research: bound causal centre sharing and audit selected family coverage

.github/workflows/p112-common-slice.yml

@@ -15,9 +15,9 @@ jobs:
     if: github.repository == 'CrocSwap/integer-mult-bounds'
     runs-on: ubuntu-24.04
     timeout-minutes: 20
-    env:
-      TMPDIR: ${{ runner.temp }}/p112-common-slice
     steps:
+      - name: Initialise P112 work directory
+        run: echo "TMPDIR=$RUNNER_TEMP/p112-common-slice" >> "$GITHUB_ENV"
       - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           fetch-depth: 0

research/centre-causal-tangent/PROOF.md

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+# First-variation gate for parameterised centre sharing
+
+Clear1nsight, substantial OpenAI assistance. Apache-2.0.
+
+## Exact contract
+
+Retain the selected 144 local calls, complete scalar words, paired bridges,
+all ordinary outputs and signed terminal exchanges. As in PR2622/2763,
+project climbs, adapters, idle factors and address translations to identity.
+Replace each of the 1,152 centre actions on its private scalar by multiplication
+by z. Also test the version with z at forward calls and z^(-1) at backward
+calls. The parameter varies near 1. A replacement must agree as a rational
+function family, not just at z=1 or a root of unity.
+
+This changed scalar oracle contract is NOT the fixed native Fourier operator.
+The theorem below cannot be used as a native recursive-call lower bound.
+
+## Time-expanded sheaf and derivative theorem
+
+A finite causal linear circuit has states x_t and constraints
+x_(t+1)=M_t(z)x_t, input injection J and final projection P. It can include
+fixed linear recoding, extra finite workspace, copies and singular clears.
+Its boundary transfer is
+
+    F(z)=P M_(s-1)(z) ... M_0(z) J.
+
+Assign state-space stalks to the chronological vertices and coordinate
+compatibility restrictions to edges. Global sections are exactly legal linear
+traces. Causality gives a unique trace for each input. This time-expanded
+sheaf language describes compatibility, not a free implementation of elimination.
+
+The product rule at z=1 gives
+
+    F'(1)=sum_t P M_(s-1)(1)...M_(t+1)(1)
+                    M'_t(1) M_(t-1)(1)...M_0(1) J.
+
+If charged gates have derivative ranks r_t and every free operation is
+independent of z, rank subadditivity gives
+
+    rank F'(1) <= sum_t r_t.
+
+Each scalar-register multiplication by z or z^(-1) has derivative rank one.
+Thus q such oracle calls imply rank F'(1)<=q. This holds for arbitrary fixed
+linear causal recoding and finite workspace, including arbitrary dirty values
+held fixed when varying payload inputs. Invertibility of every intermediate
+operation is unnecessary. A parameter-independent coordinate change cannot
+alter the required boundary derivative. A gate on d independent coordinates
+may have derivative rank d and must not be counted as a single scalar query.
+Nonlinear/adaptive branching and parameter-dependent free maps are not covered.
+
+## Application to the selected projected schedule
+
+At z=1, let C contain the actual chronological centre-input covectors and B
+the final response columns to additive centre-output perturbations. C has
+shape 1152-by-14336, and B has shape 14336-by-1152. The baseline complete
+ordinary endpoint is identity. With Sigma diagonal, all +1 or signed by call
+orientation, the product rule is equivalently
+
+    F'(1)=B Sigma C.
+
+PR2622/2763 already establish full source-history rank, and PR2622 establishes
+full final-error-response rank. Those are credited predecessor results.
+The new verifier reconstructs both from the pinned selected archive without
+importing predecessor executable code. Full modular ranks at two primes,
+with opposite pivot conventions, certify their rational ranks as 1,152.
+C is therefore surjective, B injective and Sigma invertible. Hence
+
+    rank F'(1)=1152.
+
+Combining this with the derivative theorem proves that any replacement in
+the declared parameterised linear oracle class needs at least 1,152 scalar
