Research (no κ claim): exact 38-H1 floor for the twenty-field helper block

research/helper300-twenty-field-floor/PROOF.md

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+# Exact 38-H1 floor for the twenty-field backward boundary
+
+## Result and scope
+
+For the selected backward provider's event interval **[9044,9137)**, let the
+field set be the dependency closure of roles {19,300,301}. Every exact
+replacement in the following model has the source's original per-field
+strict frame chains, hence exactly **38 rank-one paid climbs**. The original
+block attains the bound. In particular, its paid moment cannot be reduced
+for any exponent 0 < t < 1.
+
+The model permits any finite rational ADD/nonzero-SCALE factorization,
+arbitrary coefficients, arbitrary chronology, and temporary changes of
+basis. It fixes the twenty fields, their capacities, their arbitrary
+entering values, and their individual entry and exit frames. ADD operands
+must occupy the same actual frame. Paid operations are strict nested frame
+ascents of rank dim(new)-dim(old). All intermediate binary subspaces are
+allowed; the search is not restricted to the source frame catalogue.
+
+The model excludes auxiliary fields, access to fields outside this closed
+block, new warps, descents, chart-changing exchanges, cross-column
+conversions, approximate encoding, and changed input/output contracts.
+The result is a local finite-frame theorem conditional on the inherited
+native chart interpretation. It is not a new kappa, a whole-provider
+impossibility theorem, or an unconditional multiplication theorem.
+
+## Source binding and independent exact extraction
+
+Canonical checkpoint: `32890bc5de034ab8fc93ac88ead7279a6a416d04`.
+
+The actual backward tape is:
+
+`research/coupled-helper-neutral-completion/capsule/workspace/swarm-provider-audit-sol-high-20261024/repository/research/native-rank8-helper-provider/WORD.json.gz`
+
+Its compressed SHA256 is
+`7dc356001ed597ad6fc9fd3e0bc6ff7e3ec3a580a579a956827aa17fccfdd2dc`;
+its decompressed SHA256 is
+`05490717144b3e6c69c0cc9e5c5df5c69d841ca13bf29aabd1b0db180628f038`.
+Its source-payload link binds
+`research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz`, whose compressed
+and decompressed hashes are respectively
+`7679daa4ec6a40f56fc7f0202379a6f454310e94945aba4f48beaaaafb838d05` and
+`cfcd36b40326d7a5815e1db5313444379b3fb756b27d5201c561574c085cc0dd`.
+
+Indices are zero-based and the interval is half-open. Its exact field closure
+is `[18,19,39,40,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303]`.
+The 85 relevant source events are 38 H1 climbs, 43 ADDs and four SCALEs.
+Every interval event touching this set touches only this set. There is no
+warp, cache copy, clear or zero retag inside it. Intervening instructions on
+other fields commute with a replacement of the isolated block.
+
+The matrix is computed over Q by starting with twenty independent unit rows
+and applying every relevant scalar event. Its rank is twenty. The prefix
+through event 9043 has only invertible persistent ADD/SCALE, warp and climb
+operations, invertible chart adapters and phase annotations. The native
+interpretation of those frame changes is inherited. Within that model it
+imposes no relation on the twenty values; no zero-input specialization is
+used.
+
+| Role | Entry frame | Exit frame | Retained strict chain |
+|---|---:|---:|---|
+|18|20|437|20,425,437|
+|19|21|436|21,423,436|
+|39|41|436|41,426,436|
+|40|42|422|42,422|
+|288|103|432|103,422,432|
+|289|103|434|103,423,434|
+|290|56|432|56,425,432|
+|291|56|433|56,426,433|
+|292|104|434|104,427,434|
+|293|104|431|104,428,431|
+|294|57|433|57,429,433|
+|295|57|431|57,430,431|
+|296|20|435|20,427,435|
+|297|20|420|20,420|
+|298|42|435|42,429,435|
+|299|42|419|42,419|
+|300|0|437|0,21,430,437|
+|301|0|419|0,21,419|
+|302|0|428|0,41,428|
+|303|0|420|0,41,420|
+
+Here zero, line, plane and cube mean dimensions 0,1,2,3 in F_2^9. The
+boundary has thirteen line-to-cube fields, three line-to-plane fields,
+one zero-to-cube field and three zero-to-plane fields. Its invariant total
+paid rank mass is 13*2+3+3+3*2 = 38. The optimistic direct-endpoint profile
+would be {H1:3,H2:16,H3:1}, but it is not realizable under this contract.
