615143207430285493734412545214764554721618974345354908406745879634498904338764910507698162301327263393284279638423381184148102797031456262660119385165587186079208489650916496382060254548520035648467444108509028369850513811325607206912378529871509919181918922470094398386252033537015059373761755542318366132336495714184767781577135628947871135788644502998205270021119514242922845428080390878453472650466265026071870366959914516240073750386789495645204649057799109550944010363343422012340525836624507048832315952477321646801679020355095245575486055091815439803801911905090641526213972660533346548585181007206680398842501464764892200323779539265634617404770188481324305076554143209531,
+    2070465970912148396618734812346158841642385406471591226562308769346667041090487190222986810756111086506156570768230328898561059679572819345812740368548765689157482405953612075441566751926603568716532235718295868452717076124609132830801721762067740436730461226543322059734001755774653573312886732956556867023523524043696457494313825744305190122422625202506313669133947400847097839395532241493700337406441878004419970734927538613332775587531096410499109785713930409441290199790152373002455603497125910955473346928848316669893441586396046031512559015012073434526228438979101199896912638965270294174607960962283459128017626139200078924751309626030413466820883976848931162897588718349273994268940422287966510054153877835859072297965920198296507433084632314139788324946985658215429336415473999621806458236204495442257770137201423755547411206898172378391949242434053553964595646701546353222439458627406341020109150982102036154019869024771763748687804661904002962539875093064180694639492815103484657289952910558565335885287037093782783775888796410821760035118446815987040634071446317207797338295879693258223031474520244163383451873187288559732281800960678865739318194852711148803850240
+  ],
+  "proved_error_bound": [
+    7503949,
+    204963840
+  ],
+  "returned_targets": 20,
+  "scope": "Actual-reference sparse codec/consumer/return; core is destructive test double, source/consumer scalar macros explicitly priced; no new kappa",
+  "source": {
+    "gap": [
+      6255,
+      8192
+    ],
+    "identity_holes": [
+      11,
+      32,
+      53
+    ],
+    "original_entries": 4096,
+    "parameters": [
+      61,
+      3,
+      2,
+      1,
+      8,
+      32
+    ],
+    "period": 64,
+    "periodic_nonzero_coordinates": [
+      [
+        0,
+        63
+      ],
+      [
+        63,
+        0
+      ]
+    ],
+    "source_sha256": "1ed3587347dc841604ce67dac8090d5a4f403e71a0037fe453835649886a1b7a"
+  },
+  "source_height": 268,
+  "source_setup": {
+    "kernel_error_bits": 68,
+    "maximum_rational_height": 437,
+    "parameters": [
+      61,
+      3,
+      2,
+      1,
+      8,
+      32
+    ],
+    "reference_sha256": "1ed3587347dc841604ce67dac8090d5a4f403e71a0037fe453835649886a1b7a",
+    "stored_entries": 576,
+    "stored_kernels": 1,
+    "stored_weights": 61,
+    "weight_precision": 202
+  },
+  "transport": {
+    "finite_alphabet_actions": 3753372,
+    "retained_finite_alphabet_cells": 21740,
+    "roles": 30
+  }
+}