+oracle gates. Halving to 576 is impossible in this class. This is a gate on
+generic temporal sharing, NOT a complete supplier rejection or native tariff.
+
+The source-bound implementation independently derives all local endpoints,
+chronological source rows, final response columns and the complete ordinary
+baseline identity. A separately coded full dual-number interpreter over
+Q[epsilon]/epsilon^2 checks both derivative sign conventions on dense signed
+rational initial data. Each literal run retains 2,835,792 scalar instructions
+and all 1,152 centre occurrences; every final ordinary field and private clear
+must agree. This interpreter shares the input parser but calls none of the
+primary local, bridge or endpoint routines. The dense checks support code
+correctness; the rank/product-rule argument establishes the all-input result.
+
+## Fixed-operator escape and essential limitations
+
+The derivative concerns nearby parameters. A fixed native operator may obey
+special algebraic identities that no generic z obeys. For example, n independent
+coordinates transformed four times give F(z)=z^4 I_n and rank F'(1)=n. Under
+the discrete condition z^4=1, the same fixed output is identity and uses zero
+oracle calls. Consequently the generic lower bound cannot be transferred to
+a finite-order native transform. The tests execute this counterexample.
+
+A claimed free operation z I_n is another counterexample to dropping a premise:
+it supplies derivative rank n with no charged query. Fixed source/readout
+changes, weaker required endpoints, nonlinear algorithms, different native
+primitives and actual higher-dimensional oracle gates also require a new model.
+The all-output condition matters, including helper restoration. The theorem
+does not silently replace the native endpoint by a source-only observation.
+
+Eight generic test groups exercise product-rule ranks for noncommuting words,
+a tight full-rank example, singular clears, coordinate changes, lost outputs,
+fixed-operator and parameter-dependent-free-work counterexamples, and actual
+missing/corrupt source loader rejection. Both normal and optimised modes are
+registered. No independent human or formal proof-kernel acceptance is claimed.
+
+## Decision and next question
+
+NARROW to this exact generic scalar contract. A time-expanded sheaf alone
+cannot remove work required by its boundary derivative. The useful constructive
+escape is now explicit: bind one native fixed-operator identity that fails the
+generic z model but preserves the complete required dirty-input endpoint.
+Then charge source access, frames, chronological lifetimes, restoration,
+precision, hosting and the complete recursive invoice. Those costs remain
+unknown here, not zero. No paid profile, supplier or selected kappa changes.
+
+## Credit and overlap
+
+Sagar Thakrar/PR2199, after Adam Gall/PR1918, ywz65/PR2622 and the separate
+bridge-source audit PR2763 supply the projected source/response lineage.
+Tekalign/PR2814, after Arham Tahir/PR2799, already uses causal feedback and
+operator derivatives for terminal-Y error kernels; this package instead bounds
+the complete input/output derivative against fixed linear causal recoding.
+PR3151 and PR3194 retain their distinct circle/history obstructions.
+David Charbonneau/PR3147 supplies the research direction. Matarener/PR3656 and
+Ross Richardson/PR3797 retain the block and independently established full-kernel
+result; our PR3828 is not repeated. Retain Rohan Arun, Jeremy Kothe, Hao Yan,
+Aaron Wang/navinor and all selected native/compiler source attribution.
+
+The sheaf viewpoint uses established local-to-global linear algebra, including
+Hansen and Ghrist, Toward a spectral theory of cellular sheaves (2019),
+DOI10.1007/s41468-019-00038-7. The product rule and rank inequality are elementary.
+No priority claim for temporal sheaves or generic oracle lower bounds is made.
+Separate implementations share the author and source. No agent-spawning tool
+was available in this session, and no multi-agent execution is claimed.