+
+This enlarged boundary differs from the previous eight-field
+one. It has four zero-start fields, and several old plane exits have become
+cubes. In particular the old two-input/one-carrier lemma cannot simply be
+transferred. The proof below uses the freshly extracted twenty-field matrix
+and complete new frame intervals.
+
+## Why rational row equality is the right local condition
+
+Under the inherited native chart contract, a climb preserves its formal
+row while changing the invertible chart map; a coframed ADD/SCALE applies its
+rational elementary matrix to the formal rows. Therefore a legal warp-free
+window with transfer matrix T has physical map
+
+    G_exit (T tensor I) G_entry^{-1}.
+
+With fixed boundary charts, equality of T is necessary and sufficient for
+exact local equality at every positive width in that native model. Exterior
+scalar coefficients are not raised to the width. This assertion does not
+admit new unproved physical primitives or replace absolute-phase validation
+where it is required. It states the source semantics used for this local
+obstruction. A modified window containing warps would need labelled
+operator rows or additional protected boundaries.
+
+Restoring all twenty rows and charts at event 9137 preserves the unchanged
+continuation by composition. The first cached-centre copy is later, at
+11207; all eight cached copies and all eight zero retags lie after this
+boundary. No fresh full caller or physical phase replay is claimed here.
+
+## Frame-path coverage
+
+A field's strict frame path is the chain remaining after suppressing time
+spent doing scalar operations at one frame. Every possible path lies in the
+finite subspace interval [entry,exit]. The complete path counts are:
+
+- line to plane: one;
+- line to cube: four, consisting of a direct jump or one of the three
+  intermediate planes containing the entry line;
+- zero to plane: four, a direct jump or one of the three lines;
+- zero to cube: 36, consisting of one direct jump, seven line-only paths,
+  seven plane-only paths and 21 complete flags.
+
+Consequently the full domain has
+
+    4^13 * 36 * 4^3 = 154,618,822,656
+
+path assignments. This is a finite coverage count, not a claim that that
+many programs were executed. Scalar programs are not enumerated: the
+necessary conditions below apply to every factorization on a given path
+assignment.
+
+## Soundness of the optimistic station model
+
+Make a vertex for each actual binary subspace visited by a path. For each
+field ascent E -> F, add one directed edge E -> F of unit field capacity.
+Parallel edges from different fields retain their multiplicity. All vertices
+with the same subspace are merged. Grant unrestricted linear mixing and
+storage at each resulting vertex, ignoring chronological overlap and
+invertibility constraints.
+
+This graph is an optimistic relaxation. A legal ADD/SCALE can only change
+coefficients among values at the same vertex; every change of frame occurs
+along an actual field edge. Merging visits that do not occur simultaneously
+can add possible communication but cannot remove one. Each strictly ascending
+edge is traversed once by its field and carries one whole-field value. No
+field returns to that earlier frame. Repeated reads, in-place updates and
+fanout at a station do not transmit a second independent value through an
+edge already traversed.
+
+Two necessary conditions follow.
+
+**Reachability.** If T[o,i] is nonzero, there must be a directed path from
+input i's entry station to output o's exit station. A linear coefficient
+cannot appear at a station unreachable from its source.
+
+**Cut rank.** Choose input columns I and output rows O. If deleting a set
+of ascent edges disconnects every selected input station from every selected
+output station, then
+
+    rank_Q T[O,I] <= sum of the deleted edge capacities.
+
+To see this, set all other entering values to zero. The chosen outputs can
+depend on the selected inputs only through the values on the removed edges.
+Their joint information dimension is at most one field capacity per edge.
+Equivalently in formal rows their span has dimension at most the number of
+edges, with multiplicity. The graph is acyclic because every edge strictly
+increases dimension. Arbitrary mixing, feedback within a station, temporary
+bases and repeated reads cannot increase that cut dimension. At physical
+field capacity N the two sides simply acquire a common factor N.
+
+These conditions are only necessary. A path assignment that passes them is
+not automatically implementable. The final survivor below is known to be
+implementable because it is the original checked scalar word.
+
+## Domain pruning and the remaining exact certificates
+
+Initially give each field every strict path in its complete interval. To
+test one candidate path p for one field, fix that field to p and take the
+union of the edges from every currently allowed path for each other field.
+This union can contain mutually incompatible choices, and is therefore an
+even more permissive reachability relaxation. If a required nonzero matrix
+entry has no path in this union, p occurs in no feasible completion.
+
+Deleting such a path is sound. Inductively, earlier deletions have already
+preserved every feasible complete assignment. A later union of the surviving
+domains still contains every actual field edge of every remaining feasible
+assignment. Thus domain pruning does not assume a fixed scalar circuit or
+discard a solution because of an unfavorable choice of other paths.
+
+There are 68 certified deletions. They leave only four non-singleton domains:
+roles 293 and 295 have four paths each, role 300 has seven, and role 302 has
+four. The exact remaining Cartesian product has 4*4*7*4 = **448** members.
+
+All 448 members are then handled separately:
+
+- 315 have a required nonzero entry with no directed path even for that
+  assignment's optimistic merged stations;
+- 132 have an explicit cut whose capacity is below the exact rational rank
+  of the required matrix minor;
+- one survives, with exactly the retained frame chain on every field.
+
+The independent audit does not trust a max-flow solver to establish the
+cut claims. It recomputes every minor rank, verifies every listed cut edge
+and its complete parallel-edge capacity, deletes those edges, and checks
+disconnection between every selected source and sink. It also checks that
+the 447 rejected assignments are distinct, belong to the reconstructed
+remaining Cartesian product, and leave exactly the original assignment.
+
+The source assignment uses 38 H1 climbs. Thus every exact factorization in
+the stated model must use those same strict per-field frame chains and has
+the same paid profile. The floor is attained, and the normalized moment is
+38 for every 0 < t < 1. This conclusion concerns arbitrary legal scalar
+factorizations, not merely the scalar programs attempted during discovery.
+
+## Reproduction and independent evidence
+
+The primary verifier emits the source-bound certificate `RESULT.json`.
+The independently authored `check_certificate.py` validates this sibling
+certificate by default, with the repository root resolved from the package
+location. Optional `--repo` and `--certificate` flags support separate
+read-only audit locations.
+
+`check_certificate.py` imports no search code. It
+independently extracts the source matrix, frames and invoice; enumerates
+subspaces by spans of every subset of each terminal space; and validates
+all coverage and rejection certificates directly. Its observed normal run
+took less than one second under CPython 3.13. Three semantic controls run the
+actual checker on altered certificates: deleting a rejection exposes a
+coverage gap, changing a cut capacity fails the source edge count, and
+changing a target rank fails exact minor reconstruction. Normal output is a
+concise PASS receipt; `--json` provides the source hashes and detailed
+checks. The result is
+`PASS_INDEPENDENT_TWENTY_FIELD_PATH_CERTIFICATE`. No whole-provider, selected
+cold, compiler, outer, Lean or Docker replay was run for this audit.
+
+This is an exact finite certificate combined with the written station/cut
+lemmas. It is not a proof-assistant certificate or independent human review.
+
+## What a further change would have to alter
+
+Reassociation, changes of basis, different intermediate binary subspaces,
+and different scalar chronology within this boundary cannot improve the
+paid profile. Another attempt must change at least one actual hypothesis:
+for example an entry/exit chart or target interface, a transport or warp,
+or the field/capacity contract with a fully paid producer and consumer.
+Adding one H1 to save one H1+H1 -> H2 merge is already unprofitable, since
+the saving 2-2^t is less than one for 0<t<1. Such a change needs an additional
+compensating saving. A larger boundary remains untested and is not silently
+authorized by this negative result.
+
+Credit the native source and retained compiler lineage, Cao Yifan's word,
+Dillon's coupled provider, Ali Romman's cached-centre relations,
+Clear1nsight's selected completion and source validation, and GamingPuzzled's
+PR5465 bottleneck analysis. PR5471 supplies the smaller-boundary predecessor;
+its proof is not a premise of this enlarged-boundary result. Prepared by
+apoursh with substantial OpenAI Codex assistance from separate discovery and
+audit workers. Apache-2.0. No priority claim is made for elementary rank-cut
+arguments.