research/centre-causal-tangent/README.md

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+# Causal centre sharing needs operator-specific structure
+
+User-assigned priority1 in the PR3147/3828 research stream. Clear1nsight with
+substantial OpenAI assistance. Apache-2.0. A scoped scalar-oracle theorem,
+not a native sharing construction or a new multiplication exponent.
+
+## Completed mathematical result
+
+Replace each centre operation in the complete selected scalar projection by
+an unknown common multiplier z. The full boundary transfer F(z) has
+
+    rank F'(1) = 1,152,
+    rank(F(-1)-F(1)) = 1,152.
+
+The first statement blocks generic linear half-sharing near z=1. The stronger
+second statement only requires a replacement to work for BOTH scalar-oracle
+choices +1 and -1. With oracle-independent free maps, any finite fixed causal
+linear circuit requires at least1,152 rank-one oracle invocations. Arbitrary
+fixed linear recoding, finite workspace and singular clears are permitted.
+The result also covers z/z^(-1) according to forward/backward orientation.
+A576-query uniform replacement is impossible in this exact class.
+
+Neither statement proves a recursive-call lower bound for a SINGLE KNOWN native
+Fourier operator. Nonlinear/adaptive algorithms, oracle-dependent free maps,
+different required outputs and larger-rank primitives are outside the model.
+The fixed fourth-power example explicitly shows why a derivative bound cannot
+be transferred to a fixed finite-order operator. The secant bound correctly
+assigns that example zero two-point rank.
+
+PROOF.md gives the time-expanded sheaf and derivative argument. SECANT-PROOF.md
+gives the stronger discrete theorem. Causality yields the exact formula
+
+    F(z)-I = B (I-(z-1)K)^(-1) (z-1) C.
+
+The source matrix C is onto the1,152-site space, final-response B is injective,
+and feedback K is strictly lower triangular in actual chronological order.
+All source/rank facts retain PR2622/2763 credit; causal feedback retains PR2814
+credit. The new implication is the explicit uniform-oracle query bound with
+its model-discrimination controls, not rediscovery of the earlier ranks.
+
+## Source-dependent evidence
+
+At head6bc9bdb8c5b669597b6808126c5ecff5ac07a9c9 the registered fork workflow passed
+all original-source derivative and discrete checks normally and under-O:
+https://github.com/Clear1nsight/integer-mult-bounds/actions/runs/39177162360
+The final publication adds frozen SECANT.json, five generic secant tests and
+real Git-free/CLI controls. Its exact-head execution is recorded on the PR.
+
+The programme has144 calls,1,152 centre occurrences,144 paired bridges and
+14,336 ordinary outputs. Both C/B ranks are certified at two primes. The
+feedback matrix has66,146 nonzero coefficients. Each of two dense dual-number
+runs and the separate literal-minus-one run executes all2,835,792 scalar
+instructions. The minus-one replay matches all1,152 innovation cuts and all
+14,336 output coordinates. Omitting feedback changes14,330 final outputs in
+that fixture; inserting a causal cycle is rejected.
+
+The constant-frame scalar semantics are inherited. Climbs, adapters, idle and
+address operations are explicitly projected to identity. Finite dense probes
+support code correctness; the written rank/factorisation proofs establish the
+all-input scalar conclusions under these hypotheses.
+
+## Reproduction
+
+```sh
+python3.12 -I -B scripts/research_package.py --package research/centre-causal-tangent
+```
+
+The standard-library hook executes thirteen generic tests in both modes, both
+source-dependent mathematical replays in both modes, then actual Git-free
+replays and four command-line corruption controls. RESULT.json and SECANT.json
+are strictly compared, without discovery --emit flags. Original sources stay
+unchanged. The portable hook checks exported bytes and permission modes.
+
+Actual selected-archive execution was hosted on Ubuntu24.04.5/CPython3.12.15
+because local DNS was unavailable. Local CPython3.13.5 ran generic tests, not
+the original archive. Separate full literal interpreters share the source
+parser and assisted author, but not the primary local/bridge update routines.
+No spawned agents, independent human or formal proof-kernel review is claimed.
+
+## Open next questions
+
+NARROW / COMPLETE for uniform scalar-oracle sharing. A productive native route
+must identify a relation specific to the actual fixed operators, not assumed
+valid for both scalar signs. Start with one candidate identity, name all
+required source/dirty-return/output states and its legal frame interval, then
+charge all acquisition, conversion, restoration, hosting and precision costs.
+None is supplied here. This is an explicit remaining opportunity, not a claim
+that a profitable native programme has been found or that all sharing is closed.
+
+## Coordination and predecessor credit
+
+Canonical base20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision6, was
+refreshed at entry, before packaging and before the discrete refinement.
+AGENTS.md, autoresearch.md, CONTRIBUTING.md and the family gate were reread.
+Targeted live overlap was checked; complete Discussion/diff coverage was
+unavailable. The upstream claim returned403 and was not published. The fork
+record is https://github.com/Clear1nsight/integer-mult-bounds/pull/3 .
+No main branch, selected frontier, predecessor source or other PR was changed.
+
+Credit Sagar Thakrar/PR2199 after Adam Gall/PR1918; ywz65/PR2622 and PR2763's
+bridge-source audit; Tekalign/PR2814 after Arham Tahir/PR2799; David Charbonneau/
+PR3147; and the selected Rohan Arun/Jeremy Kothe/Hao Yan/Aaron Wang/native/compiler
+lineage. PR3151 and PR3194 retain distinct circle/history obstructions. Ross
+Richardson's PR3797 independently establishes the prior fixed-block full-kernel
+result and is credited alongside Matarener/PR3656; our PR3828 is not repeated.
+Classical sheaf theory retains Hansen/Ghrist attribution in PROOF.md. No
+literature-wide novelty or new native tariff is claimed.

research/centre-causal-tangent/RESULT.json

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+{"calls":144,"first_variation_rank":1152,"fixed_native_operator_bound_claimed":false,"half_share_576_rejected":true,"literal_dual_probes":[{"derivative_sha256":"4d48dde9bf90092035024c9f11e2458b342955745a7d503a85f35e1d4d88c0d7","nonzero_outputs":14316,"reciprocal":false,"scalar_operations":2835792},{"derivative_sha256":"933c5d14dec11ccaeb0fb3c63629b671974446800e12b562e4fc8dddc4d327ad","nonzero_outputs":14302,"reciprocal":true,"scalar_operations":2835792}],"live":14336,"rank_certificates":{"2147483647":{"response_rank":1152,"source_rank":1152},"2305843009213693951":{"response_rank":1152,"source_rank":1152}},"response_rows_sha256":"10681b41b6b1895a95f893dbf75e01ae164fe7bccb73a346e6c7db154c774e61","scalar_oracle_minimum":1152,"schedule_counts":{"bank_padding_cover_sweep":1,"bridge":144,"idle_cover_sweep":160,"local_cover_sweep":144,"terminal":16,"terminal_Y_translations":16,"translate_every_live_family":1},"schema_version":1,"sites":1152,"source_rows_sha256":"5c44d30669ff6d0abd456d15ae42f26aef21831490198ed7deb3a2dc3123ebca","status":"PASS_CENTRE_CAUSAL_TANGENT"}

research/centre-causal-tangent/SECANT-PROOF.md

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+# A discrete two-point strengthening
+
+Clear1nsight with substantial OpenAI assistance. Apache-2.0.
+
+The first-variation theorem requires correctness near z=1. The following
+strengthening only requires the same fixed linear programme to work for two
+scalar oracles, +1 and -1. It still does NOT prove a lower bound for one fixed
+native Fourier operator whose coefficients and identities are known.
+
+## Causal feedback factorisation
+
+Use the exact source and final-response matrices C and B from PROOF.md.
+Let K map additive centre-output errors at z=1 to later centre-input changes.
+In chronological order K is strictly lower triangular, including changes
+within a local call. Thus K is nilpotent. This is the same causal feedback
+principle used in Tekalign/PR2814, applied here to the full input/output map.
+
+For centre multiplier z and delta=z-1, the vector e of innovations satisfies
+
+    e = delta (C x + K e).
+
+Hence, with baseline endpoint F(1)=I,
+
+    F(z)-I = B (I-delta K)^(-1) delta C.
+
+Since I-delta K is unitriangular, its determinant is one for every scalar delta.
+For delta nonzero, its inverse times delta is invertible. B is injective and
+C surjective on the1152-dimensional site space. Consequently
+
+    rank(F(z)-I)=1152 for every z != 1.
+
+In particular rank(F(-1)-F(1))=1152. For the reciprocal forward/backward model,
+z=-1 also assigns -1 to every centre, so the same two-point conclusion holds.
+More generally the formula uses a diagonal Delta; invertibility requires its
+entries nonzero. Only the common -2 case is claimed by the finite certificate.
+
+## Two-point query-rank theorem
+
+For two sequences of matrices M_t(-1), M_t(1), the telescoping product identity
+expresses F(-1)-F(1) as the sum, over t, of
+
+    P M_(s-1)(-1)...M_(t+1)(-1)
+          [M_t(-1)-M_t(1)] M_(t-1)(1)...M_0(1) J.
+
+Thus the rank difference is at most the sum of the gate-difference ranks.
+Parameter-independent free maps contribute zero, and a single-register oracle
+multiplication by z or its inverse contributes rank one. Any fixed causal linear
+circuit correct for both oracle choices therefore needs at least1152 queries.
+There is no continuity assumption. Both +1 and -1 satisfy z^4=1, so merely
+restricting an otherwise unknown oracle to fourth roots of unity does not
+permit generic half-sharing for this endpoint.
+
+This does not say a fixed known operator needs1152 recursive calls. Known
+operator-dependent free maps, different required outputs, nonlinear algorithms,
+larger-rank gates and costs outside this scalar query model remain separate.
+The z^4 I example in PROOF.md has zero two-point difference and is correctly
+NOT rejected by the secant bound, even though its derivative at1 is nonzero.
+
+## Bound finite checks
+
+secant.py reconstructs every local feedback row, composes them through the
+actual owners and bridges, and verifies strict triangularity in the literal
+centre order. It solves e=-2(Cx+Ke) on a dense signed rational fixture, and
+compares all1152 innovations and all14336 final outputs against a separately
+coded full scalar execution at z=-1. Every2,835,792 scalar instruction remains
+present. The generic feedback/resolvent reconstruction and the literal point
+interpreter do not share state-update routines. They share the pinned archive,
+source parser and the credited mathematical semantics.
+
+A cycle mutation is rejected. Omitting K must change the actual selected output,
+so the feedback control cannot pass by testing an already feed-forward zero K.
+The full rank theorem follows from the written factorisation and certified
+B/C ranks, not from the single dense fixture. No extra native/source/hosting
+premise is discharged. No independent human or proof-kernel review is claimed.
+
+This is a refinement within the same conditional contract, not a selected
+frontier change. Source inputs remain pinned at canonical20ecf289. The second
+checkpoint found main unchanged; targeted centre/secant PR search found no
+match, which is not exhaustive novelty evidence.

research/centre-causal-tangent/SECANT.json

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+{"changed_outputs":14336,"chronological_order_sha256":"9daeb95e433380bf25baf88b96eb2354af2d8175b998f19072a8f247f7836746","controls":["cyclic_feedback_rejected","missing_feedback_detected"],"feedback_nonzeros":66146,"feedback_sha256":"e27b07bc9ba8023c864535f6b753574dc3fe4b480b1496c5828923d175877ace","fixed_single_native_operator_lower_bound":false,"literal_scalar_operations":2835792,"minimum_uniform_sign_oracle_queries":1152,"minus_one_innovations_sha256":"fa3ab27c0d29b170de18d1698503fa656d5bf5df46f1c2c0927e2387a9616829","minus_one_output_sha256":"aada502e49ded43e41992e95a1a595bf9ad7e9304d20d4b8b791fbfaa9fcd237","omitted_feedback_differing_outputs":14330,"schema_version":1,"secant_rank":1152,"sites":1152,"status":"PASS_CENTRE_TWO_POINT_SECANT"}